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Fragile-to-fragile Liquid Transition at Tg and Stable-Glass Phase Nucleation Rate Maximum at the Kauzmann Temperature TK

An undercooled liquid is unstable. The driving force of the glass transition at Tg is a change of the undercooled-liquid Gibbs free energy. The classical Gibbs free energy change for a crystal formation is completed including an enthalpy saving. The crystal growth critical nucleus is used as a probe to observe the Laplace pressure change Dp accompanying the enthalpy change -Vm *Dp at Tg where Vm is the molar volume. A stable glass-liquid transition model predicts the specific heat jump of fragile liquids at temperatures smaller than Tg, the Kauzmann temperature TK where the liquid entropy excess with regard to crystal goes to zero, the equilibrium enthalpy between TK and Tg, the maximum nucleation rate at TK of superclusters containing magic atom numbers, and the equilibrium latent heats at Tg and TK. Strong-to-fragile and strong-to-strong liquid transitions at Tg are also described and all their thermodynamic parameters are determined from their specific heat jumps. The existence of fragile liquids quenched in the amorphous state, which do not undergo liquid-liquid transition during heating preceding their crystallization, is predicted. Long ageing times leading to the formation at TK of a stable glass composed of superclusters containing up to 147 atoms, touching and interpenetrating, are evaluated from nucleation rates. A fragile-to-fragile liquid transition occurs at Tg without stable-glass formation while a strong glass is stable after transition.


Introduction
Vitrification is often viewed as a freezing-in process of undercooled melts instead of a phase change because there is, up to now, no intrinsic energy saving driving the formation of a new vitreous phase. A melt is seen as being stable with a well-defined viscosity and a unique temperature for the free-volume disappearance at the Vogel-Fulcher-Tammann temperature, which is deduced from the thermal variation of relaxation time or viscosity. Such a description leads to a natural freezing without thermodynamic transition because it does not include any modification of Gibbs free energy associated with molar volume thermal variation change. Nevertheless, the existence of a first-order transition near T g in triphenyl-phosphite is associated with a liquid instability [1][2][3]. Local minima in the potential energy landscape related to various local positions of all atoms are also considered to explain the equilibrium properties of amorphous substances [4].
Various models are considering the glass-liquid transition at T g as having a thermodynamic origin. Each broken chemical bond is viewed as an elementary configurational excitation called configuron. Using the Doremus' model of viscosity, the entropies and enthalpies of formation of configurons are obtained using a fitting process of experimental viscosities [5,6]. The glass-liquid transition has recently been treated within configuron percolation theory as a percolation-type phase transition with formation of dynamical fractal structures near the percolation threshold [7]. Specific heat jumps have been predicted. The formation of percolation structures made of high-density atom configurations in the glassy state was earlier suggested [8]. The nucleation of icosahedral clusters was also reported. Evteev et al studied, by molecular dynamics, atomic mechanisms of pure iron vitrification and showed that it is ensured by the formation of a percolation cluster from mutually penetrating and contacting icosahedrons with atoms at vertices and centers [9][10][11]. A fractal structure of icosahedrons incompatible with translation symmetry plays the role of binding carcass of the glassy state [10]. Wool developed the twinkling fractal theory following these initial ideas and found an explanation of relaxation phenomena near the glass transition. Clusters grow with fractal structures at the correlation length from molecular type units such as icosahedral superclusters, that are more and more frozen when the temperature decreases and lead to a disordered material [12,13]. Berthier et al showed that a growing length scale is accompanying the glass transition [14]. The twinkling dynamics of polystyrene was captured via atomic force microscopy within its glass transition region. Successive two-dimensional heights reveal that percolated clusters have lifetimes depending on their size and exist for longer time scales at lower temperatures. The computed fractal dimensions are shown to be in agreement with the theory of the fractal nature of percolating clusters [15]. All these models describe a liquid-liquid transition with liquid structures also containing superclusters below and above the glass transition and even beyond the melting temperature T m [16][17][18]. The same structure is progressively transformed and frozen by percolation instead of a new liquid phase growth from a critical nucleus. Kirpatrick and Thirumalai proposed a theory for the structural glass transition that is based on using frozen density fluctuations surrounded by surface energy to characterize it [19]. A random first-order phase transition is expected and hidden below the glass transition. The Kauzmann temperature would be the true glass transition at equilibrium [20]. Angell showed how such a theory for fragile liquids fits into a phenomenological scheme covering all glass formers [21]. A criterion analogous to the Lindemann criterion of melting was established for the glass-liquid transition [22]. It could be due to a vibrational instability of atoms located in the lattice sites. The new liquid state obtained just below T g has been still described by "means of reverse Monte Carlo (based on neutron scattering data) and molecular dynamics simulations showing that metallic glasses are composed by tiny icosahedral-like clusters, most of which are touching and/or interpenetrating yielding a microstructure of polyicosahedral clusters that follow a specific sequence of magic numbers" [23].
The glassy state relaxes more and more enthalpy when the temperature decreases below T g . Spin glasses also relax a heat maximum when their remnant magnetization is saturated after applying a high magnetic field below the phase transition [24]. There is no more relaxed energy when the magnetization is equal to zero at thermodynamic equilibrium. The relaxed enthalpy below T g seems to be saturated after a relaxation time depending on the observation temperature. Its maximum cannot be larger than the available enthalpy which is frozen between T g and the Kauzmann temperature T K where the liquid entropy excess against that of crystal goes to zero [25]. There is a need to predict the glass enthalpy reduction accompanying the stable-glass formation in this temperature window. In our model, a change of the liquid Gibbs free energy occurs at the glass transition. The change associated with the crystallization of an undercooled liquid above T g contains the classical contribution ×H m /V m but also an enthalpy saving equal to  ls ×H m /V m where is equal to (T-T m )/T m, T m being the melting temperature, H m the fusion heat per mole and V m the molar volume. This quantity corresponds in metallic liquids to the difference of conduction electron Fermi energies in crystals and their melt. At the vitreous transition temperature T g and at lower temperatures, there is a transformation of  ls in  lgs with a decrease equal to  lg = ( ls - lgs ). The enthalpy saving formulation  lg ×H m /V m of the equilibrium glass phase below T g has already been proposed and applied to Pd 43 Cu 27 Ni 10 P 20 [26]. But now, the coefficients  ls and  lgs in fragile liquids can be directly determined without using the specific heat jump at T g . This model is based on the homogeneous nucleation of new n-atom superclusters characterized by a surface energy and an energy saving  lg ×H m × n/N A driving the formation of a stable-glass phase and being proportional to n where  lg () depends on n and  N A being the Avogadro number. The Gibbs free energy change associated with the formation of these new (2) The measured isothermal relaxation time just below T g of a quenched melt can be viewed as being equal to  ns without including any contribution of t sn because only a liquid-liquid transition is considered. This is the time required for an equilibrium distribution of atoms to be established during the liquid-liquid transition preparing the steady-state nucleation of a vitreous phase [31,37]. The new liquid-phase formation is accomplished when the time-lag  ns is evolved. The coefficients K in (1-3), respectively called K ls at T n and K lgs at T g , are nearly equal in this scheme in spite of their strong thermal dependence on the melt viscosity ratio ln(/ 0 ) = B/(T-T 0 ) [38]: ( Consequently, the A value in (3) is much stronger below T g than above T g [32]. There are two timescales above and below T g . The time-lag above T g , leading to a nucleus distribution ready for steady-state nucleation, is about 10 6 times larger than the time-lag required below T g . In Turnbull and Fisher's model, lnA is nearly equal to ln(N A k B T g /V m h)f*/k B T g , where h is the Planck's constantandf*/k B T* g a thermally-activated energy barrier for atom diffusion from the melt to the homogeneously-nucleated cluster [39]. This diffusion barrier f*/k B T g is reduced at T g during cooling.
This expected change of activation enthalpy for diffusion was observed above and below T g in diffusivity measurements of impurities introduced in various glasses [40]( Figure 12).
The stable-glass formation starts by homogeneous nucleation of condensed superclusters when the time lag  ns and their steady-state nucleation time t sn depending of their n-atom number are evolved. The energy barrier for growth being much too high, the stable phase is built by percolation and interpenetration of elementary superclusters containing magic atom numbers and accelerated by reduction of the interconnected supercluster surface energy. It is shown in appendix B that a TTT diagram of the stable vitreous phase exists below T g predicting the complete transformation of the liquid in stable phase with a minimum value of t sn for a maximum value of n occurring at the Kauzmann temperature.
The viscosity above T g , in many examples of fragile glass-forming melts, perfectly obeys a scaling law, with a Vogel-Fulcher-Tammann (VFT) temperature equal to 0.77×T g [41,42]. The homogeneous nucleation temperature T 1 in a fragile melt above T g calculated without any reduction of the critical energy barrier by a small homogeneously-condensed nucleus also follows a scaling law that is a linear function of  ls0 . The comparison of theoretical and experimental scaling laws leads to the conclusion that the difference  lg0 between  lso and  lgs0 in fragile liquids is equal to 0.5× g [ [43]. The relaxed enthalpy maximum in the transformed liquid phase is equal to  lg0 ×H m in strong and fragile liquids without including the exothermic formation heat of the stable-glass phase.
The model is able to determine the boundaries separating fragile from strong liquids [29,32]. The free-volume disappearance temperature T 0m is less than T m /3 in strong liquids and greater than T m /3 in fragile liquids. The transition at T g is described by the formation at equilibrium of a new liquid phase characterized by an energy saving  lg ×H m . The derivative H m ×d( lg )/dT is used to calculate the specific heat jump per mole at T g and the enthalpy saving varying from T g to the Kauzmann temperature T K where the liquid entropy excess compared to that of crystal goes to zero [25,26]. The equilibrium enthalpy change at T g is predicted. The presence or absence of equilibrium latent heat at T g is analyzed. The value of the Kauzmann temperature T K has already been determined in some glassforming melts, observing that the specific heat jump between the undercooled liquid and the vitreous phase is nearly constant because the relaxed enthalpy has a nearly-linear decrease with the temperature increase [31,32]. The transition at T g leads to a much less fragile liquid with a temperature T 0g lower than T 0m , and consequently in all cases to a stronger behavior below T g . The following thermodynamic quantities are calculated: the coefficients  ls and  lgs above and below T g , their difference  lg (T), the specific heat jump C p (T g ), the temperatures T 0m and T 0g (T 0m > T 0g ) at which the coefficients  ls and  lgs tend to zero in the undercooled and vitreous states respectively, the frozen enthalpy H g at T g , the relaxed ultimate enthalpy H r and the frozen enthalpy below T g only knowing T g , H m and the melting temperature T m . The specific heat changes between T K and T g are predicted and used to determine the Kauzmann temperature of many fragile glass-forming melts.
About one third of fragile glass-forming melts do not follow the scaling law governing the viscosity above T m . They are characterized by a larger energy saving  lg leading to a larger specific heat jump. A reversible additional latent heat L  over that of liquids obeying scaling laws above T g is produced by heating and cooling through T g . Values of the total latent heat L + obtained by heating, including the recovered relaxed enthalpy and Lwhen it exists, are proposed assuming that the new liquid properties continue to follow a universal scaling law below T g even when the energy coefficients for crystal nucleation are not separated by a universal value of 0.5× g . The strong liquids have a specific heat jump C p (T g ) that is smaller than that of fragile liquids accompanying the decline of T om . Their transition at T g occurs without latent heat during cooling and corresponds to  lg (T g ) = 0. Their specific heat jump at T g has to be known to determine  0m ,  0g ,  lg ,  ls and  lgs .
The effective critical energy barrier determined by the energy saving of stable vitreous clusters, being smaller than the homogeneous nucleation critical barrier, controls the nucleus growth and a possible phase change. This is not the first time that the state of the undercooled liquid has been viewed as being composed of long-lived structures created in the normal-liquid structure that is locally favored by a free energy decrease when the temperature decreases. These locally-favored structures may lead to a liquid-liquid phase transition [2,43]. The model developed in this paper and completed in Appendix A demonstrates that this transition occurs in all liquids at T g and that elementary superclusters are condensed at T K in fragile liquids after a very long ageing time, leading to a stable vitreous state composed of these numerous tiny entities instead of a single supercluster inducing the condensation of the whole vitreous phase because the critical effective nucleation barrier of the melt is too large to get over it. Thermodynamics shows that stable vitreous phases have to exist in all glasses. The recent discovery of ultra-stable glasses obtained by physical vapor deposition is a strong signal in favor of such analysis [44,45].
The TTT diagrams of three bulk metallic glass formers (BMG) in crystallized phases are reproduced in Appendix B calculating the effective thermally-activated energy barrier G eff /k B T of the critical nucleus for crystal growth [29]. The critical nucleus is a 13-atom cluster having an effective energy barrier higher than that of bigger clusters. The condensation temperature of a single 13-atom cluster determines the crystallization temperatures of a whole sample along its TTT diagram. The energy saving coefficient of this type of cluster containing a small number of atoms embedded in glass-forming melts is quantified. The quantified value  ls0 of a 13-atom nucleus deduced from the experimental TTT diagram is equal to 0.7, while those for 13-atom superclusters involved in stableglass phase formation below T g are smaller.
The following plan is proposed: 2-Gibbs free energy change associated with growth nucleus formation, 3-Thermal dependence of the energy saving coefficient  nm of an n-atom condensed cluster, 4-Critical supercluster homogeneous nucleation temperature and effective nucleation temperature, 5-The two homogeneous nucleation temperatures T 1 and T 2 and the equilibrium enthalpy change of glass-forming melt at the vitreous transition T g = T 2 , 6-Scaling laws, 7-The specific heat jump at fragile-to-fragile, strong-to-strong, and strong-to-fragile liquid transitions at T g , 7.1-Fragile-to-fragile liquid transition, 7.2-Strong-to-strong liquid transition, 7.3-Strong-to-fragile liquid transition, 8-Specific heat jumps from metallic and non-metallic glasses to undercooled liquids at the vitreous transition, 9-Enthalpy thermal cycles expected in some liquids and determination of the Kauzmann temperature, 10-Stable-glass supercluster nucleation rates between T K and T g , 11 Fragile-to-fragile liquid transition at T g always occurring above T m /2, 12-Hidden freezing at T g before relaxation of quenched liquids, 13-Transformation of the stable-glass Indomethacin in undercooled liquid at T g , 14-Conclusions, Acknowledgments, Appendix A: TTT diagrams of several liquids in stable-glass phase between T K and T g , Appendix B: Superclusters of 13 atoms governing the first-crystallization time of metallic glassforming melts. References.

