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

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310 7 Modeling of YBCO Oxide Superconductors

Crystal

Diffusion Layer

Atmosphere

Δμ

Chemical Potential

211

123

Y-Concentration

Growing

Interface

Fig. 7.6. Yttrium concentration proﬁles in front of the growing interface [759] and

diﬀerences in chemical potential in the diﬀusion layer: (a) kinetics governed by inter-

face phenomena, (b) kinetics governed by both diﬀusion and interface phenomena

and (c) kinetics governed by diﬀusion

2.0

1.5

1.0

0.5

0.0

Diffusion Control

–3

–4

–5

–6

ΔT, °C

R (10

–5

/s)

0102030

Fig. 7.7. Eﬀect of kinetic coeﬃcient, k,ongrowthrate,R, as a function of the

undercooling in the case of a linear dependence on the supersaturation [759]

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 311

a quasi-linear dependence of the growth rate vs supercooling. On the contrary,

in the case of a square power law dependence vs supersaturation, two behav-

iors are possible, depending on the k value: for k<10

−4

cm/sthegrowth

rate would show a square power law dependence vs supersaturation, and for

k>10

−3

cm/s the growth rate would demonstrate a linear dependence vs

undercooling, irrespective of the square law dependence of the growth rate vs

supersaturation.

In the case of pure 123 phase, a good agreement with experiment [759]

is observed for R

and R

vs the undercooling. Hence, one can conclude

that growth of 123 phase is generally controlled simultaneously by yttrium

diﬀusion and interface kinetics processes. If a 211 properitectic phase is added

to 123 phase before the melt process, for a given undercooling the larger

amount of 211 phase can supply more yttrium diﬀusion to the growing front,

corresponding to the growth rate of solidiﬁcation process. At the same time,

the change from a quadratic dependence of R

vs supersaturation to a linear

dependence (and vice versa for R

), observed in [759], is indeed unexpected if

one assumes that the yttrium ﬂux reaching the interface is proportional to the

211 phase concentration. Obviously, this means that the growth mechanisms

in the considered directions are diﬀerent for both compositions, in spite of

identical growth conditions. Thus, one can modify the growth rate and its

anisotropy, changing solidiﬁcation process and superconductor composition.

The particle motion near an advancing solidiﬁcation front can be described

in terms of an interfacial energy relationship between a particle, solid and

liquid in the framework of the pushing/trapping mechanism of the particle by

the solidiﬁcation front (see Fig. 7.8) [1092]:

= σ

+ σ

, (7.15)

where σ

, σ

and σ

are particle/solid, particle/liquid and solid/liquid

interfacial free energies, respectively. According to this criterion, a particle

present at a growing solid phase is trapped when σ

<σ

+σ

. In contrast,

211

123 Crystal

Liquid

123 Crystal

Liquid

211

(a) Pushing (b) Trapping

> σ

+ σ

< σ

+ σ

Fig. 7.8. Schematic of interfacial energy criteria on particle pushing and trapping

312 7 Modeling of YBCO Oxide Superconductors

when σ

>σ

+ σ

, a particle is pushed out from a growing solid by the

fast diﬀusion of the liquid.

This interfacial energy criterion has been applied to melt-processed YBCO

samples in order to explain trapping of 211 particles within 123 crystals [475,

532, 534, 540]. However, this is not enough because it assumes the interaction

of the front with only one particle pushed out or trapped by the interface.

Moreover, the above criterion should be modiﬁed for anisotropically growing

crystals of melt-processed YBCO samples. The modiﬁed interfacial energy

relationship is given as [534]

PS(hkl)

= σ

+ σ

S(hkl)L

, (7.16)

where σ

PS(hkl)

, σ

and σ

S(hkl)L

are particle/(hkl) interface, particle/liquid

and (hkl) interface/liquid interfacial free energies, respectively. Then, in the

isotropically growing system or directionally growing interface, 211 particles

will be pushed out or trapped within a 123 domain with a random mode (see

Fig. 7.9). In this case, for an anisotropically growing 123 crystal, trapping

criteria for each growing plane are diﬀerent.

