Lindenmann s Rule Applied to the Melting of Crystals and Ultra-Stable

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1 Lindenann s Rule Applied to the Melting of Crystals and Ultra-Stable Glasses Robert F. Tournier Centre National de la Recherche Scientifique, Institut Neel, Université Grenoble Alpes, Consortiu de Recherches pour l Eergence de Technologies Avancées, B.P. 166, F-3804 Grenoble Cedex 09, France. E-ail address: robert.tournier@creta.cnrs.fr Abstract: The ratio of the ean square aplitude root of theral vibrations and the interatoic distance is a universal constant ls at the elting teperature T. The classical Gibbs free energy change copleted by a volue energy saving ls (or lg H that governs the liquid to solid and liquid to ultrastable glass transforations leads to a universal constant equal to ls (or lg, H being the crystal elting enthalpy. The iniu values 0.17 of ls and of ls are used to predict ultra-stable glass foration in pure etallic liquid eleents at a universal reduced teperature g = (T g T /T = Introduction The dependence of the liquid supercooling teperature on the superheating rate shows the existence of long-lived etastable nuclei surviving above the elting teperature T [1]. The classical Gibbs free energy change cannot predict the presence of such entities without introducing a copleentary negative contribution v p varying with = (T-T /T, v being the nucleus volue and p a copleentary Laplace pressure [,3]. Crystallisation and elting are initiated by the foration of solid or liquid growth nuclei accopanied by a volue change that is expected to obey to Lindenann s rule [4]. Lindeann s description shows that the ratio of the ean square aplitude root of theral vibrations and the interatoic distance is a universal constant ls at the elting teperature T. The critical copleent v p associated with crystallisation at T = T has been deterined for any pure liquid eleents and glass-foring elts as being equal to v ls0 H /V with ls0 being a nuerical critical fraction of the elting heat H. The coefficient ls0 = 0.17 is the sae for any liquid eleents [], while it is uch larger than 1 and saller than in any glass-foring elts, as shown for 84 exaples Tables and 3 in [5]. The objective of this study is to relate ls0 to the Lindeann ratio ls. The ultra-stable glass state is described as a therodynaic equilibriu between crystal and liquid states, which would be attained by supercluster foration and their percolation after a very long annealing tie at the Kauzann teperature T K [5], or by quenching the elt fro above T and annealing it at the foration teperature T sg of this phase [6]. The optiu foration teperature T sg leading to the higher density is always equal to the Kauzann teperature of strong glasses, and very often to that of fragile glasses. The enthalpy that is recovered at the glass transition teperature T g is equal to ( ls0 - lgs0 H in strong glasses and to 1.5 ( ls0 - gs0 H in fragile glasses, with gs0 being the critical fraction of elting heat leading to crystallisation of a virtual glass at T. This description agrees, in principle, with the

2 schee of a rando first-order phase transition hidden below T g occurring at the Kauzann teperature T K viewed as the true glass transition at equilibriu [7 9]. Various icroscopic odels prove the existence of a phase transition at T g [10 16]. Liquid-glass transforation is treated as a percolation-type phase transition with the foration of dynaic fractal structures near the percolation threshold. We do not exaine the proble of percolation of liquid entities, because the total volue change of all of the, depending on their nucleation rate and their ato nuber n, is liited by the effective volue change available at the glass transition. A criterion analogous to the Lindean criterion of elting was proposed by Sanditov for the softening transition fro the glass equilibriu to the liquid state [17]. We are also able to deterine Lindeann s constant fro our odel and show whether it agrees with this odel of glass elting. The universal constant ls obtained for pure etallic eleents at their elting teperatures T is used to build a odel for their vitrification. The Gibbs free energy change below T g cannot include any variation of structural relaxation enthalpy, because ls cannot be lower than this iniu value. - The application of Lindeann s rule The Gibbs free energy change for the foration of a condensed supercluster giving rise by growth to a crystal fro a liquid droplet of radius R is defined by Eq. (1: G H R H k V ln K (1 3 B ls 1/3 ls( R, ( n4 4R (1 n( V 3 V 36 S where H is the elting heat, V the olar volue, (T-T /T the reduced teperature, ls the fraction of the elting enthalpy associated with a spherical supercluster of radius R containing n atos, k B the Boltzann constant, S the elting entropy and lnk ls 90 ± in etallic liquid eleents [18]. The theral variation of n is given by Eq. (: n ( n0 (1 0 ( where n0 obeys Eq. (3 for n > 147, n0 = ls0 being the critical value, n c the critical ato nuber given by Eq. (4: n 1/3 n0 ls0 ( nc (3 n c 8(1 ls 7( ls 3 3 N A k B S ln K ls (4