Gibbs free energy change associated with growth nucleus formation
The classical Gibbs free energy change for a nucleus formation in a melt is given by (4): where R is the nucleus radius and  1ls is its surface energy. Turnbull has defined a surface energy coefficient  1ls given by (5) which is equal to (6) [38,46]: An energy saving per volume unit - ls ×H m /V m is introduced in (4) for a critical nucleus formation above T g ; the coefficient  ls being replaced by  lgs for a critical nucleus formation below T g . The coefficients  ls and  lgs have to be calculated in order to determine their difference  lg which determines the stable-glass formation enthalpy below T g when the quenched liquid escapes crystallization. The new Gibbs free energy change is given by (7), where  2ls is the new surface energy [28]: The new surface energy coefficient  2ls is given by (8): The critical radius R* 2ls in (9) and the critical thermally-activated energy barrier G* 2ls /k B T in (10) are calculated assuming d ls /dR = 0: .
They are no longer infinite at the melting temperature T m when  ls is not equal to zero. The homogeneous nucleation temperature T 2 (or  2 ) occurs when the nucleation rate J in (2) is equal to 1 and (11) is respected: The unknown surface energy coefficient  2ls in (12) is deduced from (10) and (11): The surface energy  2ls in (8) has to be minimized to obtain the homogeneous nucleation temperature T 1 (or  1 ) for a fixed value of  ls . The derivative d 2ls /d is equal to zero at the temperature T 1 (or  1 ) given by (13) assuming that ln(K) does not depend on the temperature: The enthalpy saving between solid and liquid states is determined by the knowledge of  1 .
The thermal variation of ln(K ls ), being a function of viscosity, does not modify the value of T 1 (or  1 ) as shown below [29]. The critical energy barrier in (10) is the product of a function g() and ln(K ls ). The maximum nucleation rate J still occurs at the temperature T 1 with the derivative d(lnJ)/d being equal to zero because g() is equal to 1 and dg()/d = 0: The surface energy coefficient  2ls is now given by (14) replacing by (13) in (12) The classical nucleation equation (4) has been transformed into (15) The Laplace pressure p can be calculated from the surface energy  2ls with the equation (13) [32,33]: The complement p of Laplace pressure introduced by the energy saving  ls is equal to  ls ()×H m /V m . All equations (7-16) for a critical nucleus formation can be applied below T g after replacing  ls by  lgs and T 1 (or  1 ) by T 2 (or  2 ). The equations governing the stable-glass supercluster formation below T g are developed in section 10. The Gibbs free energy change G 2ls in (15) directly depends on the cluster atom number n and the energy saving coefficient  nm of the cluster instead of depending on its molar volume V m and its radius R, as shown in (17): 9 This equation can be applied below and above T g using different values of  nm corresponding to various Laplace pressures.
The formation of superclusters having a weaker energy barrier precedes the formation of crystallized nuclei in an undercooled melt [9,16,47,48]. A supercluster containing n atoms could be easily transformed into a crystal of n atoms having the same energy saving coefficient  nm and the same surface without changing G nm . The transformation of superclusters into crystals could occur when their molar volume and consequently their coefficient  nm becomes equal to those of crystals. This condition could be not sufficient because crystals of n atoms could be facetted, less-spherical and their surface not minimized. The supercluster energy saving  nm ×H m is quantified, depending on the radius and atom number n, and can be smaller or larger than the critical energy saving  ls ×H m in glass-forming melts above T g and larger than the critical energy saving  lg ×H m of the stable-glass phase appearing below T g . The critical growth barrier G* nm of an n-atom supercluster also depends on  nm . It can be larger than the critical growth barrier G* 2ls . The effective critical energy barrier of the smallest homogeneously-condensed cluster can control the heterogeneous growth around it. It is the case for the crystallization of glass-forming melts at temperatures higher than the homogeneous nucleation temperature defined by (13).

