**3. Magnetic texturing induced by intrinsic growth nuclei surviving above the melting temperature**

### **3.1. Intrinsic nuclei surviving above the melting temperature**

182 Superconductors – Materials, Properties and Applications

J. Noudem demonstrates the possibility to form the Bi2Sr2Ca2Cu3O8+x **(**Bi2223) phase after a liquid transformation above the melting temperature [36 - 38]. As seen previously in the Bi2212 compounds, the phase diagram is complex which requires a precise control of the

**Figure 10.** Critical current density in Bi2223 textured bulk compounds measured at 77K as a function of

840 850 860 870 880 890 900 910

**T(°C)**

The texture along the c-axis is induced by means of a magnetic field. Pellets of Bi2223 were first uniaxialy pressed before annealing. The heating cycle took place in a vertical furnace placed in a superconducting magnet reaching 8T. The optimum maximum temperature lies within the range 855° and 900°C and is followed by a slow decrease in temperature of 2°C/h. A maximal value of critical current density of 1450 A/cm2 was obtained for a temperature reaching 860°C during 1 hour. Below this value, the proportion of liquid phase is too small to allow rotation of the platelets under the influence of the magnetic field and the critical current consequently sharply decreases. Above this optimum processing temperature, the transformation of Bi2223 nuclei in secondary phases (Ca/Sr)14Cu24Oy, CuO and (Ca/Sr)2CuO3 prevent the recombination of Bi2223. Thus, the critical current density gradually decreased until zero around 900°C as shown in Figure 10 in agreement with the phase diagram of

The possibility to obtain high critical current and a good homogeneity in Ag-sheathed Bi2223 tapes could follow accordingly to the previous results However, the complexity of

**2.3. Case of Bi2Sr2Ca2Cu3O8+x** 

*2.3.1. Pellets of Bi2Sr2Ca2Cu3O8+x*

temperature and composition [23].

the maximal annealing temperature [38].

0

500

1000

**Jc(A/cm2**

**)**

1500

*2.3.2. Ag-sheated tapes of Bi2Sr2Ca2Cu3O8+x*

Figure 4.

Undercooling temperatures of liquid alloys depend on the overheating rate above the melting temperature Tm of liquid alloys and elements [41-43]. The experiments presented in section 2 lead to the conclusion that intrinsic nuclei exist above Tm and are aligned in magnetic field during the growth time evolved in the solidification window between liquidus and solidus temperatures [7]. These results are in contradiction with the idea [8-10] that all crystals have to disappear at the melting temperature Tm by surface melting. The classical equation of Gibbs free energy change associated with crystal formation also predicts that the critical radius for crystal growth becomes infinite at Tm and all crystals are expected to melt [8-10]. It has been recently proposed to add an energy saving εv produced by the equalization of Fermi energies of out-of-equilibrium crystals and their melt in this equation [5,6].

Transformation of liquid-solid always induces changes of the conduction electron number per volume unit, and sometimes per atom. The equalization of Fermi energies of a spherical particle having a radius R smaller than a critical value R\*2ls (θ) produces an unknown energy saving −εv per volume unit. The εv value is equal to a fraction εls of the fusion heat ΔHm per molar volume Vm. This energy was included in the classical Gibbs free energy change Δ θ G ( ,R) 1ls associated with crystal formation in metallic melts [5,6,10]. The modified free energy change is ΔG2ls(R,θ) given by (2) and ΔG1ls(θ,R) is obtained with εls = 0:

$$
\Delta \mathbf{G}\_{2\text{is}}(\mathbf{R}, \boldsymbol{\Theta}) = \frac{\Delta \mathbf{H}\_{\text{m}}}{\mathbf{V}\_{\text{m}}} (\boldsymbol{\Theta} - \boldsymbol{\varepsilon}\_{\text{ls}}) 4\pi \frac{\mathbf{R}^3}{3} + 4\pi \mathbf{R}^2 \frac{\Delta \mathbf{H}\_{\text{m}}}{\mathbf{V}\_{\text{m}}} (1 + \boldsymbol{\varepsilon}\_{\text{ls}}) (\frac{12 \mathbf{k}\_{\text{g}} \mathbf{V}\_{\text{m}} \ln \mathbf{K}\_{\text{ls}}}{432\pi \times \Delta \mathbf{S}\_{\text{m}}})^{1/3} \tag{2}
$$

where ΔSm is the molar fusion entropy, θ = (T-Tm)/Tm the reduced temperature, R the growth nucleus radius, εls×ΔHm/Vm the energy saving εv, NA the Avogadro number, kB the Boltzmann constant. The lnKls given in (3) depends on the viscosity through the Vogel-Fulcher-Tammann (VFT) temperature T0m of the glass-forming melt which corresponds to the free volume disappearance temperature [44,45]:

$$\ln\text{ln}(\mathbf{K}\_{\mathbb{I}\_{\mathbb{K}^d}}) = \ln(\frac{\mathbf{A}\mathbf{\eta}\_0}{\eta}) = (\ln\text{A}) \pm 2 - \frac{\mathbf{B}}{(\mathbf{T} - \mathbf{T}\_{0\mathbf{m}})} \tag{3}$$

