**3.2. Melting temperature of surviving intrinsic nuclei as a function of their radius**

The radius R of liquid droplets which are created by homogeneous nucleation inside an outof-equilibrium crystal of radius Rnm as a function of the overheating rate θ is calculated from the value of ΔG2sl given by (12); (12) is obtained by replacing θ by –θ in (2) because the transformation occurs now from crystal to melt instead of melt to crystal:

$$\Delta \mathbf{G}\_{2sl}(\mathbf{R}\_{\circ}, \boldsymbol{\Theta}) = \frac{\Delta \mathbf{H}\_{\rm m}}{\mathbf{V}\_{\rm m}} (-\boldsymbol{\Theta} - \boldsymbol{\varepsilon}\_{\rm m00}) 4\pi \frac{\mathbf{R}^3}{3} + 4\pi \mathbf{R}^2 \frac{\Delta \mathbf{H}\_{\rm m}}{\mathbf{V}\_{\rm m}} (1 + \boldsymbol{\varepsilon}\_{\rm m00}) \frac{12 \mathbf{k}\_{\rm B} \mathbf{V}\_{\rm m} \ln \mathbf{K}\_{\rm k}}{432\pi \times \Delta \mathbf{S}\_{\rm m}} \tag{12}$$

where εls given by (6) is replaced by εnm0 when θ ≥ 0 because the Fermi energy difference between crystal and melt is assumed to stay constant above Tm. A crystal melts at T > Tm when its radius Rnm is smaller than the critical values 0.894 or 0.927 nm. The equation (13) of liquid homogeneous nucleation in crystals with the nucleation rate J = (1/v.tsn) is used to determine ΔG2sl/kBT as a function of Rnm [7,10,47,62]:

$$\ln(\text{J.v.t}\_{\text{sn}}) = \ln(\text{K}\_{\text{sl}}.\text{v.t}\_{\text{sn}}) - \frac{\Delta\text{G}\_{\text{sl}}(\text{R}\_{\text{nm}}, \theta)}{\text{k}\_{\text{g}}\text{T}} = 0 \tag{13}$$

Magnetic Texturing of High-Tc Superconductors 187

0 q

2

(16)

(17)

0 nm

<sup>1</sup> k [2m(U E )] = − (15)

The potential energy U0 is nearly equal to the quantified energy Eq with εls0 = 1.58 at T = Tm and Δz = 0.1073 with a critical radius of 0.894 nm, ΔHm = 5100 J/atom.g, using (14). We find Δz = 0.0994 with a critical radius of 0.927 nm and ΔHm = 4600 J. Schrödinger's equation (15) is written with wave functions ψ only depending on the distance r from the potential centre

<sup>r</sup> dr <sup>ψ</sup> <sup>+</sup> <sup>ψ</sup> <sup>=</sup> , where 1/2

n H <sup>E</sup> N ε Δ <sup>=</sup>

are given by the k value and by (16) as a function of the potential U0 associated with a crystal

The quantified values of Eq at Tm and consequently of εnm0 are calculated as a function of Rnm from the knowledge of Uo using a constant Δz. The free-energy change given by (12) is plotted versus Rnm in Figure 11 and corresponds to a liquid droplet formation of radius Rnm at various overheating rates θ. Crystals are melted when (13) is respected with ΔG2sl/kBT equal or smaller than ln(Kls.v.tsn). All crystals having a radius larger than the critical value 0.894 nm and smaller than 0.36 nm are melted at Tm while those with 0.36 < Rnm< 0.894 nm are not melted. It is predicted that, for the two ΔHm values, they would disappear at temperatures respecting θ > 0.93 assuming that all atoms are equivalent. It is known that a consistent overheating has to be applied to glass-forming melts in order to eliminate a premature crystallization during cooling [41]. A glass state is obtained when Bi-2212 melts are quenched from 1473-1523 K (0.264 < θ < 0.307) after 1800 s evolved at this temperature [60,61]. Many nuclei survive after an overheating at θ = 0.264. Bi2201 grains of diameter 10 nm are obtained after annealing the quenched undercooled state at 773 K during 600 s [61]. Consequently, the nuclei number is still larger than 1024/m3 in spite of an overheating at 1473 K (θ = 0.264) in agreement with the curves ΔG2sl/kBT which are larger than ln(Ksl.v.tsn) ≅ 20

<sup>=</sup> , where

nm0 m

A

0

n ze <sup>U</sup> 8 R <sup>Δ</sup> <sup>=</sup> πε

for a s-state electron [65]:

The quantified solutions

with t = 1800 s and v < 1 nm3 .

