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

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6.1 Transformation of Bi-2212 to Bi-2223 Phase 279

due to the presented model, the ion diﬀusion should be faster in the pore than

that in the bulk. Then, we can estimate the formation energy of Bi-2223 from

Bi-2212 phase.

The intercalation of the Ca/CuO

bi-layer into the Bi-2212 matrix can be

presented as a single jog on the otherwise straight dislocation (see Fig. 6.4a).

The climb motion of the dislocation can be accomplished by jog nucleation and

propagation along the dislocation line. When the reactant ions (Ca

, Cu

and O

2−

) are deposited at the jog, one propagates and the dislocation climbs

upward by a distance, Δx, as shown in Fig. 6.4b. Both these processes require

(a) (b)

Δ x

Bi-2223 Bi-2212

•

001

100

010

(c)

Δ x

Fig. 6.4. Schematic drawing of the dislocation model for intercalation of Ca/CuO

bi-layer into Bi-2212 matrix: (a) The insertion of a Ca/CuO

bi-layer, described as

climb motion of a dislocation in a grain with rectangular platelet shape; (b)jog

conﬁguration of the dislocation, the shading area shows the transition region; (c)

core conﬁguration of the dislocation view along the [010] direction, the channel of

diﬀusion for the reactants is shown [118]

280 6 Modeling of BSCCO Systems and Composites

net mass transport, whose rate is controlled by the diﬀusion of the reactant

ions. These ions, moving to a preexisting jog, simply translate the jog. This

translation does not change core energy and core structure of the dislocation,

as shown in Fig. 6.4c.

Considering the reactant ions as particles, suﬀering random force of their

surroundings, the equation of the jog motion can be written as

ην = F

+ F

, (6.28)

where η is the viscosity of the surrounding matrix, ν is the jog rate, F

is the

force caused by the reduction of chemical energy at Bi-2212/Bi-2223 transfor-

mation and F

is the elastic force suﬀered by the jog.

Assuming the rate of jog motion to be constant, the time t that is necessary

to insert a single Ca/CuO

bi-layer in Bi-2212 matrix of a L

×L

rectangular

sample is found as

t =

Δxv

, (6.29)

andwehavefromFig.6.4c:

| = σ

l, (6.30)

where σ

= c

is the stress, ε

is the strain, c

=14.59 ×10

dyn/cm

is the elastic constant of the Bi-2212 matrix and b

=4.4

A is Burgers vector

of the dislocation [118]. If we regard the intercalation of a single Ca/CuO

bi-layer for each Bi-2212 unit-cell, that is, a local Bi-4435 structure, which

is often observed in the early stage of the transformation, then the strain

=0.142 [66]. Moreover, we take into account that l ≈ Δx ≈ b

[119].

The bulk diﬀusion factor of O

2−

in the ab-plane of Bi-2212 phase at 850

◦

is approximately 1.1 × 10

−9

/s [118]. The diﬀusion constant for the reac-

tant ions in the pore depicted in Fig. 6.4c is at least one order of magnitude

larger than that. Using the relation between the diﬀusion constant, D,and

the viscosity constant, η,

η =

, (6.31)

where k

is Boltzmann constant and T = 850

◦

C is the temperature. Finally,

we estimate the formation energy per Ca/CuO

bi-layer E

from (6.28) and

(6.30) as

= −F

· Δx ≤ 3.5eV. (6.32)

6.1.5 Eﬀect of Deformation on Bi-2212/Bi-2223 Transformation

Obviously, it may be assumed that deformation processes render direct

inﬂuence on Bi-2212/Bi-2223 phase transformation. In the case of fabrication

of the BSCCO tapes by using oxide-powder-in-tube method, the mechanical

deformation eﬀects have been investigated at rolling the wires into tapes with

6.1 Transformation of Bi-2212 to Bi-2223 Phase 281

diﬀerent thickness reduction [625]. Each rolling step produced an approximate

20% thickness reduction. The deformation ratio R is deﬁned as the thickness

reduction of the tape:

