Crystallization of Supercooled Liquid Elements Induced by Superclusters Containing Magic Atom Numbers Abstract: Keywords: 1.

Transcription

1 Crystallization of Supercooled Liquid Eleents Induced by Superclusters Containing Magic Ato Nubers Robert F. Tournier, CRETA /CNRS, Université Joseph Fourier, B.P. 166, 804 Grenoble cedex 09, France. E-ail: Tel.: ; Fax: Abstract: A few experients have detected icosahedral superclusters in undercooled liquids. These superclusters survive above the crystal elting teperature T because all their surface atos have the sae fusion heat as their core atos and are elted by liquid hoogeneous and heterogeneous nucleation in their core, depending on superheating tie and teperature. They act as heterogeneous growth nuclei of crystallized phase at a teperature T c of the undercooled elt. They contribute to the critical barrier reduction, which becoes saller than that of crysta containing the sae ato nuber n. After strong superheating, the undercooling rate is still liited because the nucleation of 1-ato superclusters always reduces this barrier, and increases T c above a hoogeneous nucleation teperature equal to T / in liquid eleents. After weak superheating, the ost stable superclusters containing n = 1, 55, 147, 09 and 561 atos survive or elt and deterine T c during undercooling, depending on n and saple volue. The experiental nucleation teperatures T c of liquid eleents and the supercluster elting teperatures are predicted with saple volues varying by 18 orders of agnitude. The classical Gibbs free energy change is used, adding an enthalpy saving related to the Laplace pressure change associated with supercluster foration, which is quantified for n=1 and 55. Keywords: theral properties, solid-liquid interface energy, crystal nucleation, undercooling, superclusters, liquid-solid transition, overheating, non-etal to etal transition in cluster, Laplace pressure 1. Introduction An undercooled liquid develops special clusters that iniize the energy locally which are incopatible with space filling [1 ]. Such entities are hoogeneously fored in glass-foring elts, and act as growth nuclei of crysta above the glass transition [4]. The foration of icosahedral nanoclusters has often been studied by olecular dynaics siulations into or out of liquids [5 8]. Silver superclusters containing the agic ato nubers n = 1, 55, 147, 09, 561 are ore stable. Their foration teperature out of elt and their radius have been deterined [5]. Icosahedral gold nanoclusters do not preelt below their bulk elting teperature [6]. Nanoclusters have been prepared out of liquids [9 14]. The density of states of conduction electrons at the Feri energy being strongly reduced for particle diaeters saller than one nanoeter leads to a gap opening [9,10]. Growth nuclei in elts are expected to have analogous electronic properties. Superclusters containing agic ato nubers are tentatively viewed for the first tie as being the ain growth nuclei of crystallized phases in all liquid eleents. I already considered that an energy saving resulting fro the equalization of Feri energies of nuclei and elts cannot be neglected in the classical crystal

2 nucleation odel [15,16]. An enthalpy saving v per volue unit of critical radius clusters equal to H /V was introduced in the Gibbs free energy change G which gives rise to spherical clusters that transfor the critical energy barrier into a less effective energy barrier, so inducing crystal growth around the at a teperature T c uch higher than the theoretical hoogeneous nucleation teperature equal to T /. This enthalpy depends on H the elting heat per ole at the elting teperature T, V the olar volue and a nuerical coefficient. The experiental growth teperature T c is often interpreted in the literature as a hoogeneous nucleation teperature. This view is not correct because the T c of liquid eleents is highly dependent on the saple volue v [17]. The crystallization teperatures are known to be driven by an effective critical energy barrier that is strongly weakened by the Gibbs free energy change associated with ipurity clusters in the liquid [18,19]. The presence of v has for consequence to prevent the elting above T of the sallest clusters acting as intrinsic growth nuclei reducing the critical energy barrier in undercooled liquids. The critical energy saving coefficient was shown for the first tie as depending on = [(T-T )/T ] in liquid eleents with a axiu at T equal to 0.17 [15-16]. In this article, each cluster having a radius saller than the critical radius has its own energy saving coefficient n depending on, n and its radius R n. In this case too, the cluster surface energy is a linear function of n instead of a function of or T [0,1-5]. The Gibbs free energy change derivative [d(g )/dt] p = S at T continues to be equal to the entropy change whatever the particle radius is because (d /dt) T=T is equal to zero. All the surface atos of growth nuclei have the sae fusion heat as their core atos [1]. They survive for a liited tie above the elting teperature because they are not subitted to surface elting. A elt bath needs tie to attain the therodynaic equilibriu above the elting teperature T. This finding is the basic property peritting to assue for the first tie that the growth nuclei in all liquid eleents are superclusters instead of crysta. These superclusters are elted by hoogeneous nucleation of liquid in their core instead of surface elting. A prediction of superheating effects is ao presented for the first tie for 8 liquid eleents together with the predictions of undercooling rates depending on saple volues and supercluster agic ato nubers n. The undercooling teperatures of gold and titaniu have already been predicted using a continuous variation of growth nucleus radii and quantified values of v [,]. The equalization of Feri energies of liquid and superclusters is not realized by a transfer of conduction electrons fro nuclei to elts as I assued in the past [15,16,4]. I recently suggested that a Laplace pressure change p applied to conducting and nonconducting superclusters accopanied by an enthalpy saving per ole equal to V p = H is acting [5]. This quantity is proportional to 1/R n down to values of the radius R n, for which the potential energy is still equal to the quantified energy. Superclusters containing 1 and 55 atos have an energy saving coefficient n0 which is quantified. This coefficient n0 associated with an n-ato supercluster strongly depends on n up to the critical nuber n c of atos, giving rise to crystal spontaneous growth when n0 is equal to 0.17 in liquid eleents [15]. The quantified values of v are known solutions of the Schrödinger equation which are obtained assuing that the sae copleentary Laplace pressure p could be created by a virtual s-electron transfer fro the crystal to the elt or fro the elt to the crystal, creating a virtual surface charge screening associated with a spherical attractive potential [4]. All values of v for radii saller than the critical values lead to a progressive reduction of electron s-state density as a function of n []. Reduced s-state density of superclusters depending on their radius and electronic specific heat of Cu, Ag and Au n-ato superclusters are studied, iposing a relative variation of Feri energies during their foration in noble etal liquid state equal to / of the relative volue change. The radii of Ag superclusters calculated by olecular dynaics siulations in [5] are coparable with the critical radius values R* (T) deduced fro this constraint.