2

2

radius R = Rnm , n varying with the cube of Rnm:

sin kR

2

q

<sup>2</sup> nm nm 0

**3.3. Crystallization of melts induced by intrinsic nuclei at T ≤ Tm**

crystallization will occur when (17) will be respected at the temperature T:

sn sl s sn

These intrinsic nuclei act as growth nuclei during a solidification process. The first-

G() G () ln(J.v.t ) ln(K .v .t ) <sup>0</sup>

\*

Δ θ Δ θ = −− =

sl nm

B B

kT kT

kR 2mR U

<sup>1</sup> <sup>d</sup> (r ) k 0

nm

where v is the surviving crystal volume of radius Rnm and tsn the steady-state nucleation time evolved at the overheating temperature T. The radius Rnm being smaller than 1 nanometer and tsn chosen equal to 1800 s, ln(Ksl.v.tsn) is equal to 32-35 with lnKsl being given by (3).

The equalization of Fermi energies in metallic melts is accomplished through Laplace pressure. We have imagined another way of equalizing the Fermi energies of nascent crystals and melt instead of applying a Laplace pressure [5,6,47]. Free electrons are virtually transferred from crystal to melt. The quantified energy savings of crystals having a radius Rnm equal or smaller than the critical value would depend on the number of transferred free electrons. A spherical attractive potential would be created and would bound these s-state conduction electrons. This assumption implies that the calculated energy saving εnm0 at T = Tm is quantified, depends on the crystal radius Rnm which corresponds to the first energy level of one s-electron moving in vacuum in the same spherical attractive potential. These quantified values of εnm0 have been already used to predict the undercooling temperatures of gold and other liquid elements in perfect agreement with experiments [63,64]. As already shown, the attractive potential –U0 defined by (14) is a good approximation for n×Δz >> 1, n = 4πΝΑR3/3Vm being the atom number per spherical crystal of radius Rnm and Δz the fraction of electron per atom which would be transferred in vacuum from crystal to melt, e the electron charge, εnm0×ΔHm the quantified electrostatic energy saving per mole at Tm, ε<sup>0</sup> the vacuum permittivity:

$$\mathbf{U}\_{o} = \frac{\mathbf{n} \Delta \mathbf{z} \cdot \mathbf{e}^{2}}{8 \pi \mathbf{c}\_{0} \mathbf{R}\_{\text{nm}}} \ge \mathbf{E}\_{\text{q}} = \frac{\mathbf{n} \mathbf{c}\_{\text{nm}0} \Delta \mathbf{H}\_{\text{m}}}{\mathbf{N}\_{\text{A}}} \tag{14}$$

The potential energy U0 is nearly equal to the quantified energy Eq with εls0 = 1.58 at T = Tm and Δz = 0.1073 with a critical radius of 0.894 nm, ΔHm = 5100 J/atom.g, using (14). We find Δz = 0.0994 with a critical radius of 0.927 nm and ΔHm = 4600 J. Schrödinger's equation (15) is written with wave functions ψ only depending on the distance r from the potential centre for a s-state electron [65]:

$$\frac{1}{\text{r}}\frac{\text{d}^2}{\text{dr}^2}(\text{r}\psi) \, +\text{k}^2\psi = \, 0 \,, \text{where } \text{k} \, = \frac{1}{\hbar}[2\text{m}(\text{U}\_0 - \text{E}\_q)]^{1/2} \tag{15}$$

The quantified solutions

186 Superconductors – Materials, Properties and Applications

**radius** 

by (3).

the vacuum permittivity:

**3.2. Melting temperature of surviving intrinsic nuclei as a function of their** 

transformation occurs now from crystal to melt instead of melt to crystal:

Δ Δ Δ θ = −θ − ε π + π + ε

determine ΔG2sl/kBT as a function of Rnm [7,10,47,62]:

2sl nm0 nm0

sn sl sn

The radius R of liquid droplets which are created by homogeneous nucleation inside an outof-equilibrium crystal of radius Rnm as a function of the overheating rate θ is calculated from the value of ΔG2sl given by (12); (12) is obtained by replacing θ by –θ in (2) because the

3

H RH 12k V lnK G (R , ) ( )4 4 R (1 )( ) <sup>V</sup> 3 V 432 S

where εls given by (6) is replaced by εnm0 when θ ≥ 0 because the Fermi energy difference between crystal and melt is assumed to stay constant above Tm. A crystal melts at T > Tm when its radius Rnm is smaller than the critical values 0.894 or 0.927 nm. The equation (13) of liquid homogeneous nucleation in crystals with the nucleation rate J = (1/v.tsn) is used to

G (R , ) ln(J.v.t ) ln(K .v.t ) <sup>0</sup>

where v is the surviving crystal volume of radius Rnm and tsn the steady-state nucleation time evolved at the overheating temperature T. The radius Rnm being smaller than 1 nanometer and tsn chosen equal to 1800 s, ln(Ksl.v.tsn) is equal to 32-35 with lnKsl being given