Thermal dependence of the energy saving coefficient  nm of an n-atom condensed cluster
All superclusters which are formed in an undercooled melt preparing crystal formation grow first, are submitted to a complementary Laplace pressure and have a surface energy because the Gibbs free energy change contains an enthalpy saving [9,27,48]. The energy saving coefficient  nm of an natom supercluster giving rise to crystals has to be a function of  2 to survive above T m , assuming that  nm is maximum at T m , d nm /dT being equal to zero and fixing the supercluster fusion entropy as being equal to the fusion entropy S m of the bulk solid [29]: In these conditions, the cluster fusion occurs above T m and is governed by liquid droplet homogeneous nucleation above T m rather than by surface melting. This  2 thermal variation explains the undercooling rate of liquid elements [28,48].
The law (18) is expected to work also in glass-forming melts as observed in liquid elements,  nm being the quantified energy saving coefficient of an n-atom supercluster: A n-atom supercluster is submitted to a Laplace pressure which is strongly dependent of its radius. The quantified value  nm decreases with  2 and becomes equal to zero when the temperature equals T 0m (or  0m ). An atom which does not belong to a supercluster cannot be involved in the nucleation of a new phase.
The critical parameters for supercluster growth are determined by an energy saving coefficient called  ls (or  lgs ) in (19): The enthalpy saving per mole  ls ×H m tends to zero when  tends to  0m . The liquid free volume would be equal to zero at T = T 0m in the absence of transformation at T g .
A critical supercluster contains a critical number n c of atoms given by (20):

Critical supercluster homogeneous nucleation temperature and effective nucleation temperature
The thermally-activated critical energy barrier is now given by (21), where  ls is given by (19): The coefficient of ln(K ls ) in (21), called g(), becomes equal to 1 at the homogeneous nucleation temperature given in (13) and the equation (11) is respected. Homogeneously-condensed clusters of natoms act as growth nuclei at a temperature generally higher than the homogeneous nucleation temperatures T 1 and T 2 of liquids. The cluster thermally-activated critical energy barrier G* nm /k B T and its effective thermally-activated critical energy barrier are given by (22): where G nm is given by (17) and  nm by (18). These nuclei are ready to grow when the transient nucleation time  ns and the steady-state nucleation time t sn are evolved and the relation (23) respected: Three situations are met when the two critical energy barriers in (21) and (22) are compared. The growth around nuclei can be very rapid if G* 2ls /k B T given in (21) is smaller than G* nm /k B T given in (22). The steady-state nucleation time t sn (T) is calculated with (23) knowing the sample volume v. This is the case for glass-forming melt crystallization above T g . When G* 2ls /k B T is larger than G* nm /k B T, the new effective critical energy barrier is equal to (G* 2ls /k B TG nm /k B T) and has to replace G neff /k B T in (23). The effective heterogeneous nucleation temperature is strongly dependent on the volume v. This phenomenon is very important in liquid elements where the effective nucleation temperature is observed around  = 0.2 in sample volumes of few mm 3 instead of  varying from  0.58 to0.3 with much smaller samples [48,49]. The growth around clusters can be slow and shell-byshell as a function of time when G* 2ls /k B T is much larger than G* nm /k B T. It is the case for the glass phase nucleation critical barrier G* 2lg /k B T which is much larger than the critical barrier associated with an elementary n-atom cluster G* nm /k B T.

The two homogeneous nucleation temperatures T 1 and T 2 and the equilibrium enthalpy change of glass-forming melt at the vitreous transition T g = T 2
The homogeneous nucleation temperature T 1 (or  1 ) obeys the equations (13) and (19) and is a solution of the equation (24): where  0m (or T 0m ) is the temperature corresponding to  ls = 0. At the vitreous transition T g , there are changes of  lso in  lgs0,  1 in  2 (or T 1 in T 2 ) and 0m in  0g (or T 0m in T 0g ). A new equation (25) is used to calculate  2 (and T 2 ): There are two values of  1 and  2 respecting (24) or (25) for each value of  0m or  0g ; the largest value given in (26) is the chosen solution for  1 and  2 : The fragility of melts defined by Angell is larger when  0m increases [50]. In strong liquids, the temperature  1 can be calculated using (26) when  ls0 ≤ 2 and  0m ≤ 2/3 are known. There is a double solution for a fragile liquid and a minimum value of  ls0 ≥ 1 for each value of  0m ≥ 2/3 (or T 0m > T m /3) [ [51]]. The boundary separating strong from fragile liquids is  0m =2/3 (or T m /3). The values of  1 and  0m in fragile liquids are given as a function of  ls0 in (27) and (28) with "a" being unknown: The thermodynamic transition at T g is induced by a liquid-liquid transformation at the temperature T 2 (or  2 ) minimizing the energy saving  lgs of the new liquid state for a given value of  0g . Any other transition temperature leads in fragile liquids to a too large value of  lgs0 and to an energy relaxation. The thermodynamic transition temperature T g has to be also equal to T 2 . The thermodynamic parameters characterizing the new liquid state at equilibrium are  lgs0 = (1.5× g +2) and  0g , given in (27) and (28) after replacing  ls0 by  lgs0 ,  1 by  g and  0m by using  0g . The transformation of any liquid in strong liquid at T g will occur when the difference  lg = ( ls - lgs ) becomes equal to zero because there is no minimum of  lgs0 as a function of  2 in strong liquids.
A crystallization at T g = T 2 is not possible because the homogeneous nucleation rate of a critical supercluster is only equal to 1 m -3 .s -1 . In addition, it cannot start from a 13-atom supercluster because its own nucleation time is much too large. The energy saving coefficient, driving the stable glass phase formation in the new liquid state as a function of temperature, has to vary at thermodynamic equilibrium as  lg = ( ls - lgs ) as shown below. The enthalpy decrease per mole from quenched undercooled liquids, aged at temperatures less than or equal to T g to lead to stable vitreous states, is equal in (29) to the difference between the complementary energy savings for a crystal formation below T g [26] : This difference  lg occurring at T g and below T g is only due to the change of the equilibrium enthalpy between T g and the Kauzmann temperature T K . It contains a temperature-dependent positive contribution and a constant negative contribution which is viewed as the ultimate relaxed enthalpy at T K for quenched liquids respecting the scaling laws above T g . The stable-glass-to-fragile liquid transformation of indomethacin at equilibrium is accompanied by an endothermic latent heat at T g , as shown in Figure 1. A strong-glass-to-strong liquid or a strong-glass-to-fragile liquid transformation occurs without latent heat when  lg = 0 instead of being due to kinetic effects [32]. 2 . The enthalpy saving coefficients  ls and  lgs for critical nucleus formation, proportional to the complementary Laplace pressure p, for an indomethacin nucleus containing a critical atom number n c (T) respectively above and below the transition  g = 0.264, are plotted versus  2 . The differences ( ls - lgs )×H m =  lg ×H m at T g = 318 K and at the Kauzmann temperature T K determines the equilibrium latent heats involved in the stable-glass-to-fragile-liquid transition at T g and T K . The coefficient  lg given in (29) is plotted versus  2 between T m and T K. .