3 This article follows the plan below: - The supercluster foration equations leading to crystallization.1- Gibbs free energy change associated with growth nucleus foration,.- Theral dependence of the energy saving coefficient n of an n-ato condensed supercluster,.- Crystal hoogeneous nucleation teperature and effective nucleation teperature. - The odel of quantification of the energy saving of superclusters 4- Prediction of crystallization teperatures of 8 supercooled liquid eleents at constant olar volue 5- Hoogeneous nucleation of 1-ato superclusters and undercooling rate predictions 6-Maxiu overheating teperature applied to elt superclusters at constant olar volue. 7- Electronic properties of Cu, Ag and Au superclusters 8- Silver supercluster foration into and out of undercooled liquid 9- Melting of Cu, Ag and Au superclusters, varying the overheating teperatures and ties 9.1- Superheating of Cu, Ag and Au superclusters 9.- Analysis of the influence of Cu superheating tie on the undercooling rate 10- Conclusions.. Supercluster foration equations leading to crystallization.1. Gibbs free energy change associated with growth nucleus foration The classical Gibbs free energy change for a growth nucleus foration in a elt is given in (1): H 4R G1 4R 1 V, (1) where R is the nucleus radius, 1 the surface energy, H the elting heat, V the olar volue and = (TT )/T the reduced teperature. Turnbull has defined a surface energy coefficient 1 in () which is equal to () [19,6]: V ( 1 1/ 1 ) N A N A k B ln( K 6S 1 ) H V 1/, (, () where N A is the Avogadro nuber, k B the Boltzann constant, S the elting entropy and ln(k ) = 90 ±. An energy saving per volue unit H /V is introduced in (1); the new Gibbs free energy change is given by (4), where is the new surface energy [15,7]: H 4R G ( ) 4R V The new surface energy coefficient is given by (5): V 1/ ( ) N A H V. (4). (5) The critical radius R* in (6) and the critical therally-activated energy barrier G* /k B T in (7) are calculated assuing (d /dr) R=R* = 0: R V * 1/ ( ) N A, (6)

4 4. (7) They are not infinite at the elting teperature T because is no longer equal to zero [15,16]. The hoogeneous nucleation teperature T (or ) occurs when the nucleation rate J in (8) is equal to 1, lnk = 90 ± in (9) and (10) respected with G* /k B T = 90 neglecting the LnK theral variation [8]: * G J K exp( ) k BT, (8) * G ln( K ) k BT. (9) The unknown surface energy coefficient in (10) is deduced fro (7) and (9): N AkB ( ) (1 )ln( K ) 16S. (10) The surface energy in (5) has to be iniized to obtain the hoogeneous nucleation teperature T (or ) for a fixed value of. The derivative d /d is equal to zero at the teperature T (or ) given by (11), assuing that ln(k ) does not depend on the teperature: T T T. (11) The hoogeneous nucleation teperature T is equal to T / (or = /) in liquid eleents and () is equal to zero at this teperature [15,4]. The surface energy coefficient is now given by (1), replacing by (11) in (10) for each value of : 1/ ( N k ln( K ) A B 1 ) (1 ) 6S 1. (1) The classical crystal nucleation equation (4) is transfored into (1) with the introduction of the energy saving coefficient : H R H 1k BV ln K 1/ G ( R, ) ( )4 4R (1 )( ) V V 4 S. (1) The Laplace pressure p and the copleentary Laplace pressure p applied on the critical nucleus are calculated fro the surface energy with the equations (1) and (6) and p is given by (14) [1,5]: H p ( ) R V H ( p ( ) V R ),, (14) where is the copleent proportional to in the surface energy in (1). The copleent p is equal to the energy saving () H /V. The Gibbs free energy change G in (1) 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 (15): 1/ n (4 ) / Gn( n,, n) H ( n) H (n) N A N A. (15)