The equalization of Fermi energies in metallic melts is accomplished through Laplace pressure. We have imagined another way of equalizing the Fermi energies of nascent crystals and melt instead of applying a Laplace pressure [5,6,47]. Free electrons are virtually transferred from crystal to melt. The quantified energy savings of crystals having a radius Rnm equal or smaller than the critical value would depend on the number of transferred free electrons. A spherical attractive potential would be created and would bound these s-state conduction electrons. This assumption implies that the calculated energy saving εnm0 at T = Tm is quantified, depends on the crystal radius Rnm which corresponds to the first energy level of one s-electron moving in vacuum in the same spherical attractive potential. These quantified values of εnm0 have been already used to predict the undercooling temperatures of gold and other liquid elements in perfect agreement with experiments [63,64]. As already shown, the attractive potential –U0 defined by (14) is a good approximation for n×Δz >> 1, n = 4πΝΑR3/3Vm being the atom number per spherical crystal of radius Rnm and Δz the fraction of electron per atom which would be transferred in vacuum from crystal to melt, e the electron charge, εnm0×ΔHm the quantified electrostatic energy saving per mole at Tm, ε<sup>0</sup>

2

= ≥= πε

n ze n H U E 8R N Δ ε Δ

0 nm A

0 q

nm0 m

(14)

m m 2 B m ls 1/3

sl nm

Δ θ =− = (13)

B

k T

π×Δ (12)

m mm

$$\mathbf{E\_q} = \frac{\mathbf{n} \mathbf{e\_{nm\_0}} \Delta \mathbf{H\_m}}{\mathbf{N\_A}}$$

are given by the k value and by (16) as a function of the potential U0 associated with a crystal radius R = Rnm , n varying with the cube of Rnm:

$$\frac{\sin \text{k}\text{R}\_{\text{nm}}}{\text{k}\text{R}\_{\text{nm}}} = \frac{\hbar}{\sqrt{2 \text{m} \text{R}\_{\text{nm}}^2 \text{U}\_0}}, \text{ where } \text{U}\_0 = \frac{\text{n} \Delta \text{ze}^2}{8 \text{\pi} \text{\varepsilon}\_0 \text{R}\_{\text{nm}}} \tag{16}$$

The quantified values of Eq at Tm and consequently of εnm0 are calculated as a function of Rnm from the knowledge of Uo using a constant Δz. The free-energy change given by (12) is plotted versus Rnm in Figure 11 and corresponds to a liquid droplet formation of radius Rnm at various overheating rates θ. Crystals are melted when (13) is respected with ΔG2sl/kBT equal or smaller than ln(Kls.v.tsn). All crystals having a radius larger than the critical value 0.894 nm and smaller than 0.36 nm are melted at Tm while those with 0.36 < Rnm< 0.894 nm are not melted. It is predicted that, for the two ΔHm values, they would disappear at temperatures respecting θ > 0.93 assuming that all atoms are equivalent. It is known that a consistent overheating has to be applied to glass-forming melts in order to eliminate a premature crystallization during cooling [41]. A glass state is obtained when Bi-2212 melts are quenched from 1473-1523 K (0.264 < θ < 0.307) after 1800 s evolved at this temperature [60,61]. Many nuclei survive after an overheating at θ = 0.264. Bi2201 grains of diameter 10 nm are obtained after annealing the quenched undercooled state at 773 K during 600 s [61]. Consequently, the nuclei number is still larger than 1024/m3 in spite of an overheating at 1473 K (θ = 0.264) in agreement with the curves ΔG2sl/kBT which are larger than ln(Ksl.v.tsn) ≅ 20 with t = 1800 s and v < 1 nm3 .

### **3.3. Crystallization of melts induced by intrinsic nuclei at T ≤ Tm**

These intrinsic nuclei act as growth nuclei during a solidification process. The firstcrystallization will occur when (17) will be respected at the temperature T:

$$\ln(\text{J.v.t}\_{\text{sn}}) = \ln(\text{K}\_{\text{sl}}, \text{v.t}\_{\text{sn}}) - \left[\frac{\Delta \text{G}\_{\text{sl}}^{\ast}(\Theta)}{\text{k}\_{\text{B}} \text{T}} - \frac{\Delta \text{G}\_{\text{nm}}(\Theta)}{\text{k}\_{\text{B}} \text{T}}\right] = 0 \tag{17}$$

where ΔG\*2ls/kBT is given by (5) and ΔGnm/kBT is the contribution of unmelted crystal of radius Rnm to the reduction of ΔG\*2ls/kBT given by (2).

Magnetic Texturing of High-Tc Superconductors 189

**Figure 12.** The difference (ΔG\*2ls/kBTm−ΔGnm/kBTm=ΔGeff/kBTm), the ln(Kls.vs.t) = 70.1 with v = 10-6 m3, Kls= 76.4, tsn = 1800 s, and ln(Kls.v.t) = 28.6 with Kls = 76.4, v = 10<sup>−</sup>24 m3 and tsn = 1800 s are represented as a function of the surviving nucleus radius Rnm. All nuclei with 0.45 < Rnm <0.894 nm can grow at T = Tm in

The calculation of the first-crystallization nucleation total time t has to take into account not only the steady-state nucleation time tsn but also the time-lag τns in transient nucleation [44]