3 where S is the elting entropy of crystals and ls the critical enthalpy saving coefficient, given as a function of the reduced teperature by Eq. (5 []: ( ls ls0 (1 0 (5 Eq. (1 is written as a function of the ato nuber n instead of the supercluster radius, with N A being the Avogadro nuber: G nls 1/ 3 1/ 3 n (4 N Ak B ln( K lg / 3, nls H ( nls H (1 nls (3 (6 N A N A 36S ( n, n The entropy of fusion given by (dg nls /dt p is equal to n S /N A because d nls /dt is equal to zero for = 0, and then the surface energy is constant during the solid-liquid transforation. The elting heat of supercluster surface atos is not weakened. Superclusters are viewed as superatos. The free electrons in a superato occupy orbitals that are defined by the entire group of atos of the supercluster, rather than by each individual ato separately [19,0]. We consider that the elting heat is proportional to the nuber n of atos foring a superato. These equations have been successfully applied to the crystallisation of pure etallic undercooled liquids. Such nuclei containing agic ato nubers govern the undercooling rate [3] Lindenann s rule predicts that the supercluster radius R is increased by the root of the ean square aplitude of ato vibrations when it is elted. An increase R of the radius R is applied to the surface energy given by Eq. (1. Eq. (7 is obtained assuing that the surface energy does not vary during the transforation fro solid to liquid: 1 R n0 (1 R R (1 1/ ls ls0 1 (7 R The coefficient n0 is always larger than its critical value ls0 at T when n < n c. Then, the corresponding Lindeann ratio is larger than its critical value. For the weakest values of ls0, we have ls0 R/R. The critical enthalpy saving associated with the Laplace pressure change accopanying a supercluster condensation having the critical radius for crystal growth is given below T g by Eq. (8: gs H ( V gs0 (1 0g H V (8 The enthalpy change at T g transfors Eq. ( into Eq. (8. The indices ls and gs are related to the undercooled liquid crystallisation and to the glass crystallisation, respectively. The nuerical coefficient

4 ls0 in Eq. ( above T g becoes weaker below T g and is transfored into gs0 while the teperature T 0 deduced fro 0 = (T 0 T /T is reduced and is equal to T 0g in 0g. These enthalpy saving coefficients ls and gs are equal to zero at T T 0 and T T 0g and to ls0 and gs0 respectively at the crystal elting teperature T. The enthalpy change, with the index lg, is associated with the ultra-stable glassto-liquid transforation and is equal to the difference between Eqs. ( and (8 given by Eq. (9 [5,6]: ( H lg ( ls H gs ls0 H (1 0 gs0 H (1 0g (9 The endotheric latent heat irr recovered at T g during the transition fro ultra-stable glass state to liquid is the axiu value ( ls0 gs0 H of lg at T. It is enhanced by the enthalpy available between the teperature where lg ( in Eq. (9 is equal to zero and T g which is equal to 0.5 ( ls0 gs0 H for fragile glasses and to zero for strong glasses. The Gibbs free energy change is given at T g by Eq. (10 before the transforation of the ultra-stable glass into a liquid state: G n (4 ln( K 1/ 3 A B lg / 3 n lg ( n,, n lg H ( irr H (1 irr 3n N A N A 36S N k (10 Eq. (7 can be applied to the elting of ultra-stable glasses replacing n0 by the critical coefficient ls0 for crystal elting, by irr = ( ls0 - gs0 for devitrification of ultra-stable strong glasses and by irr = 1.5 ( ls0 - gs0 for devitrification of fragile glasses [6]. These last coefficients define the fraction irr of H associated with the endotheric latent heat accopanying these transforations at T g. A softening transition occurs at T g and the Lindeann constant lg of the liquid-glass transforation depends on the difference of Lindeann s constants ls and gs of two liquid states of the sae substance above and below T g. Eqs. (11 and (1 are respectively followed by strong and fragile glass-foring elts: lg ls gs (11 lg 1.5 ( ls gs (1 3- Typical values of the Lindeann constant for crystal and ultrastable glass elting 3.1 The Lindeann constant of pure liquid eleents The energy saving coefficient ls0 was deterined by us in 007 as being constant and equal to 0.17 for about 30 pure etallic liquid eleents []. We did not propose any explanation at that tie. Applying Eq. (7, ls0 corresponds to a Lindeann constant equal to 0.103, which is in good agreeent with other deterinations [1 3]. We conclude that the enthalpy saving coefficient cannot be saller than This property is used to propose, in Section 4, that a new faily of ultra-stable glasses coposed of all pure liquid eleents exists.