R =(t

− t

)/t

, (6.33)

where t

and t

are the original and ﬁnal thicknesses of the superconductor

core, respectively. The obtained test dependencies of Bi-2223 phase concen-

tration on deformation and duration of ﬁnal annealing at T = 833

◦

Care

shown in Fig. 6.5. These results demonstrate that the mechanical deforma-

tion strongly aﬀects the phase transformation of Bi-2212 to Bi-2223 in the

early stages of annealing, implying that diﬀerent mechanisms may dominate

the transformation kinetics above and below deﬁnite value of the deformation

ratio (here 60%). This unusual behavior, demonstrating minimum value of

transformation kinetics at R = 60%, states an existence of two major eﬀects

for BSCCO tapes during the mechanical preparation, namely (i) a part of

deformation energy is absorbed by the oxide core as fracture energy, splitting

the Bi-2212 crystal and producing new interfaces; (ii) a part of the energy is

stored in the crystals as the result of lattice distortion of the Bi-2212 crys-

tals. The ﬁrst eﬀect takes place when the mechanical deformation ratio, R,is

relatively low, while the second eﬀect occurs when higher mechanical defor-

mation is applied to the BSCCO tapes [625]. Both eﬀects have diﬀerent con-

tributions to the Bi-2212/Bi-2223 phase transformation process. Obviously,

the area of the fractured surfaces increases as the mechanical deformation

ratio enhances, increasing the surface energy. This surface energy causes re-

crystallization of the fractured Bi-2212 crystals, reducing the energy of the sys-

tem during ﬁnal annealing. At the same time, the lattice distortion increases

the internal energy and the phase balance, facilitating phase transformation,

forming Bi-2223 instead of the re-crystallized Bi-2212 crystals. Hence, there

are two possible types of annealing behavior for the Bi-2212 phase, either the

1.0

0.8

0.6

0.4

0.2

0.0

45 hours

25 hours

15 hours

Mechanical Strain (%)

Relative Content of

Bi-2223 Phase

30 40 50 60 70 80

Fig. 6.5. Bi-2223 phase content versus the mechanical deformation ratios [625]

282 6 Modeling of BSCCO Systems and Composites

Bi-2212 re-crystallization to minimize the surface area or reaction leading to

the phase balance and forming the Bi-2223 phase. Hence, the annealing be-

havior of the deformed Bi-2212 crystals is determined either by the surface

energy or by the chemical potential and internal energy of the Bi-2212 crys-

tals. When the surface energy is higher than the chemical potential and the

internal energy, re-crystallization of the Bi-2212 crystals occurs. At the same

time, when the chemical potential and internal energy dominate the system,

the chemical reaction takes place. Thus, it could be assumed in this case that

re-crystallization is probably a favorable process at R<60% and the trend

increases with the deformation ratio. The surface area rapidly increased with

the increasing of R, until the surface area reached a “saturation” value. This

slowed down the Bi-2223 formation with increasing mechanical deformation.

However, at R>60%, the lattice distortion of the Bi-2212 crystals increased

rapidly, causing an increase of the internal energy with increasing mechanical

deformation

Figure 6.6 is a schematic illustration, showing that the driving force for

the Bi-2223 phase formation is enhanced by the internal energy. Annealing

the sample at temperatures above the equilibrium temperature of the Bi-2212

and Bi-2223 phases, T

(T>T

), allows the Bi-2223 phase to form because of

the lower free energy. At the same time, the formation of the Bi-2212 phase

occurs at lower annealing temperatures (T<T

). As the internal energy

of the Bi-2212 phase is increased by the mechanical deformation, the free

energy of Bi-2212 phase becomes G



Bi−2212

= G

Bi−2212

+ E

int

. One is pre-

sented by the curve parallel to the original curve, deﬁning free energy of the

Bi-2212 phase (G

Bi−2212

) without the mechanical deformation, but is shifted

ΔG

Bi-2212

= G

Bi-2212

+ E

int

Bi-2212

Bi-2223

’

