Parinov I.A. Microstructure and Properties of High-Temperature Superconductors

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6.2 Modeling of Preparation Processes for BSCCO/Ag Tapes 299

–5

–10

300 K[1] 300 K[3] 77 K[3]

Width (mm)

(MPa)

0.0 0.3 0.5 0.8 1.0 1.3

Fig. 6.18. FEM, showing the variation of σ

with thermal cycling across the width

of BSCCO core. Cycles are denoted by the numbers in square brackets [776]

where E is Young’s modulus, ν is Poisson’s ratio, K

is the slope of the

stress–strain curve after yielding and σ

is the yield stress. Figure 6.18 shows

the variation of σ

with thermal cycling across the width of the tape. The

stress state is fairly uniform in the center of the core except at the edges,

where the curvature of the core can act as a stress concentrator.

Modeling of YBCO Oxide Superconductors

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid

7.1.1 Heterogeneous Mechanism

Initially, for peritectic reaction (basis of melt processing), heterogeneous mech-

anism has been suggested, that is, a formation of 123 phase in direct contact

of melt and properitectic (“green”) 211 phase followed by the growth of 123

crystallites in account of the component diﬀusion through product layer [489].

Shortcoming of this mechanism results in the following causes:

(1) Calculated growth rate of 123 phase, limited by diﬀusion of slowest compo-

nent (D

3+ = 10

−11

/s), should be approximately 4 μm/h [54], which

is some order of magnitude less than test data [54, 890, 891]. In this case,

the peritectic reaction, which occurs in diﬀusion mechanism, should be

completed after formation of 123 layer around 211 particles.

(2) It is impossible to describe YBCO platelet structure, because diﬀusion of

components should mostly impede in the direction of dominant growth,

that is, in ab-plane.

(3) Parallel disposition of platelets is not explained, because arbitrary initial

orientation of particles of the 211 properitectic phase into liquid should

lead to the absence of anisotropy and to arbitrary orientation of 123 crys-

tallites, presented in Fig. 7.1a.

7.1.2 Models Based on Yttrium Diﬀusion in Liquid

In realization of melt-processing for preparation of YBCO large-granular sam-

ples with improved superconducting properties, it is important a priori to

obtain a grain growth in the direction of the gradient to maximize the ratio

of “external” thermal gradient (G) to solidiﬁcation rate (R). From this point

of view, it seems better to transfer the sample across a large thermal gradi-

ent. However, in this case a large part of the sample stays for a long time at

elevated temperature, where Y

BaCuO

,BaCuO

and CuO

co-exist. More-

over, an undesirable change of the geometric shape of the sample is possible

302 7 Modeling of YBCO Oxide Superconductors

by liquid ﬂow. Thus, a good compromise between G and R is required, be-

cause the G/R ratio governs the stability of the solidiﬁcation process. The

thermodynamic aspects of this problem considered in [155, 475] have led to a

model of 211 phase solubility in liquid and homogeneous formation of nuclei

(see Fig. 7.1b). This model is based on the following assumptions:

(1) Ion concentration of Y

attains a maximum in liquid near the surface

of 211 particle and decreases sharply at the mid-distance of neighbor 211

inclusions. When the distance exceeds this length, the ion concentration

of Y

decreases asymptotically down to a value corresponding to the

peritectic melt.

(2) Increasing of yttrium concentration in the liquid enhances a possibility of

123 phase solidiﬁcation because of metastability of the melt and existence

of ﬁgurative point in the two-phase region “123 solid phase–melt.”

(3) Constitutional undercooling in the system, leading to supersaturation of

the melt by yttrium, is found (i) by “external” thermal gradient, (ii) by

undercooling in comparison with saturated melt and (iii) by decreasing

of solidiﬁcation temperature for melt, saturated by yttrium relatively to

peritectic melt.

211

Liquid

(a)

Liquid

211

(b)

Fig. 7.1. Models of heterogeneous formation of nuclei at 211 particles [489] (a)and

of solubility of 211 phase into liquid, assisted by homogeneous formation of nuclei

(b) [155, 475]

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 303

The theoretical analysis leads us to introduce a criterion connected to

conception of undercooling and to compare the G/R ratio with the relation,

−C

)/D

[739], where C

) is the ion concentration of Y

in the

solid (liquid) phase at the interface level; D

is the diﬀusion coeﬃcient of the

ion of Y

and m

is the slope of the liquidus curve in the temperature–

composition phase diagram. Depending on the various situations presented in

Fig. 7.2 two cases are possible:

