Fragile-to-Fragile Liquid Transition at T g and Stable-Glass Phase Nucleation Rate Maximum at the Kauzmann Temperature T K

Transcription

1 Fragile-to-Fragile Liquid Transition at T g and Stable-Glass Phase Nucleation Rate Maxiu at the Kauzann Teperature T K Robert F. Tournier Centre National de la Recherche Scientifique, Université Joseph Fourier, Consortiu de Recherches pour l Eergence de Technologies Avancées, B.P. 166, 3804 Grenoble-Cedex 09, France; robert.tournier@creta.cnrs.fr; Tel: +33-(0) ; Fax : +33-(0) Abstract: An undercooled liquid is unstable. The driving force of the glass transition at T g is a change of the undercooled-liquid Gibbs free energy. The classical Gibbs free energy change for a crystal foration is copleted including an enthalpy saving. The crystal growth critical nucleus is used as a probe to observe the Laplace pressure change p accopanying the enthalpy changev p at T g where V is the olar volue. A stable glass-liquid transition odel predicts the specific heat jup of fragile liquids at T T g, the Kauzann teperature T K where the liquid entropy excess with regard to crystal goes to zero, the equilibriu enthalpy between T K and T g, the axiu nucleation rate at T K of superclusters containing agic ato nubers, and the equilibriu latent heats at T g and T K. Strongto-fragile and strong-to-strong liquid transitions at T g are also described and all their therodynaic paraeters are deterined fro their specific heat jups. The existence of fragile liquids quenched in the aorphous state, which do not undergo liquid-liquid transition during heating preceding their crystallization, is predicted. Long ageing ties leading to the foration at T K of a stable glass coposed of superclusters containing up to 147 atos, touching and interpenetrating, are evaluated fro nucleation rates. A fragile liquid-liquid transition occurs at T g without stable-glass foration while a strong glass is stable after transition. Keywords: kj glasses ; P glass transitions ; pe etallic glasses ; ph. nonetallic glasses ; pj polyers ; Q-nucleation 1. Introduction Vitrification is often viewed as a freezing-in process of undercooled elts instead of a phase change because there is, up to now, no intrinsic energy saving driving the foration of a new vitreous phase. A elt is seen as being stable with a well-defined viscosity and a unique teperature for the free-volue disappearance at the Vogel-Fulcher-Taann teperature, which is deduced fro the theral variation of relaxation tie or viscosity. Such a description leads to a natural freezing without therodynaic transition because it does not include any odification of Gibbs free energy associated with olar volue theral variation change. Nevertheless, the existence of a first-order transition near T g in triphenyl-phosphite is associated with a liquid instability [1-3]. Local inia in the potential energy landscape related to various local positions of all atos are also considered to explain the equilibriu properties of aorphous substances [4]. Various odels are considering the glass-liquid transition at T g as having a therodynaic origin. Each broken cheical bond is viewed as an eleentary configurational excitation called configuron. Using the Doreus odel of viscosity, the entropies and enthalpies of foration of configurons are obtained using a fitting process of experiental viscosities [5,6]. The glass-liquid transition has recently been treated within configuron percolation theory as a percolation-type phase transition with foration of dynaical fractal structures near the percolation threshold [7]. Specific heat jups have been predicted. The foration of percolation structures ade of high-density ato configurations in the glassy state was earlier suggested [8]. The nucleation of icosahedral clusters was also reported. Evteev et al studied, by olecular dynaics, atoic echaniss of pure iron vitrification and showed that it is ensured by the foration of a percolation cluster fro utually

2 penetrating and contacting icosahedrons with atos at vertices and centers [9-11]. A fractal structure of icosahedrons incopatible with translation syetry 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 phenoena near the glass transition. Clusters grow with fractal structures at the correlation length fro olecular type units such as icosahedral superclusters, that are ore and ore frozen when the teperature decreases and lead to a disordered aterial [1,13]. Berthier et al showed that a growing length scale is accopanying the glass transition [14]. The twinkling dynaics of polystyrene was captured via atoic force icroscopy within its glass transition region. Successive two-diensional heights reveal that percolated clusters have lifeties depending on their size and exist for longer tie scales at lower teperatures. The coputed fractal diensions are shown to be in agreeent with the theory of the fractal nature of percolating clusters [15]. All these odels describe a liquid-liquid transition with liquid structures also containing superclusters below and above the glass transition and even beyond the elting teperature T [16-18]. The sae structure is progressively transfored and frozen by percolation instead of a new liquid phase growth fro a critical nucleus. Kirpatrick and Thirualai 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 rando first-order phase transition is expected and hidden below the glass transition. The Kauzann teperature would be the true glass transition at equilibriu [0]. Angell showed how such a theory for fragile liquids fits into a phenoenological schee covering all glass forers [1]. A criterion analogous to the Lindeann criterion of elting was established for the glass-liquid transition []. It could be due to a vibrational instability of atos located in the lattice sites. The new liquid state obtained just below T g has been still described by eans of reverse Monte Carlo (based on neutron scattering data) and olecular dynaics siulations showing that etallic glasses are coposed by tiny icosahedral-like clusters, ost of which are touching and/or interpenetrating yielding a icrostructure of polyicosahedral clusters that follow a specific sequence of agic nubers [3]. The glassy state relaxes ore and ore enthalpy when the teperature decreases below T g. Spin glasses also relax a heat axiu when their renant agnetization is saturated after applying a high agnetic field below the phase transition [4]. There is no ore relaxed energy when the agnetization is equal to zero at therodynaic equilibriu. The relaxed enthalpy below T g sees to be saturated after a relaxation tie depending on the observation teperature. Its axiu cannot be larger than the available enthalpy which is frozen between T g and the Kauzann teperature T K where the liquid entropy excess against that of crystal goes to zero [5]. There is a need to predict the glass enthalpy reduction accopanying the stable-glass foration in this teperature window. In our odel, 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 /V but also an enthalpy saving equal to ls H /V where is equal to (T-T )/T, T being the elting teperature, H the fusion heat per ole and V the olar volue. This quantity corresponds in etallic liquids to the difference of conduction electron Feri energies in crystals and their elt. At the vitreous transition teperature T g and at lower teperatures, there is a transforation of ls in lgs with a decrease equal to lg = ( ls - lgs ). The enthalpy saving forulation lg H /V of the equilibriu glass phase below T g has already been proposed and applied to Pd 43 Cu 7 Ni 10 P 0 [6]. But now, the coefficients ls and lgs in fragile liquids can be directly deterined without using the specific heat jup at T g. This odel is based on the hoogeneous nucleation of new n-ato superclusters characterized by a surface energy and an energy saving lg H n/n A driving the foration of a stable-glass phase and being proportional to n where lg () depends on n and N A being the Avogadro nuber. The Gibbs free energy change associated with the foration of these new

3 3 superclusters is equal to lg H /V and does not contain the classical contribution H /V associated with crystallization. The nucleation rate of these new superclusters has a axiu at T K instead of T g as shown in part 10. The existence of a liquid-liquid transition at T g is confired without nucleation of these entities. An enthalpy is always relaxed below T g in all liquids. The foration of the stable-glass phase in fragile glasses is accopanied by an exotheric latent heat which does not exist in strong glasses. The glass transition in strong liquids is a true liquid liquid transforation without hidden latent heat as already shown in various odels [6-8,10-15,19-]. The classical odel of crystal nucleation is copleted by adding an unknown enthalpy saving in the Gibbs free energy change equal to ls H = V p for a supercluster foration, where p is a copleentary Laplace pressure acting on the growth critical nucleus [7]. A new equation for a nucleus foration has been established and the new hoogeneous nucleation teperature corresponds to a iniu value of the surface energy for each value of ls. The energy saving coefficient ls is a function of, as already shown in liquid eleents [8-30]. The derivative of the Gibbs free energy change (dg ls /dt) T for a critical nucleus foration at the elting teperature T is equal to the bulk fusion entropy S. The coefficient ls has a axiu value ls0 at T = T which tends to zero at T = T 0. The inias of ls0 and ls occur at a hoogeneous nucleation teperature equal to ( ls )/3. They depend on 0 (or 0 ) when the quenched liquid has escaped crystallization which is induced at higher teperatures by a tiny intrinsic nucleus reducing the effective energy barrier for crystal growth [9]. The change of ls in lgs at T g with ls > lgs, leads, far below the crystallization teperature, to the existence of two hoogeneous nucleation teperatures T 1 and T, corresponding to two values ls and lgs above and below T g respectively, and to a change of T 0 (or 0 =(T 0 -T )/T )) in T 0g (or 0g =(T 0g -T )/T )) [31].The hoogeneous nucleation teperature T (or ) equals T g (or g ) because the liquid transforation has to lead to the iniu value of lgs [3,33]. When the transition takes places at a teperature lower or larger than T,the lgs is too large and leads to a relaxation towards its value at equilibriu. The coefficients ls and lgs tend to zero at the teperatures T 0 and T 0g with T 0 > T 0g and are used as Vogel-Fulcher-Taann teperatures T 0 characterizing the liquids above and below T g. The transition at T g = T (or g = ) follows a scaling law that is a linear function of the energy saving coefficient lgs0 of the new liquid phase extrapolated at T [3]. The value of T g is used to deterine lgs below T g and the teperature T 0g (or 0g ), where lgs would be extrapolated to zero. A teperature tie transforation (TTT) diagra describes the crystallization at teperatures uch higher than T g with a nucleation tie t of about seconds at the nose teperature T n [34-36]. These crystallization teperatures are higher than the new hoogeneous nucleation teperature T 1 calculated fro the critical energy barrier because a tiny intrinsic nucleus reduces the effective energy barrier for crystal growth [8,37]. The isotheral nucleation total tie t contains two added contributions /6 ns and t sn, ns being the tie-lag for the transient nucleation and t sn the steady-state nucleation tie [37]. The two contributions to t at the nose teperature T n being siilar, ns is of the order of 50 s at T n [9,31]. The tie lag ns is inversely proportional to K, as shown in (1), K being a coefficient in the exponential dependence of the supercluster nucleation rate J with the therallyactivated energy barrier G eff /k B T in (), and being the Zeldovich factor in (1) which is weakly dependent on the teperature, N A the Avogadro nuber and k B the Boltzann constant [37]: N V ns A, (1) 1 K