5 3. The Lindeann constant of ultra-stable fragile glasses at T g The endotheric enthalpy coefficients of ultra-stable glasses have been deterined for any fragile glassforing elts. They are equal to 1.5 ( ls0 - gs0 and known for 49 non-etallic and 3 etallic glassforing elts. The coefficients ls0 and lg0 are listed in Tables and 3 in [5]. The corresponding Lindeann constants lg given by Eqs. (7 and (1 are plotted versus the reduced glass transition teperature g in Figure 1. The values calculated by Sanditov vary between [17]. Our results cover these values. Figure 1: The Lindeann constant lg is calculated with Eqs. (7 and (1. The coefficient nls0 in Eq. (7 has been replaced by irr = 1.5 ( ls0 - gs0.the lg of bulk etallic and non-etallic glasses are equal for the sae reduced value of T g. 3.3 The Lindeann constant of fragile glass-foring elts at their elting teperature T The enthalpy saving coefficient ls0 is used to calculate ls at the elting teperature of glass-foring elts. The ls follows a linear law as a function of g in Figure : ls = g

6 Figure : The Lindeann constant ls calculated with Eq. (7 at the elting teperature T of glassforing elts, plotted as a function of the reduced glass transition teperature g. These values of ls are about 6 ties larger than the iniu value and increase with g. The vibration aplitude attains 66% of the characteristic interatoic distance for the largest ratio T g /T. We know fro Inoue s work that alloys having the highest T g /T have three features in their alloy coponents, i.e. ulticoponent systes, significant atoic size ratios above 1% and negative heats of ixing [4,5]. These high values of ls could be due to a strong coupling of pairs of different atos in the liquid, which does not exist in pure liquid eleents. 3.4 The Lindeann constant of strong glass-foring elts at T g The endotheric enthalpy coefficient of a strong ultra-stable glass at T g is equal to ( ls0 - gs0, while the specific heat jup at T g is equal to S ( ls0 - gs0 / g. The Lindeann constant lg is then proportional to the specific heat jup. This jup is known to be uch saller than that of a fragile glass. It is iportant to verify whether these glasses obey Eq. (7, and the values of lg can be saller than the iniu value. Nine strong glasses have been studied and soe values of ( ls0 - gs0 are reported in Table 1 in [5]. Five of the (CaAl Si O 8, As Te 3.13, CaMgSi O 6, Zr Ti 8.5 Cu 7.5 Ni 10 Be 7.5, Au 77 Ge 13.6 Si 9.4 have < lg < 0.13; and four of the (SiO, BeF, NaAlSi 3 O 8, GeO have 0.09< lg < The endotheric latent heat at T g corresponds to a softening transition and the lg of the glass state is the difference of Lindeann s constants of two liquid states of the sae substance, above and below T g as shown by Eq. (11. Sanditov s proposal is verified. 4- Application of Lindeann s rule to the vitrification of pure etallic liquid eleents Recent work renews earlier findings of glass foration in pure etals of sall size and thickness [6 36]. There is a need for a fundaental understanding of the resistance to crystallisation of these glasses [37]. The glass transition teperature T g is unknown. Our odel needs in principle to know T g in order to be applied. Lindeann s rule is used to deterine this teperature. The glass transition transfors ls0 in Eq.

7 (5 to gs0 in Eq. (8. In pure liquid eleents, Lindeann s rule shows that this change is not possible. Eq. (8 can be used below T g with gs0 = ls0 = There is no structural relaxation enthalpy because ( ls0 gs0 is equal to zero [5,6]. The liquid eleents are fragile with 0 = /3 (T 0 = T /3 and becoe strong glasses below g. The reduced teperature 0 = /3 in Eq. (5 is changed in 0g = 1 in Eq. (8 as it occurs in any strong glasses having a Vogel-Fulcher-Taann teperature equal to zero below T g. The reduced glass transition teperature g is then given by Eq. (13 [5,6]: gs0 3 g g 1 0g (13 The variation of gs0 with g is plotted in Figure 3. The value of g corresponding to gs0 = 0.17 is This is a universal value of the reduced glass transition teperature of pure liquid eleents corresponding to T g = T. The reduced hoogeneous nucleation teperature of crystals is equal to /3 in the liquid state. The glass transition could be asked during heating by crystallisation of the liquid in its neighbourhood. Figure 3: Eq. (13 is used to represent gs0 as a function of the reduced glass transition teperature g for 0g = 1. The point corresponding to the universal value 0.17 of gs0 corresponds to g = Eq. (9 is transfored in Eqs. (14 and (15, which are universal equations describing the enthalpy change lg below g :