(1) If G/R ratio is high, that is, G is high, the slope of the curve giving the

temperature at the interface level is high, whatever the position of this

interface. The comparative behavior of the temperature at the interface

level and the temperature of the liquidus is given in Fig. 7.2c. The equation

G/R ≥ m

− C

)/D

(7.1)

Temperature

Distance

actual

(c)

liquidus

∗

(a)

∗

Liquid

Solid

Temperature

Distance

∗

Temperature

Distance

liquidus

Constitutionally

Supercooled

Region

(d)

actual

∗

(b)

Y-Concentration

Distance

∞

Solute Enriched

Layer in Front of

Liquid-Solid

Interface

∗

Fig. 7.2. (a) Temperature and (b) concentration proﬁles, showing the constitutional

undercooling; (c) conditions for a plane front; (d) unstable case, where imposed

temperatures are lower than the temperature of the liquidus [286]. T

∗

is the initial

interface temperature

304 7 Modeling of YBCO Oxide Superconductors

is continuously satisﬁed. The interface temperature evolution is stable in

time, and the solidiﬁcation front is ﬂat because the interface temperature

follows the temperature of the liquidus.

(2) If the G/R ratio is relatively small, inequality (7.1) is not satisﬁed, causing

instability of the interface temperature (see Fig. 7.2d), because it should

be continuously re-adjusted to that of the liquidus. Therefore, the solidi-

ﬁcation front is not ﬂat, but has cellular (dendritic) shape or consists of

equiaxed blocky (see Fig. 7.3), because the interface temperature ﬂuctu-

ates between the two curves, leading alternatively to a liquid or a solid

phase.

The above results have been used to explain directed solidiﬁcation of

YBCO after peritectic reaction [890].

The diﬀusion of yttrium is necessary to re-compose 123 phase in the liq-

uid, that is, for a non-classical peritectic reaction, which is governed by dif-

fusion through melt and additionally demands to attain triple point between

primary, secondary and liquid phases [493]. The driving forces inside the dif-

fusion zone, leading to a migration of yttrium, correspond to concentration

gradients. Two phenomena, presented in Fig. 7.4, have been established for

these concentration gradients [156, 741]: (i) the change of the chemical po-

tential at the “211 phase-liquid” interface caused by the curvature of the 211

particles and (i) constitutional undercooling (ΔT

), which corresponds to the

temperature diﬀerence between the actual temperature of the solidiﬁcation

interface and the temperature of the “211 phase-liquid” interface.

The ﬁrst phenomenon explains a change of 211 particles during melt trans-

formation in 123 crystal. The change of chemical potential, caused by the

Growth Rate, R [

µm/s]

Temperature Gradient, lg (G[K/m])

123

123 123 123

Faceted

Plane

Front

Cellular

(Dendritic)

Solidification

Equaxed

Blocky

02 8

Fig. 7.3. Eﬀects of temperature gradient and growth rate on the morphology of the

solidiﬁcation interface [155]

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 305

r = rr = ∞

L, r = ∞

≡ C

ΔT

211

ΔT

ΔC

123

Y-Concentration

Temperature

211

123

L, 211

211

123

211

123

Fig. 7.4. Schematic phase diagram, showing the undercooling and the concentration

diﬀerences as the driving force for the diﬀusion of yttrium during the peritectic

re-composition of 123 phase [759]

curvature of the 211 particles, relates to the undercooling, ΔT

, derived from

the well-known Gibbs–Thomson relationship:

ΔT

=2Γ

211

/r , (7.2)

where Γ

211

is Gibbs–Thomson coeﬃcient (= σ/ΔS), σ is the interface energy

between the 211 particle with radius, r, and the liquid, ΔS is the volumetric

entropy of melt. Then, the composition diﬀerence of yttrium, ΔC

,whichis

considered to be a driving force for diﬀusion, is expressed using the phase

diagram (see Fig. 7.4), as

ΔC

=ΔC

= C

211

L(r)

− C

211

L(r=∞)

2Γ

211

, (7.3)

where C

211

L(r)

and C

211

L(r=∞)

are the yttrium concentrations of liquid at the 211

particle and at the 123 interface, respectively; m

211

is the liquidus slope of

the 211 phase.