5 5 The foration of superclusters having a weaker effective energy barrier than that of crysta precedes the foration of crystallized nuclei in an undercooled elt [5,9]. A spherical surface containing n atos being iniized, a supercluster having a radius saller than the critical radius cannot be easily transfored into a non-spherical crystal of n atos because the surface energy would increase. The critical radius of superclusters could be larger than that of crysta because the supercluster density could be saller, as already predicted for Ag [5] and confired in part 7. In these conditions, the transforation of a supercluster into a crystal is expected to occur above the critical radius for crystal growth when the Gibbs free energy change begins to decrease with the radius, while that of a supercluster increases up to its critical radius. It is shown in parts and 4 that the supercluster energy saving n H is quantified, depends on cluster radius R and ato nuber n, and is larger than the critical energy saving H. The cluster s previous foration during undercooling deterines the spontaneous growth teperature T c reducing the effective critical energy barrier. The sallest hoogeneously-condensed cluster contro the heterogeneous growth of crysta at teperatures higher than the hoogeneous nucleation teperature T / ( = /) even in liquids which are at therodynaic equilibriu at T before cooling... Theral dependence of the energy saving coefficient n of an n-ato condensed cluster All growth nuclei which are fored in an undercooled elt are subitted to a copleentary Laplace pressure. The energy saving coefficient n of an n-ato cluster given in (16), being a function of as already shown [15], is axiu at T, with (d n /dt) T=T equal to zero: H H v n0 n0(1 ) V V 0, (16) where n0 is the quantified energy saving coefficient of an n-ato cluster at T depending on the spherical nucleus radius R [4]. This theral variation has for consequence that the fusion entropy per ole of a cluster of radius R is equal to the fusion entropy S of the bulk solid [15,4] d( G 4R dt n ) T T S V In these conditions, cluster surface atos having the sae fusion heat as core atos, the cluster elts above T by liquid droplet hoogeneous nucleation above T rather than by surface elting as expected for superclusters [6]. This theral variation has already been used to predict the undercooling rate of soe liquid eleents [,]. The critical paraeters for spontaneous supercluster growth are deterined by an energy saving coefficient called in (17): H H v l s0(1 ) V V 0., (17) where 0 = 0.17 is the critical value at T and 0 =.5 in liquid eleents[15,4]. A critical supercluster contains a critical nuber n c of atos given by: n c ( ). (18)

6 6.. Crystal hoogeneous nucleation teperature and effective nucleation teperature The therally-activated critical energy barrier is now given by (19): * G 1(1 ) ln( K ) k BT 81( ) (1 ), (19) where is given by (17). The coefficient of ln(k ) in (19) becoes equal to 1 at the hoogeneous nucleation teperature T / and the equation (9) and (11) are respected. Hoogeneously-condensed superclusters of n-atos act as growth nuclei at a teperature generally higher than the hoogeneous nucleation teperatures T / of liquid eleents because they reduce the critical energy barrier as shown in (0) [18]: ln( J v t sn ) ln( K v t sn * G ) ( k T B G k T B n, (0) ) 0 where v is the saple volue, J the nucleation rate, t sn the steady-state nucleation tie, lnk = 90 ±, G* /k B T defined in (19) and G n in (15). The equation (0) is applied, assuing that n-ato superclusters preexist in elts when they have not been elted by superheating above T. It can ao be applied when the hoogeneous condensation tie of an n-ato supercluster is evolved and its own critical energy barrier is crossed. The cluster therally-activated critical energy barrier G* n /k B T and the effective therally-activated critical energy barrier G neff /k B T of an n-ato supercluster are given by (1) and (): * Gn 1(1 n ) ln( K ) k BT 81( n ) (1 ), (1) G * neff Gn Gn 1(1 n ) ln( K ) Gn kbt kbt kbt 81( kbt n ) (1 ), () where G n is given by (15), n by (16) and lnk = 90 ±. The quantified value n0 H of the cluster energy saving at T is defined in the next part. The transient nucleation tie being neglected, the growth around these nuclei is only possible when the steady-state nucleation tie t sn is evolved and the relation () is respected: ln( J n G vtsn ) ln( K v tsn ) k T B neff, () where v is the saple volue and t sn the steady-state nucleation tie. The crystallization follows this cluster foration tie when, in addition, (0) is respected. The effective nucleation teperature deduced fro (0) does not result fro a hoogeneous nucleation because it strongly depends on the saple volue v. This phenoenon explains why the effective nucleation teperature in liquid eleents is observed around = 0. in saple volues of a few instead of varying fro 0.58 to 0. in uch saller saples [17,0].. Quantification of energy saving associated with supercluster foration The potential energy saving per nucleus volue unit H /V is equal to the Laplace pressure change p = /R accopanying the transforation of a liquid droplet into a nucleus. The quantified energy is saller than /R at low radius R for n = 1 and 55. The calculation is ade by creating a Laplace pressure on the surface of a spherical nucleus containing n atos which would result fro a virtual transfer of n z electrons in

7 7 s-states fro the nucleus to the elt, z being the fraction of transferred electrons per supercluster ato [1]. The potential energy U 0 would be equal to (4) and to zero beyond the nucleus radius R: n z e U 0 8 0R, (4) where e is the electron charge, and 0 the vacuu perittivity [, p.15]. The quantified energy E q at T is given by (5): E q n n0 H (5) where N A is the Avogadro nuber. The quantified energy saving is given by (16) as a function of with ( 0 ) - =.5 when n0 is known. The Schrödinger equation depends only on the distance R of an s-state electron fro the spherical potential center. The quantified solutions E q leading to the values of n0 are given by the two equations in (6), depending on U 0 which is equal to the copleentary Laplace pressure p acting on an n-ato cluster having a volue equal to 4R / through an interediate paraeter called k: k sin( k R) k R 1 N A ( U 1/ 0 E q ),, (6) 1/ ( 0 0 R ) where 0 is the electron rest ass and ħ Planck s constant divided by p. The critical radii of liquid eleents are sufficiently large at T to assue that U 0 is equal to E q and to deduce the values ofz fro the relation (4) = (5) with R = R* (=0) and n0 = 0 = The potential energy U 0 given by (4) is ao equal to4r / p. Consequently, the z in (4) does not depend on R at T. The value of U 0 is deduced fro the ato nuber n which depends on olar volues V of solid eleents extrapolated at T fro published tables of theral expansion [16,]. The values of n0 are calculated as a function of R using (7) instead of (6) for n 147 because U 0 is assued to be equal to E q : * 0 R n0 R (7) The condensed-cluster energy savings n0 H of 1 and 55 atos are quantified and calculated fro (5). The theral variation of n is given in (16) using these quantified values of n0. 4. Prediction of crystallization teperatures T c of 8 undercooled liquid saples of various diaeters The quantified and the potential energy saving coefficients n0 of silver clusters have been calculated using (6) and (7) and are represented in Figure 1 as a function of supercluster radius R n which is assued to continuously vary. These coefficients are equal for n 147. This last approxiation is used in all liquid eleents. Figure 1. The energy saving coefficient n0 versus the supercluster radius R n. The quantified (square points) and non-quantified (diaond points) energy saving coefficients n0 are plotted versus the silver cluster radius. This coefficient is strongly weakened when R < 0.5 n. Quantification is necessary for an ato nuber n < 147.