4 4 J G K exp( k T B eff. () ) The easured isotheral relaxation tie just below T g of a quenched elt 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 tie required for an equilibriu distribution of atos to be established during the liquid-liquid transition preparing the steady-state nucleation of a vitreous phase [31,37]. The new liquid-phase foration is accoplished when the tie-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 schee in spite of their strong theral dependence on the elt viscosity ratio ln(/ 0 ) = B/(T-T 0 ) [38]: A0 B ln( K) ln( ) (ln A) ( T T 0. (3) ) Consequently, the A value in (3) is uch stronger below T g than above T g [3]. There are two tiescales above and below T g. The tie-lag above T g, leading to a nucleus distribution ready for steady-state nucleation, is about 10 6 ties larger than the tie-lag required below T g. In Turnbull and Fisher s odel, lna is nearly equal to ln(n A k B T g /V h)f*/k B T g, where h is the Planck s constantandf*/k B T* g a therally-activated energy barrier for ato diffusion fro the elt to the hoogeneously-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 easureents of ipurities introduced in various glasses [40](Figure 1). The stable-glass foration starts by hoogeneous nucleation of condensed superclusters when the tie lag ns and their steady-state nucleation tie t sn depending of their n-ato nuber are evolved. The energy barrier for growth being uch too high, the stable phase is built by percolation and interpenetration of eleentary superclusters containing agic ato nubers and accelerated by reduction of the interconnected supercluster surface energy. It is shown in appendix B that a TTT diagra of the stable vitreous phase exists below T g predicting the coplete transforation of the liquid in stable phase with a iniu value of t sn for a axiu value of n occurring at the Kauzann teperature. The viscosity above T g, in any exaples of fragile glass-foring elts, perfectly obeys a scaling law, with a Vogel-Fulcher-Taann (VFT) teperature equal to 0.77 T g [41,4]. The hoogeneous nucleation teperature T 1 in a fragile elt above T g calculated without any reduction of the critical energy barrier by a sall hoogeneously-condensed nucleus also follows a scaling law that is a linear function of ls0. The coparison of theoretical and experiental 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 axiu in the transfored liquid phase is equal to lg0 H in strong and fragile liquids without including the exotheric foration heat of the stable-glass phase. The odel is able to deterine the boundaries separating fragile fro strong liquids [9,3]. The free-volue disappearance teperature T 0 is less than T /3 in strong liquids and greater than T /3 in fragile liquids. The transition at T g is described by the foration at equilibriu of a new liquid phase characterized by an energy saving lg H. The derivative H d( lg )/dt is used to calculate the specific heat jup per ole at T g and the enthalpy saving varying fro T g to the Kauzann

5 5 teperature T K where the liquid entropy excess copared to that of crystal goes to zero [5,6]. The equilibriu enthalpy change at T g is predicted. The presence or absence of equilibriu latent heat at T g is analyzed. The value of the Kauzann teperature T K has already been deterined in soe glassforing elts, observing that the specific heat jup between the undercooled liquid and the vitreous phase is nearly constant because the relaxed enthalpy has a nearly-linear decrease with the teperature increase [31,3]. The transition at T g leads to a uch less fragile liquid with a teperature T 0g lower than T 0, and consequently in all cases to a stronger behavior below T g. The following therodynaic quantities are calculated: the coefficients ls and lgs above and below T g, their difference lg (T), the specific heat jup C p (T g ), the teperatures T 0 and T 0g (T 0 > 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 ultiate enthalpy H r and the frozen enthalpy below T g only knowing T g, H and the elting teperature T. The specific heat changes between T K and T g are predicted and used to deterine the Kauzann teperature of any fragile glass-foring elts. About one third of fragile glass-foring elts do not follow the scaling law governing the viscosity above T. They are characterized by a larger energy saving lg leading to a larger specific heat jup. 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 L - when it exists, are proposed assuing 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 jup C p (T g ) that is saller than that of fragile liquids accopanying the decline of T o. Their transition at T g occurs without latent heat during cooling and corresponds to lg (T g ) = 0. Their specific heat jup at T g has to be known to deterine 0, 0g, lg, ls and lgs. The effective critical energy barrier deterined by the energy saving of stable vitreous clusters, being saller than the hoogeneous nucleation critical barrier, controls the nucleus growth and a possible phase change. This is not the first tie that the state of the undercooled liquid has been viewed as being coposed of long-lived structures created in the noral-liquid structure that is locally favored by a free energy decrease when the teperature decreases. These locally-favored structures ay lead to a liquid-liquid phase transition [,43]. The odel developed in this paper and copleted in Appendix A deonstrates that this transition occurs in all liquids at T g and that eleentary superclusters are condensed at T K in fragile liquids after a very long ageing tie, leading to a stable vitreous state coposed of these nuerous tiny entities instead of a single supercluster inducing the condensation of the whole vitreous phase because the critical effective nucleation barrier of the elt is too large to get over it. Therodynaics 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 diagras of three bulk etallic glass forers (BMG) in crystallized phases are reproduced in Appendix B calculating the effective therally-activated energy barrier G eff /k B T of the critical nucleus for crystal growth [9]. The critical nucleus is a 13-ato cluster having an effective energy barrier higher than that of bigger clusters. The condensation teperature of a single 13-ato cluster deterines the crystallization teperatures of a whole saple along its TTT diagra. The energy saving coefficient of this type of cluster containing a sall nuber of atos ebedded in glass-foring elts is quantified. The quantified value ls0 of a 13-ato nucleus deduced fro the

6 6 experiental TTT diagra is equal to 0.7, while those for 13-ato superclusters involved in stableglass phase foration below T g are saller. The following plan is proposed: - Gibbs free energy change associated with growth nucleus foration, 3- Theral dependence of the energy saving coefficient n of an n-ato condensed cluster, 4- Critical supercluster hoogeneous nucleation teperature and effective nucleation teperature, 5- The two hoogeneous nucleation teperatures T 1 and T and the equilibriu enthalpy change of glass-foring elt at the vitreous transition T g = T, 6- Scaling laws, 7- The specific heat jup at fragile-to-fragile, strong-to-strong, and strong-to-fragile liquid transitions at T g, 7.1- Fragile-to-fragile liquid transition, 7.- Strong-to-strong liquid transition, 7.3-Strong-to-fragile liquid transition, 8- Specific heat jups fro etallic and non-etallic glasses to undercooled liquids at the vitreous transition, 9- Enthalpy theral cycles expected in soe liquids and deterination of the Kauzann teperature, 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 /, 1- Hidden freezing at T g before relaxation of quenched liquids, 13- Transforation of the stable-glass Indoethacin in undercooled liquid at T g, 14- Conclusions, Acknowledgents, Appendix A: TTT diagras of several liquids in stable-glass phase between T K and T g, Appendix B: Superclusters of 13 atos governing the first-crystallization tie of etallic glassforing elts. References.. Gibbs free energy change associated with growth nucleus foration The classical Gibbs free energy change for a nucleus foration in a elt is given by (4): H 4R H, (4) 3 G1 ls 4 R 1ls V 3 V 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]: V H 3 1/ 1 ls ( ) 1 ls, (5) N A V 1/ 3 N Ak B ln( Kls ) 1ls. (6) 36S

7 7 An energy saving per volue unit - ls H /V is introduced in (4) for a critical nucleus foration above T g ; the coefficient ls being replaced by lgs for a critical nucleus foration below T g. The coefficients ls and lgs have to be calculated in order to deterine their difference lg which deterines the stable glass foration enthalpy below T g when the quenched liquid escapes crystallization. The new Gibbs free energy change is given by (7), where ls is the new surface energy [8]: 3 H 4R Gls ( ls ) 4R ls V 3 The new surface energy coefficient ls is given by (8): V ( 1/ 3 ls ) N A ls H V. (7). (8) The critical radius R* ls in (9) and the critical therally-activated energy barrier G* ls /k B T in (10) are calculated assuing d ls /dr = 0: R V, (9) * ls 1/ 3 ls ( ) ls N A. (10) They are no longer infinite at the elting teperature T when ls is not equal to zero. The hoogeneous nucleation teperature T (or ) occurs when the nucleation rate J in () is equal to 1 and (11) is respected: * Gls ln( K k T B ls ). (11) The unknown surface energy coefficient ls in (1) is deduced fro (10) and (11): 3 ls 3N AkB ( ls ) (1 16S )ln( K ls ). (1) The surface energy ls in (8) has to be iniized to obtain the hoogeneous nucleation teperature T 1 (or 1 ) for a fixed value of ls. The derivative d ls /d is equal to zero at the teperature T 1 (or 1 ) given by (13) assuing that ln(k) does not depend on the teperature: d 3 ls d 3 ls d ls d ( ls )(3 ls ),

8 8 T1 T 1 T ls 3. (13) The enthalpy saving between solid and liquid states is deterined by the knowledge of 1. The theral variation of ln(k ls ), being a function of viscosity, does not odify the value of T 1 (or 1 ) as shown below [9]. The critical energy barrier in (10) is the product of a function g() and ln(k ls ). The axiu nucleation rate J still occurs at the teperature T 1 with the derivative d(lnj)/d being equal to zero because g() is equal to 1 and dg()/d = 0: ln J ln( K ls * G ) k T B ls ln( K ls ) g( )ln( K ls ), d ln J d d ln( K d ls ls ). ls g( ) 0 ) dg( ) ln( K d d ln( K ) d The surface energy coefficient ls is now given by (14) replacing by (13) in (1) for each value of ls : 1/ 3 N k ln( K ) (. (14) 1ls A B ls ls 1 ls ) (1 ls ) 36S The classical nucleation equation (4) has been transfored into (15) introducing the energy saving coefficient ls : G H R H 1k ln K 3 B ls 1/ 3 ls ( R, ) ( ls )4 4R (1 ls )( ) V 3 V 43 S V. (15) The Laplace pressure p can be calculated fro the surface energy ls with the equation (13) [3,33]: ls H p ( ls ). (16) R V The copleent p of Laplace pressure introduced by the energy saving ls is equal to ls () H /V. All equations (7-16) for a critical nucleus foration can be applied below T g after replacing ls by lgs and T 1 (or 1 ) by T (or ). The equations governing the stable-glass supercluster foration below T g are developed in section 10. The Gibbs free energy change G ls in (15) directly depends on the cluster ato nuber n and the energy saving coefficient n of the cluster instead of depending on its olar volue V and its radius R, as shown in (17): n (4 ) 1/ 3 / 3 Gn( n,, n) H ( n) H ls (3n). (17) N A N A