The constitutional undercooling is stated using the second composition

change ΔC

, given by extending towards low temperatures the “211 phase-

liquid” liquidus line [476]:

ΔC

=ΔC



123

−

211



ΔT

. (7.4)

306 7 Modeling of YBCO Oxide Superconductors

Introduction of a third parameter, ΔC

, taking into account the imposed

gradient, completes this model [476]. This term is directly given by the dis-

tance, z, between the solidiﬁcation interface and the considered 211 particle:

ΔC

=ΔC

123

, (7.5)

where m

123

is the liquidus slope of the 123 phase. Thus, total diﬀerence of the

yttrium concentration, governing the solidiﬁcation process of the 123 phase is

equal to

ΔC =ΔC

+ΔC

. (7.6)

The next development of the model is connected with the assumption of

the 211 particles to be spheres and consideration of the concentration dif-

ference in yttrium between the 211 phase and the solidiﬁcation front as the

driving force for the diﬀusion of yttrium in the liquid phase (see Fig. 7.5) [476].

Assume that the distance between 211 particles is equal to 2δ,andthe

211 phase remains in ﬁnal structure. The concentration diﬀerences of yttrium

near solidiﬁcation front of 123 phase due to solution of 211 particles are shown

in Fig. 7.5b. Moreover, we assume that the 211 particles are in interfacial

equilibrium with their surrounding liquid and are large enough that the eﬀect

of radius of curvature on melting point is negligible.

The phase diagram in

the peritectic region, presented in Fig. 7.4, shows that the temperature of

the growing 123 front is undercooled by an amount ΔT

below the peritectic

temperature, T

,where

ΔT

=ΔT

+ΔT

, (7.7)

where ΔT

is the depression of the integrated temperature, resulting from the

temperature gradient, G;ΔT

is the maximum constitutional supercooling

ahead of the interface (i.e., temperature diﬀerence between the equilibrium

liquidus and the actual temperature at x = δ)andΔT

is the temperature

depression, resulting from the deviation in yttrium concentration at the 211

particle interface from that of the peritectic liquid composition, C

.From

Fig. 7.5b

ΔT

= Gδ . (7.8)

The yttrium concentration in equilibrium with the 123 interface (at x =0,

T = T

123

)isC

123

, and that at x = δ and T = T

123

+ΔT

in equilibrium

with the 211 particles is C

L,211

. The dependence of the yttrium distribution

for x>δis given by the equilibrium 211 liquidus. Equating yttrium solvent

rejected from the growing 123 front to that diﬀusing into the liquid gives

If the particles are suﬃciently small, this radius of curvature eﬀect could be a

signiﬁcant, even a major, deﬁning driving force of mass transport, leading to 123

phase solidiﬁcation [477].

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 307

123 Phase Liquid + 211 Phase

(a)

123 211

Liquid

Distance

Y-Concentration

ΔC

123

211

123

(b)

2δ

Fig. 7.5. (a) Model of peritectic solidiﬁcation and (b) yttrium concentration proﬁle

near solidiﬁcation front of 123 phase due to the dissolution of 211 particles. J is the

ﬂux of yttrium ions, which is necessary for growth of the 123 phase

−R(C

123

− C

123

)=−D



L,211

− C

123



, (7.9)

where R is the growth rate of the solidiﬁcation front. A relation between

interface undercooling (ΔT

) and growth rate, R, can be derived, assuming

123

−C

123

≈ C

123

−C

. Moreover, linear liquidus lines, shown in Fig. 7.4,

result in C

123

= C

− ΔT

123

,andC

L,211

= C

− (ΔT

− ΔT

)/m

211

Substitution of these expressions in (7.9) and making use of (7.8) gives

ΔT



Rδ

123

− C

) −

Gδ

211



123

211

− m

123

. (7.10)

308 7 Modeling of YBCO Oxide Superconductors

Thus, ΔT

is linearly proportional to R,whenG is suﬃciently small.

Similarly, the next relation can be written between ΔT

and R:

ΔT

= m

123

Rδ

123

− C

) − Gδ . (7.11)

The quantity ΔT

is the constitutional supercooling at x = δ, and it is the

maximum constitutional supercooling in the semisolid region. In solidiﬁcation

of this type, there must always be ﬁnite constitutional supercooling in front

of the growing interface in order to create the compositional driving force for

diﬀusion from the particle surface to the crystal interface. Assuming an exis-

tence of maximum undercooling [155] and noting that the distance between

the solidiﬁcation front and considered 211 particle, z = δ,weobtainfrom

(7.11) a relation deﬁning the maximum growth rate, R

max

, compatible with

a plane solidiﬁcation front:

max

z(C

123

− C

)