123

Random 211

Trapping

(a) Isotropic Growth

211 + Liquid

(b) Anisotropic Growth

(010)

211 + Liquid

123

211 Segregation

(110)

(100)

Fig. 7.9. Schematic of particle trapping modes in isotropically (a) and anisotropi-

cally (b) growing systems

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 313

The crystal structure of 123 domains at the peritectic temperature has

a tetragonal symmetry. Therefore, there are three main growth interfaces of

(100), (010) and (001) planes (see Fig. 7.10). In the tetragonal crystal struc-

ture, the atomic arrangement of a (100) plane coincides with that of the (010)

plane. But the atomic arrangement of a (001) plane is diﬀerent from that of

the (100)/(010) plane. The boundary condition of particle trapping is inferred

from the interfacial energy diﬀerence among growing (hkl) planes. The criteria

for 211 particle trapping in a melt-processed YBCO system are given as [530]

Δσ

(hkl)

= σ

PS(hkl)

− (σ

+ σ

S(hkl)L

); (7.17)

Δσ

(100)

=Δσ

(010)

=Δσ

(001)

; (7.18)

Δσ

(101)

=Δσ

(011)

=Δσ

(110)

=Δσ

(111)

. (7.19)

From the interfacial energy model and experimental observations of 211

patterns [481, 532, 534, 1105], it is followed that the crystallographic planes of

X-like 211 tracks are the six diagonal planes of 123 domain, which divide the

cubic space into three pairs of pyramids (see Fig. 7.11). These diagonal planes

may vary with the aspect ratio of grown 123 domains. For example, there

may be [110] planes for a cubic domain or [103] planes for an orthorhombic

domain [534] (see Fig. 7.12). In the case of planar segregation of 211 phase,

a pair of pyramids having (001) growth fronts corresponds to 211-free spaces,

while the other two pairs of pyramids having (100) and (010) growth fronts

are ﬁlled with 211 particles.

(010)

(111)

(001)

(011)

(110)

(100)

Fig. 7.10. Boundary planes for 211 particle trapping, based on an interfacial energy

model

314 7 Modeling of YBCO Oxide Superconductors

[100]

[001]

[010]

(001)

(010)

(100)

(a) (b)

(c)

(d)

Fig. 7.11. (a) Three-dimensional demonstration of the planes, where 211 particles

are trapped in a stoichiometric YBCO system, (b) the 211-free spaces and (c, d)

the 211-ﬁlled spaces in a 211 phase excess system

(c)

(a)

(b)

Fig. 7.12. Variation of diagonal planes as a function of grain anisotropy

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 315

The crystallographic alignment of 211 particles may also be explained in

terms of an interfacial energy relationship between the (uvw) plane of a 211

particle, (hkl) plane of interface and liquid [530]. The corresponding criterion

is given as

P(uvw)S(hkl)

= σ

P(uvw)L

+ σ

S(hkl)L

, (7.20)

where σ

P(uvw)

is an interfacial energy of the (uvw) plane of a 211 particle. In

this case, these particles have a polygonal shape with faceted surfaces with

interfacial anisotropy.

An alternative way to express the interfacial energy criteria is the liquid

wetting angle (φ) at advancing 123 interfaces. The trapping criterion of the

211 particle can be expressed with the liquid wetting angle and has been

proposed for a melt-inﬁltrated YBCO system [481]. As schematically shown

in Fig. 7.13, two diﬀerent situations can exist for particle trapping. In the

case of φ =0

◦

, a 211 particle should be pushed toward the liquid, because

the liquid ﬁlm is present between the 211 particle and the growth front. In

contrast, when φ>0

◦

and the 211 particle is not dissolved completely in the

liquid, the particle is expected to be easily trapped within the 123 domain.

7.1.4 Models of Platelets-Like Growth of 123 Phase

The model of platelets-like growth of 123 phase, which is more faster in the

ab-plane compared with the c-axis direction in the limit of single-crystalline

material and taking into account interaction with 211 particle, is presented in

[10]. The 211 particle may contribute to the gap-formation process, when the

growth front bipasses this particle, and the 123 material does not envelop the

211 particle completely by growing at nearly identical rates on either side of

the particle (see Fig. 7.14d). In this case, the resultant microstructure consists

(a)

(b)

211

Liquid

Film

211

123 123

Liquid

Film

211

123 123123

123

Fig. 7.13. Schematic of two diﬀerent cases for 211 particle trapping within growing

123 interface [481]. Dihedral angle, φ, between 211 and 123 phases is (a) φ =0

◦

and

(b) φ>0

◦

316 7 Modeling of YBCO Oxide Superconductors

Growth

Direction

211

Macroscopic Growth

Direction

211

Macroscopic

Growth Direction

211

(a)