9 9 This equation can be applied below and above T g using different values of n corresponding to various Laplace pressures. The foration of superclusters having a weaker energy barrier precedes the foration of crystallized nuclei in an undercooled elt [9,16,47,48]. A supercluster containing n atos could be easily transfored into a crystal of n atos having the sae energy saving coefficient n and the sae surface without changing G n. The transforation of superclusters into crystals could occur when their olar volue and consequently their coefficient n becoes equal to those of crystals. This condition could be not sufficient because crystals of n atos could be facetted, less-spherical and their surface not iniized. The supercluster energy saving n H is quantified, depending on the radius and ato nuber n, and can be saller or larger than the critical energy saving ls H in glass-foring elts above T g and larger than the critical energy saving lg H of the stable-glass phase appearing below T g. The critical growth barrier G* n of an n-ato supercluster also depends on n. It can be larger than the critical growth barrier G* ls. The effective critical energy barrier of the sallest hoogeneously-condensed cluster can control the heterogeneous growth around it. It is the case for the crystallization of glass-foring elts at teperatures higher than the hoogeneous nucleation teperature defined by (13). 3. Theral dependence of the energy saving coefficient n of an n-ato condensed cluster All superclusters which are fored in an undercooled elt preparing crystal foration grow first, are subitted to a copleentary Laplace pressure and have a surface energy because the Gibbs free energy change contains an enthalpy saving [9,7,48]. The energy saving coefficient n of an n- ato supercluster giving rise to crystals has to be a function of to survive above T, assuing that n is axiu at T, d n /dt being equal to zero and fixing the supercluster fusion entropy as being equal to the fusion entropy S of the bulk solid [9]: 3 d( G 3 4R dt n ) T T S V. In these conditions, the cluster fusion occurs above T and is governed by liquid droplet hoogeneous nucleation above T rather than by surface elting. This theral variation explains the undercooling rate of liquid eleents [8,48]. The law (18) is expected to work also in glass-foring elts as observed in liquid eleents, n being the quantified energy saving coefficient of an n-ato supercluster: H H n n0 n0 ( 1 ). (18) V V 0 A n-ato supercluster is subitted to a Laplace pressure which is strongly dependent of its radius. The quantified value n decreases with and becoes equal to zero when the teperature equals T 0 (or 0 ). An ato which does not belong to a supercluster cannot be involved in the nucleation of a new phase.

10 10 The critical paraeters for supercluster growth are deterined by an energy saving coefficient called ls (or lgs ) in (19): v l s0( 1 0 H H ls ) V V. (19) The enthalpy saving per ole ls H tends to zero when tends to 0. The liquid free volue would be equal to zero at T = T 0 in the absence of transforation at T g. A critical supercluster contains a critical nuber n c of atos given by (0): n c 3. (0) 3( 3 ls 3 ls ) 4. Critical supercluster hoogeneous nucleation teperature and effective nucleation teperature The therally-activated critical energy barrier is now given by (1), where ls is given by (19): G k B * ls T 3 1(1 ls ) ln( Kls ). (1) 81( ) (1 ) ls The coefficient of ln(k ls ) in (1), called g(), becoes equal to 1 at the hoogeneous nucleation teperature given in (13) and the equation (11) is respected. Hoogeneously-condensed clusters of n- atos act as growth nuclei at a teperature generally higher than the hoogeneous nucleation teperatures T 1 and T of liquids. The cluster therally-activated critical energy barrier G* n /k B T and its effective therally-activated critical energy barrier are given by (): G k B neff T G k T B * n G k T B n 3 1(1 n ) ln( K) G, () n 81( ) (1 ) k T n B where G n is given by (17) and n by (18). These nuclei are ready to grow when the transient nucleation tie ns and the steady-state nucleation tie t sn are evolved and the relation (3) respected: ln( J n vt sn ) ln( K ls vt sn G ) k T B neff. (3) 0 Three situations are et when the two critical energy barriers in (1) and () are copared. The growth around nuclei can be very rapid if G* ls /k B T given in (1) is saller than G* n /k B T given in (). The steady-state nucleation tie t sn (T) is calculated with (3) knowing the saple volue v. This is the case for glass-foring elt crystallization above T g. When G* ls /k B T is larger than G* n /k B T, the new effective critical energy barrier is equal to (G* ls /k B TG n /k B T) and has to replace G neff /k B T in (3). The effective heterogeneous nucleation teperature is strongly dependent on the volue v. This phenoenon is very iportant in liquid eleents where the effective nucleation

11 11 teperature is observed around = 0. in saple volues of few 3 instead of varying fro 0.58 to0.3 with uch saller saples [48,49]. The growth around clusters can be slow and shell-byshell as a function of tie when G* ls /k B T is uch larger than G* n /k B T. It is the case for the glass phase nucleation critical barrier G* lg /k B T which is uch larger than the critical barrier associated with an eleentary n-ato cluster G* n /k B T. 5. The two hoogeneous nucleation teperatures T 1 and T and the equilibriu enthalpy change of glass-foring elt at the vitreous transition T g = T The hoogeneous nucleation teperature T 1 (or 1 ) obeys the equations (13) and (19) and is a solution of the equation (4): 1 ls ls0 0, (4) where 0 (or T 0 ) is the teperature corresponding to ls = 0. At the vitreous transition T g, there are changes of lso in lgs0, 1 in (or T 1 in T ) and 0 in 0g (or T 0 in T 0g ). A new equation (5) is used to calculate (and T ): 0g lg s0 3 lg s0 0. (5) There are two values of 1 and respecting (4) or (5) for each value of 0 or 0g ; the largest value given in (6) is the chosen solution for 1 and : 3 9 4( ) / 1/ ls0 ls (6) 0 ls0 The fragility of elts defined by Angell is larger when 0 increases [50]. In strong liquids, the teperature 1 can be calculated using (6) when ls0 and 0 /3 are known. There is a double solution for a fragile liquid and a iniu value of ls0 1 for each value of 0 /3 (or T 0 > T /3) [ [51]]. The boundary separating strong fro fragile liquids is 0 =/3 (or T /3). The values of 1 and 0 in fragile liquids are given as a function of ls0 in (7) and (8) with a being unknown: 0) 1.5 a, (7) ls0 ls ( 1 g ls (8) ls0 The therodynaic transition at T g is induced by a liquid-liquid transforation at the teperature T (or ) iniizing the energy saving lgs of the new liquid state for a given value of 0g. Any other transition teperature leads in fragile liquids to a too large value of lgs0 and to an energy relaxation. The therodynaic transition teperature T g has to be also equal to T. The

12 1 therodynaic paraeters characterizing the new liquid state at equilibriu are lgs0 = (1.5 g +) and 0g, given in (7) and (8) after replacing ls0 by lgs0, 1 by g and 0 by using 0g. The transforation of any liquid in strong liquid at T g will occur when the difference lg = ( ls lgs ) becoes equal to zero because there is no iniu of lgs0 as a function of in strong liquids. A crystallization at T g = T is not possible because the hoogeneous nucleation rate of a critical supercluster is only equal to 1-3.s -1. In addition, it cannot start fro a 13-ato supercluster because its own nucleation tie is uch too large. The energy saving coefficient, driving the stable glass phase foration in the new liquid state as a function of teperature, has to vary at therodynaic equilibriu as lg = ( ls - lgs ) as shown below. The enthalpy decrease per ole fro quenched undercooled liquids, aged at teperatures less than or equal to T g to lead to stable vitreous states, is equal in (9) to the difference between the copleentary energy savings for a crystal foration below T g [6] : lg s0 ls0 H H lg H ls0 lg s0 ( ). (9) 0g 0 This difference lg occurring at T g and below T g is only due to the change of the equilibriu enthalpy between T g and the Kauzann teperature T K. It contains a teperature-dependent positive contribution and a constant negative contribution which is viewed as the ultiate relaxed enthalpy at T K for quenched liquids respecting the scaling laws above T g. The stable-glass-to-fragile liquid transforation of indoethacin at equilibriu is accopanied by an endotheric latent heat at T g, as shown in Figure 1. A strong-glass-to-strong liquid or a strong-glass-to-fragile liquid transforation occurs without latent heat when lg = 0 instead of being due to kinetic effects [3]. Figure 1. Enthalpy saving coefficients versus = [(T-T )/T ]. The enthalpy saving coefficients ls and lgs for critical nucleus foration, proportional to the copleentary Laplace pressure p, for an indoethacin nucleus containing a critical ato nuber n c (T) respectively above and below the transition g = 0.64, are plotted versus. The differences ( ls - lgs ) H = lg H at T g = 318 K and at the Kauzann teperature T K deterines the equilibriu latent heats involved in the stable-glass-to fragile-liquid transition at T g and T K. The coefficient lg given in (9) is plotted versus between T and T K..

13 13 An exaple is given in Figure to illustrate the change of energy saving at T = T g in fragile-tofragile liquid transforations. The coefficients ls0 and lgs0 of the etallic glass-foring elt Pd 43 Ni 10 Cu 7 P 0 are calculated using T 0 = 430 K and T 0g = 365 K as a function of the hoogeneous nucleation teperature T. The energy saving coefficient ls0 for a crystal foration is equal to for 1 = The coefficient lgs0 has a iniu equal to at = 0.8. All values of lgs0 below and above = -0.8 are larger. A liquid quench down to T a is followed by enthalpy relaxation fro ls0 = down to lgs0 = The therodynaic glass transition has to occur at g = because there is no ore enthalpy relaxation at T g = T [31-33]. The existence of the tie-lag ns, strongly varying with the teperature, has for consequence that the glass transition can occur in a wide window of teperatures. Figure. Miniizing the energy saving coefficients. The energy saving coefficients ls0 (triangles) and lgs0 (diaonds) of Pd 43 Ni 10 Cu 7 P 0 have been calculated as a function of the hoogeneous nucleation teperature T using T 0 = 430 K, T 0g = 365 K, T = 80 K and (5). Note the iniu values of ls0 and lg0 given by (7) and (8) for a = 1. The transition at the teperature T* g of the iniu transfors ls0 into lg0 and T 0 into T og during the relaxation tie. A quench down to T a also leads to a transforation of ls0 into lg0 due to the existence of a iniu relaxation tie between the two liquid states of about 100 s. Slow physical vapor deposition rates produce vitreous saples having a new local packing arrangeent as copared to that of bulk saples below T g and being at therodynaic equilibriu. Indoethacin has been deposited on substrates cooled around the Kauzann teperature in a highly stable vitreous state. This phenoenon of ultra-stable glass foration could be induced by a firstorder transition separating the norally observed high teperature liquid fro a new low-teperature equilibriu supercooled liquid [44]. The liquid-to-glass transforation is obtained at T g after 30,000 s instead of 100 s and accopanied by a strong change of local packing arrangeent, as observed by wide-angle X-ray scattering easureents. These observations are copatible with the existence of a true liquid-to-glass transition which is produced at equilibriu by an endotheric latent heat during heating at T g after an exotheric latent heat at T K. The progressive increase of the enthalpy saving cannot continue below the Kauzann teperature [5]. The enthalpy excess of the undercooled liquid