> \* 2ls 1/2 2 B c <sup>1</sup> <sup>G</sup> ( ) 3 kT j Δ

<sup>π</sup> ,

The time lag τns is proportional to the viscosity η while the steady-state nucleation rate J is proportional to η-1; the Jτns in (19) is not dependent on η [44]; Γ is the Zeldovitch factor, a\*0 = π2/6, N the atom number per volume unit, jc the atom number in the critical nucleus; Kls is defined by (3). Equation (17) is also applied when tsn is small compared with the time lag τns. A very small value of tsn/τns undervalues the time t by a factor 2 or 3 and has a negligible effect in a logarithmic scale. The first crystallization occurs when J = (v.tsn)-1 in the presence of one intrinsic growth nuclei in the sample volume v. The critical energy barrier is obtained using (5)

Assuming that the intrinsic nuclei are very numerous regardless the sample volume v, the

<sup>Δ</sup> = − , where \* \*

\* eff

B

2 ns t 6 π

τ =

\* ns 0

; tsn is deduced from (20) with ln(Kls) given by (3).

eff 2ls nm B BB G G G kT kT kT

Δ Δ <sup>Δ</sup> = − (20)

ls a N 2 K

π Γ

>> τ (18)

3 2ls

c 3 nm

<sup>32</sup> <sup>j</sup> 3( ) π×α <sup>=</sup> θ−ε (19)

and

2 sn ns t t 6 π =+τ , when

Γ =

steady-state nucleation rate J per volume unit and per second is given by:

sn ls sn

<sup>G</sup> ln(J.v.t ) ln(K v.t ) k T

a sample volume of 10-24 m3 using our model of identical atoms.

with:

\* \* ns 0 eff

aN G J exp( ) 2 kT

τ= −

B

πΓ , where

Δ

and the values of εnm as a function of θ2

**Figure 11.** Values of ΔG2sl given by (12) divided by kBT and calculated with a radius R equal to an intrinsic nucleus radius Rnm smaller than the critical radius equal to 0.894 nanometer are plotted versus Rnm for θ = (T-Tm)/Tm = 0, 0.1, 0.15, 0.264 and 0.93. The straight line nearly parallel to the Rnm axis represents ln(Kls.v.tsn) with v equal to the volume of a sphere having a radius Rnm. All nuclei with ΔG2sl/kBT > ln(Kls.v.tsn) are not melted. The lower radius below which all nuclei melt slowly increases with θ. All nuclei would disappear after 1800 s at T = 2248 K (θ = 0.93).

The lnKls given by (3) is equal to 76.4 at Tm while the volume sample vs is assumed to be equal to 10-6 and 10-24 m3 respectively leading to ln(Kls.vs.tsn) = 70.1 and ln(Kls.vs.t) = 28.6 with tsn = 1800 s. In Figure 12, (ΔG\*2ls/kBT−ΔGnm/kBT) = ΔGeff/kBT, ln(Kls.vs.tsn) = 70.1 and 28.6 are plotted as a function of Rnm at T = Tm. Nuclei with 0.45 < Rnm< 0.894 nm could act as growth nuclei at Tm. Nuclei with Rnm smaller than 0.45 nm contain about one or two layers of atoms surrounding a centre atom. All metallic glass-forming melts contain the same type of clusters that govern their time-temperature-transformation (TTT) diagram above Tg [47]. The Bi2212 TTT diagram has to exist even if it is not known. In contradiction with our model of identical atoms, intrinsic nuclei having a radius Rnm respecting 0.45 < Rnm < 0.894 nm have to be melted after an overheating up to 1473 K (θ = 0.264) because a weak cooling rate leads to a vitreous state in this type of glass-forming melt after applying an overheating rate θ = 0.264 [61].

radius Rnm to the reduction of ΔG\*2ls/kBT given by (2).

where ΔG\*2ls/kBT is given by (5) and ΔGnm/kBT is the contribution of unmelted crystal of

**Figure 11.** Values of ΔG2sl given by (12) divided by kBT and calculated with a radius R equal to an intrinsic nucleus radius Rnm smaller than the critical radius equal to 0.894 nanometer are plotted versus Rnm for θ = (T-Tm)/Tm = 0, 0.1, 0.15, 0.264 and 0.93. The straight line nearly parallel to the Rnm axis represents ln(Kls.v.tsn) with v equal to the volume of a sphere having a radius Rnm. All nuclei with ΔG2sl/kBT > ln(Kls.v.tsn) are not melted. The lower radius below which all nuclei melt slowly increases

The lnKls given by (3) is equal to 76.4 at Tm while the volume sample vs is assumed to be equal to 10-6 and 10-24 m3 respectively leading to ln(Kls.vs.tsn) = 70.1 and ln(Kls.vs.t) = 28.6 with tsn = 1800 s. In Figure 12, (ΔG\*2ls/kBT−ΔGnm/kBT) = ΔGeff/kBT, ln(Kls.vs.tsn) = 70.1 and 28.6 are plotted as a function of Rnm at T = Tm. Nuclei with 0.45 < Rnm< 0.894 nm could act as growth nuclei at Tm. Nuclei with Rnm smaller than 0.45 nm contain about one or two layers of atoms surrounding a centre atom. All metallic glass-forming melts contain the same type of clusters that govern their time-temperature-transformation (TTT) diagram above Tg [47]. The Bi2212 TTT diagram has to exist even if it is not known. In contradiction with our model of identical atoms, intrinsic nuclei having a radius Rnm respecting 0.45 < Rnm < 0.894 nm have to be melted after an overheating up to 1473 K (θ = 0.264) because a weak cooling rate leads to a vitreous state in this type of glass-forming melt after applying an overheating rate θ =

with θ. All nuclei would disappear after 1800 s at T = 2248 K (θ = 0.93).

