Crystallization of Supercooled Liquid Elements Induced by Superclusters Containing Magic Atom Numbers

A few experiments have already detected the presence of icosahedral superclusters in undercooled liquids, confirming a possible homogeneous nucleation of such entities as suggested by Franck. These superclusters survive in melts above the crystal melting temperature Tm because all their surface atoms have the same fusion heat as their core atoms and are melted by homogeneous nucleation of liquid in their core, depending on overheating time and temperature. In complete contrast to current ideas, a long time is necessary to melt them and to attain the thermodynamic equilibrium above Tm. They act as heterogeneous growth nuclei of crystallized phase at a temperature Tc of the undercooled melt when they are not melted. They contribute to the reduction of the critical barrier, which becomes smaller than that of crystals containing the same atom number n. The undercooling rate is always limited, even in a liquid at thermodynamic equilibrium, because the homogeneous nucleation of 13-atom superclusters reduces the energy barrier, and increases Tc above the homogeneous nucleation temperature equal to Tm/3 in liquid elements. After weak superheating, the most stable superclusters containing n = 13, 55, 147, 309 and 561 atoms survive or melt and determine Tc during undercooling, which depends on n and on the sample volume. The experimental nucleation temperatures Tc of 32 liquid elements and the melting temperatures of superclusters are predicted without any adjustable parameter using a sample volume varying by nearly 18 orders of magnitude. The classical Gibbs free energy change is used, adding an enthalpy saving related to the Laplace pressure change associated with supercluster formation, which is quantified and strongly weakened for n = 13 and 55.


Introduction
An undercooled liquid develops special clusters that minimize the energy locally which are incompatible with space filling [1][2][3]. Such entities are homogeneously formed in glass-forming melts, and act as growth nuclei of crystals above the glass transition [4]. The formation of icosahedral nanoclusters has often been studied by molecular dynamics simulations into or out of liquids [5][6][7][8]. Silver superclusters containing the magic atom numbers n = 13, 55, 147, 309, 561 are more stable. Their formation temperature out of melt and their radius have been determined [5]. Icosahedral gold nanoclusters do not premelt below their bulk melting temperature [6]. Nanoclusters have been prepared out of liquids [9][10][11][12][13][14]. The density of states of conduction electrons at the Fermi energy being strongly reduced for particle diameters smaller than one nanometer leads to a gap opening [9,10]. Growth nuclei in melts are expected to have analogous electronic properties.
Superclusters containing magic atom numbers are tentatively viewed for the first time as being the main growth nuclei of crystallized phases in all liquid elements. I already considered that an energy saving resulting from the equalization of Fermi energies of nuclei and melts cannot be neglected in the classical crystal nucleation model [15,16]. An enthalpy saving  v per volume unit of critical radius clusters equal to  ls ×H m /V m was introduced in the Gibbs free energy change G 2ls which gives rise to spherical clusters that transform the critical energy barrier into a less effective energy barrier, so inducing crystal growth around them at a temperature T c much higher than the theoretical homogeneous nucleation temperature equal to T m /3. This enthalpy depends on H m the melting heat per mole at the melting temperature T m , V m the molar volume and  ls a numerical coefficient. The experimental growth temperature T c is often interpreted in the literature as a homogeneous nucleation temperature. This view is not correct because the T c of liquid elements is highly dependent on the sample volume v [17]. The crystallization temperatures are known to be driven by an effective critical energy barrier that is strongly weakened by the Gibbs free energy change associated with impurity clusters in the liquid [18,19]. The presence of  v has for consequence to prevent the melting above T m of the smallest clusters acting as intrinsic growth nuclei reducing the critical energy barrier in undercooled liquids.
The critical energy saving coefficient  ls was shown for the first time as depending on  2 = [(T-T m )/T m ] 2 in liquid elements with a maximum at T m equal to 0.217 [15][16].
In this article, each cluster having a radius smaller than the critical radius has its own energy saving coefficient  nm depending on  2 , n and its radius R nm . In this case too, the cluster surface energy is a linear function of  nm instead of a function of  or T [20,[21][22][23][24][25]. The Gibbs free energy change derivative [d(G 2ls )/dT] p = S m at T m continues to be equal to the entropy change whatever the particle radius is because (d ls /dT) T=Tm is equal to zero.
All the surface atoms of growth nuclei have the same fusion heat as their core atoms [21]. They survive for a limited time above the melting temperature because they are not submitted to surface melting. A melt bath needs time to attain the thermodynamic equilibrium above the melting temperature T m . This finding is the basic property permitting to assume for the first time that the growth nuclei in all liquid elements are superclusters instead of crystals. These superclusters are melted by homogeneous nucleation of liquid in their core instead of surface melting. A prediction of superheating effects is also presented for the first time for 38 liquid elements together with the predictions of undercooling rates depending on sample volumes and supercluster magic atom numbers n. The undercooling temperatures of gold and titanium have already been predicted using a continuous variation of growth nucleus radii and quantified values of  v [22,23].
The equalization of Fermi energies of liquid and superclusters is not realized by a transfer of conduction electrons from nuclei to melts as I assumed in the past [15,16,24]. I recently suggested that a Laplace pressure change p applied to conducting and nonconducting superclusters accompanied by an enthalpy saving per mole equal to V m ×p =  ls × H m is acting [25]. This quantity is proportional to 1/R nm down to values of the radius R nm , for which the potential energy is still equal to the quantified energy. Superclusters containing 13 and 55 atoms have an energy saving coefficient  nm0 which is quantified. This coefficient  nm0 associated with an n-atom supercluster strongly depends on n up to the critical number n c of atoms, giving rise to crystal spontaneous growth when nm0 is equal to 0.217 in liquid elements [15].
The quantified values of  v are known solutions of the Schrödinger equation which are obtained assuming that the same complementary Laplace pressure p could be created by a virtual s-electron transfer from the crystal to the melt or from the melt to the crystal, creating a virtual surface charge screening associated with a spherical attractive potential [24]. All values of  v for radii smaller than the critical values lead to a progressive reduction of electron s-state density as a function of n [23]. Reduced s-state density of superclusters depending on their radius and electronic specific heat of Cu, Ag and Au n-atom superclusters are studied, imposing a relative variation of Fermi energies during their formation in noble metal liquid state equal to 2/3 of the relative volume change. The radii of Ag superclusters calculated by molecular dynamics simulations in [5]