14 14 can only relax in a window of teperatures extending fro T g to T K. A saturation of the relaxed enthalpy has already been observed at T K [5]. The existence up to T g of highly-stable glasses, when they are prepared at equilibriu by physical vapor deposition, has the result that the current transforation at T g, observed up to now without latent heat, is a liquid-to-liquid transition occurring in a pseudo-equilibriu state without enthalpy relaxation at a well-defined teperature T g. The isotherally-relaxed enthalpy decreases when the annealing teperature increases up to T g and the enthalpy recovery easured at T g has to be equal to zero. This pseudo-equilibriu liquid state is not the equilibriu vitreous state and is obtained after an isotheral relaxation at a teperature T a higher than T K and lower than T g. Equation (9) shows that the ultiate relaxed enthalpy to attain the new undercooled liquid state at T K is equal to lg0 H = ( ls0 - lgs0 ) H and is recovered and included in the endotheric latent L + at T g. A copleentary exotheric enthalpy has to be relaxed at T K after a long steady-state nucleation tie to give rise to the ultra-stable vitreous state. This exotheric latent heat produced at T K has to be equal to the endotheric enthalpy at T g given in (9) due to the therodynaic character of the liquid-tostable-glass transition. 6. Scaling laws Equations (8-30) are scaling laws deterining the new energy saving coefficient lgs0 occurring at = g and the teperatures T 0g (and 0g ) and T 0 (and 0 ) of fragile liquids fro the knowledge of the therodynaic transition teperature, which is defined as the disappearance teperature of the relaxed enthalpy [31,33]. The hoogeneous nucleation teperature T 1 (or 1 ) is deterined as a function of ls0 using (7) and follows a scaling law because, in any exaples of fragile liquids, the viscosity is scaled by a VFT teperature T 0 = 0.77 T g with 0.35< g < -0. [41,4]. Scaling laws are easily obtained fro (30-3), with a = 1 leading to ls0 = g +, lg0 = 0.5 g and to a latent heat during heating equal to 0.5 g due to the transforation of the stable glass in an undercooled liquid state: H ( ) lg H 0.5 ( 0) a, (30) : s0 ls g lg0 0.5 g ( a 1) g, (31) 9 1. (3) g ( a 1) g ( ) H L (1 a) g H 4 a 3 g The equilibriu enthalpies at = K and g contain in (3) a contribution independent of teperature and a contribution proportional to. This equilibriu enthalpy has a iniu value at K when the undercooled liquid is the ost unstable. It contains the axiu relaxed enthalpy at T K which is equal to 0.5 g H and recovered at T g, the endotheric latent heat (a-1) g H delivered at g when a is less than 1 and a contribution depending on. The latent heat L + ( g ) becoes endotheric by heating at g and equal to H ( g ). The latent heat L g obtained by cooling is equal to zero for aand only exists when a is less than 1 because the liquid is ore fragile than a liquid with a = 1 with its higher value of T 0. Measureents of As Se 3 and As S 3 showing the existence of a glass-foration heat obtained by cooling at T g deonstrates the existence of L [5]. The total endotheric latent heat produced by heating at T g would be equal to L + for a << 1

15 15 and to the enthalpy recovery L + = 0.5 g H for a = 1 if the stable glass phase has been previously fored by a long ageing at T K. There is no latent heat associated with a stable-glass foration 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 is still produced at T K when the transient relaxation tie ns is not evolved after quenching. The transforation in a stable glass is realized when this tie-lag is evolved. 7. Specific heat jups at the fragile-to-fragile, strong-to-strong, strong-to-fragile liquid transitions at T g 7.1. Fragile-to-fragile liquid transition A fragile-to-fragile liquid transition induces a new liquid state. The enthalpy derivative [d(h)/dt] calculated using (9) or (3) is equal to the specific heat difference C p (T) between a quenched fragile undercooled liquid and its new equilibriu state given by (33), as already shown in 008, without knowing at that tie the exact values of therodynaic paraeters [6]: d( ls lg s ) lg s0 ls C p ( T) H S dt 0g 0 The equilibriu specific heat jup below T g is defined by (34): 0 g. (33) T T 9 9 C p ( T ) S T T, (34) 4a 6 When a = 1, the scaling law is obeyed and the jup at T g is equal to (35): p g g C T ) 1. 5 S (. (35) Equation (35) is respected in any glass-foring elts as shown by [37](p. 48) with any C p (T g ) jups extending Wunderlich s previous finding [53]. The experiental values of C p (T g ) are used in part 8 to calculate the nuber a when (35) is not respected, including the experiental uncertainties on the easureents of various therodynaic paraeters. 7.. Strong-to-fragile liquid transition A strong-to-fragile liquid transition also induces a new liquid state. This phenoenon occurs when the fragile-to-fragile liquid transition teperature is expected to be a little lower than 0.5 T ( g < 0.5). The relation (8) is obeyed because the liquid is fragile above T g with 0 > /3 and becoes strong below T g with og./3. Equations (8, 36-38) are still used in part 7.3 to calculate ( ls0 - lgs0 ), o, T 0, C p (T g ), ls0, lgs0 and 0g, which is now respecting the inequality -1 < 0g < /3 (and 0 < T 0g < T /3). The Fe 50 Co 50 glass-foring elt transition is a good exaple of this phenoenon [54]. The therodynaic paraeters of several glasses undergoing this type of transition are presented in Table 1-1. The quantities T 0, T 0g, = ( ls0 - lgs0 ) given in Table 1-1 are calculated only knowing the experiental values of C plg (T g ), H, T g and T. All T 0 values of these fragile liquids are larger than T /3, whereas their T 0g values are saller than T /3. The teperatures T 0 are not very different fro the extrapolated VFT teperatures: 334/335, 768/768, 650/716, 417/37, 17 /41 [ [55]].

16 16 Table 1.1. Strong-to-fragile liquid transforations at the vitreous transition. Table 1.. Strong-to-strong liquid transforations at the vitreous transition. Heat capacity units are joules per gra.ato K. Fusion heat H are given in kilojoules per gra.ato. The energy saving coefficient is equal to the difference ( ls0 - lgs0 ). Materials T T g g H C p ls0 lgs0 T 0 T 0g Ref. 1-1 Fragile-to-strong CaAl Si O [55,60] As Te [61,6] CaMgSi O [55,63,67] Zr Ti 8.5 Cu 7.5 Ni 10 Be [64,65] Au 77 Ge 13.6 Si [66] 1- Strong-to-strong NaAlSi 3 O [55,67] SiO [55,68] BeF [56, 69,70] GeO [68,70,71] 7.3. Strong-to-strong liquid transition A strong-to-strong liquid transition also induces a new stronger liquid state assuing that lg in (31) is equal to zero at T g in the absence of iniu values of ls0 and lg0 when T 0 is less than T /3. In this case, the specific heat jup becoes saller than (35) and equal to (36): p g ls0 lg s0 g C ( T ) S. (36) Equation (6) applied at the vitreous transition is used to deterine lgs0 with (37): lg s0 3 g. (37) g 1 0g The specific heat jup is uch saller 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-foring elt, the saller the ( ls0 - lgs0 ) value. Equations (36,38) are used to calculate ( ls0 - lgs0 ), o, T 0, C p (T g ), ls0 and lgs0 : ls0 0 lg s0 0g ls0 g lg s0. (38)

17 17 The transforation paraeters of strong liquids are given in Table 1-. The teperatures T og are chosen equal to 0 K and T 0 equal to the VFT teperatures [55,56]. The known values H, T g and T are used to calculate the specific heat jup C p (T g ), the energy saving coefficients ls0, lg0 above and below T g and their difference. The jups C p (T g ) per g.ato are very sall copared to a crystal specific heat equal to 5 J/at.g.K. The calculated and experiental values are.1 and.05, 1 and.6, 1 and 0, 1.5 and.09 respectively. They are in good agreeent considering a easureent uncertainty of about 0.5 to 1 J/at.g.K. 8. Specific heat jups fro etallic and non-etallic glasses to undercooled liquids at the vitreous transition The specific heat differences C plx (T g ) between soe fragile etallic liquids and crystals are indicated in colun 8 of Table. The specific heat jups C plg (T g ) at the fragile-to-fragile liquid transition given in Table and Table 3 with a = 1 are equal to 1.5 S as predicted by (35) and in agreeent with other reports, within the easureent uncertainties of specific heat, fusion heat and elting teperature of all liquids. It has been recently found that the jups C plg of any etallic glasses are equal to 13.7 ± 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 any etallic liquid eleents [46]. Values of C plg (T g ) of aterials N 3 and N 6 in Table are deduced fro the slope of the axiu relaxed enthalpy versus teperature [31]. Liquids with a << 1 in Tables and 3 have a larger specific heat jup. The a values in Figure 3 are calculated at T g with (34) using the experiental values of C plg (T g ) and represented as a function of T g /T. The transition teperatures T g, which are used in all tables to calculate therodynaic paraeters, are close to the therodynaic transition teperatures where the relaxed enthalpy is equal to zero [31,33]. This approxiation has a weak influence on the. Values of a larger than 1 are used to define in Figure 3 an experiental uncertainty of ± 6.5% and of ± 13% on the specific heat jup in this odel. The a values equal to 1 in Table and Table 3 correspond to this uncertainty. The fusion enthalpies of ZnCl N 50 and B 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 polyorphs [58,59]. There are 36 glass-foring elts aong 49 following the scaling law (35), with the 13 others following (34) with values of a saller than 1. Table. Specific heat jups at the vitreous transition of etallic glass-foring elts. The units are Kelvin and Joule/at.g/K. N Materials T g g lg0 a ls0 C plx C plg 1.5S T 0 T 0g Ref. 1 Pd 40 Ni 10 Cu 30 P [7,73] Pd 43 Ni 10 Cu 7 P [7,74,75] 3 Zr 44 Ti 11 Ni 10 Cu 10 Be [76] 4 Zr 41. Ti 13.8 Cu 1.5 Ni 10 Be [36,77,78] 5 Pd 40 Ni 40 P [7,80,81] 6 Ti 40 Zr 5 Ni 8 Cu 9 Be [8]