(b)

(c)

(d)

211

Fig. 7.14. Schematic of possible eﬀect of 211 particles on the growth of melt-

processed YBCO crystal: (a) 123 platelet abuts a growing platelet; (b) heterogeneous

nucleation occurs at the “platelet–211 particle–liquid” junction; (c) once suﬃcient

growth along the c direction occurs such that the 211 particle can be bypassed,

rapid lateral growth will occur; (d) this process results in a gap, which may not heal

completely [10]

of platelets of identical orientation separated by gaps, which are ﬁlled in with

the rejected liquid.

The generalization of this model [334] describes the formation of linear and

plane gaps between 123 platelets and rejection of liquid at domain boundaries

in the case of 211 particle trapping by 123 matrix. As is well known, there is a

potential possibility of formation of not only the YBa

7−x

(123) phase,

but the BaCuO

(011) phase from liquid, which does not demonstrate super-

conducting properties and has solidiﬁcation temperature about 1015

◦

Cthat

is near to start formation of the 123 phase [34]. Thus, solidiﬁcation of either

123 phase or 011 phase is possible [317]. The situation, connected with faster

solidiﬁcation of barium cuprate, may be explained on the basis of the phase

diagram and taking into account the above-described mechanism of the 123

phase formation [10, 334]. The relatively small (compared with the process of

solidiﬁcation of the 011 phase) rate of establishment of equilibrium between

liquid (L) and solid 123 phase can lead to thermodynamic equilibrium in each

of these components. The liquid will demonstrate independence because of

low concentration of yttrium that is caused by low rate of change between

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 317

the solid and liquid phases. In this case, a homogeneous formation of 123

nuclei is impossible, and the ﬁgurative point, describing melt, displaces into

L +BaCuO

+ CuO region, resulting in the solidiﬁcation of barium cuprate.

There are two causes for breaking of usual equilibrium between liquid and

solid 123 phase: (i) a growth of the 211 particles (due to their diﬀerent sizes

and well-known Ostwald ripening), reducing interphase interface with melt,

decreasing yttrium ﬂux into liquid, leading to envelopment of these particles

by solidiﬁcation front with rejection of the yttrium-depleted melt at the inter-

granular boundaries and ﬁnally initiating a de-lamination of the system; (ii)

a decreasing of the thermal stability of 123 nuclei in the liquid, leading to ab-

sence of dominating 123 solidiﬁcation even at suﬃcient yttrium concentration

[318]. These causes change the stoichiometry of liquid in the case of trapping

of large 211 particles [334], and even solidiﬁcation of pure 011 phase [318].

The development of the model [475] for the case of low G/R ratio is carried

out in [936]. The microstructure close to the quenched solid–liquid interface

exhibits bridges of 123 material between the solidifying 123 interface and

211 particles. In order to describe their morphologies, a combination of both

phenomena, namely a peritectic reaction being mediated by the liquid and a

peritectic transformation of the 211 particles, being linked to the solidiﬁcation

front via bridges of 123 phase, is necessary. Note that all the above models

neglect peritectic transformation [494]. The principal diﬀerence of the model

[936] from the models [156, 475, 741] is the account of inﬂuence on the process

of the 123 phase formation of Lifschitz–Zlyozov boundary eﬀects [291, 629,

997, 1090], and action of the capillary attraction forces between moving front

of the 123 phase solidiﬁcation and 211 particles into liquid.

The entire local process of the engulfment of 211 particles into the so-

lidifying interface can be explained, considering four steps: (i) liquid-phase

diﬀusion-controlled growth, following the temperature gradient, if the 211

particle is far away from the phase boundary (see Fig. 7.15a); (ii) bridge for-

mation, when the 123 interface faces an increased Y

concentration gradient,

when being approached by a 211 particle (see Fig. 7.15b); (iii) peritectic sur-

face reaction during the engulfment process (see Fig. 7.15c); and (iv) peritectic

transformation (negligible eﬀect compared to previous steps) (see Fig. 7.15d).