833°C

Temperature (°C)

Gibbs Free Energy

ΔG’

Fig. 6.6. A schematic illustration shows the mechanism, in that the driving force

of the Bi-2223 phase formation is enhanced by the internal energy [625]

6.2 Modeling of Preparation Processes for BSCCO/Ag Tapes 283

up to G



Bi−2212

(see Fig. 6.6). Obviously, it results in a decrease in the equilib-

rium temperature of the Bi-2212 and Bi-2223 phases (T



). As a result of

the reduced equilibrium temperature, the driving force for the Bi-2223 phase

formation from the overheating (833

◦

C − T



) increases from ΔG to ΔG



,as

shown in Fig. 6.6. In this case, the chemical potential and the internal energy

replace the surface energy as the dominant energies of the system. Therefore,

after annealing, the content of the Bi-2223 phase increased together with the

deformation ratio, when R>60% [625]. Moreover, apart from the pointed

eﬀects of mechanical deformation, there are other parameters deﬁning the

Bi-2212/Bi-2223 phase transformation, namely the eﬀective diﬀusivity, con-

centration gradient and grain size. For example, the ﬁne grain size leads to an

increase in the grain boundary (interface) area. In this case, the eﬀect of the

surface diﬀusion would be stronger than that of the lattice diﬀusion during

the phase transformation and grain growth processes. This implies that the

formation of Bi-2223 phase should increase together with the parameter of R.

In addition, the mechanical deformation might also inﬂuence the number of

nuclei and the nucleation rate. Generally, the nucleation rate is independent

of the nucleation sites. The dependence between the number (or volume) of

nuclei, N , the nucleation rate, v, the area of the nucleation sites, A,andthe

annealing time, t, can be expressed as [625]

N = νAt . (6.34)

Obviously, for a certain nucleation rate, higher mechanical deformation

exposes more surfaces, providing a large amount of Bi-2223 nuclei that facili-

tate the phase transformation from Bi-2212 to Bi-2223. However, the various

mechanical deformation processes may result in diﬀerent grain boundary ar-

eas, deﬁning value of R, corresponding to the minimum formation of Bi-2223

phase. Thus, mechanical deformation causes both the thermodynamics and

kinetics of the BSCCO phase formation.

6.2 Modeling of Preparation Processes

for BSCCO/Ag Tapes

As it has been noted in the previous chapters, HTSC tapes are subjected

to multi-staged and complex thermal, mechanical and magnetic treatments

during preparation. Below, we consider several model and computational ap-

proaches directed to the process optimization of BSCCO/Ag tape fabrication

and to attainment of the improved structure-sensitive and superconducting

properties.

6.2.1 Sample Texturing by External Magnetic Field

When superconductor is placed in a magnetic ﬁeld, the axis of maximum

susceptibility for each grain aligns with the magnetic ﬁeld direction, that is,

284 6 Modeling of BSCCO Systems and Composites

the grains should align with the c-axis parallel to the external magnetic ﬁeld.

For quantitative estimation of the material texturing degree under magnetic

ﬁeld, consider the rotation of superconductor grains into liquid in the early

stages of their growth (i.e., when the particles can rotate without interacting)

[275]. We assume in this model that the magnetic ﬁeld does not inﬂuence on

the processes of grain nucleation and growth into liquid.

Let an anisotropic grain with volume V be placed in a magnetic ﬁeld H .

Then, the change in magnetic energy of the grain with a change in magnetic

ﬁeld can be written as

= −MV dH = −(M

cos θ + M

sin θ)V dH, (6.35)

where M is the magnetic moment per unit volume, which can be resolved in

the two directions c and ab and θ is the angle between the magnetic ﬁeld and

the c-axis of the grain. For HTSC in their normal state, the magnetic moments

and M

are paramagnetic moments. Then, (6.35) can be reduced to the

form:

= −(χ

cos

θ + χ

sin

θ)VHdH, (6.36)

where χ

and χ

are the paramagnetic susceptibilities along the c-axis and

in the ab-plane, respectively. Integrating (6.36), we obtain the expression for

the magnetic energy of a grain:



= −(χ

cos

θ + χ

sin

θ)VH

/2 (6.37)

(θ, H)=−(χ

+Δχ cos

θ)VH

/2 , (6.38)

where Δχ is the diﬀerence in the volume susceptibilities of the grain.