18 18 7 Pt 57.3 Cu 14.6 Ni 5.3 P [83,84] 8 Zr 5.5 Al 10 Ni 14.6 Cu 17.9 Ti [85] 9 Zr46Cu46Al (86] 10 Zr 57 Al 10 Ni 1.6 Cu 15.4 Nb [85] 11 Zr 58.5 Cu 15.6 Ni 1.8 Al 10.3 Nb [83,87] 1 Pr 55 Ni 5 Al [88] 13 Pd 77.5 Cu 6 Si [80,89,90] 14 Zr 45 Cu 39.3 Al 7 Ag [86,91] 15 Cu 47 Ti 34 Zr 11 Ni [85,9] 16 Mg 65 Cu 5 Y [93] 17 Zr 65 Cu 17.5 Ni 10 Al [83,94,98] 18 La 55 Al 5 Ni 5 Cu 10 Co [96,97] 19 Zr 65 Cu 7.5 Al [95,98] 0 La 55 Al 5 Ni 10 Cu [96,97] 1 La 55 Al 5 Ni 15 Cu [96,97] La 55 Al 5 Ni 5 Cu [96,97] 3 La 55 Al 5 Ni [96,97] Figure 3. The nuber a versus T g /T. The nuber a of fragile non-polyeric glass-foring liquids defined in (30) at T g is plotted as a function of T g /T Table 3. A collection of specific heat jups easured in fragile liquids and selected by Wang, Angell and Richert [ [70]] are copared to 1.5 S. The units are Kelvin and J/ole/K. The fusion entropy of N 17 has

19 19 been changed using new easureents [99]. The ZnCl and B O 3 fusion entropies are decreased to respect C plg = 1.5 S because of the existence of crystallographic instabilities under pressure and hidden polyorphs which are also acting under Laplace pressure [ [58], [59]]. N Materials T g g lg0 a ls0 C plg 1.5S T 0g T 0 Ref. 1 -D-fructose [100,101] o-terphenyl [10,103] 3 -Toluidine [104,105] 4 Flopropione [70,106,107] 5 Maltitol [108,109] 6 Probucol [106,107,110] 7 Griseofulvin [106,107] 8 Indoethacin [70,111] 9 D-glucose [70,100,101] 10 PMS [70,11] 11 Sucrose [70,100,110] 1 Glibenclaide [106,107] 13 Propylene Carbonate [113] 14 Sorbitol [70,110] 15 Li-Acetate [70] 16 Triphenylethene [114] 17 As Se [5,99] 18 1,3,5-tri--Naphtylbenzene [70,115] 19 Phenobarbital [106,107] 0 Isopropylbenzene [116] 1 Hydro-chloro-thiazide [106,107] 3-Methylpentane [117] 3 Salol [113,114] 4 -Cresol [104,105] 5 Ca(NO 3 ) -4H O [118] 6 Xylitol [119] 7 Phenolphthalein [70] 8 9-Broo phenanthrene [113] 9 Triphenyl phosphite [70] 30 α-phenil -cresol [10] 31 H SO 4-3H O [11,1] 3 Diethylphthalate [13] 33 -Fluorotoluene [69,104,105] 34 -ethyl tetrahydrofuran [14,15] 35 n-butene [16,17] 36 Toluene [18] 37 Glycerol [113] 38 -Methyl pentane [50,19] 39 Ethylbenzene [18] 40 n-propanol [130]

20 Broopentane [14] 4 -Methyl-1-propanol [131] 43 Seleniu [103,13] 44 Butyronitrile [133] 45 Cis-/trans-Decalin [113] 46 Ethanol [69,70, ] 47 Methanol [50,136,137] 48 Ethylene glycol [138] 49 -Xylene [104,105] 50 ZnCl [58,139] 51 B O [59,140,141] 9. Enthalpy theral cycles expected in soe liquids and deterination of the Kauzann teperature The ultiate relaxed enthalpy is the axiu relaxed enthalpy occurring at equilibriu at the Kauzann teperature. The total equilibriu enthalpy changes at T g equal to the latent heat L + and L - as defined by (3) are given in Tables 4 and 5. The relaxed enthalpy of As Se 3 is saturated at the Kauzann teperature, equal to 6.4 J/g corresponding to.48 kj/ole and is approxiately equal to that given by the scaling law 0.5 g H = -.1 kj/ole [5]. This glass-foring elt does not follow the scaling law above T g, as shown in Table 3 N 17. Its therodynaic paraeters given in Table 3 and 5 are T = 645 K, T 0 = 380 K, T 0g = 93 K, g = 0.83, a = 0.753, H = 15.6 kj/ole, L + = 0.1 H = 3.8 kj/ole and L - = 0.07 H = 1.09 kj/ole. An endotheric enthalpy L + of 3.5 kj/ole (9.04 J/g) corresponding to lg = 0.4 has been easured at T g by transforing the liquid state below T g into the undercooled liquid state above T g after a long ageing tie near T K equal to 166 hours. The application of (3) using a = leads to a latent heat L = 0.07 H = 1.09 kj/ole and L + = 0.10 H = 3.8 kj/ole which are approxiately equal to experiental values of 1.07 kj/ole and 3.5 kj/ole respectively [5](Table 1). These experiental results confir that the ultiate relaxed enthalpy always respects the scaling law whatever the nuber a ay be and that an exotheric latent heat can be observed at T g while cooling a glass-foring elt through the vitreous transition when the nuber a is uch saller than 1 [33]. The ultiate enthalpy recovery of butyronitrile has also been studied by easuring the relaxed enthalpy after vapor deposition at 40 K, far below T K. It is equal to 1.3 kj/ole and larger than 0.5 g H = 1 kj/ole and saller than the equilibriu enthalpy of the stable glass phase equal to 1.5 kj/ole expected at T K including the latent heat 0.5 g H associated with a stable glass foration when a = 1 [133]. The out-of-equilibriu entropy of the undercooled liquid reains larger than that of crystals. In these conditions, the ultiate relaxed enthalpy H r equal to 0.5 g in all liquids can be used to calculate T K considering that all therodynaic properties are obeying scaling laws below T g. The axiu relaxed enthalpy decreases with teperature. 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 Kauzann teperature T K is calculated using the ean theoretical specific heat below T g deduced fro (33) and iposing the ultiate enthalpy recovery to be given by (39). The K is finally given by (40) with the nuber a deterined by iposing the theoretical specific heat jup to be equal to the experiental one:

21 1 K TK T ( T ) C p lg ( T Tg ) ( TK ) 0.5 g H, (39) ls0 g 0.5 g ( 0 lg s0 0g ) 1 4a g 0.5 g. (40) (9 6a) 5 In any cases for which a = 1, the scaling law K g is respected. 3 The equilibriu enthalpies divided by the fusion heat are equal to lg and represented in Figures 4, 5 and 6 for Pd 43 Ni 10 Cu 7 P 0 BMG N (a = 1), indoethacin Glass N 8 (a = 1) and As Se 3 Glass N 17 (a = 0.776) as a function of teperature below T g. There is no latent heat expected by cooling with a = 1 because the teperature T g only corresponds to a liquid-liquid transition which attains a pseudo-equilibriu after relaxation. On the contrary, a partial latent heat equal to L is expected for a << 1. The out-of-equilibriu liquid quenched below T g fro high teperatures to T K can be totally transfored into a stable glass phase after ageing at T K when the nucleation tie is iniu. The ultiate enthalpy has to be relaxed to attain first the pseudo-equilibriu liquid state and, after a uch longer tie, the stable glass phase producing an exotheric latent heat. An endotheric latent heat L + is needed to transfor the stable glass phase into an undercooled liquid at T g. Figure 4. The stable glass phase foration at T K in Pd 43 Ni 10 Cu 7 P 0. The enthalpy variation fro the quenched to the equilibriu state of the stable glass phase obtained at T K divided by H is represented by lg as a function of teperature. The irreversible and endotheric latent heats at T g are equal to 0.5 g H (a = 1).

22 Figure 5. The stable glass phase foration between T K and T g in Pd 43 Ni 10 Cu 7 P 0 and indoethacin. The enthalpy saving of indoethacin below the vitreous transition T g represented by the coefficient lg is plotted versus teperature and copared to that of Pd 43 Cu 3 Ni 10 P 0. 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-equilibriu enthalpies of undercooled elts are obtained during cooling after relaxation at the annealing teperature T. After a long ageing at T K, the transitions to the stable glass states would be accopanied by an exotheric latent heat and the stable glass enthalpies would increase up to T g. An irreversible endotheric latent heat equal to 0.5 g H is needed to return to the equilibriu undercooled liquid state above T g. Figure 6. The foration of stable glass phase in As Se 3. The enthalpy variation divided by H represented by - lg as a function of the teperature. A reversible latent heat occurs along AB and BA at T g during cooling and heating because a << 1. There is no ore enthalpy change along BC when the undercooled liquid is rapidly cooled without being relaxed. The structural relaxation progressively transfors BC into BD and the undercooled liquid is transfored into a new liquid state, producing an exotheric relaxed enthalpy. After a long ageing along DE, the undercooled liquid is transfored into a stable glass phase. The glass phase enthalpy increases along EF by heating. An irreversible endotheric enthalpy is produced by heating along FB. The teperature difference T K = (T g T K ) is copared in any of the exaples given in Tables 4 and 5 to experiental values obtained by entropy extrapolations fro teperatures above T g using specific

23 3 heat laws easured fro T g to T. These quantities are nearly equal for a = 1 in any 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 ethod. In addition, the liquid specific heat excess below T g is nearly constant, as deonstrated by a axiu relaxed enthalpy linearly decreasing with teperature up to T g as shown in Figures 4, 5 and 6. The presence of a hidden freezing before relaxation at T g is deonstrated in part 11 and the existence of a copleentary 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 g H ) and always saller than the available enthalpy H T Tg as already predicted [103]. The frozen entropy is also saller than the available entropy T C pls S dt T Tg C because C plg is always saller than the specific heat difference C plx between undercooled liquid and crystal. pls dt Table 4. Therodynaics paraeters of soe bulk etallic glasses: The equilibriu endotheric latent heats L + and L divided by H, the difference T K = (T g T K ) and the frozen enthalpy H g /H = ( g +L + ) below T g divided by H of soe fragile etallic glass-foring elts. Fusion heat units are kilojoules per g.ato. The axiu value of the relaxed enthalpy is equal to 0.5 g H ; T K = T g -T K ; L + and L are equilibriu latent heats. N Materials T H a H g /H L -/ H L + /H T Kcalc T Kexp Ref 1 Pd 40 Ni 10 Cu 30 P [7] Pd 43 Ni 10 Cu 7 P [7] 3 Zr 44 Ti 11 Ni 10 Cu 10 Be [33,76] 4 Zr 41. Ti 13.8 Cu 1.5 Ni 10 Be [77,78] 5 Pd 40 Ni 40 P [81] 6 Ti 40 Zr 5 Ni 8 Cu 9 Be Pt 57.3 Cu 14.6 Ni 5.3 P [84] 8 Zr 5.5 Al 10 Ni 14.6 Cu 17.9 Ti [85] 9 Zr 46 Cu 46 Al [86] 10 Zr 57 Al 10 Ni 1.6 Cu 15.4 Nb [85] 11 Zr 58.5 Cu 15.6 Ni 1.8 Al 10.3 Nb [87] 1 Pr 55 Ni 5 Al [88] 13 Pd 77.5 Cu 6 Si [80] 14 Zr 45 Cu 39.3 Al 7 Ag [91] 15 Cu 47 Ti 34 Zr 11 Ni [85] 16 Mg 65 Cu 5 Y [93]