Gibbs free energy change associated with growth nucleus formation
The classical Gibbs free energy change for a growth nucleus formation in a melt is given in (1): where R is the nucleus radius,  1ls the surface energy, H m the melting heat, V m the molar volume and  = (TT m )/T m the reduced temperature. Turnbull has defined a surface energy coefficient  1ls in (2) which is equal to (3) [19,26]: where N A is the Avogadro number, k B the Boltzmann constant, S m the melting entropy and ln(K ls ) = 90 ± 2.
An energy saving per volume unit  ls ×H m /V m is introduced in (1); the new Gibbs free energy change is given by (4), where  2ls is the new surface energy [15,27]: (4) The new surface energy coefficient  2ls is given by (5): The critical radius R* 2ls in (6) and the critical thermally-activated energy barrier G* 2ls /k B T in (7) are calculated assuming (d ls /dR) R=R*2ls = 0: .
They are not infinite at the melting temperature T m because  ls is no longer equal to zero [15,16]. The homogeneous nucleation temperature T 2 (or  2 ) occurs when the nucleation rate J in (8) is equal to 1, lnK ls = 90 ± 2 in (9) and (10) respected with G* 2ls /k B T = 90 neglecting the LnK ls thermal variation [28]: ) ln( The unknown surface energy coefficient  2ls in (10) is deduced from (7) and (9): The surface energy  2ls in (5) has to be minimized to obtain the homogeneous nucleation temperature T 2 (or  2 ) for a fixed value of  ls . The derivative d 2ls /d is equal to zero at the temperature T 2 (or  2 ) given by (11), assuming that ln(K ls ) does not depend on the temperature: The homogeneous nucleation temperature T 2 is equal to T m /3 (or  2 = 2/3) in liquid elements and  ls () is equal to zero at this temperature [15,24].
The surface energy coefficient  2ls is now given by (12), replacing by (11) in (10) The classical crystal nucleation equation (4) The Laplace pressure p and the complementary Laplace pressure p applied on the critical nucleus are calculated from the surface energy  2ls with the equations (13) and (6) and p is given by (14) [21,25]: where  2ls is the complement proportional to  ls in the surface energy in (13). The complement p is equal to the energy saving  ls ()×H m /V m . The Gibbs free energy change G 2ls in (13) directly depends on the cluster atom number n and the energy saving coefficient  nm of the cluster instead of depending on its molar volume V m and its radius R as shown in (15): The formation of superclusters having a weaker effective energy barrier than that of crystals precedes the formation of crystallized nuclei in an undercooled melt [5,29]. A spherical surface containing n atoms being minimized, a supercluster having a radius smaller than the critical radius cannot be easily transformed into a non-spherical crystal of n atoms because the surface energy would increase. The critical radius of superclusters could be larger than that of crystals because the supercluster density could be smaller, as already predicted for Ag [5] and confirmed in part 7. In these conditions, the transformation 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 3 and 4 that the supercluster energy saving nm ×H m is quantified, depends on cluster radius R and atom number n, and is larger than the critical energy saving ×H m . The cluster's previous formation during undercooling determines the spontaneous growth temperature T c reducing the effective critical energy barrier. The smallest homogeneously-condensed cluster controls the heterogeneous growth of crystals at temperatures higher than the homogeneous nucleation temperature T m /3 ( 2 = 2/3) even in liquids which are at thermodynamic equilibrium at T m before cooling.