Magnetic Texturing of High-Tc Superconductors 185

The equalization of Fermi energies of out-of-equilibrium crystals and melts does not produce any charge screening induced by a transfer of electrons from the crystal to the melt as assumed in the past [5,6,47]. It is realized by a Laplace pressure p given by (8) [56,57] depending on the critical radius R\*2ls and the surface energy σ defined by the coefficient of

> m \* ls 2ls m 2 H <sup>p</sup> [ ( )] <sup>R</sup> <sup>V</sup>

> > \*

\*

The free-volume disappearance reduced temperature θ0m of the melt above T\*g which is

2 2 0m ls0 ls0 8 4 9 9

These formulae are applied to Bi2212 with θ\*g = −0.42 (T\*g = 676 K); T\*g is assumed to be 50 K below the first-crystallization temperature observed at 723 K [58]. Using (10), we find εls0 = 1.58 and T0m = 532 K with a liquidus temperature of 1165 K [59]. The lnKls is equal to 76.4 with lnA = 85 and B/(Tm-T0m) = 8.6. The fusion heat ΔHm is chosen to be equal to the mean value of 11 measurements depending on composition minus the heat produced by an oxygen loss of 1/3×mole [22,25,60]. We obtain ΔHm = 7100 -2000 = 5100 J/at.g, a molar volume Vm = 144\*10-6 m3 and a fusion entropy 4.38 J/K/at.g. Another evaluation of ΔHm leads to 4600 J using a measured oxygen loss of 2.1% of the sample mass in a magnetic field gradient maximum at the liquidus temperature T = 1165 K and a measured heat of 8000 J/at.g for a composition 2212 [60]. The critical radius R\*2ls (θ = 0) in the two cases are respectively equal to 8.94×10-10 and 9.27×10-10 m; these critical nuclei contain 188 and 209 atoms (Bi, Sr, Ca, Cu and O). Surviving nuclei have a smaller radius and contain less atoms. The energy saving coefficient εnm0 of non-melted crystals has to be determined in order to explain why growth nuclei are of the order of 1024-1025 per m3 as shown by nano-crystallization of glass-forming melts and are able to induce solidification at Tm when the applied overheating rate remains

lg 0 lg <sup>g</sup> ε =ε θ= = θ + ( 0) 1.5 \* 2 (9)

ls0 ls <sup>g</sup> ε =ε θ= =θ + ( 0) 2 (10)

θ =ε−ε (11)

The energy saving coefficient εls and consequently p can be determined from the knowledge of the vitreous transition T\*g in metallic and non-metallic glass-forming melts [56,57]. The liquidglass transformation is accompanied by a weakening of the free volume disappearance temperatures from θ0m to θog, of energy saving coefficients at Tm from εls0 to εlg0 and consequently of VFT temperatures from T0m to T0g [56]. The vitreous transition T\*g occurs at a crystal homogeneous nucleation temperature T2lg giving rise, in a first step, to vitreous clusters during a transient nucleation time equal to the glass relaxation time. The energy saving

coefficients εls0 and εlg0 are determined by θ\*g respectively given by (9) and (10):

equal to its VFT temperature [45] is given by (11)]:

× σ <sup>Δ</sup> = = θ−ε θ (8)

4πR2 in (2):

weak [47,61].

where lnA is equal to 87±2, B/(Tg-T0m) = 36-39, Tg being the vitreous transition temperature [46,47]. The critical radius and the critical energy barrier are respectively given versus temperature by (4) and (5) assuming that a minimum value of εv exists at each temperature T ≤ Tm and εls = 0 for R > R\*2ls:

$$\mathbf{R}\_{2\text{ls}}^{\*}\left(\Theta\right) = \frac{-\mathfrak{D}(1+\mathfrak{e}\_{\text{ls}})}{\mathfrak{G}-\mathfrak{e}\_{\text{ls}}} (\frac{\mathrm{V}\_{\text{m}}}{\mathrm{N}\_{\text{A}}})^{1/3} (\frac{12\mathbf{k}\_{\text{B}}\mathbf{N}\_{\text{A}}\ln\mathbf{K}\_{\text{ls}}}{432\pi \times \Delta\mathbf{S}\_{\text{m}}})^{1/3} \tag{4}$$

$$\frac{\Delta \mathbf{G}\_{2\text{ls}}^{\ast}}{\mathbf{k}\_{\text{g}} \, \mathrm{T}} = \frac{12(1 + \mathfrak{e}\_{\text{ls}})^3 \ln \mathbf{K}\_{\text{ls}}}{81(\Theta - \mathfrak{e}\_{\text{ls}})^2 (1 + \Theta)} \tag{5}$$