24 4 17 Zr 65 Cu 17.5 Ni 10 Al [98,14] 18 La 55 Al 5 Ni 5 Cu 10 Co [96] 19 Zr 65 Cu 7.5 Al La 55 Al 5 Ni 10 Cu [96] 1 La 55 Al 5 Ni 15 Cu [96] La 55 Al 5 Ni 5 Cu [96] 3 La 55 Al 5 Ni [96] Table 5. Therodynaics paraeters of soe glasses: The equilibriu endotheric latent heats L + and L divided by H, the difference T K = (T g T K ) and the frozen enthalpy H g /H = ( g +L + ) between T K and T g divided by H of soe fragile etallic glass-foring elts. The relaxed ultiate enthalpy is equal to 0.5 g H. The fusion enthalpy of N 17 is changed using new easureents [5]. The ZnCl and B O 3 fusion enthalpies are also changed to respect C plg = 1.5 S because there exists a crystallographic instability under pressure [58,59] and then under Laplace pressure. Materials T H a L + /H L - /H H g /H T K T K Ref (K) (kj/ol) calc calc exp 1 -D-fructose [50] o-terphenyl [103] 3 -Toluidine [50] 4 Flopropione Maltitol Probucol Griseofulvin Indoethacin [111] 9 D-glucose [50] 10 PMS [50] 11 Sucrose [50] 1 Glibenclaide Propylene Carbonate Sorbitol [110] 15 Li-Acetate Triphenylethene As Se [5,99] 18 1,3,5-tri Naphtylbenzene 19 Phenobarbital

25 5 0 Isopropyl benzene Hydro-chloro-thiazide Methylpentane Salol [50] 4 -Cresol [105] 5 Ca(NO 3 ) -4H O [118] 6 Xylitol Phenolphthalein [50] 8 9-Broophenanthrene Triphenyl phosphite Phenil -cresol H SO 4-3H O [50] 3 Diethylphthalate Fluorotoluene ethyl tetrahydrofuran 35 n-butene [17] 36 Toluene [104,105] 37 Glycerol [50] 38 -Methylpentane [19] 39 Ethylbenzene n-propanol [131] 41 3-Broopentane ethyl-1-propanol Seleniu [13] 44 Butyronitrile [143] 45 cis-/trans-decalin Ethanol [50] 47 Methanol [50] 48 Ethylene glycol [50] 49 -Xylene [105] 50 ZnCl B O [111] 10. Stable-glass supercluster nucleation rates between T K and T g The Gibbs free energy change for a stable-glass nucleus foration is no longer given by (15) and is equal to (41) because the quantity H /V * is eliinated fro the Gibbs free energy change leading to this new liquid state: G 3 H R H 1kBV ln Kls 1/ 3 lg ( R, ) ( lg )4 4R (1 lg )( ) V 3 V 43 S. (41)

26 6 The critical radius and the therally-activated critical barrier are given by (4) and (43) instead of (9) and (10): R (1 ) V k ln( K ), (4) * lg B lg 1/ 3 lg ( ) lg 36S G k B * lg T 1(1 lg ) ln( Klg ) 81( ) (1 ) lg 3. (43) The critical barrier in is high because the coefficient lg is always sall in all liquids. Then, the stable glass phase cannot directly grow fro the critical radius and is fored by hoogeneous foration of nuerous tiny superclusters percolating, interpenetrating, and then growing by reduction of their surface energy with the tie increase. The Gibbs free energy change associated with the foration of a stable-glass nucleus of radius R containing n atos is equal to (44) instead of (17): G 1/ 3 1/ 3 n (4 ) N AkB ln( Klg ) / 3 nlg ( n,, nlg ) H ( nlg ) H (1 nlg ) (3n) N A N A 36S. (44) (45): The energy saving coefficient nlg of a glass nucleus of radius R containing n atos is given by, (45) 9 1 n lg n lg 0 1 ( a 1) ( ) 3 a g where the coefficient lg0 = 0.5 g in (3) has been replaced by nlg0,which is proportional to the copleentary Laplace pressure and to 1/R when the inequality n 147 is respected. The value of nlg0 is predicted using (46) at = 0: nlg0 lg0 R R * lg lg0 nc n 1/ 3, (46) 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 [6,144]. In this particular case, the exact olar volue of superclusters being unknown, nlg0 is better deterined fro the nucleation teperature of the stableglass phase which occurs at the Kauzann teperature 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 (0) or (48) and G*/k B T by (1) or (43), and that the transient ties of nucleation at T g or at the nose teperature T n of the TTT diagra of crystallization above T g are close to 50 s:

27 7 n c 4 ( R 3 1 ( 3n * lg ) 3 N V A c 8N * 1/ G ) k T B A (1 ) 7 lg, (47) 3 k B S ln( K lg ( ) lg ) 3. (48) The theral variations of lnk ls and lnk lg in (3) respectively depend on B /(T-T 0 ) and B g /(T-T 0g ), which are equal and deduced at T g fro easureents of viscosity above and below T g. The glass and crystal steady-state nucleation ties t sn given by (3) depending on the K value can be calculated as a function of the teperature when the effective therally-activated energy barrier G neff /k B T given by () or (49) is known: G k B neff T 3 1(1 n lg ) ln( K lg ) G, (49) n lg 81( ) (1 ) k T n lg B where G nlg is given in (44). The crystallized or vitreous superclusters can grow beyond their own initial radius R when (3) or (50) is respected: Gneff. (50) ln( J nvtsn ) ln( Klg vtsn ) kbt The n-ato supercluster forations occur in the saple when their nucleation tie is evolved. The ato nuber n in spherical superclusters is chosen equal to the following stable agic nubers which are considered in an icosahedral structure for etals with face-centered cubic lattices: 13, 55, 147, 309, and 561 [47]. The nucleation rate logarith of n-ato superclusters lnj n = ln(v.t sn ) is calculated without knowing the stable-glass doain volue v and the nucleation tie t sn because the axia of nucleation rates occur at the Kauzann teperature T K and leads to a whole transforation of the liquid. The hoogeneous nucleation rates lnj n of n-ato superclusters ready for growth in three BMG; Pd 43 Ni 10 Cu 7 P 0 N, Pt 57.3 Cu 14.6 Ni 5.3 P.8 N 7, Cu 47 Ti 34 Zr 11 Ni 8 N 15, and four glasses; indoethacin G. N 8, A Se 3 G. N 17, diethylphthalate G. N 3 and seleniu N 43 represented in Figure 7 are calculated using the paraeters given in Table 6 and (50). The enthalpy saving is given by (3) and all results presented here are obtained without introducing a constant equilibriu enthalpy saving below T K, in order to show that the odel directly leads to the value of T K. The sae axiu at T K is still observed when a constant enthalpy change is introduced below T K. A nucleation rate equal to exp (0.7)/ 3 /s would transfor a liquid volue of 1 3 into a stable glass in an additional tie of 1 s if the glass doain could attain this volue by nucleus growth beyond the critical radius. It is not possible because the critical energy barrier given in (43) is always too high. Eleentary clusters containing 13, 55, 147, 309, 561 and 93 atos were studied. The nubers n = 147 or 309 lead directly to the highest axia of lnj n at T K using nlg0 values obeying (46). The odel applied to four glasses works without using any adjustable paraeter for n = 147 and 309. The ato nubers n have been chosen in Figure 7 as the nuber n inducing the largest nucleation rate at T K. The steady-state nucleation rate depends on the fusion enthalpy H as shown by (44) and (17). The exaples given in Figure 7 cover a broad distribution of fusion heats and, consequently, various nucleation rates in glassforing elts. There is no stable glass nucleus being fored at T g. The supercluster nucleation rate has a axiu at T K in all these exaples. The transition at T g cannot be described by supercluster

28 8 nucleation having a surface energy in spite of the knowledge of the enthalpy difference lg H /V between undercooled liquid and stable-glass phase and the specific heat jup prediction. The transition at T g is a liquid-liquid transition characterized by the enthalpy change that is predicted in (3). Other liquid-liquid transition odels involving superclusters of liquid nature are ore successful to describe the at the icroscopic scale [6-8,10-15]. Table 6. Paraeters used to calculate the nucleation rates lnj n, the nucleation ties t and t sn of stable vitreous phases. The units are based on eter, kelvin, joule and second. The entropy S is given per g.ato and V in 3 per ole. These droplets give rise to very tiny stable-glass doains and to axia of nucleation rates at T K, as shown in Figure 7. The critical nuber of cluster atos n c at the elting teperature T, the nucleation rate logarith ln(j n / 3 /s) at T K of n-ato eleentary superclusters, the extrapolated negative critical energy saving coefficient lg0 at T as shown in Figure 1, and the energy saving coefficient nlg0 at T ( = 0) of n-ato superclusters are given. The lnk lg value is deterined assuing that the transient relaxation tie at T g is equal to 50 s. Glass lnk lg B/(T g -T 0g ) lg0 T 0g nlg0 S V n n c T K T g lnj n T= T g 10 6 Pd 40 Ni 10 Cu 30 P Pt 57.3 Cu 14.6 Ni 5.3 P Cu 47 Ti 34 Zr 11 Ni Indoethacin As Se Diethylphthalate Seleniu Figure 7. The n-ato supercluster nucleation rate logariths lnj n of seven glass-foring elts versus = (T-T )/T. They are plotted versus the reduced teperature below g. The unit of J n is -3.s -1. The nucleation rates of stable-glass superclusters are negligible at T g, while they are high and axiu at T K. A liquid-toliquid transition occurs at T g without stable-glass supercluster foration in Pd 43 Ni 10 Cu 7 P 0 BMG N, Cu 47 Ti 34 Zr 11 Ni 8 BMG N 15, indoethacin G. N 8, seleniu G. N 43, diethylphthalate G. N 3, As Se 3 G. N 17, and Pt 57.3 Cu 14.6 Ni 5.3 P.8 BMG N 7.