Thermal dependence of the energy saving coefficient  nm of an n-atom condensed cluster
All growth nuclei which are formed in an undercooled melt are submitted to a complementary Laplace pressure. The energy saving coefficient  nm of an n-atom cluster given in (16), being a function of  2 as already shown [15], is maximum at T m , with (d nm /dT) T=Tm equal to zero: where  nm0 is the quantified energy saving coefficient of an n-atom cluster at T m depending on the spherical nucleus radius R [24].
This thermal variation has for consequence that the fusion entropy per mole of a cluster of radius R is equal to the fusion entropy S m of the bulk solid [15,24] m m Tm T In these conditions, cluster surface atoms having the same fusion heat as core atoms, the cluster melts above T m by liquid droplet homogeneous nucleation above T m rather than by surface melting as expected for superclusters [6]. This  2 thermal variation has already been used to predict the undercooling rate of some liquid elements [22,23].
The critical parameters for spontaneous supercluster growth are determined by an energy saving coefficient called  ls in (17): where  ls0 = 0.217 is the critical value at T m and  2 0m = 2.25 in liquid elements [15,24]. A critical supercluster contains a critical number n c of atoms given by:

Crystal homogeneous nucleation temperature and effective nucleation temperature
The thermally-activated critical energy barrier is now given by (19): where  ls is given by (17). The coefficient of ln(K ls ) in (19) becomes equal to 1 at the homogeneous nucleation temperature T m /3 and the equation (9) and (11) are respected.
Homogeneously-condensed superclusters of n-atoms act as growth nuclei at a temperature generally higher than the homogeneous nucleation temperatures T m /3 of liquid elements because they reduce the critical energy barrier as shown in (20) [18]: where v is the sample volume, J the nucleation rate, t sn the steady-state nucleation time, lnK ls = 90 ± 2, G* 2ls /k B T defined in (19) and G nm in (15). The equation (20) is applied, assuming that n-atom superclusters preexist in melts when they have not been melted by superheating above T m . It can also be applied when the homogeneous condensation time of an n-atom supercluster is evolved and its own critical energy barrier is crossed. The cluster thermally-activated critical energy barrier G* nm /k B T and the effective thermally-activated critical energy barrier G neff /k B T of an n-atom supercluster are given by (21) and (22): T k where G nm is given by (15),  nm by (16) and lnK ls = 90 ± 2. The quantified value  nm0 ×H m of the cluster energy saving at T m is defined in the next part. The transient nucleation time being neglected, the growth around these nuclei is only possible when the steady-state nucleation time t sn is evolved and the relation (23) is respected: where v is the sample volume and t sn the steady-state nucleation time. The crystallization follows this cluster formation time when, in addition, (20) is respected. The effective nucleation temperature deduced from (20) does not result from a homogeneous nucleation because it strongly depends on the sample volume v. This phenomenon explains why the effective nucleation temperature in liquid elements is observed around  = 0.2 in sample volumes of a few mm 3 instead of  varying from  0.58 to 0.3 in much smaller samples [17,30].