Due to the peritectic character of the 123 phase creation, this phase needs

a Y-concentration that is not provided by the melt, being in equilibrium with

the 211 phase. Therefore, as in classical nucleation theory, a depletion zone

arises and the growth of the 123 phase is driven by a concentration gradient

in the depletion zone, δ, close to the 123 interface. At the same time, the

dissolving 211 particles maintain a medium yttrium concentration in liquid,

, corresponding to the Ostwald ripening theory.

Bridge formation starts when the depletion zone ahead of the 123 phase

boundary and dissolution region of the 211 particle begin to overlap (see

Fig. 7.15b). The increased concentration gradient leads to an accelerated

growth of the 123 phase toward the 211 particles, resulting in a bridge.

318 7 Modeling of YBCO Oxide Superconductors

123

211

123

Liquid

123

Liquid

211

123

211

123

Δc

Distance, ΔT

)

c(x, T)

––

Δc

––

211

(T)

(a)

123

––

Δc*

––

δ*

Δc*

δ*

c(x, T)

211

(b)

(c)

(d)

Fig. 7.15. Model of local inﬂuence of 211 particles on the growth morphology of

123 phase. The parameter c(x, T ) corresponds to the local yttrium concentration, c

represents the equilibrium solubility, and c

the mean concentration, corresponding

to the Ostwald ripening theory; dc/dx denotes the concentration gradient at the 123

interface. (a) Liquid diﬀusion-controlled growth, (b) bridge formation, (c) peritectic

reaction and (d) peritectic transformation [936]

7.1 Modeling of 123 Phase Solidiﬁcation from Liquid 319

The growing bridge, reaching the 211 interface, deﬁnes the start of the peritec-

tic reaction (in its original sense) [422], which subsequently covers the surface

of the 211 particle with solid 123 material (see Fig. 7.15c). Once the 211 sur-

face is covered, any further formation of 123 phase is governed by peritectic

transformation, that is, strongly limited by diﬀusion in the solid and negligible

(Fig. 7.15d).

As example of the above model, we explain the 1:1 correlation between 211

particle size and thickness of the 123 platelets, observed in [488]. A possible

explanation is based on two assumptions: (i) the plate-like growth of the 123

phase is caused by the strong anisotropic growth rates [10], ν

>> ν

,and

(ii) the engulfment of the 211 particles is provided by the growing 123 matrix.

The anisotropy of the growth rates leads to a preferred growth of the ab-

planes parallel to the temperature gradient. At the same time, the low growth

rate along the c-axis direction results in a morphological instability, leading to

residual melt enclosed in planar defects between the ab-platelets. The platelet

A closest to the 211 particle (see Fig. 7.16a) is the ﬁrst to be inﬂuenced by the

spherical diﬀusion region of the 211 particle and therefore will form a bridge

as explained above. Next growth in all (a, b and c) directions of this platelet A

is governed by the peritectic reaction, occurring along the 211 particle surface.

In particular, the c-axis growth of platelet A is in competition with the fast

ab-plane growth of the adjacent platelet B (see Fig. 7.16b). While the growth

of platelet B is additionally accelerated, when approaching the 211 particle,

the c-axis growth of platelet A is more and more decelerated because supply

of yttrium is more and more hindered by the growing peritectic reaction layer.

Considering the outer platelets C, note there is only competitive growth with

one platelet (e.g., platelet B), giving rise to enhanced growth as well as be-

ing parallel to the ab-planes and the c-axis. This leads to an enlarged platelet

thickness of the outer platelets after passing the particle (see Fig. 7.16c and d).

This process automatically results after some iterations in a platelet thickness,

corresponding to the mean particle diameter, which is supported by exper-

imental observation presented in Fig. 7.17b. Once the thickness is reached,

further growth of the 123 phase will be determined either by two platelets,

each passing one side of the 211 particle (see Fig. 7.18a) or one platelet passing

the whole particle, respectively (see Fig. 7.18b). Note that even a hypothetic

planar interface would change to a cellular morphology, when interacting with

211 particles, also yielding platelet dimensions, according to the 211 particle

diameter (see Fig. 7.19). The zipper-like mechanism, acting in this process,

results in an oriented growth of multiple connected plateletstoaquasi-single

crystal. Thus, the considered mechanism of solidiﬁcation explains both a very

high rate of the 123 phase creation and an existence of suﬃciently sharp

boundary between 211 inclusions and 123 matrix [741]. The models [10, 936]

also explain a coincidence of 211 particle sizes with thickness of 123 platelets,

observed in [488].