29 9 11. Fragile-to-fragile liquid transition at T g always occurring above T / All the fragile glass-foring elts in Tables and 3 have a transition teperature T g larger than T /. A strong-to-fragile liquid transition only exists when lgs0 is saller than 1.5, as shown in Table 1.1. In all the given exaples, the ideal glass transition teperature T 0g is always lower than T K. The Kauzann teperature cannot be lower than T og. A transition at T g = T / would lead to T og = T K = T, T 0 = 0.43 T and liiting values equal to 1.5 for lg0 and 1.5 for lso. This property explains why soe fragile glass-foring liquids do not undergo a visible liquid-to-liquid transition before being crystallized by heating the at teperatures a little higher than T / [145,146]. When a visible transition teperature T g is lower than T /, as shown in Table 1-1 for Au 77 Ge 13.6 Si 9.4, a fragile-to-strong liquid transition exists for T /3 < T 0 < T. There is no liquid-liquid transition in any undercooled elt for T < T 0 < 0.43 T and an aorphous state is observed below the crystallization teperature. 1. Hidden freezing at T g before relaxation of quenched liquids The specific heat of a quenched liquid is always assued as being continuous below T g before relaxation. The Kauzann teperature is extrapolated using the specific heat theral variation easured fro T g to T. This extrapolation is in contradiction with the linear decrease of the relaxed enthalpy with teperature 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 sall for saples obeying the scaling law above T g and often large in Table 5 for saples having a latent heat L delivered at T g by cooling. A signature of a freezing transition at T g exists without being fully accoplished before enthalpy relaxation, as shown in Figures 8, 9 and 10. The Pd 43 Cu 7 Ni 10 P 0 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 5 and altitol are increasing. The theoretical and experiental Kauzann teperatures of altitol are not the sae 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. Figure 8. Specific heat of the undercooled liquid Pd 43 Cu 7 Ni 10 P 0. The undercooled liquid specific heat C pl of BMG N is plotted versus teperature above T g = 576 K using experiental results [79]. The specific heat

30 30 between T g and T K is calculated by adding the specific heat change given by (34) to the experiental values C pg of the glass phase. There is a weak change of the slope at T g without exotheric latent heat. This explains why the calculated and extrapolated values of T K are about the sae. Figure 9. Specific heat of the undercooled liquid Zr 44 Ti 11 Ni 10 Cu 10 Be 5. The undercooled liquid specific heat of BMG N 3 is plotted versus teperature above T g using known experiental 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 copleentary slope corresponding to the difference of specific heat between the glass and crystallized states (C pg -C px ). The calculated and experiental values of T K are equal. Figure 10. The undercooled liquid specific heat of altitol. The undercooled liquid specific heat of Glass N 5 is plotted versus teperature above and below T g using known experiental results (continuous line) [146]. An extrapolation of C pl below T g leads to a Kauzann teperature of 61 K. The specific heat below T g is calculated by adding (34) to the easured glass specific heat (point line). There is a slope increase accopanying the freezing transition before enthalpy relaxation. The Kauzann teperature deduced fro (3) is saller and equal to 85 K. 13. Transforation of the stable-glass indoethacin in undercooled liquid at T g

31 31 The specific heat of four indoethacin saples has been easured and copared [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 isotherally aged below T g during 7 onths and 37 days. The enthalpy of the aged and stable glass saples below T g are saller than that of an ordinary glass [10,13]. The authors have claied that their ageing ties have not transfored indoethacin into stable glass. The enthalpy difference between an ordinary glass and the stable glass is equal to 4000 ± 400 J/ole and the predicted latent heat L + in Table 5 N 8 is saller and equal to H = 600 J/ole. The specific heat difference between ordinary glasses and those subitted to ageing increases slightly with ageing tie. The specific heat of a stable glass is also a little less than that of an ordinary glass. These observations show that the pseudo-equilibriu obtained when ns is evolved after structural relaxation is not fully attained as expected fro theoretical considerations, which have shown that the steady-state and transient nucleation ties are ixed and not siply added [37]. The odel 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-equilibriu state. The sall experiental increase of C plg, which is equal to 19 ± 10 J/ole.K, increases the enthalpy difference, and a corrected value of 19 T K = 67 J/ole.K reduces the observed latent heat at T g fro 4000 to 3373 ± 400 kj/ole.k [148]. The transition teperature T g observed after ageing is 7 K higher, as shown by specific heat jups 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 elt at roo teperature during ageing is equal to about C plg 0 K. This enthalpy has to be reinjected in the saple at the therodynaic glass transition. The endotheric latent heat of this aged saple is equal to 900 J/ole, in agreeent with the observed enthalpy excess. The therodynaic transition teperature is close to 35 K for both aged, relaxed saples and stable glass saples instead of 318 K as easured for ordinary glasses. It is so because the nucleation tie of the liquid phase in the glass phase sees to be iniu at this teperature and equal to about 4000 s in agreeent with nanoscale specific heat easureents [148]. This relaxation tie is larger than 50 s because the stable-glass density and consequently the energy barrier for ato diffusion fro the vitreous state to the undercooled elt is larger [39]. Figure 11. The transient nucleation tie-lag ns of undercooled liquid droplets containing the critical ato nuber in the stable glass phase. This calculated transient nucleation tie-lag ns is plotted versus

32 3 teperature. Experiental points noted Kearns are found in [148] (Fig. 6). The saples are thin fils of stable glass phase which have a thickness of 900 n, a volue v = and are subitted to annealing at teperatures lower than T g and equal to 318, 30, 34 and 35 K. The new therodynaic paraeters are: lg0 = ls0 lg0 = 1.68, 0g = 0.518, 0 = The Gibbs free energy change has reversed its sign and is given inwith ( ls0 lg0) = lg0. All stable-glass superclusters have a negligible steady-state nucleation tie at T g.the transient nucleation tie-lag ns in (1) is chosen as being equal to 3910 s at T g = 35 K corresponding to lnk gl = in (3), B/(T g -T 0 )= in (3), the Zeldovitch factor = , G* lg /k B T = 754 and V = for 41 atos per indoethacin olecule. A good agreeent with experiental values of ns is obtained as shown in Figure 11 [147](figure 6). The tie dependence of the elting teperature of the stable glass is due to the theral variation of the transient nucleation tie-lag ns. 14. Conclusions A new odel introduces an enthalpy saving at T g describing the equilibriu property changes of any fragile glass-foring elts below T g only knowing T g, T the elting teperature, and H the elting heat. The specific heat jup at T g, the reduction fro T 0 to T 0g of the teperatures at which the enthalpy savings associated with crystal nucleus foration would be extrapolated to zero fro above and below T g, the Kauzann teperature T K, the enthalpy saving between T g and T K, the relaxed enthalpy, its ultiate value at T K, the latent heats associated with liquid-stable-glass transition and TTT diagras 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-equilibriu tie which strongly increases when the teperature decreases below T g. The tie dependence of T g easured varying cooling and heating rates is due to this incubation tie which is viewed as a transient nucleation tie ns in undercooled liquids preparing the phase transforation. The incubation tie at T g is equal to about 50 s and is the sae as that observed in TTT diagras at the crystallization nose teperature above T g in spite of a large change of the viscosity between these two teperatures. It is proportional to the reverse of the constant K defining the transient nucleation tie ns in (1). A ean 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 transfored 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 tie-lag of transient nucleation. The sall specific heat jups at T g in strong liquids are used to deterine their teperature T 0 lower than T /3, assuing that the new value of T 0g becoes equal to zero at 0 K. The odel also predicts the absence of fragile-to-fragile liquid transition near T / when T 0 is larger than T and saller than 0.43 T. The energies savings ls () H /V and lgs () H /V are enthalpy excesses per unit volue of undercooled liquids above and below T g as copared to that of the crystallized state while lg () H /V is the enthalpy excess of the undercooled liquid as copared to that of the stableglass phase with lg () being equal to [ ls () lgs ()]. These quantities induce different Laplace pressures on the superclusters which ay lead to the condensation of a new phase fro the

33 33 undercooled elt. The energy saving ls () H giving rise to crystallization above T g is transfored below T g in lgs () H respecting lgs < ls with a iniized value of lgs. The new liquid phase below T g cannot be transfored in crystals because the nucleation tie of a critical supercluster is uch too long. The stable glass phase foration is driven by an enthalpy change at T g equal to lg () H /V. The specific heat difference between an undercooled elt and the stable-glass phase is predicted between T K and T g. The supercluster hoogeneous nucleation teperatures T 1 above T g and T = T g occur far below the crystallization teperature when the energy saving coefficients ls0 and lg0 at T are iniu, lg0 being always equal to 1.5 g + and ls0 to g + in about 70% of all fragile glass-foring elts where g = (T g -T )/T. Their teperatures T 0g and T 0 follow scaling laws depending on lg0 and ls0, their specific heat jup at T g being equal to 1.5 H /T, and their reduced Kauzann teperature square respecting K = 5 g/3. The critical stable-glass nucleus is subitted to a Laplace pressure change p accopanying the enthalpy changev p = lg H per ole below T g, The critical energy saving coefficients ls, lgs and lg are linear functions of instead of = (T-T )/T. The energy saving coefficients nlgs0 and nls0 at T of a n-ato supercluster are proportional to its reverse radius when n 147 and is quantified for n < 147. Any n-ato supercluster fored below T is not subitted to preelting because it has an energy saving coefficient depending on and consequently its surface atos have the sae elting teperature T than the core ones. The critical energy barriers of elts are too high below T g for having a single supercluster giving rise by growth to an infinite supercluster at any teperature saller than T g. The equilibriu glass phase is obtained by hoogeneous foration, percolation and interpenetration of eleentary supercluster ultitude containing agic ato nubers filling all the space during very long nucleation ties. The critical energy barrier associated with these eleentary superclusters depends on their ato nuber n and is reduced by the large Gibbs free energy change associated with their foration. Their nucleation ties and TTT diagras are predicted between T K and T g in appendix A. The stable vitreous phase is ainly fored at the Kauzann teperature T K because the nucleation rate of superclusters which are ready to grow has a pronounced axiu at this teperature where the nucleation ties are iniu values. The liquid-to-liquid transition at T g is not induced by stableglass supercluster foration and is a direct consequence of the enthalpy change. The therodynaic 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 teperatures between T K and T g. An endotheric latent heat is recovered in fragile liquids at T g during heating which depends on the annealing teperature below T g and is only equal to the relaxed enthalpy when a stable glass phase has not been fored. The stable glass phase foration also produces an exotheric enthalpy at T K which is recovered at T g by a copleentary endotheric latent heat. In strong liquids, there is no latent heat accopanying the stable-glass foration and the transition at T g is a liquid-liquid transition after a iniu of relaxation tie below T g in agreeent with descriptions of the foration of dynaical fractal structures near a percolation threshold. In any glass-foring elts, there is no first-order transition because there is no exotheric latent heat produced at T g during cooling in spite of the presence of an endotheric 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-foring elt is soeties so fragile that a reversible latent heat exists at T g which is only a