Quantification of energy saving associated with supercluster formation
The potential energy saving per nucleus volume unit  ls ×H m /V m is equal to the Laplace pressure change p = 2× ls /R accompanying the transformation of a liquid droplet into a nucleus. The quantified energy is smaller than 2× ls /R at low radius R for n = 13 and 55. The calculation is made by creating a Laplace pressure on the surface of a spherical nucleus containing n atoms which would result from a virtual transfer of n×z electrons in s-states from the nucleus to the melt, z being the fraction of transferred electrons per supercluster atom [31].
The potential energy U 0 would be equal to (24) and to zero beyond the nucleus radius R: where e is the electron charge, and 0 the vacuum permittivity [32, p.135]. The quantified energy E q at T m is given by (25): where N A is the Avogadro number. The quantified energy saving is given by (16) where m 0 is the electron rest mass and ħ Planck's constant divided by 2p. The critical radii of liquid elements are sufficiently large at T m to assume that U 0 is equal to E q and to deduce the values ofz from the relation (24) = (25) with R = R* 2ls (=0) and  nm0 =  ls0 = 0.217. The potential energy U 0 given by (24) is also equal to4R 3 /3×p. Consequently, the z in (24) does not depend on R at T m . The value of U 0 is deduced from the atom number n which depends on molar volumes V m of solid elements extrapolated at T m from published tables of thermal expansion [16,33]. The values of  nm0 are calculated as a function of R using (27) instead of (26) for n ≥ 147 because U 0 is assumed to be equal to E q : The condensed-cluster energy savings  nm0 ×H m of 13 and 55 atoms are quantified and calculated from (25).
The thermal variation of  nm is given in (16) using these quantified values of  nm0 .

Prediction of crystallization temperatures T c of 38 undercooled liquid samples of various diameters
The quantified and the potential energy saving coefficients  nm0 of silver clusters have been calculated using (26) and (27) and are represented in Figure 1 as a function of supercluster radius R nm which is assumed to continuously vary. These coefficients are equal for n ≥ 147. This last approximation is used in all liquid elements. (square points) and non-quantified (diamond points) energy saving coefficients  nm0 are plotted versus the silver cluster radius. This coefficient is strongly weakened when R < 0.5 nm.
Quantification is necessary for an atom number n < 147.
Properties of 38 elements are classified in Table 1: Column 6-the supercluster radius R nm in nanometers deduced from the molar volume V m using the relation (28): Column 7-the energy saving coefficient  nm0 associated with the n-atom supercluster calculated using (28) for n ≥ 147 and (26) for n = 13 and 55, with z given in Table 2  Column 9-the reduced crystallization temperature  c calc calculated using (20), Column 10-the thermally-activated effective energy barrier G eff /k B T given in (20) and (19) leading to the crystallization of the corresponding liquid element, Column 11-the calculated diameter D calc in mm of the liquid droplet of volume v submitted to crystallization at  c calc using (20) and v×t sn  v = /6×D 3 assuming that t sn = 1 s, Column 12-the experimental diameter D exp in millimeters of the liquid droplet crystallizing at  cexp , Column 13-references. The experimental reduced crystallization temperatures  c exp are plotted in Figure 2 versus the calculated  c calc using the supercluster atom-number n leading to about the same droplet diameter D calc as the experimental one D exp . A good agreement is obtained between these values in 32 liquid elements in Figure 2 and Table 1. There is no good agreement for Hg, Sn, Al, Cd, V, and Cr because these elements are known to contain impurities or oxides. Their undercooling rates are too low compared to the calculated ones.
In Figure 3, the calculated droplet diameter logarithms are plotted as a function of those of experimental droplets used to study the undercooling rate. Six orders of magnitude are studied, corresponding to 18 orders of volume magnitude. Figure 3 shows that the model is able to describe the crystallization temperature dependence on the volume sample.  Table 1. The smaller the atom number n, the smaller is the undercooling temperature, as shown in Table 1. The experimental undercooling reduced temperature  c of gallium is the lowest of all the liquid elements and is equal to 0.58 and is a little higher than2/3, corresponding to a crystallization temperature T c equal to 129K [17] and to a melting temperature of the  phase equal to 303 K. The gallium  phase is crystallized after undercooling. Its melting temperature is 257 K instead of 303 K for the  phase and its fusion entropy is 10.91 J/K/mole, as shown in Table 1, instead of 18.4 J/K/mole [54]. Its crystallization temperature of 129 K occurs in fact at  c = 0.5. The calculated value is equal to the experimental one due to a previous condensation of 13atom cluster which weakens the critical energy barrier. The model works without any adjustable parameter, and is able also to predict the nucleation rate of 13-atom clusters and the diameter of gallium droplets obtained with the liquid dispersion technique.