0.264 [61].

**Figure 12.** The difference (ΔG\*2ls/kBTm−ΔGnm/kBTm=ΔGeff/kBTm), the ln(Kls.vs.t) = 70.1 with v = 10-6 m3, Kls= 76.4, tsn = 1800 s, and ln(Kls.v.t) = 28.6 with Kls = 76.4, v = 10<sup>−</sup>24 m3 and tsn = 1800 s are represented as a function of the surviving nucleus radius Rnm. All nuclei with 0.45 < Rnm <0.894 nm can grow at T = Tm in a sample volume of 10-24 m3 using our model of identical atoms.

The calculation of the first-crystallization nucleation total time t has to take into account not only the steady-state nucleation time tsn but also the time-lag τns in transient nucleation [44] with:

$$\mathbf{t} = \mathbf{t}\_{\mathrm{sn}} + \frac{\boldsymbol{\pi}^2}{6} \boldsymbol{\pi}\_{\mathrm{ns}}, \text{ when } \mathbf{t} >> \frac{\boldsymbol{\pi}^2}{6} \boldsymbol{\pi}\_{\mathrm{ns}} \tag{18}$$

$$\mathbf{J}\mathbf{\tau}^{\rm ns} = \frac{\mathbf{a}\_{\rm \rm s}^{\rm \rm N}}{2\pi\Gamma} \exp(-\frac{\Delta\mathbf{G}\_{\rm eff}^{\rm \rm s}}{\mathbf{k}\_{\rm g}\Gamma}), \text{ where } \Gamma = \left(\frac{1}{3\pi\mathbf{k}\_{\rm g}\Gamma}\frac{\Delta\mathbf{G}\_{\rm 2k}^{\rm \rm s}}{\mathbf{j}\_{\rm c}^{\rm 2}}\right)^{1/2}, \text{ } \mathbf{\tau}^{\rm ns} = \frac{\mathbf{a}\_{\rm \rm s}^{\rm \rm N}}{2\pi\mathbf{K}\_{\rm \rm s}\Gamma} \text{ and } \mathbf{j}\_{\rm c} = \frac{32\pi \times \mathbf{a}\_{2\rm s}^{\rm 3}}{3(\Theta - \mathbf{e}\_{\rm nm})^{3}} \tag{19}$$

The time lag τns is proportional to the viscosity η while the steady-state nucleation rate J is proportional to η-1; the Jτns in (19) is not dependent on η [44]; Γ is the Zeldovitch factor, a\*0 = π2/6, N the atom number per volume unit, jc the atom number in the critical nucleus; Kls is defined by (3). Equation (17) is also applied when tsn is small compared with the time lag τns. A very small value of tsn/τns undervalues the time t by a factor 2 or 3 and has a negligible effect in a logarithmic scale. The first crystallization occurs when J = (v.tsn)-1 in the presence of one intrinsic growth nuclei in the sample volume v. The critical energy barrier is obtained using (5) and the values of εnm as a function of θ2 ; tsn is deduced from (20) with ln(Kls) given by (3).

Assuming that the intrinsic nuclei are very numerous regardless the sample volume v, the steady-state nucleation rate J per volume unit and per second is given by:

$$\ln(\text{J.v.t}\_{\text{sa}}) = \ln(\text{K}\_{\text{ls}}\text{v.t}\_{\text{sa}}) - \frac{\Delta\text{G}^{\circ}\_{\text{eff}}}{\text{k}\_{\text{g}}\text{T}}, \text{ where } \frac{\Delta\text{G}^{\circ}\_{\text{eff}}}{\text{k}\_{\text{g}}\text{T}} = \frac{\Delta\text{G}^{\circ}\_{\text{2s}}}{\text{k}\_{\text{g}}\text{T}} - \frac{\Delta\text{G}\_{\text{nm}}}{\text{k}\_{\text{g}}\text{T}}\tag{20}$$

where tsn is the steady-state nucleation time and ΔGnm the free-energy change associated with the previous solidification of non-melted crystals [8-10,44,62]].