Figure 1. Enthalpy saving coefficients versus 2 = [(T-T m )/T m ]
An example is given in Figure 2 to illustrate the change of energy saving at T 2 = T g in fragile-tofragile liquid transformations. The coefficients  ls0 and  lgs0 of the metallic glass-forming melt Pd 43 Ni 10 Cu 27 P 20 are calculated using T 0m = 430 K and T 0g = 365 K as a function of the homogeneous nucleation temperature T 2 . The energy saving coefficient  ls0 for a crystal formation is equal to 1.718 for  1 = 0.188. The coefficient  lgs0 has a minimum equal to 1.577 at  2 = 0.282. All values of  lgs0 below and above  2 = -0.282 are larger. A liquid quench down to T a is followed by enthalpy relaxation from  ls0 = 1.718 down to  lgs0 = 1.577. The thermodynamic glass transition has to occur at  g =  2 because there is no more enthalpy relaxation at T g = T 2 [31][32][33]. The existence of the time-lag  ns , strongly varying with the temperature, has for consequence that the glass transition can occur in a wide window of temperatures. The energy saving coefficients  ls0 (triangles) and  lgs0 (diamonds) of Pd 43 Ni 10 Cu 27 P 20 have been calculated as a function of the homogeneous nucleation temperature T 2 using T 0m = 430 K, T 0g = 365 K, T m = 802 K and (25). Note the minimum values of  ls0 and  lg0 given by (27) and (28) for "a" = 1. The transition at the temperature T* g of the minimum transforms  ls0 into  lg0 and T 0m into T og during the relaxation time. A quench down to T a also leads to a transformation of  ls0 into  lg0 due to the existence of a minimum relaxation time between the two liquid states of about 100 s.
Slow physical vapor deposition rates produce vitreous samples having a new local packing arrangement as compared to that of bulk samples below T g and being at thermodynamic equilibrium. Indomethacin has been deposited on substrates cooled around the Kauzmann temperature in a highly stable vitreous state. This phenomenon of ultra-stable glass formation could be induced by "a firstorder transition separating the normally observed high temperature liquid from a new low-temperature equilibrium supercooled liquid" [44]. The liquid-to-glass transformation is obtained at T g after 30,000 s instead of 100 s and accompanied by a strong change of local packing arrangement, as observed by wide-angle X-ray scattering measurements. These observations are compatible with the existence of a true liquid-to-glass transition which is produced at equilibrium by an endothermic latent heat during heating at T g after an exothermic latent heat at T K . The progressive increase of the enthalpy saving cannot continue below the Kauzmann temperature [25]. The enthalpy excess of the undercooled liquid can only relax in a window of temperatures extending from T g to T K . A saturation of the relaxed enthalpy has already been observed at T K [52].
The existence up to T g of highly-stable glasses, when they are prepared at equilibrium by physical vapor deposition, has the result that the current transformation at T g , observed up to now without latent heat, is a liquid-to-liquid transition occurring in a pseudo-equilibrium state without enthalpy relaxation at a well-defined temperature T g . The isothermally-relaxed enthalpy decreases when the annealing temperature increases up to T g and the enthalpy recovery measured at T g has to be equal to zero. This pseudo-equilibrium liquid state is not the equilibrium vitreous state and is obtained after an isothermal relaxation at a temperature T a higher than T K and lower than T g . Equation (29) shows that the ultimate relaxed enthalpy to attain the new undercooled liquid state at T K is equal to  lg0 ×H m = ( ls0 - lgs0 )×H m and is recovered and included in the endothermic latent L + at T g . A complementary exothermic enthalpy has to be relaxed at T K after a long steady-state nucleation time to give rise to the ultra-stable vitreous state. This exothermic latent heat produced at T K has to be equal to the endothermic enthalpy at T g given in (29) due to the thermodynamic character of the liquid-tostable-glass transition.

Scaling laws
Equations (28)(29)(30) are scaling laws determining the new energy saving coefficient  lgs0 occurring at  2 =  g and the temperatures T 0g (and  0g ) and T 0m (and  0m ) of fragile liquids from the knowledge of the thermodynamic transition temperature, which is defined as the disappearance temperature of the relaxed enthalpy [31,33]. The homogeneous nucleation temperature T 1 (or  1 ) is determined as a function of  ls0 using (27) and follows a scaling law because, in many examples of fragile liquids, the viscosity is scaled by a VFT temperature T 0 = 0.77 T g with 0.35<  g < -0.2 [41,42]. Scaling laws are easily obtained from (30)(31)(32), with a = 1 leading to  ls0 =  g +2,  lg0 = 0.5× g and to a latent heat during heating equal to 0.25× g due to the transformation of the stable glass in an undercooled liquid state: The equilibrium enthalpies at  =  K and  g contain in (32) a contribution independent of temperature and a contribution proportional to  2 . This equilibrium enthalpy has a minimum value at  K when the undercooled liquid is the most unstable. It contains the maximum relaxed enthalpy at T K which is equal to 0.5× g ×H m and recovered at T g , the endothermic latent heat (a-1) g ×H m delivered at  g when "a" is less than 1 and a contribution depending on  . The latent heat L + ( g ) becomes endothermic by heating at  g and equal to H ( g ). The latent heat L   g   obtained by cooling is equal to zero for aand only exists when "a" is less than 1 because the liquid is more fragile than a liquid with a = 1 with its higher value of T 0m . Measurements of As 2 Se 3 and As 2 S 3 showing the existence of a "glass-formation heat" obtained by cooling at T g demonstrates the existence of L  [52]. The total endothermic latent heat produced by heating at T g would be equal to L + for a << 1 and to the enthalpy recovery L + = 0.25× g ×H m for a = 1 if the stable glass phase has been previously formed by a long ageing at T K . There is no latent heat associated with a stable-glass formation at T K in a strong liquid. It does not exist because  lg is equal to zero at T g while a relaxed enthalpy equal to  lg0 ×H m is still produced at T K when the transient relaxation time  ns is not evolved after quenching. The transformation in a stable glass is realized when this time-lag is evolved.

7.
Specific heat jumps at the fragile-to-fragile, strong-to-strong, strong-to-fragile liquid transitions at T g

Fragile-to-fragile liquid transition
A fragile-to-fragile liquid transition induces a new liquid state. The enthalpy derivative [d(H)/dT] calculated using (29) or (32) is equal to the specific heat difference C p (T) between a quenched fragile undercooled liquid and its new equilibrium state given by (33), as already shown in 2008, without knowing at that time the exact values of thermodynamic parameters [26]: The equilibrium specific heat jump below T g is defined by (34): When a = 1, the scaling law is obeyed and the jump at T g is equal to (35): Equation (35) is respected in many glass-forming melts as shown by [37](p. 48) with many C p (T g ) jumps extending Wunderlich's previous finding [53]. The experimental values of C p (T g ) are used in part 8 to calculate the number "a" when (35) is not respected, including the experimental uncertainties on the measurements of various thermodynamic parameters.

Strong-to-fragile liquid transition
A strong-to-fragile liquid transition also induces a new liquid state. This phenomenon occurs when the fragile-to-fragile liquid transition temperature is expected to be a little lower than 0.5×T m ( g < 0.5). The relation (28) is obeyed because the liquid is fragile above T g with  0m > 2/3 and becomes strong below T g with  og ≤.2/3. Equations (28,(36)(37)(38) are still used in part 7.3 to calculate ( ls0 - lgs0 ),  om , T 0m , C p (T g ),  ls0 ,  lgs0 and  0g , which is now respecting the inequality -1 <  0g < 2/3 (and 0 < T 0g < T m /3). The Fe 50 Co 50 glass-forming melt transition is a good example of this phenomenon [54]. The thermodynamic parameters of several glasses undergoing this type of transition are presented in Table  1-1. The quantities T 0m , T 0g ,  = ( ls0 - lgs0 ) given in Table 1

Strong-to-strong liquid transition
A strong-to-strong liquid transition also induces a new stronger liquid state assuming that  lg in (31) is equal to zero at T g in the absence of minimum values of  ls0 and  lg0 when T 0m is less than T m /3. In this case, the specific heat jump becomes smaller than (35) and equal to (36): Equation (26) applied at the vitreous transition is used to determine  lgs0 with (37): The specific heat jump is much smaller in strong glasses because the glass viscosity has to follow an Arrhenius law with T 0g = 0 K and  0g = 1 instead of a negative value of T 0g which would increase C p ; the stronger the glass-forming melt, the smaller the ( ls0 - lgs0 ) value. Equations (36,38) are used to calculate ( ls0 - lgs0 ),  om , T 0m , C p (T g ),  ls0 and  lgs0 : 17 The transformation parameters of strong liquids are given in Table 1-2. The temperatures T og are chosen equal to 0 K and T 0m equal to the VFT temperatures [55,56]. The known values H m , T g and T m are used to calculate the specific heat jump C p (T g ), the energy saving coefficients  ls0,  lg0 above and below T g and their difference. The jumps C p (T g ) per g.atom are very small compared to a crystal specific heat equal to 25 J/at.g.K. The calculated and experimental values are 2.1 and 2.05, 1 and 2.6, 1 and 0, 1.5 and 2.09 respectively. They are in good agreement considering a measurement uncertainty of about 0.5 to 1 J/at.g.K.