8 8 Properties of 8 eleents are classified in Table 1: Colun 1- the liquid eleents are classified as a function of their olar fusion entropy S, Colun - the olar volue of solid eleents at T in, Colun - their fusion entropy S in J/K/ole, n Colun 4- their elting teperature T in Kelvin, Colun 5- the ato agic nuber n of the supercluster inducing crystallization of the supercooled liquid at the closest teperature to the experiental crystallization teperature, Colun 6- the supercluster radius R n in nanoeters deduced fro the olar volue V using the relation (8): 4Rn N A, (8) n V Colun 7- the energy saving coefficient n0 associated with the n-ato supercluster calculated using (8) for n 147 and (6) for n = 1 and 55, with z given in Table colun, Colun 8- the experiental reduced crystallization teperature c exp = (T c -T )/T of a liquid droplet having a diaeter D exp, Colun 9- the reduced crystallization teperature c calc calculated using (0), Colun 10- the therally-activated effective energy barrier G eff /k B T given in (0) and (19) leading to the crystallization of the corresponding liquid eleent, Colun 11- the calculated diaeter D calc in of the liquid droplet of volue v subitted to crystallization at c calc using (0) and v t sn v = /6 D assuing that t sn = 1 s, Colun 1- the experiental diaeter D exp in illieters of the liquid droplet crystallizing at cexp, Colun 1- references. Table 1. Reduced crystallization teperatures of 8 supercooled liquid eleents induced by condensed superclusters containing n = 1, 55, 147, 09 or 561 atos

9 9 V 10 6 S T n R n n0 c c Geff D calc D exp Ref kbt J./K. K n Exp. Calc. Fe [4,5] In [6,7] Ti [8] Zr [9,40] Mn [0,41] Pb [17] Co [6,4] Ag [4] Au [0,44] Tc [8] Cr [8] Re [45] Ir [8,46] Mo [8,47] Os [8,48] Pd [0,49] Pt [8] Cu [50-5] Rh [8] Ta [8,5] Nb [8] Hg [17,4] V [8] Ni [4,6] Ru [8] Hf [8] Ga [17,54] Cd [17] Zn [8] Al [17,55] W [6,56] Sn [17,7] Bi [17,57] Sb [17] Te [17] Se [58] Si [59-6] Ge [6,4,5,6] The experiental reduced crystallization teperatures c exp are plotted in Figure versus the calculated c calc using the supercluster ato-nuber n leading to about the sae droplet diaeter D calc as the experiental one D exp. A good agreeent is obtained between these values in liquid eleents in Figure and Table 1. There is no good agreeent for Hg, Sn, Al, Cd, V, and Cr because these eleents are known to contain ipurities or oxides. Their undercooling rates are too low copared to the calculated ones. In Figure, the calculated droplet diaeter logariths are plotted as a function of those of experiental droplets used to study the undercooling rate. Six orders of agnitude are studied, corresponding to 18 orders of volue agnitude. Figure shows that the odel is able to describe the crystallization teperature dependence on the volue saple. Figure. Experiental undercooling teperatures versus calculated undercooling teperatures. The experiental reduced crystallization teperatures c exp = (T c -T )/T are

10 10 plotted versus the calculated ones c calc of 8 liquid eleents listed in Table 1. The saller the ato nuber n, the saller is the undercooling teperature, as shown in Table 1. Figure. Calculated liquid droplet diaeters versus experiental liquid droplet diaeters. The calculated and experiental droplet diaeters being crystallized are copared in a logarithic scale. Saller liquid droplets lead to lower undercooling teperatures, as shown in Table1. The ato nubers n of growth nuclei in liquid eleents are represented in Figure 4 as a function of the reduced experiental crystallization teperature c exp of liquid droplets having various diaeters. The undercooling rate change is two ties greater when the diaeter varies fro 0.06 to 8.4 and fro to 5 for n = 1 and n = 09, respectively. The experiental undercooling reduced teperature c of galliu is the lowest of all the liquid eleents and is equal to 0.58 and is a little higher than/, corresponding to a crystallization teperature T c equal to 19K [17] and to a elting teperature of the phase equal to 0 K. The galliu phase is crystallized after undercooling. Its elting teperature is 57 K instead of 0 K for the phase and its fusion entropy is 10.91