Homogeneous nucleation of 13-atom superclusters and undercooling rate predictions
Equations (20)(21)(22)(23) are now used to calculate the homogeneous formation reduced temperature  13c of 13-atom clusters in a melt cooled below T m from thermodynamic equilibrium state at T m and the crystallization reduced temperature  c that they induce in liquid droplets of 10 micrometers in diameter. In Table 2, 38 liquid elements are considered. In 33 of them, the 13-atom cluster formation temperature is much larger than the crystallization temperature ( 13c >>  c ). On the contrary, in indium, mercury, gallium , cadmium and zinc, the two reduced temperatures are equal within the uncertainty on the energy saving coefficient value  13m0 given in Table 2. The crystallization temperatures of bismuth, selenium, tellurium, antimony, silicon and germanium with a growth around 13-atom clusters are predicted in good agreement with experimental values obtained with various sizes of droplets, as shown in Table 1.
In Figure 5, the homogeneous condensation reduced temperatures of 13-atom superclusters are compared with the reduced spontaneous growth temperatures which induce crystallization. The growth is organized around these 13-atom clusters which are formed, at temperatures higher than that of spontaneous crystallization. These homogeneous and heterogeneous crystallization temperatures depend on the droplet diameters. Their values given in Table 2, Column 10 are the lowest undercooling temperatures which can be obtained with 10 micrometer droplets. In Table 2, the liquid elements are still classified as a function of their fusion entropy S m given in Table 1 (Column 3): Column 1-List of liquid elements,

Superheating and melting of n-atom superclusters by liquid homogeneous nucleation
N-atom superclusters survive above the melting temperature T m up to an superheating temperature which is time-dependent. They can be melted by liquid homogeneous nucleation in their core instead of surface melting.
The Gibbs free energy change associated with their melting at a temperature T > T m is given by (29): where the energy saving coefficient  nm is given in (16) even for > 0. The fusion enthalpy has changed sign as compared to (15) and the equalization of Fermi energies always still leads to an energy saving. An n-atom supercluster melts when (30) is respected: where v n is the n-atom supercluster volume deduced from its radius given in Table 1, (28) and t sn is the superheating time at its own melting temperature because the supercluster radius is much smaller than its critical radius. The time t sn is chosen equal to 600 seconds and lnK sl to 90.

Overheating and melting of n-atom superclusters by liquid heterogeneous nucleation
Melting temperatures of superclusters are reduced by previous melting of a 13-atom droplet in their core.
These entities melt when (31) is respected for n =13: where the critical energy barrier G* nm /k B T no longer exists and is replaced by G nm /k B T,  nm in (16),  nm0 in Table 2, t sn = 600 s and lnK ls = 90. The critical barrier is not involved in (31) because the n-atom supercluster radius is much smaller than the critical radius for liquid growth and G* nm >> G nm .