Magnetic Texturing of High-Tc Superconductors 191

Surviving intrinsic nuclei have been called non-melted crystals. These clusters exist in the melt above Tm and they have a stability only expected in super-clusters which could correspond to a complete filling of electron shells [51,52]. This assumption might be correct if the structural change of surviving crystals in less-dense superclusters occurs without complementary Gibbs free energy. The Gibbs free energy change (12) induced by a volume increase would remain constant because it only depends on R/(Vm/NA)1/3. An increase of the cluster molar volume would be followed by an increase of R without change of εnm0 and U0 in (15), (16) and (8). Then, surviving intrinsic nuclei could be super-clusters already observed in liquid aluminium by high energy X-rays [55]. We develop here the idea that the

The model described here also predicts the undercooling temperatures of liquid elements [64]. It only requires the knowledge of the melting temperature of each nucleus as a function of their radius, the fusion heat of a bulk multicomponent material, its molar volume, its melting temperature and its vitreous transition temperature. Adjustable parameters are not required. There is an important limitation to its use because all atoms are considered as being identical. Thus, it cannot describe the behaviour at high temperatures of clusters which have a composition strongly different from that of the melt. We did not find any publication about the existence of a vitreous state of Y123. This could be due to a much smaller viscosity of this melt. Nevertheless, this material can be magnetically textured due to the presence of intrinsic clusters above Tm. Its unknown energy saving coefficient εls0 could be smaller than 1 and could not be determined using equations only governing fragile glass-forming melts. The growth of nuclei occurs inside the temperature interval between liquidus and solidus. The magnetic field can give a direction to the free crystals when their size depending on their magnetic anisotropy is sufficiently large; the processing time below

Magnetic texturing experiments using pre-reacted powders of Bi2212, Bi2223 or Y123 show that crystal alignment in magnetic field is successful when the annealing temperature is slightly superior to the critical temperature Tm. This temperature corresponds to the liquidus temperature Tm above which there is no solid structure except isolated growth nuclei imbedded in the melt having radii smaller than a critical value. The growth around nuclei occurs in a window of solidification extending from Tm and leads to crystals which become free to align in the magnetic field when their anisotropy energy becomes larger than kBT. The thermal cycle has to use an annealing temperature a little larger than Tm in order to be sure that crystallisation will start from surviving nuclei of superconducting phases and not from polycrystalline entities. The overheating temperature has to be limited to a few degrees Celsius up to a few tens of degrees above Tm depending on the amplitude of the magnetic field. This will insure that the number of nuclei per volume unit remains important and that the composition of nuclei is not progressively transformed into non-superconducting

super-clusters are solid residues instead of belonging to the melt structure.

the liquidus temperature has to be adapted to their growth velocity.

secondary phase nuclei accompanying the oxygen loss.

**4. Conclusion** 

**Figure 13.** The logarithm of the nucleation full time t is plotted as a function of the nucleus radius Rnm using three values of lnA, ΔHm = 5100 J. and the corresponding values of lnKls at Tx = 773 K. The shortest nucleation times are equal to 584, 79 and 11 s as compared to an experimental value of 600 s. This nucleation time is dominated by the transient time τns which becomes much larger at Tg = 676 K and equal to 2×1018 s. These results lead to (Tx –Tg) ≅ 97 K measured with a heating rate of 10K/mn. Similar curves are obtained using ΔHm = 4600 J and are not represented.

The steady–state homogeneous-nucleation time tsn (ΔGnm = 0) necessary to observe a first crystallization would be equal to 1024 seconds with a sample volume v =10-24 m3. The energy barrier ΔGnm of intrinsic nuclei is calculated using (2); the radius Rnm of intrinsic nuclei is temperature-independent down to the growth temperature; εnm (θ) is assumed to vary as εls (θ) given by (3) with εls0 replaced by εnm0 which is calculated using (14-16). A unique nucleus in a volume vs = 10-24 m3 still acts as growth nucleus and produces a nanocrystallization process of the undercooled melt as shown in Figure 13.

The Bi2212 melt contains more than 1024 nuclei per m3 in full agreement with nanocrystallization experiments [61]. The values of lnA are equal to 87 ± 2 as already shown for glass-forming melts having about the same Tg/Tm and Vm. The total nucleation time t is calculated by adding tsn and τns with (18). The shortest nucleation time is equal to 584, 79 and 11 s for lnA = 85, 87 and 89 respectively. The nanocrystallization has been observed at T = 773 K after 600 s in perfect agreement with the curve lnA = 85. The model developed above has already predicted the time-temperature transformation diagram of several glassforming melts [47].

Surviving intrinsic nuclei have been called non-melted crystals. These clusters exist in the melt above Tm and they have a stability only expected in super-clusters which could correspond to a complete filling of electron shells [51,52]. This assumption might be correct if the structural change of surviving crystals in less-dense superclusters occurs without complementary Gibbs free energy. The Gibbs free energy change (12) induced by a volume increase would remain constant because it only depends on R/(Vm/NA)1/3. An increase of the cluster molar volume would be followed by an increase of R without change of εnm0 and U0 in (15), (16) and (8). Then, surviving intrinsic nuclei could be super-clusters already observed in liquid aluminium by high energy X-rays [55]. We develop here the idea that the super-clusters are solid residues instead of belonging to the melt structure.