Specific heat jumps from metallic and non-metallic glasses to undercooled liquids at the vitreous transition
The specific heat differences C plx (T g ) between some fragile metallic liquids and crystals are indicated in column 8 of Table 2. The specific heat jumps C plg (T g ) at the fragile-to-fragile liquid transition given in Table 2 and Table 3 with a = 1 are equal to 1.5×S m as predicted by (35) and in agreement with other reports, within the measurement uncertainties of specific heat, fusion heat and melting temperature of all liquids. It has been recently found that the jumps C plg of many metallic glasses are equal to 13.7 ± 2 J/K/at.g [57]. Their fusion entropy is expected to be equal to 9.13 ± 1.3 J/K/g.at applying (35), as already observed in many metallic liquid elements [46]. Values of C plg (T g ) of materials N°3 and N°6 in Table 2 are deduced from the slope of the maximum relaxed enthalpy versus temperature [31].
Liquids with a << 1 in Tables 2 and 3 have a larger specific heat jump. The "a" values in Figure 3 are calculated at T g with (34) using the experimental values of C plg (T g ) and represented as a function of T g /T m . The transition temperatures T g , which are used in all tables to calculate thermodynamic parameters, are close to the thermodynamic transition temperatures where the relaxed enthalpy is equal to zero [31,33]. This approximation has a weak influence on them. Values of "a" larger than 1 are used to define in Figure 3 an experimental uncertainty of ± 6.5% and of ± 13% on the specific heat jump in this model. The "a" values equal to 1 in Table 2 and Table 3 correspond to this uncertainty. The fusion enthalpies of ZnCl 2 N° 50 and B 2 O 3 N°51 have been reduced to respect (35) because of the existence of crystallographic instabilities under pressure and then under Laplace pressure and of hidden polymorphs [58,59]. There are 36 glass-forming melts among 49 following the scaling law (35), with the 13 others following (34) with values of "a" smaller than 1.

Enthalpy thermal cycles expected in some liquids and determination of the Kauzmann temperature
The ultimate relaxed enthalpy is the maximum relaxed enthalpy occurring at equilibrium at the Kauzmann temperature. The total equilibrium enthalpy changes at T g equal to the latent heat L + and Las defined by (32) are given in Tables 4 and 5. The relaxed enthalpy of As 2 Se 3 is saturated at the Kauzmann temperature, equal to 6.4 J/g corresponding to 2.48 kJ/mole and is approximately equal to that given by the scaling law 0.5× g ×H m = -2.21 kJ/mole [52]. This glass-forming melt does not follow the scaling law above T g , as shown in Table 3 N°17. Its thermodynamic parameters given in Table 3 and 5 are T m = 645 K, T 0m = 380 K, T 0g = 293 K,  g = 0.283, a = 0.753, H m = 15.6 kJ/mole, L + = 0.21×H m = 3.28 kJ/mole and L -= 0.07×H m = 1.09 kJ/mole. An endothermic enthalpy L + of 3.5 kJ/mole (9.04 J/g) corresponding to  lg = 0.224 has been measured at T g by transforming the liquid state below T g into the undercooled liquid state above T g after a long ageing time near T K equal to 166 hours. The application of (32) using a = 0.753 leads to a latent heat L  = 0.07×H m = 1.09 kJ/mole and L + = 0.210×H m = 3.28 kJ/mole which are approximately equal to experimental values of 1.07 kJ/mole and 3.5 kJ/mole respectively [52] (Table 1). These experimental results confirm that the ultimate relaxed enthalpy always respects the scaling law whatever the number "a" may be and that an exothermic latent heat can be observed at T g while cooling a glass-forming melt through the vitreous transition when the number "a" is much smaller than 1 [33].
The ultimate enthalpy recovery of butyronitrile has also been studied by measuring the relaxed enthalpy after vapor deposition at 40 K, far below T K . It is equal to 1.3 kJ/mole and larger than 0.5× g ×H m = 1 kJ/mole and smaller than the equilibrium enthalpy of the stable glass phase equal to 1.5 kJ/mole expected at T K including the latent heat 0.25× g ×H m associated with a stable glass formation when a = 1 [133]. The out-of-equilibrium entropy of the undercooled liquid remains larger than that of crystals. In these conditions, the ultimate relaxed enthalpy H r equal to 0.5× g in all liquids can be used to calculate T K considering that all thermodynamic properties are obeying scaling laws below T g . The maximum relaxed enthalpy decreases with temperature. Its derivative dH r /dT is equal to a specific heat difference C plg (T), being nearly constant below T g and nearly equal to C plg (T g ).
The Kauzmann temperature T K is calculated using the mean theoretical specific heat below T g deduced from (33) and imposing the ultimate enthalpy recovery to be given by (39). The  K is finally given by (40) with the number "a" determined by imposing the theoretical specific heat jump to be equal to the experimental one: In many cases for which a = 1, the scaling law The equilibrium enthalpies divided by the fusion heat are equal to  lg and represented in Figures 4, 5 and 6 for Pd 43 Ni 10 Cu 27 P 20 BMG N° 2 (a = 1), indomethacin Glass N°8 (a = 1) and As 2 Se 3 Glass N°17 (a = 0.776) as a function of temperature below T g . There is no latent heat expected by cooling with a = 1 because the temperature T g only corresponds to a liquid-liquid transition which attains a pseudo-equilibrium after relaxation. On the contrary, a partial latent heat equal to L  is expected for "a" << 1. The out-of-equilibrium liquid quenched below T g from high temperatures to T K can be totally transformed into a stable glass phase after ageing at T K when the nucleation time is minimum. The ultimate enthalpy has to be relaxed to attain first the pseudo-equilibrium liquid state and, after a much longer time, the stable glass phase producing an exothermic latent heat. An endothermic latent heat L + is needed to transform the stable glass phase into an undercooled liquid at T g . Figure 4. The stable glass phase formation at T K in Pd 43 Ni 10 Cu 27 P 20 . The enthalpy variation from the quenched to the equilibrium state of the stable glass phase obtained at T K divided by H m is represented by  lg as a function of temperature. The irreversible and endothermic latent heats at T g are equal to 0.25 × g ×H m (a = 1). 43 Ni 10 Cu 27 P 20 and indomethacin. The enthalpy saving of indomethacin below the vitreous transition T g represented by the coefficient - lg is plotted versus temperature and compared to that of Pd 43 Cu 23 Ni 10 P 20 . There is no reversible latent heat in the liquid-toliquid transition at T g even if a latent heat L + is expected during the heating of the stable glass phase. These two liquids undergo a phase transition at T g characterized by a change in the enthalpy slope at T g because "a" = 1. Pseudo-equilibrium enthalpies of undercooled melts are obtained during cooling after relaxation at the annealing temperature T. After a long ageing at T K , the transitions to the stable glass states would be accompanied by an exothermic latent heat and the stable glass enthalpies would increase up to T g . An irreversible endothermic latent heat equal to 0.25× g ×H m is needed to return to the equilibrium undercooled liquid state above T g . Figure 6. The formation of stable glass phase in As 2 Se 3 . The enthalpy variation divided by H m represented by - lg as a function of the temperature. A reversible latent heat occurs along AB and BA at T g during cooling and heating because "a" << 1. There is no more enthalpy change along BC when the undercooled liquid is rapidly cooled without being relaxed. The structural relaxation progressively transforms BC into BD and the undercooled liquid is transformed into a new liquid state, producing an exothermic relaxed enthalpy. After a long ageing along DE, the undercooled liquid is transformed into a stable glass phase. The glass phase enthalpy increases along EF by heating. An irreversible endothermic enthalpy is produced by heating along FB.