Prediction of melting temperatures of superclusters in 38 liquid elements by melt superheating above T m
The reduced melting temperatures = (T-T m )/T m of superclusters depending on their atom number n are given in several columns of Table 3 and in Figure 6. They are calculated assuming that the molar volume is constant, t sn = 600 s. and lnK sl = 90. The liquid elements having fusion entropy S m larger than 20 J/K/mole have a melting temperature which is determined by liquid homogeneous nucleation because the 13-atom clusters melt at higher temperatures while those with S m < 20 J/K/mole are submitted to chain-melting. These melting temperatures are given in columns 10, 11, 12 and 13 of Table 3 versus S m .
. . In Table 3,  All these results have been obtained assuming that the superheating time at their own melting temperature is 600 seconds. The time effects on copper supercluster melting are examined in part 9 in relation with detailed experimental studies [51]. Hom.

Electronic properties of Cu, Ag and Au superclusters
Electronic properties of superclusters can be calculated from the enthalpy saving associated with their formation temperature in noble metallic liquids because this energy is due to Fermi energy equalization of liquid and superclusters [23]. The Fermi energy difference E F between condensed superclusters of radius R nm containing n atoms and liquid state at T m can be directly evaluated for noble metals using (32) and assuming that z is small, as shown in Table 2: where m is the ratio of electron masses m*/m 0 , m 0 being the electron rest mass and m* the effective electron mass which is assumed to be the same in superclusters and liquid states, and z being calculated at variable temperature using the known quantified energy saving  ls in (17) and (25,26). The Fermi energy difference E F is plotted in Figure 7 as a function of 1/R* 2ls , where R* 2ls is given in (6) for Cu, Ag and Au assuming that the molar volume does not depend on temperature and a continuous variation of R* 2ls . The quantified value  ls is given in (17) and the U 0 and z values are calculated with (25). For R* > 1 nm, E F is proportional to the Laplace pressure, while for R << 1 nm there is a gap opening in the conduction electron band accompanying the quantification of the energy saving. This analysis is able to detect well-known properties of clusters out of the melt which become much less conducting at very low radii [10]. A strong variation of E F at constant molar volume V m is observed in Figure 7. In principle, the E F has to obey (33) in the liquid state because the Fermi energy E F depends on (V m ) 2/3 : where E F is the Fermi energy of the liquid, V m the molar volume of a supercluster of infinite radius, V m is the variation of the molar volume with the radius decrease. The supercluster molar volume V m has to depend on the particle radius instead of being constant. Equation (33) is respected when the formation temperature T of superclusters corresponding to the critical atom number n c in (18) and to a molar volume V m depending on R* 2ls is introduced. The formation temperatures of superclusters with magic atom numbers are indicated in Figure 8 using a special molar volume thermal variation V m (T) given in Figure 9 for each liquid element.


The following laws are used in Figure 8 and They correspond to an infinite radius for superclusters in the absence of crystallization [15]. The molar volume V m of bulk superclusters would be attained when  nm becomes equal to zero using the critical radius as a hidden variable becoming infinite instead of the temperature.
The Fermi energy change E F depends on z in (32); z is calculated with (24)(25)(26) for each radius R = R* 2ls (T) in (6), determining n from (28). Equation (33) is now respected for Cu, Ag and Au, as shown in Figure   10. The Fermi energy E F is defined in (34), assuming that there is one conduction electron per atom in Cu, Ag and Au: where V m is the liquid molar volume at T m which is equal to 7.95×10 -6 , 11.5×10 -6 , and 11.3×10 -6 m 3 /mole for Cu, Ag and Au respectively [64]. The molar electronic specific heat C el =  el ×T of Cu, Ag and Au superclusters can be obtained from the knowledge of their electronic density of states D(E F ) at the Fermi level, calculated with (35) and (36) [64]: The z values have been previously determined from (24)(25)(26). Each n-atom supercluster has its own molar volume V m and its own z at T m is determined with (24) = (25) with R = R* 2ls (T m ) depending on V m and  nm0 = 0.217. The electronic specific heat coefficient  el is plotted in Figure 11 as a function of the supercluster molar volume.
The electronic specific heat coefficient  el of superclusters falls when their molar volume V m and their radius decrease below T m . The coefficients  el of Cu, Ag and Au crystals at 4 K are a little larger, being equal to 0.695, 0.646 and 0.729 instead of 0.48, 0.547 and 0.599 mJ/K 2 /mole at T m respectively [64]. Small crystals are known to become insulating for radii smaller than 5 nm when they are studied out of their melt [10]. This electronic transformation is also present in superclusters and is very abrupt below their critical growth volume at T m as shown in Figure 11. The  el at T m is also calculated as a function of the supercluster radius R and represented in Figure 12. The coefficient z is obtained at T m with (24) = (25) and  nm =  ls0 = 0.217. Then, the potential energy U 0 depending on R is known for each value of R and the quantified coefficient  nm0 of an n-atom supercluster of radius R is deduced from (25,26). In Figure 12, the highest points are calculated at T m while the lowest are already shown in Figure 11. The smallest superclusters are still metallic at T m , while they become insulating when the temperature is close to T m /3. All these predictions are in good agreement with many properties of divided metals. They are only based on an enthalpy saving equal to 0.217×(12.25× 2 )×H m for the supercluster formation in all liquid elements.