The critical radius at the melting temperature is no longer infinite at Tm (θ = 0) because εls0 is not nil. Out-of-equilibrium crystals having a radius smaller than R\*2ls are not melted because their surface energy barrier is too high. The thermal variation of εls is an even function of θ given by (6):

$$\mathfrak{e}\_{\rm v} = \mathfrak{e}\_{\rm ls} \frac{\Delta \mathbf{H}\_{\rm m}}{\mathbf{V}\_{\rm m}} = \mathfrak{e}\_{\rm lso} (1 - \frac{\Theta^2}{\Theta\_{0\rm m}^2}) \frac{\Delta \mathbf{H}\_{\rm m}}{\mathbf{V}\_{\rm m}} \tag{6}$$

where θ0m corresponds to the free-volume disappearance temperature T0m [5]. The fusion entropy is always equal to the one of the bulk material regardless of the crystal radius R and of εls0 value as shown in (7) because dεls/dT = 0 at the melting temperature Tm of the crystal :

$$\frac{3}{4\pi\text{R}^3} \left| \frac{\text{d} \left[ \Delta\text{G}\_{2\text{ls}}(\Theta) \right]}{\text{dT}} \right|\_{\text{T}=\text{T}\_{\text{mc}}} = -\frac{\Delta\text{S}\_{\text{m}}}{\text{V}\_{\text{m}}} \tag{7}$$

All crystals outside a melt generally have a fusion temperature depending on their radius; they melt by surface melting. Here, they melt by homogeneous nucleation of liquid droplets. These properties are analogous to that of liquid droplets coated by solid layers that are known to survive above Tm [48,49]. The crystal stability could be enhanced by an interface thickness of several atom layers [50] or these entities could be super-clusters which could contain magic numbers of atoms [51,52]. They could melt by liquid homogeneous nucleation instead of surface melting. The presence of super-clusters in melts being able to act as growth nuclei was recently confirmed by simulation and observation in glass-forming melts and liquid elements [53-55].

The equalization of Fermi energies of out-of-equilibrium crystals and melts does not produce any charge screening induced by a transfer of electrons from the crystal to the melt as assumed in the past [5,6,47]. It is realized by a Laplace pressure p given by (8) [56,57] depending on the critical radius R\*2ls and the surface energy σ defined by the coefficient of 4πR2 in (2):

184 Superconductors – Materials, Properties and Applications

T ≤ Tm and εls = 0 for R > R\*2ls:

and liquid elements [53-55].

given by (6):

the free volume disappearance temperature [44,45]:

2ls

lg s

− +ε θ =

where ΔSm is the molar fusion entropy, θ = (T-Tm)/Tm the reduced temperature, R the growth nucleus radius, εls×ΔHm/Vm the energy saving εv, NA the Avogadro number, kB the Boltzmann constant. The lnKls given in (3) depends on the viscosity through the Vogel-Fulcher-Tammann (VFT) temperature T0m of the glass-forming melt which corresponds to

0m

θ−ε π×Δ (4)

Δ +ε <sup>=</sup> θ−ε +θ (5)

θ (6)

(7)

(3)

0

<sup>A</sup> <sup>B</sup> ln(K ) ln( ) (ln A) 2 (T T ) <sup>η</sup> = = ±−

where lnA is equal to 87±2, B/(Tg-T0m) = 36-39, Tg being the vitreous transition temperature [46,47]. The critical radius and the critical energy barrier are respectively given versus temperature by (4) and (5) assuming that a minimum value of εv exists at each temperature

\* 1 ls m / 3 1 B A ls / 3

2(1 ) 12k N lnK <sup>V</sup> R () ( ) ( ) N 432 S

\* 3 2ls ls ls 2

G 12(1 ) lnK k T 81( ) (1 )

The critical radius at the melting temperature is no longer infinite at Tm (θ = 0) because εls0 is not nil. Out-of-equilibrium crystals having a radius smaller than R\*2ls are not melted because their surface energy barrier is too high. The thermal variation of εls is an even function of θ