Subsequent to the nucleation event, nuclei will start their growth under

the inﬂuence of a magnetic ﬁeld. In the early stages of growth, the grains

are completely surrounded by a liquid phase and thus they can be regarded

as small particles, rotating in a free medium without intergrain interactions.

Consider the probability, f(θ), that a grain has an orientation with angle θ

under the inﬂuence of a magnetic ﬁeld. It can be expressed, according to a

classic Boltzmann statistics, as

f(θ)dθ =

exp[−E(T,H,θ)/kT ]dθ

π/2

(

exp[−E(T,H,θ)/kT ]dθ

. (6.39)

Then, for total number of grains, n, the mean number of grains with an

orientation between θ and θ +dθ can be given as

n(θ)dθ = nf(θ)dθ = n

exp[−E

(T,H,θ)/kT ]dθ

π/2

(

exp[−E

(T,H,θ)/kT ]dθ

. (6.40)

6.2 Modeling of Preparation Processes for BSCCO/Ag Tapes 285

Hence, the distribution, n(θ), can be related to an alignment parameter,

which is used to quantify the degree of texture in melt-processed supercon-

ductors under the inﬂuence of a magnetic ﬁeld. This alignment parameter, F ,

such that F = 1 for a completely alignment structure and F = 0 for a totally

random structure, can be found as

F =1−

H=0

, (6.41)

where s

is the variance of the grain distribution for a particular processing

condition and s

H=0

is the variance of the distribution in the absence of a mag-

netic ﬁeld. It is known that the anisotropy of molar magnetic susceptibility

Δχ

molar

≈ 22.5 × 10

−5

/mol [53, 1099]. Then, the anisotropy of volume

magnetic susceptibility, Δχ, used in (6.38), is 1.5 × 10

−6

, if we assume a

density 5.5g/cm

[862] for the superconductor. Figure 6.7 shows the depen-

dence of the parameter F on external magnetic ﬁeld for various grain sizes at

the temperature, T = 875

◦

C, which is approximately 5

◦

C smaller than the

melting temperature of Bi-2212 phase.

As it is followed from Fig. 6.7, when the magnetic ﬁeld increases, there

is a trend for the texture to increase, except in cases where the grain size is

small. Therefore, a high degree of alignment can be obtained by increasing

the magnetic ﬁeld and the grain size. In the case of larger grain sizes, the

magnetic ﬁeld tends to saturate and thus increasing the magnetic ﬁeld has

only a negligible eﬀect on the degree of texture.

1.0

0.8

0.6

0.4

0.2

0.0

Magnetic Field (T)

Alignment Parameter, F

0246810

Fig. 6.7. Alignment parameter as a function of magnetic ﬁeld for various grain

sizes. The processing temperature is 875

◦

C [275]

286 6 Modeling of BSCCO Systems and Composites

6.2.2 Deformation at Tape Cooling

Deformations caused by compressing stresses, forming into ceramic core

during sample cooling are deﬁned. Known estimations of mechanical proper-

ties of the silver matrix and oxide superconducting ﬁlaments at all considered

temperature intervals are used.

Young’s modulus. Following [60], we assume a linear dependence of Young’s

modulus on temperature, E

(T ); for BSCCO use data obtained in [321].

Plastic deformation. The linear relation between yielding stress, σ

,and

temperature, suggested in [60], uses for adaptation to the room-temperature

results [394] and to value σ

= 0, corresponding to the melting temperature.

The yielding strain is found from Hooke’s law. The behavior of the pressed

multi-phase ceramic core is suggested to be elastic.