Figure 5. The stable glass phase formation between T K and T g in Pd
The temperature difference T K = (T g T K ) is compared in many of the examples given in Tables 4 and  5 to experimental values obtained by entropy extrapolations from temperatures above T g using specific heat laws measured from T g to T m . These quantities are nearly equal for "a" = 1 in many cases. For a < 1, they cannot be equal, as shown in Tables 4 and 5, because the contribution of the latent heat L  to the available entropy is not subtracted in the extrapolation method.
In addition, the liquid specific heat excess below T g is nearly constant, as demonstrated by a maximum relaxed enthalpy linearly decreasing with temperature up to T g as shown in Figures 4, 5 and  6. The presence of a hidden freezing before relaxation at T g is demonstrated in part 11 and the existence of a complementary reduction of T K for a < 1 is explained. The frozen enthalpy values H g given in Tables 4 and 5 are equal to (L + 0.5× g ×H m ) and always smaller than the available enthalpy pls m dT C H as already predicted [103]. The frozen entropy is also smaller than the available entropy because C plg  is always smaller than the specific heat difference C plx  between undercooled liquid and crystal. Table 4. Thermodynamics parameters of some bulk metallic glasses: The equilibrium endothermic latent heats L + and L  divided by H m , the difference T K = (T g T K ) and the frozen enthalpy H g /H m = (× g +L + ) below T g divided by H m of some fragile metallic glass-forming melts. Fusion heat units are kilojoules per g.atom. The maximum value of the relaxed enthalpy is equal to 0.5× g ×H m ; T K = T g -T K ; L + and L  are equilibrium latent heats.   Table 5. Thermodynamics parameters of some glasses: The equilibrium endothermic latent heats L + and L  divided by H m , the difference T K = (T g T K ) and the frozen enthalpy H g /H m = (× g +L + ) between T K and T g divided by H m of some fragile metallic glass-forming melts. The relaxed ultimate enthalpy is equal to 0.5× g ×H m . The fusion enthalpy of N°17 is changed using new measurements [52]. The ZnCl 2 and B 2 O 3 fusion enthalpies are also changed to respect C plg = 1.5×S m because there exists a crystallographic instability under pressure [58,59] and then under Laplace pressure.

Stable-glass supercluster nucleation rates between T K and T g
The Gibbs free energy change for a stable-glass nucleus formation is no longer given by (15) and is equal to (41) because the quantity H m /V m * is eliminated from the Gibbs free energy change leading to this new liquid state: The critical radius and the thermally-activated critical barrier are given by (42) and (43) instead of (9) and (10): The critical barrier in is high because the coefficient  lg is always small in all liquids. Then, the stable glass phase cannot directly grow from the critical radius and is formed by homogeneous formation of numerous tiny superclusters percolating, interpenetrating, and then growing by reduction of their surface energy with the time increase. The Gibbs free energy change associated with the formation of a stable-glass nucleus of radius R containing n atoms is equal to (44) instead of (17): The energy saving coefficient  nlg of a glass nucleus of radius R containing n atoms is given by (45): where the coefficient  lg0 = 0.5× g in (32) has been replaced by  nlg0 ,which is proportional to the complementary Laplace pressure and to 1/R when the inequality n ≥ 147 is respected. The value of  nlg0 is predicted using (46) at  = 0: where  lg0 is equal to 0.5× g below T g . At lower values of n, the energy saving is weakened by quantification and  nlg0 is strongly reduced [26,144]. In this particular case, the exact molar volume of superclusters being unknown,  nlg0 is better determined from the nucleation temperature of the stableglass phase which occurs at the Kauzmann temperature T K . The values of lnK lg or lnK ls are calculated with (1) knowing that the Zeldovich factor  is defined by (47), n c by (20) or (48) and G*/k B T by (21) or (43), and that the transient times of nucleation at T g or at the nose temperature T n of the TTT diagram of crystallization above T g are close to 50 s: The thermal variations of lnK ls and lnK lg in (3) respectively depend on B m /(T-T 0m ) and B g /(T-T 0g ), which are equal and deduced at T g from measurements of viscosity above and below T g . The glass and crystal steady-state nucleation times t sn given by (23) depending on the K value can be calculated as a function of the temperature when the effective thermally-activated energy barrier G neff /k B T given by (22) or (49) is known: where G nlg is given in (44). The crystallized or vitreous superclusters can grow beyond their own initial radius R when (23) or (50) is respected: The n-atom supercluster formations occur in the sample when their nucleation time is evolved. The atom number n in spherical superclusters is chosen equal to the following stable magic numbers which are considered in an icosahedral structure for metals with face-centered cubic lattices: 13, 55, 147, 309, and 561 [47]. The nucleation rate logarithm of n-atom superclusters lnJ n = ln(v.t sn ) is calculated without knowing the stable-glass domain volume v and the nucleation time t sn because the maxima of nucleation rates occur at the Kauzmann temperature T K and leads to a whole transformation of the liquid. The homogeneous nucleation rates lnJ n of n-atom superclusters ready for growth in three BMG; Pd 43 Ni 10 Cu 27 P 20 N°2, Pt 57.3 Cu 14.6 Ni 5.3 P 22.8 N°7, Cu 47 Ti 34 Zr 11 Ni 8 N°15, and four glasses; indomethacin G. N°8, A 2 Se 3 G. N°17, diethylphthalate G. N°32 and selenium N°43 represented in Figure 7 are calculated using the parameters given in Table 6 and (50). The enthalpy saving is given by (32) and all results presented here are obtained without introducing a constant equilibrium enthalpy saving below T K , in order to show that the model directly leads to the value of T K . The same maximum at T K is still observed when a constant enthalpy change is introduced below T K . A nucleation rate equal to exp (20.7)/m 3 /s would transform a liquid volume of 1 mm 3 into a stable glass in an additional time of 1 s if the glass domain could attain this volume by nucleus growth beyond the critical radius. It is not possible because the critical energy barrier given in (43) is always too high. Elementary clusters containing 13, 55, 147, 309, 561 and 923 atoms were studied. The numbers n = 147 or 309 lead directly to the highest maxima of lnJ n at T K using  nlg0 values obeying (46). The model applied to four glasses works without using any adjustable parameter for n = 147 and 309. The atom numbers n have been chosen in Figure 7 as the number n inducing the largest nucleation rate at T K . The steady-state nucleation rate depends on the fusion enthalpy H m as shown by (44) and (17). The examples given in Figure 7 cover a broad distribution of fusion heats and, consequently, various nucleation rates in glassforming melts. There is no stable glass nucleus being formed at T g . The supercluster nucleation rate has a maximum at T K in all these examples. The transition at T g cannot be described by supercluster 28 nucleation having a surface energy in spite of the knowledge of the enthalpy difference  lg ×H m /V m between undercooled liquid and stable-glass phase and the specific heat jump prediction. The transition at T g is a liquid-liquid transition characterized by the enthalpy change that is predicted in (32). Other liquid-liquid transition models involving superclusters of liquid nature are more successful to describe them at the microscopic scale [6][7][8][10][11][12][13][14][15]. Table 6. Parameters used to calculate the nucleation rates lnJ n , the nucleation times t and t sn of stable vitreous phases. The units are based on meter, kelvin, joule and second. The entropy S m is given per g.atom and V m in m 3 per mole. These droplets give rise to very tiny stable-glass domains and to maxima of nucleation rates at T K , as shown in Figure 7. The critical number of cluster atoms n c at the melting temperature T m , the nucleation rate logarithm ln(J n /m 3 /s) at T K of n-atom elementary superclusters, the extrapolated negative critical energy saving coefficient  lg0 at T m as shown in Figure 1, and the energy saving coefficient  nlg0 at T m ( = 0) of n-atom superclusters are given. The lnK lg value is determined assuming that the transient relaxation time at T g is equal to 50 s.  They are plotted versus the reduced temperature below  g . The unit of J n is m -3 .s -1 . The nucleation rates of stable-glass superclusters are negligible at T g , while they are high and maximum at T K . A liquid-toliquid transition occurs at T g without stable-glass supercluster formation in Pd 43

Fragile-to-fragile liquid transition at T g always occurring above T m /2
All the fragile glass-forming melts in Tables 2 and 3 have a transition temperature T g larger than T m /2. A strong-to-fragile liquid transition only exists when  lgs0 is smaller than 1.25, as shown in Table  1.1. In all the given examples, the ideal glass transition temperature T 0g is always lower than T K . The Kauzmann temperature cannot be lower than T og . A transition at T g = T m /2 would lead to T og = T K = 0.3545×T m , T 0m = 0.423 T m and limiting values equal to 1.25 for  lg0 and 1.5 for  lso . This property explains why some fragile glass-forming liquids do not undergo a visible liquid-to-liquid transition before being crystallized by heating them at temperatures a little higher than T m /2 [145,146]. When a visible transition temperature T g is lower than T m /2, as shown in Table 1-1 for Au 77 Ge 13.6 Si 9.4 , a fragile-to-strong liquid transition exists for T m /3 < T 0m < 0.3715×T m . There is no liquid-liquid transition in any undercooled melt for 0.3715×T m < T 0m < 0.423×T m and an amorphous state is observed below the crystallization temperature.