Silver supercluster formation into and out of undercooled liquid
The formation of icosahedral silver clusters with magic numbers n of atoms equal to 13,55,147,309,561,923, 1415 and 2057 has been already studied out of liquid by molecular dynamics in the temperature range 0-1300 K. Icosahedral clusters of 13, 55 and 147 are formed below room temperature and larger clusters with n= 309, 561, 923, 1415 are formed from 300 to 1000 K. The radii of these Ag stable superclusters have been found to be equal to 2.74, 5.51, 8.32, 11.14 and 14.94 Ǻ for n = 13, 55, 147, 309, 561 respectively [5]. The Ag radii have also been calculated in the liquid using the molar volume shown in Figure 9 and their formation temperature as deduced from the critical radius. Their values for n = 13, 55, 147, 309, 561 are nearly equal to those predicted by molecular dynamics, as shown in Table 4 and Figure 13. Table 4. The Ag supercluster radii with magic atom numbers. The radius R is deduced from molar volume V m and equal to critical radius R* 2ls (T) given in (6). For T > T m /3 = 411.33 K, the energy saving coefficient  ls in (17) Figure 13. The critical atom number in blue versus the critical radius and R MD the radius calculated by molecular dynamics simulations in red square [5].
9. Melting of Cu, Ag and Au superclusters varying the superheating times 9.1.

Overheating of Cu, Ag and Au superclusters
The melting temperatures are now calculated using the molar volume associated with the supercluster radius as shown in Figure 9. The superheating time continues to be equal to 600 seconds. The supercluster radius variation is continuous while the radius of magic number 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 temperature  = (T-T m )/T m .
The points labeled "homogeneous" are calculated assuming that supercluster melting is produced by liquid homogeneous nucleation using (29,30). The triangles labeled (n-13) are calculated assuming that the supercluster melting is induced by previous formation of liquid droplets of 13 atoms into superclusters using (31). The homogeneous nucleation temperatures are much too high compared to the (n-13) temperatures. The undercooling temperatures depend on the volume sample v. The square points are determined for ln(K ls .v.t sn ) = 71.8 corresponding to v.t sn = 12×10 -9 m 3 .s and a heterogeneous nucleation induced by superclusters of radius R when the applied superheating temperature is smaller than those indicated by triangles. Another supercooling temperature represented by triangle points is added in Figures 15 and 16. In Figure 15, v.t sn is equal to 7.08×10 22 m 3 .s while, in Figure 16, v.t sn = 15×10 -7 m 3 .s. These three figures show that an undercooling rate of about 20% is generally obtained when the sample volume is of the order of a few mm 3 and the applied superheating rate is less than about 25%. The undercooling temperature is very stable when the superheating is less than 25%. Larger undercooling rates are obtained using much smaller volume samples [17].