The model described here also predicts the undercooling temperatures of liquid elements [64]. It only requires the knowledge of the melting temperature of each nucleus as a function of their radius, the fusion heat of a bulk multicomponent material, its molar volume, its melting temperature and its vitreous transition temperature. Adjustable parameters are not required. There is an important limitation to its use because all atoms are considered as being identical. Thus, it cannot describe the behaviour at high temperatures of clusters which have a composition strongly different from that of the melt. We did not find any publication about the existence of a vitreous state of Y123. This could be due to a much smaller viscosity of this melt. Nevertheless, this material can be magnetically textured due to the presence of intrinsic clusters above Tm. Its unknown energy saving coefficient εls0 could be smaller than 1 and could not be determined using equations only governing fragile glass-forming melts. The growth of nuclei occurs inside the temperature interval between liquidus and solidus. The magnetic field can give a direction to the free crystals when their size depending on their magnetic anisotropy is sufficiently large; the processing time below the liquidus temperature has to be adapted to their growth velocity.

### **4. Conclusion**

190 Superconductors – Materials, Properties and Applications

where tsn is the steady-state nucleation time and ΔGnm the free-energy change associated

**Figure 13.** The logarithm of the nucleation full time t is plotted as a function of the nucleus radius Rnm using three values of lnA, ΔHm = 5100 J. and the corresponding values of lnKls at Tx = 773 K. The shortest nucleation times are equal to 584, 79 and 11 s as compared to an experimental value of 600 s. This nucleation time is dominated by the transient time τns which becomes much larger at Tg = 676 K and equal to 2×1018 s. These results lead to (Tx –Tg) ≅ 97 K measured with a heating rate of 10K/mn.

The steady–state homogeneous-nucleation time tsn (ΔGnm = 0) necessary to observe a first crystallization would be equal to 1024 seconds with a sample volume v =10-24 m3. The energy barrier ΔGnm of intrinsic nuclei is calculated using (2); the radius Rnm of intrinsic nuclei is temperature-independent down to the growth temperature; εnm (θ) is assumed to vary as εls (θ) given by (3) with εls0 replaced by εnm0 which is calculated using (14-16). A unique nucleus in a volume vs = 10-24 m3 still acts as growth nucleus and produces a nano-

The Bi2212 melt contains more than 1024 nuclei per m3 in full agreement with nanocrystallization experiments [61]. The values of lnA are equal to 87 ± 2 as already shown for glass-forming melts having about the same Tg/Tm and Vm. The total nucleation time t is calculated by adding tsn and τns with (18). The shortest nucleation time is equal to 584, 79 and 11 s for lnA = 85, 87 and 89 respectively. The nanocrystallization has been observed at T = 773 K after 600 s in perfect agreement with the curve lnA = 85. The model developed above has already predicted the time-temperature transformation diagram of several glass-

Similar curves are obtained using ΔHm = 4600 J and are not represented.

crystallization process of the undercooled melt as shown in Figure 13.

forming melts [47].

with the previous solidification of non-melted crystals [8-10,44,62]].

Magnetic texturing experiments using pre-reacted powders of Bi2212, Bi2223 or Y123 show that crystal alignment in magnetic field is successful when the annealing temperature is slightly superior to the critical temperature Tm. This temperature corresponds to the liquidus temperature Tm above which there is no solid structure except isolated growth nuclei imbedded in the melt having radii smaller than a critical value. The growth around nuclei occurs in a window of solidification extending from Tm and leads to crystals which become free to align in the magnetic field when their anisotropy energy becomes larger than kBT. The thermal cycle has to use an annealing temperature a little larger than Tm in order to be sure that crystallisation will start from surviving nuclei of superconducting phases and not from polycrystalline entities. The overheating temperature has to be limited to a few degrees Celsius up to a few tens of degrees above Tm depending on the amplitude of the magnetic field. This will insure that the number of nuclei per volume unit remains important and that the composition of nuclei is not progressively transformed into non-superconducting secondary phase nuclei accompanying the oxygen loss.

Short and long lengths of Bi2212 conductors obtained from pre-reacted powders of Bi2212 were textured using the same thermal cycle with a maximal annealing temperature a little larger than Tm and a relatively small annealing time. This thermal cycle is to be compared to the usual long annealing time imposed at lower temperatures in order to form the superconducting phase from non-reacted precursors. This process is successful even without magnetic field because the melt already contains surviving nuclei of this phase acting as growth nuclei inside the silver-sheathed filaments having a section sufficiently small to accommodate the platelets obtained after growth. The alignment of these platelets and consequently the superconducting critical current are improved by using a dynamic process reproducing the thermal cycle presented above where the wire moves through the magnetic field.

Magnetic Texturing of High-Tc Superconductors 193

The authors thank the Institute of Physics Publishing, IOP publishing, Taylor and Francis Group Journals, Elsevier for giving the authorization to reproduce Figure 1 and 2 as published in [16], Figure 4 as published in [23], Figure 6 as published in [26], Figure 7 as published in [34], Figure 8 and 9 as published in [35] and Figure 10 as published in [38].