34 34 fraction of the available enthalpy. In these liquids, the total endotheric latent heat at T g is uch larger than the exotheric latent heat obtained by cooling when it exists. Appendix A TTT diagras of several liquids in stable-glass phase between T K and T g The paraeters used to calculate the nucleation ties are given in Table 6 for three BMG: Pd 43 Ni 10 Cu 7 P 0 N, Pt 57.3 Cu 14.6 Ni 5.3 P.8 N 7 and Cu 47 Ti 34 Zr 11 Ni 8 N 15 and four glasses: indoethacin N 8, A Se 3 N 17, diethylphthalate N 3 and seleniu N 43. The nucleation ties t and t sn of these seven glasses are plotted as a function of teperature in Figures A1, A, A3, A4, A5, A6, and A7 respectively. The total tie t = ns + /6 t sn includes the transient relaxation tie ns starting fro the quenched liquid state, while t sn is the steady-state nucleation tie of the stable-glass doain of volue v having a iniu at T K. The critical energy barrier being uch too high, the volue v is chosen equal to the volue of the n-ato cluster instead of the saple volue in order to have the iniu of t very close to T K. This assuption is in agreeent with the description of the glass-state by olecular dynaics siulations as coposed of tiny icosahedral-like clusters, ost of which touching or interpenetrating yielding a icrostructure of polyicosahedral clusters that follow a specific sequence of agic nubers [3]. The Zeldovitch factor is calculated with (47) using G* n /k B T instead of G* lg /k B T and n c equal to the nuber n. The ain uncertainty on the nucleation tie t coes fro the uncertainty on the value of B/(T g -T 0g ) chosen at T g, given in Table 6 and fro the transient nucleation tie-lag ns of stable-glass superclusters which could be uch larger than 50 s as shown for indoethacin in part 1. It explains why the iniu of t soeties occurs at a teperature a little larger than T K. It is shown here that the nucleation rates of these eleentary clusters are sufficiently large after ageing to produce this type of icrostructure knowing that the growth around these nuclei could be strongly accelerated when the surface energy declines with touching and interpenetrating superclusters. Figure A1. TTT diagras of Pd 43 Ni 10 Cu 7 P 0. The TTT diagras of BMG N represented by the logariths of the total nucleation tie t (s) and the steady-state nucleation tie t sn (s) are plotted versus teperature. The

35 35 inia of lnt sn and lnt occur at 510 K and 51 K respectively. The Kauzann teperature is equal to 51 K. The supercluster volue logarith ln(v/ 3 ) is equal to It contains 147 atos. Figure A. TTT diagras of Pt 57.3 Cu 14.6 Ni 5.3 P.8. The logariths of the steady-state nucleation tie t sn (s) and of the total nucleation tie t(s) of BMG N 7 are plotted versus teperature. The eleentary cluster contains 55 atos. The iniu of t sn (s) occurs at T K = 419 K. The iniu of t(s) occurs at T = 46 K. The supercluster volue logarith is ln (v/ 3 ) = 6.. Figure A3. TTT diagras of Cu 47 Ti 34 Zr 11 Ni 8 versus T. The logariths of the steady-state nucleation tie t sn and the total nucleation tie t of BMG N 15 are plotted versus teperature. The eleentary cluster contains 55 atos. The iniu of t sn occurs at T K = 541 K while that of t occurs at 545 K. The supercluster volue logarith is ln (v/ 3 ) = 6.3.

36 36 Figure A4. TTT diagras of Indoethacin. The logariths of the steady-state nucleation tie t sn and the total nucleation tie t of Glass N 8 are plotted versus teperature. The eleentary supercluster contains 147 atos. The iniu of t ns (s) occurs at T = 8 K while that of t(s) occurs at T = 88 K. The Kauzann teperature is equal to 85 K. The supercluster volue logarith is ln(v/ 3 ) = The indoethacin olecule contains 41 atos. Figure A5. TTT diagras of As Se 3. The logariths of the steady-state nucleation tie t sn and the total nucleation tie t of Glass N 17 are plotted versus teperature. The eleentary clusters contain 309 atos. The iniu of t sn (s) and t(s) occurs at T = 430 K for n = 309. The supercluster volue logarith is ln(v/ 3 ) = The calculated Kauzann teperature is T K = 434 K.

37 37 Figure A6. TTT diagras of diethylphthalate. The logariths of the steady-state nucleation tie t sn and of the total nucleation tie t of Glass N 3 are plotted versus teperature. The supercluster contains 147 and 309 atos. The inia of t sn (s) and t(s) for n=309 occur at 15 K and T K = 154 K respectively. The supercluster volue logarith is ln(v/ 3 ) = The diethylphthalate olecule contains 30 atos. Figure A7. TTT diagras of Se. The logariths of the steady-state nucleation tie t sn and the total nucleation tie t of Glass N 43 are plotted versus teperature. The cluster contains 55 atos. The inia of t sn (s) and t (s) occur at T K = 54 K and T= 59 K respectively. The supercluster volue logarith is ln(v/3) = The nucleation rate of eleentary clusters becoes larger at teperatures uch saller than T g and attains a axiu value at T K, as shown in Figure 7. The foration of a stable glass phase by ageing is possible when the long foration tie of an eleentary supercluster is evolved and when (50) is respected authorizing its growth. All the liquids studied in Figure 7 could be transfored between T K and T g in ties equal to the su of structural relaxation tie ns and the foration tie of the eleentary supercluster. Studies are necessary to deterine whether the transforation ties by ageing at T K can becoe sufficiently sall in soe glasses. Soe liquids such as As Se 3, Pd 43 Cu 7 Ni 10P 0 and Indoethacin having the largest nucleation rates are good candidates for such studies. The nucleation ties of As Se 3 are the sallest and an ageing tie of about 35 hours (within an uncertainty of about 35 7 hours) at T = 430 K could be sufficient to induce the stable-glass phase. This rough

38 38 estiation is in agreeent with the copleentary latent heat in agreeent predicted and easured after an ageing of 166 hours at 418 K [51]. The substrate teperature used for indoethacin physical vapor deposition has been varied fro 65 to 305 K [44](Fig. 5). There is no ore stable glass foration above 97 K in agreeent 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 logariths are negative in all liquids. A liquid-to-liquid transition is nevertheless easily observed at T g because the tie ns necessary to attain the pseudo-equilibriu of the new liquid phase is of the order of 50 s. The enthalpy recovery of a quenched elt increases during cooling down to T K and is relaxed at each annealing teperature between T K and T g when the undercooled elt attains its pseudoequilibriu, as shown in Figs 4-6. The whole undercooled liquid is transfored into a glass phase at T K. The axia of the eleentary supercluster nucleation rate at T K are obtained without using any adjustable paraeter for n 147 with nlg0 being calculated with (46). The coefficient nlg0 for n = 55 is adjusted by fixing the axiu nucleation rate at T K. An exotheric latent heat has to be produced to undergo the transition at T K and the glass survives in this new equilibriu state fro T K to T g arked by hysteresis cycles. The glass phase is elted at T g with the help of the endotheric heat L +. The enthalpies of Pd 43 Ni 10 Cu 7 P 0, As Se 3 and indoethacin undercooled liquids at equilibriu 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-equilibriu value after an increase by a factor of to 3 fro 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 tie dependence of T g is observed during heating of rapidly quenched elts which have not had the tie to attain the pseudoequilibriu during cooling through T g because a iniu tie of about 50 s is needed to undergo the transition [5,76,144]. The calculations of the enthalpy saving coefficients lg are only based on the knowledge of the glass transition T g, the elting teperature T and therodynaic considerations related to scaling laws. The fusion enthalpy H has to be known to calculate the latent heats and the ultiate enthalpies of the stable-glass phase. Appendix B Superclusters of 13 atos governing the first-crystallization tie of etallic glass-foring elts Superclusters subitted 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. Paraeters used to calculate the Tie-Teperature Transforation diagras above T g. The unit of entropy and volue are J/K/ato.g. and 3 respectively. Crystal lna T 0 (K) B (K) ls0-0g - 0 n0 S V 3 n V 3 Ref BMG N [34,73] BMG N [35,74,75,83] BMG N [36,77-79]

39 39 Figure B1. The calculated TTT diagras of Pd 40 Ni 10 Cu 30 P 0, Pd 43 Ni 10 Cu 7 P 0, and Zr 41. Ti 13.8 Cu 1.5 Ni 10 Be.5. They are in agreeent with the experiental observations of the crystallization nucleation. A study of isotheral nucleation tie is ade after quenching the elt fro above T down to the annealing teperature. The tie-lag ns for transient nucleation has to be included in the calculation of the total nucleation tie t which is equal to ( ns + /6 t sn ) [37]. The calculated liquid crystal TTT diagras of three glass-foring elts Pd 40 Ni 10 Cu 30 P, Pd 43 Ni 10 Cu 7 P 0, and Zr 41. Ti 13.8 Cu 1.5 Ni 10 Be.5 are represented in Figure B1 in agreeent with the experiental studies [34-36]. The paraeters given in Table B1 are used. The 13-ato supercluster is the iniu size entity which is fored by hoogeneous 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 uch larger than the critical value G* ls /k B T: G13 G13 ln( J13 v tsn ) ln( Kls v tsn ) k T k T B * B, (B1) 0 where G 13 and G* 13 /k B T are given by (17) and () with n = 13. The 13-ato cluster nucleation rate J 13 is equal to (v.t sn ) because ln(j 13.v.t sn ) = 0, where v is the saple volue for crystallization and t sn the steady-state nucleation tie of the 13-ato supercluster. The 130 and lna values have been varied to reproduce the total nucleation tie t and the nose teperatures of the experiental TTT diagras [9,4]. Acknowledgents The first steps of this work have been also sponsored by the sino-french Laboratory for the Application of Superconductors and Magnetic Materials involving the Northwestern Polytechnical University (NPU) in Xi an, the Institut Polytechnique de Grenoble and the Centre National de la Recherche Scientifique in Paris. These initial works were presented at the Workshop of Solidification Processing-1: Melt Structure and Nucleation in 011 in Xi an (P. R. China), at the 5 th International Workshop on Materials Analysis and Processing in 01 in Autrans (France) and at the eeting of the Magneto-Science Society of Japan in 01 in Kyoto. The author thanks Professor Lian Zhou and Professor Wanqi Jie in Xi an, Professor Eric Beaugnon in Grenoble and Professor Tsunehisa Kiura