Thermal expansion. The behavior of parameters α

(T )andα

BSCCO

(T )

at low temperatures is stated from results of [781] and is ﬁtted to the value

19 ×10

−6

−1

for Ag and to the room-temperature data for BSCCO [321]. At

more high temperatures, we suggest a linear dependence up to 22 ×10

−6

−1

for Ag and 16.5 × 10

−6

−1

for BSCCO (at 1110 K).

At high temperatures, a diﬀerent thermal compression leads to elastic

deformations, both matrix and core. In this case, the deformations at tem-

perature change on the value of ΔT are estimated, using the law “action–

counteraction” for two-component tape, consisting of ﬁlaments and matrix.

From Hooke’s law,

Δε

BSCCO

(T )=



BSCCO

(T )

(1 − F )E

(T )



−1

(α

− α

BSCCO

)ΔT ; (6.42)

Δε

(T )=



(1 − F )E

(T )

BSCCO

(T )



−1

(α

BSCCO

− α

)ΔT, (6.43)

where F = %BSCCO/(%BSCCO + %Ag) is the ﬁlling factor. The values of

BSCCO

(T )andε

(T ) are deﬁned, integrating (6.42) and (6.43):

BSCCO

(T )=



Δε

BSCCO

ΔT

dt ; ε

(T )=



Δε

ΔT

dt, (6.44)

where T

is the temperature of the tape annealing. A transition from elastic

to plastic regime (i.e., from linear to non-linear dependence) occurs, when

(T ) attains the yielding strain, ε

(T ). Below corresponding transition

temperature, the silver matrix deforms plastically (non-linearly), leading to

a loading of the superconductor by constant stress, σ

[835]. Finally, the

required core deformation is found as

BSCCO

(T )=ε

(T )

(1 − F )E

(T )

BSCCO

(T )

. (6.45)

The obtained numerical results are presented in Fig. 6.8 [835].

6.2 Modeling of Preparation Processes for BSCCO/Ag Tapes 287

0.5

0.4

0.3

0.2

0.1

0.0

10 15 20 25 30 35 40

77 K

293 K

Filling Factor (%)

Deformation (%)

Fig. 6.8. Deformation of BSCCO core as a function of ﬁlling factor for tapes, cooled

down to room temperature and cryogenic temperature (77 K)

6.2.3 Eﬀects of Mechanical Loading

The introduction of “freedom parameter,” Δ

, estimates qualitatively the

inﬂuence of diﬀerent constraint factors on the mass-ﬂow behavior and also

presents the models of re-distribution of mass and ﬂux in the powder compact

due to plastic deformation of composite [385].

Freedom Parameter

Consider the compression of a two-dimensional slab of an ideal plastic solid

between two ﬂat anvils as shown in Fig. 6.9. If there is friction between the

slab and the anvils, “barreling” occurs during the process (see Fig. 6.9b). In

this case, there is inhomogeneous stress–strain state of material. The pres-

sure at the interface between the anvil and the slab needed to induce plastic

deformation is given as [441]

P =2k exp



2μx



, 0 ≤ x ≤

. (6.46)

The mean pressure is

≈ 2k



μL



, (6.47)

288 6 Modeling of BSCCO Systems and Composites

(b)

(a)

Fig. 6.9. Processes of two-dimensional compression: (a) two sides are free, and there

is no friction between the slab and the anvils; (b) two sides are free, but there is

substantial friction between the slab and the anvils

where 2k is the ﬂow yield, coinciding with the mean pressure at μ = 0 and/or

L<<h; μ is the sliding friction factor; h is the sample height and L is the

contact size. If there is sticking friction at the interface, then the pressure

is [441]

P =2k





, 0 ≤ x ≤

, (6.48)

and the mean pressure is

=2k





. (6.49)

The increasing pressure, or “friction hill,” is caused by friction and reaches

a maximum in the center (see Fig. 6.10). The higher the friction, the higher the

pressure needed to cause mass ﬂow. Obviously, the pressure strongly depends

on the ratio between the height and the contact size, h/L.Iftheratioh/L