[1] Beaugnon E, Bourgault D, Braithwaite D, de Rango P, Perrier de la Bâthie R, Sulpice et al. Material processing in high static magnetic field: a review of an experimental study of levitation, phase separation, convection and texturation. J de Physique,1993; 3:399-

[2] Yamaguchi M, Tanimoto Y. Magneto-Science: Magnetic Field Effects on Materials: Fundamentals and Applications. Ed: Springer Series in Materials Science vol 89, Tokyo:

[3] Tournier R. Method for preparing an oriented and textured magnetic material, U.S.A.

[5] Tournier, RF. Presence of intrinsic growth nuclei in overheated and undercooled liquid

[6] Tournier RF. Tiny crystals surviving above the melting temperature and acting as growth nuclei of the high –Tc superconductor microstructure. Mat. Sc. Forum, 2007;

[7] Tournier R F, Beaugnon E. Texturing by cooling a metallic melt in a magnetic field. Sci.

[8] Turnbull D. Kinetics of solidification of supercooled liquid mercury droplets. J. Chem.

[12] Mikelson AE, Karklin YK. Control of crystallization processes by means of magnetic

[13] Ferreira JM, Maple M B, Zhou H, Hake RR, Lee BW, Seaman CL, Kuric M V et al. Magnetic field alignment of high-Tc Superconductors RBa2Cu3O7-δ. Appl. Phys. A ,

[14] Lees MR, de Rango P, Bourgault D, Barbut JM, Braithwaite D, Lejay P et al. Bulk textured rare-earth-Ba2Cu3O7-δ prepared by solidification in a magnetic field.

[9] Kelton K F. Crystal nucleation in liquids and glasses, Solid State Phys. 1995; 45:75 -177. [10] Vinet B, Magnusson L, Fredriksson H, Desré PJ. Correlations between surface and interface energies with respect to crystal nucleation. J. Coll.Interf. Sc. 2002; 255:363-374. [11] Tong HY, Shi FG. Abrupt discontinuous relationships between supercooling and melt

[4] Tournier R, Crystal making method. U.S.A. patent, 1993; june 8, N° 5,217,944.

Technol. Adv. Mater. 2009; 10:014501 (10pp), available from: URL:

**Acknowledgments** 

**5. References** 

421.

Kodansha. 2006.

546-549:1827-40.

Phys.1952; 20: 411-24.

1988; 47:105-10.

Patent, 1992; Dec. 1, N° 5,168,096.

elements. Physica B, 2007; 392:79-91.

http://stacks.iop.org/1468-6996/10/014501.

overheating. Appl. Phys. Lett. 1997; 70:841-43

fields. J. Cryst. Growth, 1981; 52:524-29

Supercond. Sci. Technol.1992; 5:362-7.

We have used a model which predicts glass-forming melt properties without using adjustable parameters and only knowing the fusion heat of a bulk multicomponent material, its molar volume, its melting temperature and its vitreous transition temperature Tg. The Bi2212 and Bi2223 ceramics are known to lead to a vitreous state after quenching. The knowledge of Tg leads to the homogeneous nucleation temperature of crystal and to the energy saving associated with the complementary Laplace pressure acting on a critical radius crystal which is associated with the Fermi energies equalization of this crystal and melt at all temperatures T larger than Tg. The melting temperature of each nucleus above Tm depends on its radius which is smaller than its critical value at Tm. A simple method of quantification is used to determine the complementary Laplace pressure depending on the crystal radius. This energy saving of surviving crystals is calculated assuming that the equalization of Fermi energies between crystal and melt virtually occurs by free electron transfer from crystal to melt. A bound state of a s-state electron would be produced in the electrostatic spherical potential of radius Rnm; the energy saving depending on Rnm would be equal to its first energy level. This energy is constant above Tm at constant radius and decreases with θ2 = (T-Tm)2/Tm2 below Tm. The transient and steady-state nucleation times of the Bi2212 nano-crystallized state are predicted in agreement with the observed values. There is a limitation to the model since we consider all atoms to be identical. Consequently, it cannot describe the behaviour at too high temperatures of clusters which have a composition strongly different from that of the melt. The model predicts that the highest melting temperature of surviving crystals occurs at θ = 0.93. This value is too high because the vitreous state is obtained by quenching the melt from θ ≅ 0.3. In addition, Bi2212 and, Bi2223 nuclei are rapidly transformed in secondary phase clusters which govern the nanocrystallization of the melt. We did not find any publication about the existence of a vitreous state of YBCO. This could be due to a much smaller viscosity of this melt. Nevertheless, this material can be magnetically textured due to the presence of intrinsic clusters above Tm. Its unknown energy saving coefficient εls0 could be smaller than 1 and could not be determined using equations only governing fragile glass-forming melts.

### **Author details**

Laureline Porcar, Patricia de Rango, Daniel Bourgault and Robert Tournier *Institut Néel/CRETA/CNRS/ University Joseph Fourier/Grenoble/France* 