B ls

v ls ls0 2

ε =ε =ε −

3

ls A m

2 m m

mc 2ls m

<sup>m</sup> T T

m m 0m H H (1 ) V V Δ Δ θ

where θ0m corresponds to the free-volume disappearance temperature T0m [5]. The fusion entropy is always equal to the one of the bulk material regardless of the crystal radius R and of εls0 value as shown in (7) because dεls/dT = 0 at the melting temperature Tm of the crystal :

> 3 d G () S 4 R dT <sup>V</sup> <sup>=</sup>

All crystals outside a melt generally have a fusion temperature depending on their radius; they melt by surface melting. Here, they melt by homogeneous nucleation of liquid droplets. These properties are analogous to that of liquid droplets coated by solid layers that are known to survive above Tm [48,49]. The crystal stability could be enhanced by an interface thickness of several atom layers [50] or these entities could be super-clusters which could contain magic numbers of atoms [51,52]. They could melt by liquid homogeneous nucleation instead of surface melting. The presence of super-clusters in melts being able to act as growth nuclei was recently confirmed by simulation and observation in glass-forming melts

 Δ θ <sup>Δ</sup> = − <sup>π</sup>

η −

$$\mathbf{p} = \frac{\mathbf{2} \times \boldsymbol{\sigma}}{\mathbf{R}\_{2\text{ks}}^{\*}} = \frac{\Delta \mathbf{H}\_{\text{m}}}{\mathbf{V}\_{\text{m}}} [\boldsymbol{\Theta} - \boldsymbol{\varepsilon}\_{\text{ls}}(\boldsymbol{\Theta})] \tag{8}$$

The energy saving coefficient εls and consequently p can be determined from the knowledge of the vitreous transition T\*g in metallic and non-metallic glass-forming melts [56,57]. The liquidglass transformation is accompanied by a weakening of the free volume disappearance temperatures from θ0m to θog, of energy saving coefficients at Tm from εls0 to εlg0 and consequently of VFT temperatures from T0m to T0g [56]. The vitreous transition T\*g occurs at a crystal homogeneous nucleation temperature T2lg giving rise, in a first step, to vitreous clusters during a transient nucleation time equal to the glass relaxation time. The energy saving coefficients εls0 and εlg0 are determined by θ\*g respectively given by (9) and (10):

$$
\mathfrak{e}\_{\text{lg }0} = \mathfrak{e}\_{\text{lg}}(\Theta = 0) = 1.5 \, ^\ast \Theta\_\text{g} \, ^\ast + 2 \, \tag{9}
$$

$$\mathfrak{e}\_{\rm ls0} = \mathfrak{e}\_{\rm ls}(\Theta = 0) = \mathfrak{e}\_{\rm g}^{\rm \, \, \,} + \mathfrak{2} \tag{10}$$

The free-volume disappearance reduced temperature θ0m of the melt above T\*g which is equal to its VFT temperature [45] is given by (11)]:

$$
\Theta\_{0\text{m}}^2 = \frac{8}{9}\mathfrak{e}\_{\text{ls0}} - \frac{4}{9}\mathfrak{e}\_{\text{ls0}}^2 \tag{11}
$$

These formulae are applied to Bi2212 with θ\*g = −0.42 (T\*g = 676 K); T\*g is assumed to be 50 K below the first-crystallization temperature observed at 723 K [58]. Using (10), we find εls0 = 1.58 and T0m = 532 K with a liquidus temperature of 1165 K [59]. The lnKls is equal to 76.4 with lnA = 85 and B/(Tm-T0m) = 8.6. The fusion heat ΔHm is chosen to be equal to the mean value of 11 measurements depending on composition minus the heat produced by an oxygen loss of 1/3×mole [22,25,60]. We obtain ΔHm = 7100 -2000 = 5100 J/at.g, a molar volume Vm = 144\*10-6 m3 and a fusion entropy 4.38 J/K/at.g. Another evaluation of ΔHm leads to 4600 J using a measured oxygen loss of 2.1% of the sample mass in a magnetic field gradient maximum at the liquidus temperature T = 1165 K and a measured heat of 8000 J/at.g for a composition 2212 [60]. The critical radius R\*2ls (θ = 0) in the two cases are respectively equal to 8.94×10-10 and 9.27×10-10 m; these critical nuclei contain 188 and 209 atoms (Bi, Sr, Ca, Cu and O). Surviving nuclei have a smaller radius and contain less atoms. The energy saving coefficient εnm0 of non-melted crystals has to be determined in order to explain why growth nuclei are of the order of 1024-1025 per m3 as shown by nano-crystallization of glass-forming melts and are able to induce solidification at Tm when the applied overheating rate remains weak [47,61].