Hidden freezing at T g before relaxation of quenched liquids
The specific heat of a quenched liquid is always assumed as being continuous below T g before relaxation. The Kauzmann temperature is extrapolated using the specific heat thermal variation measured from T g to T m . This extrapolation is in contradiction with the linear decrease of the relaxed enthalpy with temperature which reveals that the specific heat is nearly constant below T g . In fact, there is a slope change of the specific heat at T g . This change is very often small for samples obeying the scaling law above T g and often large in Table 5 for samples having a latent heat L  delivered at T g by cooling. A signature of a freezing transition at T g exists without being fully accomplished before enthalpy relaxation, as shown in Figures 8, 9 and 10. The Pd 43 Cu 27 Ni 10 P 20 undercooled liquid which follows the scaling law above T g has a specific heat slope decreasing at T g , whereas that of Zr 44 Ti 11 Ni 10 Cu 10 Be 25 and maltitol are increasing.
The theoretical and experimental Kauzmann temperatures of maltitol are not the same as shown in Figure 10 [147]. The calculated one is weakened by a reduction of the available entropy below T g due to the latent heat L  delivered during cooling at T g . Many discrepancies between calculated and extrapolated values of T K are explained by a reduction of the available entropy due to the existence of a latent heat L  at T g . The undercooled liquid specific heat C pl of BMG N°2 is plotted versus temperature above T g = 576 K using experimental results [79]. The specific heat between T g and T K is calculated by adding the specific heat change given by (34) to the experimental values C pg of the glass phase. There is a weak change of the slope at T g without exothermic latent heat. This explains why the calculated and extrapolated values of T K are about the same. The undercooled liquid specific heat of BMG N°3 is plotted versus temperature above T g using known experimental results [76]. Below T g down to T K , the specific heat of the new liquid state has been calculated adding (34) and 0.36 J/at.g.K to the crystallized phase specific heat instead of introducing a complementary slope corresponding to the difference of specific heat between the glass and crystallized states (C pg -C px ). The calculated and experimental values of T K are equal. Figure 10. The undercooled liquid specific heat of maltitol. The undercooled liquid specific heat of Glass N°5 is plotted versus temperature above and below T g using known experimental results (continuous line) [146]. An extrapolation of C pl below T g leads to a Kauzmann temperature of 261 K. The specific heat below T g is calculated by adding (34) to the measured glass specific heat (point line). There is a slope increase accompanying the freezing transition before enthalpy relaxation. The Kauzmann temperature deduced from (32) is smaller and equal to 285 K.

31
The specific heat of four indomethacin samples has been measured and compared [148,149,150]. The first one is a stable glass which has been obtained by physical vapor deposition. The second one is an ordinary glass and the two others have been isothermally aged below T g during 7 months and 37 days. The enthalpy of the aged and stable glass samples below T g are smaller than that of an ordinary glass [102,123]. The authors have claimed that their ageing times have not transformed indomethacin into stable glass. The enthalpy difference between an ordinary glass and the stable glass is equal to 4000 ± 400 J/mole and the predicted latent heat L + in Table 5 N°8 is smaller and equal to 0.066×H m = 2600 J/mole. The specific heat difference between ordinary glasses and those submitted to ageing increases slightly with ageing time. The specific heat of a stable glass is also a little less than that of an ordinary glass. These observations show that the pseudo-equilibrium obtained when  ns is evolved after structural relaxation is not fully attained as expected from theoretical considerations, which have shown that the steady-state and transient nucleation times are mixed and not simply added [37]. The model used here predicts the specific heat difference between a quenched undercooled liquid and its stable glass phase instead of the difference between a quenched undercooled liquid and the new liquid phase in a pseudo-equilibrium state. The small experimental increase of C plg , which is equal to 19 ± 10 J/mole.K, increases the enthalpy difference, and a corrected value of 19×T K = 627 J/mole.K reduces the observed latent heat at T g from 4000 to 3373 ± 400 kJ/mole.K [148]. The transition temperature T g observed after ageing is 7 K higher, as shown by specific heat jumps which are as large as that of the stable-glass phase [150] (Fig. 6). The enthalpy recovery at T g which has been previously relaxed by the undercooled melt at room temperature during ageing is equal to about C plg × 20 K. This enthalpy has to be reinjected in the sample at the thermodynamic glass transition. The endothermic latent heat of this aged sample is equal to 2900 J/mole, in agreement with the observed enthalpy excess. The thermodynamic transition temperature is close to 325 K for both aged, relaxed samples and stable glass samples instead of 318 K as measured for ordinary glasses. It is so because the nucleation time of the liquid phase in the glass phase seems to be minimum at this temperature and equal to about 4000 s in agreement with nanoscale specific heat measurements [148]. This relaxation time is larger than 50 s because the stable-glass density and consequently the energy barrier for atom diffusion from the vitreous state to the undercooled melt is larger [39].  [148] (Fig. 6). The samples are thin films of stable glass phase which have a thickness of 2900 nm, a volume v = 1.13 mm 3 and are submitted to annealing at temperatures lower than T g and equal to 318, 320, 324 and 325 K.
The new thermodynamic parameters are:  lg0 =  ls0  lg0 = 1.628,  0g = 0.518,  0m = 0.439. The Gibbs free energy change has reversed its sign and is given inwith ( ls0  lg0) =  lg0 . All stable-glass superclusters have a negligible steady-state nucleation time at T g .The transient nucleation time-lag  ns in (1)

Conclusions
A new model introduces an enthalpy saving at T g describing the equilibrium property changes of many fragile glass-forming melts below T g only knowing T g , T m the melting temperature, and H m the melting heat. The specific heat jump at T g , the reduction from T 0m to T 0g of the temperatures at which the enthalpy savings associated with crystal nucleus formation would be extrapolated to zero from above and below T g , the Kauzmann temperature T K , the enthalpy saving between T g and T K , the relaxed enthalpy, its ultimate value at T K , the latent heats associated with liquid-stable-glass transition and TTT diagrams of vitreous and crystallized phase nucleation are predicted. The transition at T g is, in a first step, a liquid-liquid transition with a pseudo-equilibrium time which strongly increases when the temperature decreases below T g .
The time dependence of T g measured varying cooling and heating rates is due to this incubation time which is viewed as a transient nucleation time  ns in undercooled liquids preparing the phase transformation. The incubation time at T g is equal to about 50 s and is the same as that observed in TTT diagrams at the crystallization nose temperature above T g in spite of a large change of the viscosity between these two temperatures. It is proportional to the reverse of the constant K defining the transient nucleation time  ns in (1). A mean value of lnK = 63.9 ± 1.3 is obtained at T g for all liquids listed in Table 6.
Fragile-to-fragile and strong-to-fragile liquid transitions are observed. Strong liquids are transformed into stronger liquids and stable-glasses by cooling below T g without latent heat. In all liquids, an enthalpy is always relaxed below T g during the time-lag of transient nucleation . The small specific heat jumps at T g in strong liquids are used to determine their temperature T 0m lower than T m /3, assuming that the new value of T 0g becomes equal to zero at 0 K. The model also predicts the absence of fragile-to-fragile liquid transition near T m /2 when T 0m is larger than 0.3715×T m and smaller than 0.423×T m .
The energies savings  ls ()×H m /V m and  lgs ()×H m /V m are enthalpy excesses per unit volume of undercooled liquids above and below T g as compared to that of the crystallized state while  lg ()×H m /V m is the enthalpy excess of the undercooled liquid as compared to that of the stableglass phase with  lg () being equal to [ ls ()   lgs ()]. These quantities induce different Laplace pressures on the superclusters which may lead to the condensation of a new phase from the undercooled melt. The energy saving  ls ()×H m giving rise to crystallization above T g is transformed below T g in lgs ()×H m respecting  lgs <  ls with a minimized value of  lgs . The new liquid phase below T g cannot be transformed in crystals because the nucleation time of a critical supercluster is much too long. The stable glass phase formation is driven by an enthalpy change at T g equal to lg ()×H m /V m . The specific heat difference between an undercooled melt and the stable-glass phase is predicted between T K and T g . The supercluster homogeneous nucleation temperatures T 1 above T g and T 2 = T g occur far below the crystallization temperature when the energy saving coefficients  ls0 and  lg0 at T m are minimum,  lg0 being always equal to 1.5× g +2 and  ls0 to  g +2 in about 70% of all fragile glass-forming melts where  g = (T g -T m )/T m . Their temperatures T 0g and T 0m follow scaling laws depending on  lg0 and  ls0 , their specific heat jump at T g being equal to 1.5×H m /T m , and their reduced Kauzmann temperature square respecting  2 K = 5× 2 g /3.
The critical stable-glass nucleus is submitted to a Laplace pressure change p accompanying the enthalpy changeV m ×p =  lg ×H m per mole below T g , The critical energy saving coefficients  ls ,  lgs and  lg are linear functions of  2 instead of  = (T-T m )/T m . The energy saving coefficients  nlgs0 and  nls0 at T m of a n-atom supercluster are proportional to its reverse radius when n ≥ 147 and is quantified for n < 147. Any n-atom supercluster formed below T m is not submitted to premelting because it has an energy saving coefficient depending on    and consequently its surface atoms have the same melting temperature T m than the core ones. The critical energy barriers of melts are too high below T g for having a single supercluster giving rise by growth to an infinite supercluster at any temperature smaller than T g . The equilibrium glass phase is obtained by homogeneous formation, percolation and interpenetration of elementary supercluster multitude containing magic atom numbers filling all the space during very long nucleation times. The critical energy barrier associated with these elementary superclusters depends on their atom number n and is reduced by the large Gibbs free energy change associated with their formation. Their nucleation times and TTT diagrams are predicted between T K and T g in appendix A. The stable vitreous phase is mainly formed at the Kauzmann temperature T K because the nucleation rate of superclusters which are ready to grow has a pronounced maximum at this temperature where the nucleation times are minimum values. The liquid-to-liquid transition at T g is not induced by stableglass supercluster formation and is a direct consequence of the enthalpy change.
The thermodynamic transitions at T g cannot be described as second-order or first-order transitions. All fragile and strong liquids produce relaxed enthalpy after quenching followed by annealing temperatures between T K and T g . An endothermic latent heat is recovered in fragile liquids at T g during heating which depends on the annealing temperature below T g and is only equal to the relaxed enthalpy when a stable glass phase has not been formed. The stable glass phase formation also produces an exothermic enthalpy at T K which is recovered at T g by a complementary endothermic latent heat. In strong liquids, there is no latent heat accompanying the stable-glass formation and the transition at T g is a liquid-liquid transition after a minimum of relaxation time below T g in agreement with descriptions of the formation of dynamical fractal structures near a percolation threshold. In many glass-forming melts, there is no first-order transition because there is no exothermic latent heat produced at T g during cooling in spite of the presence of an endothermic latent heat during heating. A second-order phase transition character exists because the specific heat difference between liquid and glass is successfully calculated in all liquids using the first-derivative of the enthalpy saving. The glass-forming melt is sometimes so fragile that a reversible latent heat exists at T g which is only a fraction of the available enthalpy. In these liquids, the total endothermic latent heat at T g is much larger than the exothermic latent heat obtained by cooling when it exists.