11 11 J/K/ole, as shown in Table 1, instead of 18.4 J/K/ole [54]. Its crystallization teperature of 19 K occurs in fact at c = 0.5. The calculated value is equal to the experiental one due to a previous condensation of 1- ato cluster which weakens the critical energy barrier. The odel works without any adjustable paraeter, and is able ao to predict the nucleation rate of 1-ato clusters and the diaeter of galliu droplets obtained with the liquid dispersion technique. Figure 4. N-ato superclusters acting as growth nuclei versus the experiental reduced undercooling teperatures. The nuber n of atos of superclusters is plotted versus the experiental reduced teperature of crystallization c exp. 5. Hoogeneous nucleation of 1-ato superclusters and undercooling rate predictions Equations (0-) are now used to calculate the hoogeneous foration reduced teperature 1c of 1- ato clusters in a elt cooled below T fro therodynaic equilibriu state at T and the crystallization reduced teperature c that they induce in liquid droplets of 10 icroeters in diaeter. In Table, 8 liquid eleents are considered. In of the, the 1-ato cluster foration teperature is uch larger than the crystallization teperature ( 1c >> c ). On the contrary, in indiu, ercury, galliu, cadiu and zinc, the two reduced teperatures are equal within the uncertainty on the energy saving coefficient value 10 given in Table. The crystallization teperatures of bisuth, seleniu, telluriu, antiony, silicon and geraniu with a growth around 1-ato clusters are predicted in good agreeent with experiental values obtained with various sizes of droplets, as shown in Table 1. In Figure 5, the hoogeneous condensation reduced teperatures of 1-ato superclusters are copared with the reduced spontaneous growth teperatures which induce crystallization. The growth is organized around these 1-ato clusters which are fored, at teperatures higher than that of spontaneous crystallization. These hoogeneous and heterogeneous crystallization teperatures depend on the droplet diaeters. Their values given in Table, Colun 10 are the lowest undercooling teperatures which can be obtained with 10 icroeter droplets.

12 1 Figure 5. Condensation teperatures of 1-ato superclusters and crystallization teperatures in 10 icroeter droplets versus the elting entropy in J/K/ole. The squares are the reduced foration teperatures of 1-ato superclusters which are ready to grow. These teperatures are larger than or equal to the reduced teperatures of spontaneous crystallization around the represented by diaond points. In Table, the liquid eleents are still classified as a function of their fusion entropy S given in Table 1 (Colun ): Colun 1- List of liquid eleents, Colun - Critical radius for spontaneous growth, Colun - The nuber z per ato of s-electrons virtually transferred fro superclusters to elt at T, Colun 4- The energy saving coefficient n0 of 1-ato superclusters calculated with (4-6), Colun 5- The energy saving coefficient n0 of 55-ato superclusters calculated with (4-6),, Colun 6- The energy saving coefficient n0 of 147-ato superclusters calculated with (7) for n 147, Colun 7- The energy saving coefficient n0 of 09-ato superclusters, Colun 8- The energy saving coefficient n0 of 561-ato superclusters, Colun 9- The condensation reduced teperature of 1-ato superclusters in 10 icroeter droplets, Colun 10- The spontaneous growth reduced teperature around 1-ato superclusters in 10 icroeter droplets. Table. Energy saving coefficients n0 of n-ato superclusters, and reduced condensation teperatures 1c of 1-ato superclusters inducing spontaneous growth at c = (T c -T )/T in 10 icroeter droplets. 1 6

13 1 R* z n0 n0 n0 n0 n0 1c (10) c (10) n n=1 n=55 n=147 n=09 n=561 n=1 n=1 Fe In Ti Zr Mn Pb Co Ag Au Tc Cr Re Ir Mo Os Pd Pt Cu Rh Ta Nb Hg V Ni Ru Hf Ga Cd Zn Al W Sn Bi Sb Te Se Si Ge Maxiu superheating teperatures of superclusters at constant olar volue 6.1. Superheating and elting of n-ato superclusters by liquid hoogeneous nucleation N-ato superclusters survive above the elting teperature T up to an superheating teperature which is tie-dependent. They can be elted by liquid hoogeneous nucleation in their core instead of surface elting. The Gibbs free energy change associated with their elting at a teperature T > T is given by (9):

14 14 G n n (4 ) ( n,, n ) H ( n ) H (n) N N A 1/ A /, (9) where the energy saving coefficient n is given in (16) even for > 0. The fusion enthalpy has changed sign as copared to (15) and the equalization of Feri energies always still leads to an energy saving. An n-ato supercluster elts when (0) is respected: G ln( K sl.vn.tsn) k B T n, (0) where v n is the n-ato supercluster volue deduced fro its radius given in Table 1, (8) and t sn is the superheating tie at its own elting teperature because the supercluster radius is uch saller than its critical radius. The tie t sn is chosen equal to 600 seconds and lnk sl to Overheating and elting of n-ato superclusters by liquid heterogeneous nucleation Melting teperatures of superclusters are reduced by previous elting of a 1-ato droplet in their core. These entities elt when (1) is respected for n =1: G ln( K sl.vn.tsn) k T B n G k T where the critical energy barrier G* n /k B T no longer exists and is replaced by G n /k B T, n in (16), n0 in Table, t sn = 600 s and lnk = 90. The critical barrier is not involved in (1) because the n-ato supercluster radius is uch saller than the critical radius for liquid growth and G* n >> G n. B 1 (1) 6.. Prediction of elting teperatures of superclusters in 8 liquid eleents by elt superheating above T The reduced elting teperatures = (T-T )/T of superclusters depending on their ato nuber n are given in several coluns of Table and in Figure 6. They are calculated assuing that the olar volue is constant, t sn = 600 s. and lnk sl = 90. The liquid eleents having fusion entropy S larger than 0 J/K/ole have a elting teperature which is deterined by liquid hoogeneous nucleation because the 1-ato clusters elt at higher teperatures while those with S < 0 J/K/ole are subitted to chain-elting. Figure 6. The elting teperatures of superclusters containing 1, 55, 147, 09 and 561 atos. These elting teperatures are given in coluns 10, 11, 1 and 1 of Table versus S.