8 lg. 5 1 with /3 <0.63 (14 lg 0.17 (1 with1 <</3 (15 A latent heat equal to H is predicted at the glass transition teperature. These equations are represented in Figure 4 together with Eqs. ( and (5: Figure 4: The critical enthalpy saving coefficients ls and gs associated with crystal foration given by Eqs. (5 and (8 are plotted versus the reduced teperature between = 1 and 0. The enthalpy saving coefficient lg associated with the ultra-stable glass state foration and given by Eq. (9 is also plotted versus. At the glass transition, an exotheric heat is produced at equilibriu and probably reported at the final teperature after quenching. The Lindeann constant lg at T g deduced fro Eq. (7 is equal to 0.051, because the latent heat is equal to H. Eq. (14 is nearly a straight line and can be approxiated by a linear function of in Eq. (16: lg (1 (16 The exotheric latent heat at g is not produced during a rapid quench, as observed in any exaples of glasses having a specific heat jup larger than 1.5 S [5], and has to occur at the final roo teperature leading to the glass state. The lg cannot vary with at low teperatures because the entropy associated with the latent heat calculated at g = 0.63 has to stay equal to H /(1/T at teperatures lower than T /3. This condition respects Eq. (15. The enthalpy jup at roo teperature (T= 300 K is represented for several liquid etals in Figure 5.

9 Figure 5: The enthalpy saving coefficient lg plotted versus. The reduced final teperature = (300- T /T obtained after quenching is indicated together for the following liquid eleents: W (-0.918, Ta ( , V (-0.86, Pd (-0.836, Ag ( and Al ( The enthalpy saving coefficient lg calculated with Eq. (15 is represented by a dashed line. The latent heat per ole is H. The enthalpy calculated fro 0 K up to /3 is equal to H while that fro /3 up to g is equal to H. lg has to be equal to zero below the Kauzann teperature. The reduced Kauzann teperature K is then equal to and the frozen enthalpy fro T K to T g is now equal to H. The total frozen entropy below T g in all etals is equal to 0.78 S where S is the elting entropy of crystals. The factor exists because the specific heat change starts at T g while the latent heat is produced at roo teperature after quenching. The specific heat jup is equal to H d lg /dt and is defined by Eq. (17: ( T T C p 0.17 ( 1 T 4 9 S in Joule/K/ole fro T /3 to T g, C p S in Joule/K/ole fro 0 K to T /3 (17 These universal values are represented in Figure 6:

10 Figure 6: The universal value C p /S of the reduced specific heat jup divided by the crystal elting entropy plotted versus = (T-T /T for any liquid eleents. The glass transition teperature of liquid eleents is T. 5- Conclusions The introduction, in the classical Gibbs free energy change, of an enthalpy saving associated with foration, in pure etallic liquid eleents, of solid superclusters, reveals the applicability of Lindeann s rule and leads to the expected theoretical value of the universal constant ls = This new equation works because the elting heat of superclusters, acting as growth nuclei, and containing n atos, is proportional to n, as expected in superatos. The critical supercluster containing n c atos elts at the teperature T of crystals. The crystallisation could be induced by hoogeneous or heterogeneous nucleation of superclusters containing agic ato nubers. Soe of the survive above T when their enthalpy saving coefficient varying as (n c /n 1/3 is uch larger than that of the critical supercluster containing n c atos, and when the liquid superheating rate is too sall. Lindeann s rule is used to predict the glass transition teperature T g = T of liquid eleents and universal therodynaic properties of these glasses. Lindeann s constants ls of glass-foring elts at T are uch higher than when the ratio T g /T increases. This inforation could be the sign of strong pairing of atos of different nature. The devitrification of ultra-stable glasses at T g is associated with Lindeann s constant lg that is equal in strong glasses to the difference of Lindeann s constants ls and gs of two liquid states of the sae substance above and below T g. This finding is agreeent with Sanditov s recent work, which considers that the softening transition at T g is soewhat siilar to elting.

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