Analysis of the influence of Cu superheating time on the undercooling rate
The superheating time has a strong influence on supercluster melting, as shown by studies of Cu undercooling [51]. It has been found that a minimum superheating temperature of 40 K is required in order to achieve any undercooling prior to crystallization nucleation. This phenomenon is also observed in many magnetic texturing experiments [65]. In Table 3, the first Cu supercluster to be melted at  = 0.033 in 600 s, corresponding to a superheating of 44.7 K, contains 13 atoms in perfect agreement with the observation. There is no other supercluster melting. A temperature below which no small supercluster melts is predicted in this model. The lowest value of the undercooling temperature is obtained when 6 thermal cycles are applied prior to nucleation after 6 steps of 2400 s at 1473 K. The total time evolved at 1473 K is 14400 s. In Figure 17

Conclusions
The undercooling temperatures of 32 of the 38 liquid elements are predicted for the first time in good agreement with experimental values depending on the sample volume, without using any adjustable barrier energy, and only assuming the existence of growth nuclei containing stable magic atom numbers n equal to 13, 55, 147, 309 and 561 that are generally devoted to icosahedral structures. The model is based on a volume enthalpy saving  v previously determined to be equal to 0.217×H m /V m at T m and added to the classical Gibbs free energy change for a critical nucleus formation in a melt. This enthalpy is due to the Laplace pressure change p acting on the growth nuclei and equalizing the Fermi energies of liquid and nuclei in metallic liquids.
The Gibbs free energy change has to contain a contributionV m ×p which has been neglected up to now because its magnitude was unknown. This missing enthalpy has serious consequences because the critical radius for crystal growth is considered, in the classical view, as being infinite at the melting temperature and all solid traces being eliminated in melts. This is in contradiction with many experiments on the superheating influence on undercooling rates and on magnetic texturing efficiency [51,[65][66][67][68][69]. Nuclei having radii smaller than the critical radius at T m are melted at higher temperatures depending on the superheating time and on their atom number.
Some growth nuclei survive above T m because they are superclusters that are not melted by surface melting.
This new property of superclusters is a consequence of the thermal variation of  v , which is a unique function of  2 = [(T-T m )/T m ] 2 being maximum at T m , and a fusion heat equal to that of bulk crystals. The surface atom fusion heat is not weakened and there is no premelting of these entities depending on their radius. This thermal variation was established, for the first time, from our study of the maximum undercooling rate of the same liquid elements. In addition, it is the only law validating the existence of non-melted 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 m is calculated by creating a virtual s-electron transfer from the nucleus of radius R to the melt and an electrostatic spherical potential induced by the surface charges and also varying with R -1 .
The Schrödinger equation solutions are known and used to predict the condensation temperatures of 13-atom superclusters in undercooled melts which govern the crystallization temperatures of liquids having fusion entropy larger than 20 J/K/mole.
The superclusters are melted by homogeneous or heterogeneous liquid nucleation in their core. The liquid homogeneous nucleation is effective in all superclusters when S m ≥ 20 J/K/mole while a chain melting is produced, starting with a 13-atom droplet induced in the core of the supercluster and being magnified with the time increase at the superheating temperature. The model is able to predict an approximate value of the minimum time necessary to melt superclusters and to attain the true thermodynamic equilibrium of the melt at any superheating temperature.
The electronic specific heat of superclusters submitted to Laplace pressure in metals is determined for the first time from the enthalpy saving deduced from undercooling experiments. It strongly declines with radius as compared to that of a bulk metal, in agreement with the conductance properties of tiny clusters having radii smaller than 5 Ǻ. The electronic s-state density of superclusters is greatly weakened compared to that of bulk crystals when their radius decreases. The supercluster critical radii deduced from the nucleation model are in quantitative agreement with recent molecular dynamics simulations devoted to Ag cluster radii.
The transformation of superclusters in crystals occurs for a radius between the critical radius for crystal growth and that for supercluster growth because the superclusters have a much lower density than crystals. The Gibbs free energy change from the liquid state to crystal becomes smaller than that of the supercluster just above its maximum at the crystal critical radius.

Acknowledgments:
Thanks Fermi energies without electron transfer. A virtual transfer is nethertheless considered to quantify the energy saving in agreement with undercooling rates. The author thanks Dr Andrew Mullis for the confirmation that the copper sample weights used in [51] are equal to 0.6 to 1 g.