15 15.. In Table, Colun 1- The liquid eleents are classified as a function of the fusion entropy S. Colun - The elting teperature of 1-ato superclusters is high for large fusion entropies of Bi, Sb, Te, Se, Si and Ge. The highest value = 0.89 is obtained for Si; the lowest = 0 is obtained in 8 liquid eleents. Hoogeneous liquids do not contain any condensed cluster. They are crystallizing during undercooling because 1-ato clusters are condensed as shown in Figure 5. Their previous presence at T does not have any influence on the undercooling rate. Colun - The calculated elting teperatures of 55-ato clusters induced by previous foration in their core of a droplet of 1 atos are often lower than those of the 1-ato clusters. Then, they elt at the sae teperature as the 1-ato clusters. Colun 4- The calculated elting teperatures of 147-ato clusters induced by previous foration in their core of a droplet of 1 atos are larger than those of the 1-ato clusters fro Fe to Al. They are nearly equal for W and saller fro Sn to Ge. Colun 5- The calculated elting teperatures of 09-ato clusters induced by previous foration in their core of a droplet of 1 atos are larger those of the 1-ato clusters fro Fe to Sn. They are saller fro Bi to Ge Colun 6- The calculated elting teperatures of 561-ato clusters induced by previous foration in their core of a droplet of 1 atos are larger than those of the1-ato clusters fro Fe to Sn except W. They are saller fro Bi to Ge. Coluns 7, 8, 9 and 10- The elting teperatures of 55-, 147-, 09- and 561-ato superclusters are obtained considering hoogeneous liquid nucleation without introducing heterogeneous nucleation fro 1-ato droplets. Coluns 11, 1, 1 and 14- The expected elting teperatures of 55-, 147-, 09- and 561-ato superclusters are selected in order to be coherent between the. The hoogeneous nucleation teperature of soe superclusters having a large fusion entropy are soeties saller than those of the 1-ato superclusters, as shown in Figure 6. All these results have been obtained assuing that the superheating tie at their own elting teperature is 600 seconds. The tie effects on copper supercluster elting are exained in part 9 in relation with detailed experiental studies [51].

16 16 Table. The elting teperatures of superclusters. The final elting teperatures are given in Coluns, 11, 1, 1 and 14. The teperatures in Coluns, 4, 5 and 6 are calculated assuing that the elting starts fro 1-ato droplets acting as heterogeneous nuclei in the core of superclusters. Those in Coluns 7, 8, 9 and 10 correspond to a liquid hoogeneous nucleation f f (n- f (n- f (n- f (n- f (n) f (n) f (n) f (n) f (n) f (n) f (n) f (n) 1) 1) 1) 1) Ho. Ho. Ho. Ho. n=1 n=55 n=147 n=09 n=561 n=55 n=147 n=09 n=561 n=55 n=147 n=09 n=561 Fe In Ti Zr Mn Pb Co Ag Au Tc Cr Re Ir Mo Os Pd Pt Cu Rh Ta Nb Hg V Ni Ru Hf Ga Cd Zn Al W Sn Bi Sb Te Se Si Ge

17 17 7. Electronic properties of Cu, Ag and Au superclusters Electronic properties of superclusters can be calculated fro the enthalpy saving associated with their foration teperature in noble etallic liquids because this energy is due to Feri energy equalization of liquid and superclusters []. The Feri energy difference E F between condensed superclusters of radius R n containing n atos and liquid state at T can be directly evaluated for noble eta using () and assuing that z is sall, as shown in Table : EF n H nz N A, () where is the ratio of electron asses */ 0, 0 being the electron rest ass and * the effective electron ass which is assued to be the sae in superclusters and liquid states, and z being calculated at variable teperature using the known quantified energy saving in (17) and (5,6). The Feri energy difference E F is plotted in Figure 7 as a function of 1/R*, where R* is given in (6) for Cu, Ag and Au assuing that the olar volue does not depend on teperature and a continuous variation of R*. The quantified value is given in (17) and the U 0 and z values are calculated with (5). For R* > 1 n, E F is proportional to the Laplace pressure, while for R << 1 n there is a gap opening in the conduction electron band accopanying the quantification of the energy saving. This analysis is able to detect well-known properties of clusters out of the elt which becoe uch less conducting at very low radii [10]. Figure 7. Feri energy difference E F between liquid and superclusters. The E F in ev/ole is plotted as a function of the reverse of the critical radius R* in n -1. A strong variation of E F at constant olar volue V is observed in Figure 7. In principle, the E F has to obey () in the liquid state because the Feri energy E F depends on (V ) / : EF V, () E F where E F is the Feri energy of the liquid, V the olar volue of a supercluster of infinite radius, V is the variation of the olar volue with the radius decrease. The supercluster olar volue V has to depend on the particle radius instead of being constant. Equation () is respected when the foration teperature T of superclusters corresponding to the critical ato nuber n c in (18) and to a olar volue V depending on R* V