(ΔT

)

max

+ Gz

123



. (7.12)

Thus, the highest solidiﬁcation rates compatible with a steady-state growth

are obtained theoretically at the following conditions: (i) with larger under-

cooling, (ii) with higher temperature gradient and (iii) with smaller size of

211 particles after peritectic decomposition (z is directly dependent on the

particle size). Equation (7.12) shows that the maximum solidiﬁcation rate is

inversely proportional to the distance z between the 211 particles, resulting

from the peritectic decomposition of 123 phase. This suggests a decrease in

the 211 particle size and the addition of very ﬁne 211 powder to the starting

123 precursor, because an increase of the volume ratio of the 211 phase in

the melt for a given particle size results in a reduction of the interparticle

spacing z. In fact, such an addition of 211 to 123 oﬀers multiple advantages:

– It allows the geometric shape of the sample to be kept at high temperature

owing to the formation of a solid skeleton.

– Diﬀusion of yttrium favors the supply of this species at solidiﬁcation.

– It compensates the loss of yttrium in the recombination process, where

entrapment of 211 particles takes place.

– Addition of properitectic 211 dispersion reﬁnes the grain size of the 211

precipitates, coming from the peritectic decomposition of the 123 phase

(properitectic 211 particles act as nucleation centers).

– It improves the mechanical properties of the textured 123 sample.

– The 211 inclusions, in excess trapped in the textured 123 material, act as

pinning centers, increasing the critical current.

Compare the terms (ΔT

)

max

and Gz, taking into account that in a clas-

sical furnace z ≈ 1 μmandG<100 K/cm [190]. Then, it is obvious that

Gz << (ΔT

)

max

and the solidiﬁcation rate appears practically insensitive

to the temperature gradient and is proportional to the undercooling, ΔT

It should be noted that the case of YBCO system, where the slopes of the

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 309

liquidus curves are quite diﬀerent for 211 and 123 phases, leading to relatively

large driving forces, appears to be favorable for sample texturing.

The directed observations of 123 phase solidiﬁcation using an IR camera

have found a rate of solidiﬁcation close to 10

−7

m/s and estimated the diﬀusion

coeﬃcient of yttrium in the liquid by the value 6 × 10

−11

/s [310]. From a

practical point of view this solidiﬁcation rate appears to be quite low. From

(7.12), it is governed by two parameters: ΔT

and z. During a slow cooling

rate, r, in a thermal gradient, G, because of the translation, x, of the “liquid-

solid” interface in this gradient the undercooling is given vs time as [190]

ΔT

=ΔT

+ rt − Gx . (7.13)

This equation shows that solidiﬁcation at a constant temperature (r =0)is

possible but is limited by the thermal gradient. In this case, only a large ini-

tial undercooling, ΔT

, can lead to relatively large superconducting crystals

(monodomains).

7.1.3 Models Based on Interface Phenomena

The models [156, 475, 741] considered in the previous section seem in good

agreement with test results, but they do not take into account the interface ki-

netics processes, assuming the crystallization of a pure 123 phase even though

some 211 inclusions are always entrapped in the textured 123 material. More-

over, addition of properitectic 211 dispersion to 123 phase can provoke yttrium

supersaturation at the solidiﬁcation interface. Then, the diﬀusion rate of yt-

trium is rapid compared with the propagation of the solidiﬁcation front, so

the diﬀusion of yttrium is no longer the limiting factor for the growth of 123

phase. Therefore, it is followed to introduce in consideration interface kinetics

phenomena. The supersaturation can be presented as

σ =

− C

123

, (7.14)

where C

is the yttrium concentration at the “123 phase-liquid” interface

and C

123

is the yttrium concentration at equilibrium. Three cases can be

distinguished (see Fig. 7.6): (i) the growth rate of 123 phase is governed by

the yttrium diﬀusion (curve c); (ii) the growth rate of 123 phase is under

mixed control conditions (curve b) and (iii) the growth rate of 123 phase is

governed by the interface reaction (curve a).

A model taking into account the interface phenomena [759] introduces two

rates of solidiﬁcation (in the ab-plane and in the c-axis direction): R

= k

and R

= k

σ,wherek

and k

are the kinetic coeﬃcients for the a-(or

b-) and c-directions, respectively. The eﬀect of the kinetic coeﬃcient on the

growth rate as a function of the undercooling can be taken into account, us-

ing either square power law dependence or linear one on the supersaturation

[759]. As followed in Fig. 7.7, a linear dependence vs supersaturation leads to