Appendix A TTT diagrams of several liquids in stable-glass phase between T K and T g
The parameters used to calculate the nucleation times are given in Table 6 Figures A1, A2, A3, A4, A5, A6, and A7 respectively. The total time t =  ns +  2 /6×t sn includes the transient relaxation time  ns starting from the quenched liquid state, while t sn is the steady-state nucleation time of the stable-glass domain of volume v having a minimum at T K . The critical energy barrier being much too high, the volume v is chosen equal to the volume of the n-atom cluster instead of the sample volume in order to have the minimum of t very close to T K . This assumption is in agreement with the description of the glass-state by molecular dynamics simulations "as composed of tiny icosahedral-like clusters, most of which touching or interpenetrating yielding a microstructure of polyicosahedral clusters that follow a specific sequence of magic numbers" [23]. The Zeldovitch factor is calculated with (47) using G* nm /k B T instead of G* 2lg /k B T and n c equal to the number n. The main uncertainty on the nucleation time t comes from the uncertainty on the value of B/(T g -T 0g ) chosen at T g , given in Table 6 and from the transient nucleation time-lag  ns of stable-glass superclusters which could be much larger than 50 s as shown for indomethacin in part 12. It explains why the minimum of t sometimes occurs at a temperature a little larger than T K . It is shown here that the nucleation rates of these elementary clusters are sufficiently large after ageing to produce this type of microstructure knowing that the growth around these nuclei could be strongly accelerated when the surface energy declines with touching and interpenetrating superclusters.       The nucleation rate of elementary clusters becomes larger at temperatures much smaller than T g and attains a maximum value at T K , as shown in Figure 7. The formation of a stable glass phase by ageing is possible when the long formation time of an elementary supercluster is evolved and when (50) is respected authorizing its growth. All the liquids studied in Figure 7 could be transformed between T K and T g in times equal to the sum of structural relaxation time  ns and the formation time of the elementary supercluster. Studies are necessary to determine whether the transformation times by ageing at T K can become sufficiently small in some glasses. Some liquids such as As 2 Se 3 , Pd 43 Cu 27 Ni estimation is in agreement with the complementary latent heat in agreement predicted and measured after an ageing of 166 hours at 418 K [51]. The substrate temperature used for indomethacin physical vapor deposition has been varied from 265 to 305 K [44] (Fig. 5). There is no more stable glass formation above 297 K in agreement with the rapid fall of the nucleation rate above T K , as shown in Figure 7.
The stable glass phase cannot be obtained at T g by cooling because the nucleation rate logarithms are negative in all liquids. A liquid-to-liquid transition is nevertheless easily observed at T g because the time  ns necessary to attain the pseudo-equilibrium of the new liquid phase is of the order of 50 s. The enthalpy recovery of a quenched melt increases during cooling down to T K and is relaxed at each annealing temperature between T K and T g when the undercooled melt attains its pseudoequilibrium, as shown in Figs 4-6. The whole undercooled liquid is transformed into a glass phase at T K . The maxima of the elementary supercluster nucleation rate at T K are obtained without using any adjustable parameter for n ≥ 147 with  nlg0 being calculated with (46). The coefficient  nlg0 for n = 55 is adjusted by fixing the maximum nucleation rate at T K . An exothermic latent heat has to be produced to undergo the transition at T K and the glass survives in this new equilibrium state from T K to T g marked by hysteresis cycles.
The glass phase is melted at T g with the help of the endothermic heat L + . The enthalpies of Pd 43 Ni 10 Cu 27 P 20 , As 2 Se 3 and indomethacin undercooled liquids at equilibrium are represented below T g in Figs 4-6. Along the recovered enthalpy line obtained by relaxation below T g , the viscosity attains its pseudo-equilibrium value after an increase by a factor of 2 to 3 from its value in the quenched-liquid state before relaxation [64,76,78,93]. This viscosity relaxation shows that the undercooled liquid has already undergone a change into a frozen liquid state before relaxation. A time dependence of T g is observed during heating of rapidly quenched melts which have not had the time to attain the pseudoequilibrium during cooling through T g because a minimum time of about 50 s is needed to undergo the transition [52,76,144].
The calculations of the enthalpy saving coefficients  lg are only based on the knowledge of the glass transition T g , the melting temperature T m and thermodynamic considerations related to scaling laws. The fusion enthalpy H m has to be known to calculate the latent heats and the ultimate enthalpies of the stable-glass phase.

Appendix B Superclusters of 13 atoms governing the first-crystallization time of metallic glass-forming melts
Superclusters submitted to various Laplace pressure give rise to stable-glass phase and others having different energy saving coefficients are acting as growth crystal nuclei far above T g . Table B1. Parameters used to calculate the Time-Temperature Transformation diagrams above T g. The unit of entropy and volume are J/K/atom.g. and m 3 respectively. A study of isothermal nucleation time is made after quenching the melt from above T m down to the annealing temperature. The time-lag  ns for transient nucleation has to be included in the calculation of the total nucleation time t which is equal to ( ns +  2 /6×t sn ) [37]. The calculated liquidcrystal TTT diagrams of three glass-forming melts Pd 40 Ni 10 Cu 30 P 2 , Pd 43 Ni 10 Cu 27 P 20 , and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 are represented in Figure B1 in agreement with the experimental studies [34][35][36]. The parameters given in Table B1 are used. The 13-atom supercluster is the minimum size entity which is formed by homogeneous nucleation. These superclusters are condensed when (B1) is respected with n = 13 inducing a spontaneous supercluster growth up the critical size and beyond it because the cluster energy barrier G* 13 /k B T is much larger than the critical value G* 2ls /k B T: where G 13m and G* 13m /k B T are given by (17) and (22) with n = 13. The 13-atom cluster nucleation rate J 13 is equal to (v.t sn )  because ln(J 13 .v.t sn ) = 0, where v is the sample volume for crystallization and t sn the steady-state nucleation time of the 13-atom supercluster. The  13m0 and lnA values have been varied to reproduce the total nucleation time t and the nose temperatures of the experimental TTT diagrams [29,42].