18 18 is introduced. The foration teperatures of superclusters with agic ato nubers are indicated in Figure 8 using a special olar volue theral variation V (T) given in Figure 9 for each liquid eleent. Figure 8. The foration teperatures of superclusters containing n atos. The foration teperatures of Ag critical superclusters are plotted versus the critical nuber n c of atos that they contain. The superclusters with agic ato nubers are represented by squares. Figure 9. The supercluster olar volue change of Cu, Ag and Au. The supercluster olar volue change with the critical radius R* (being a hidden variable) is plotted as a function of their foration teperature in Kelvin up to T. Each point corresponds to a supercluster of radius R* and to an n-ato nuber. The following laws are used in Figure 8 and Figure 9: 6 6 V ( ) for Cu, 6 6 V ( ) for Ag and Au, where is equal to (T-T )/T. All superclusters containing agic ato nubers have their olar volue obeying these laws at their critical foration teperature. The olar volues V for = are axiu and equal to , , and /ole, for Cu, Ag and Au respectively.

19 19 They correspond to an infinite radius for superclusters in the absence of crystallization [15]. The olar volue V of bulk superclusters would be attained when n becoes equal to zero using the critical radius as a hidden variable becoing infinite instead of the teperature. The Feri energy change E F depends on z in (); z is calculated with (4 6) for each radius R = R* (T) in (6), deterining n fro (8). Equation () is now respected for Cu, Ag and Au, as shown in Figure 10. The Feri energy E F is defined in (4), assuing that there is one conduction electron per ato in Cu, Ag and Au: E F ( 0 V ) /, (4) where V is the liquid olar volue at T which is equal to , , and /ole for Cu, Ag and Au respectively [64]. Figure 10. Relative Feri energy change between superclusters and liquid. The E F /E F is plotted as a function of the relative volue change V /V of superclusters, where V is the olar volue of the supercluster having an infinite radius. The olar electronic specific heat C el = el T of Cu, Ag and Au superclusters can be obtained fro the knowledge of their electronic density of states D(E F ) at the Feri level, calculated with (5) and (6) [64]: EF D( EF ) N Az (5) el D( EF ) kb (6) The z values have been previously deterined fro (4 6). Each n-ato supercluster has its own olar volue V and its own z at T is deterined with (4) = (5) with R = R* (T ) depending on V and n0 = The electronic specific heat coefficient el is plotted in Figure 11 as a function of the supercluster olar volue. The electronic specific heat coefficient el of superclusters fal when their olar volue V and their radius decrease below T. The coefficients el of Cu, Ag and Au crysta at 4 K are a little larger, being equal to 0.695, and 0.79 instead of 0.48, and J/K /ole at T respectively [64]. Sall crysta are known to becoe insulating for radii saller than 5 n when they are studied out of their elt [10]. This electronic

20 0 transforation is ao present in superclusters and is very abrupt below their critical growth volue at T as shown in Figure 11. The el at T is ao calculated as a function of the supercluster radius R and represented in Figure 1. The coefficient z is obtained at T with (4) = (5) and n = 0 = Then, the potential energy U 0 depending on R is known for each value of R and the quantified coefficient n0 of an n-ato supercluster of radius R is deduced fro (5,6). In Figure 1, the highest points are calculated at T while the lowest are already shown in Figure 11. The sallest superclusters are still etallic at T, while they becoe insulating when the teperature is close to T /. All these predictions are in good agreeent with any properties of divided eta. They are only based on an enthalpy saving equal to 0.17 (1.5 ) H for the supercluster foration in all liquid eleents. Figure 11. Electronic specific heat of superclusters. The supercluster electronic specific heat coefficient in J/K /ole is plotted versus their olar volue in when the critical radius is saller and saller below T. Figure 1. Electronic specific heat coefficient of Cu, Ag, and Au superclusters as a function of supercluster radius R at T = T (colored points) and for T < T when R is the critical radius (black points). 8. Silver supercluster foration into and out of undercooled liquid

21 1 The foration of icosahedral silver clusters with agic nubers n of atos equal to 1, 55, 147, 09, 561, 9, 1415 and 057 has been already studied out of liquid by olecular dynaics in the teperature range K. Icosahedral clusters of 1, 55 and 147 are fored below roo teperature and larger clusters with n= 09, 561, 9, 1415 are fored fro 00 to 1000 K. The radii of these Ag stable superclusters have been found to be equal to.74, 5.51, 8., and Ǻ for n = 1, 55, 147, 09, 561 respectively [5]. The Ag radii have ao been calculated in the liquid using the olar volue shown in Figure 9 and their foration teperature as deduced fro the critical radius. Their values for n = 1, 55, 147, 09, 561 are nearly equal to those predicted by olecular dynaics, as shown in Table 4 and Figure 1. Table 4. The Ag supercluster radii with agic ato nubers. The radius R is deduced fro olar volue V and equal to critical radius R* (T) given in (6). For T > T / = 411. K, the energy saving coefficient in (17) is used with 0 = 0.17 and ( 0 ) - =.5. For T < 411. K, is equal to zero. The radii R MD result fro olecular dynaics siulations [5]. n R (Ǻ) R MD (Ǻ) T (K) Figure 1. The critical ato nuber in blue versus the critical radius and R MD the radius calculated by olecular dynaics siulations in red square [5]. 9. Melting of Cu, Ag and Au superclusters varying the superheating ties Overheating of Cu, Ag and Au superclusters The elting teperatures are now calculated using the olar volue associated with the supercluster radius as shown in Figure 9. The superheating tie continues to be equal to 600 seconds. The supercluster radius variation is continuous while the radius of agic nuber clusters is indicated in Figures 14, 15 and 16. In these three figures, the Cu, Ag, and Au supercluster radius is plotted versus the reduced teperature = (T-T )/T. The points labeled hoogeneous are calculated assuing that supercluster elting is produced by liquid

22 hoogeneous nucleation using (9,0). The triangles labeled (n-1) are calculated assuing that the supercluster elting is induced by previous foration of liquid droplets of 1 atos into superclusters using (1). The hoogeneous nucleation teperatures are uch too high copared to the (n-1) teperatures. The undercooling teperatures depend on the volue saple v. The square points are deterined for ln(k.v.t sn ) = 71.8 corresponding to v.t sn = s and a heterogeneous nucleation induced by superclusters of radius R when the applied superheating teperature is saller than those indicated by triangles. Another supercooling teperature represented by triangle points is added in Figures 15 and 16. In Figure 15, v.t sn is equal to s while, in Figure 16, v.t sn = s. These three figures show that an undercooling rate of about 0% is generally obtained when the saple volue is of the order of a few and the applied superheating rate is less than about 5%. The undercooling teperature is very stable when the superheating is less than 5%. Larger undercooling rates are obtained using uch saller volue saples [17]. Figure 14. Supercooling teperatures of liquid copper controlled by unelted superclusters having elting teperatures depending on overheating rate applied during 600 s. Figure 15. Supercooling teperatures of liquid silver depending on saple volue and controlled by unelted superclusters having elting teperatures depending on overheating rate applied during 600 s.

23 Figure 16. Supercooling teperatures of liquid gold depending on saple volue and controlled by unelted superclusters having elting teperatures depending on overheating rate applied during 600 s. 9.. Analysis of the influence of Cu superheating tie on the undercooling rate The superheating tie has a strong influence on supercluster elting, as shown by studies of Cu undercooling [51]. It has been found that a iniu superheating teperature of 40 K is required in order to achieve any undercooling prior to crystallization nucleation. This phenoenon is ao observed in any agnetic texturing experients [65]. In Table, the first Cu supercluster to be elted at = 0.0 in 600 s, corresponding to a superheating of 44.7 K, contains 1 atos in perfect agreeent with the observation. There is no other supercluster elting. A teperature below which no sall supercluster elts is predicted in this odel. The lowest value of the undercooling teperature is obtained when 6 theral cycles are applied prior to nucleation after 6 steps of 400 s at 147 K. The total tie evolved at 147 K is s. In Figure 17, the tie necessary to elt all superclusters surviving in copper elt is calculated. With lnk = 89.6 instead of 90, the tie to elt all the 1-ato clusters is 141 s, while that to elt 55-ato clusters is s, which is in very good agreeent with the easureents [51]. The other superclusters are elted in very short ties after the elting of the 55- ato clusters. The reduced undercooling teperature becoes equal to = 0.59 after these theral treatents of a saple of 5.7 in diaeter. A hoogeneous nucleation of 1- and 55-ato clusters leads to = 0.5. These results show that superclusters can be chain elted with a weaker superheating if the tie evolved at the overheating teperature is increased substantially beyond 600 s. Our odel can be used to evaluate the approxiate superheating tie leading to a therodynaic equilibriu of a elt without condensed superclusters at any teperature above T.

24 4 Figure 17. Chain elting of superclusters versus superheating tie. The first cluster to be elted at = contains 1 atos, the next ones 55, 147, 09 and 561 because the liquid droplet grows in the core of the largest particles. 10. Conclusions The undercooling teperatures of of the 8 liquid eleents are predicted for the first tie in good agreeent with experiental values depending on the saple volue, without using any adjustable barrier energy, and only assuing the existence of growth nuclei containing stable agic ato nubers n equal to 1, 55, 147, 09 and 561 that are generally devoted to icosahedral structures. The odel is based on a volue enthalpy saving v previously deterined to be equal to 0.17 H /V at T and added to the classical Gibbs free energy change for a critical nucleus foration in a elt. This enthalpy is due to the Laplace pressure change p acting on the growth nuclei and equalizing the Feri energies of liquid and nuclei in etallic liquids. The Gibbs free energy change has to contain a contributionv p which has been neglected up to now because its agnitude was unknown. This issing enthalpy has serious consequences because the critical radius for crystal growth is considered, in the classical view, as being infinite at the elting teperature and all solid traces being eliinated in elts. This is in contradiction with any experients on the superheating influence on undercooling rates and on agnetic texturing efficiency [51,65 69]. Nuclei having radii saller than the critical radius at T are elted at higher teperatures depending on the superheating tie and on their ato nuber. Soe growth nuclei survive above T because they are superclusters that are not elted by surface elting. This new property of superclusters is a consequence of the theral variation of v, which is a unique function of = [(T-T )/T ] being axiu at T, and a fusion heat equal to that of bulk crysta. The surface ato fusion heat is not weakened and there is no preelting of these entities depending on their radius. This theral variation was established, for the first tie, fro our study of the axiu undercooling rate of the sae liquid eleents. In addition, it is the only law validating the existence of non-elted intrinsic entities. The energy saving is proportional to the supercluster reverse radius R -1 when n 147 and is quantified for n < 147. The quantified energy at T is calculated by creating a virtual s-electron transfer fro the nucleus of radius R to the elt and an electrostatic spherical potential induced by the surface charges and ao varying with R -1. The Schrödinger equation solutions are known and used to predict the condensation teperatures of 1-ato