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

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8.1 YBCO Ceramic Sintering and Fracture 341

maximum free surface energy) [1034]. The breaks, covering these facets, causes

a possibility of new particles addition practically in any site of the surface.

Number of breaks does not limit the re-crystallization rate, and the facets

grow self-perpendicular.

Numerical algorithm for modeling of the abnormal grain growth includes

the following procedures:

(1) the deﬁnition of all nearest neighbors for each grain;

(2) the statement of the nearest neighbors pair (i, j)with

max

1≤i,j≤n

|1/R

− 1/R

(3) the growth of larger from grains (i, j)attheexpenseofother;

(4) the veriﬁcation of the conditions: |1/R

− 1/R

|≤I

/2; j =1, 2 ...,n

and 1/R

min

− 1/R

max

≤ I

;

(5) the end of grain growth in the case of fulﬁllment, at least of one of the con-

ditions (4) or corresponding change of the parameters: n

with the following repetition of the steps (1)–(5).

An example of model microstructure fragment before and after abnormal

grain growth is presented in Fig. 8.2b and c.

8.1.2 Ceramic Cracking During Cooling

The residual stresses form around 211 inclusions into large-grain 123 matrix

during cooling due to thermal mismatch in their behavior. Big internal stresses

can be created at tetragonal–orthorhombic phase transition in the large-grain

matrix (the size of single 123 crystallites can attain some millimeter [431])

that leads to fracture of 123 matrix and to formation of defects. In this case,

a density of damage and of defect structures is the function of annealing time

at transition from tetragonal to orthorhombic phase.

Thus, the peculiarities of the YBCO ceramic microcracking are connected

with the tetragonal–orthorhombic phase transition, causing a spontaneous

deformation due to the phase mismatch, and also the thermal expansion

anisotropy (TEA) of grains. Due to macroscopic heterogeneity of 211 par-

ticles in 123/211 composite, the thermal expansion factors of 123 (α

123

)and

211 (α

211

) phases diﬀer signiﬁcantly (see Table 8.2). As a result, the 123/211

Table 8.2. Some properties of 123/211 composite

Parameter YBa

(c-axis) YBa

(ab-plane) Y

BaCuO

References

E (GPa) 143 182 213 [910]

ν 0.255 0.255 0.25

[338]

α 3.2 × 10

−5b

0.86 × 10

−5b

1.24 × 10

−5

[612, 757]

(MPa m

1/2

) 0.8 0.32 – [227]

The typical value for oxides.

This includes thermal expansion and oxygen increase.

342 8 Computer Simulation of HTSC Microstructure and Toughening

composite has diﬀerent average values of thermal expansion factors in various

parts of the sample:

123/211

= α

123

+ α

211

. (8.3)

These heterogeneities lead to complicated picture of residual stresses af-

ter sample cooling from sintering temperature. Microcracks form in 123/211

polycrystalline structure under the inﬂuence of internal stresses between 123

grains during sintering due to anisotropy in their thermal expansion [335].

The 123 grain has a larger compression along the c-axis direction compared

to the a-andb-axes during cooling (owing to both thermal expansion and

oxygen content increase). This creates residual stresses, leading to tension of

the sample along the c-axis.

In contrast to ferroelectric ceramics, where deformation phase mismatch

is the main cause of microcracking [796], we consider both eﬀects in the pro-

posed model of HTSC ceramics. The tetragonal–orthorhombic phase transi-

tion (occurred for YBa

7−x

at T ∼ 600 − 700

◦

C and accompanied by

oxygen diﬀusion [551, 774]) leads to alignment of twinning platelet grains in

ab-plane, which is perpendicular to c-axis. However, twinning is practically

absent, leading to alignment along the c-axis. The shrinkage along this direc-

tion is not compensated by twinning, causing microcracking parallel to (001)

[27]. Moreover, a change of elastic stiﬀness components C

at phase

transformation leads to spontaneous deformation of crystalline lattice [345].

Due to this, eﬀect of TEA may be estimated on cooling and spontaneous de-

formation, caused by the phase transition on the formation of intergranular

microcracks. The critical length of cracked boundary, l

, can be estimated

as [294]

= β

(1 + ν)/(Eε)]

. (8.4)

Here, K

is the fracture toughness of grain boundary; β

≈ 3.5isthetest

constant; E is Young’s modulus; ν is Poisson’s ratio; ε is the deformation of

intergranular boundary due to TEA or deformation phase mismatch. In the

case of the TEA eﬀect, ε =ΔαΔT ,whereΔα is the thermal expansion factors

diﬀerence, deﬁned by extreme values of α;ΔT is the temperature diﬀerence

at cooling. In the case of deformation phase mismatch, ε is the spontaneous

deformation caused by change of the elastic stiﬀness components.

Selecting Δα =5× 10

−6

−1

[294] and ΔT = 625 K (in order to exclude

the tetragonal–orthorhombic phase transition) [217], we have for the case of

TEA ε ∼ 3.1 ×10

−3

. Comparing with deformation caused by the phase tran-

sition (ε ∼ 10

−4

[346]), it may be concluded that the main factor deﬁning a

microcracking of superconducting ceramic is the TEA. Therefore, we consider

below only this case.

The criterion of miqrocrack formation at a boundary with length, l,is

selected, taking into account a grain misorientation [295]:

l/l

≥ 2/[1 + cos(2Θ

− 2Θ

)] , (8.5)

8.1 YBCO Ceramic Sintering and Fracture 343

where Θ

(i =1, 2) is the angle between the axis of maximum compression in

i-grain and the grain boundary plane. In order to estimate these angles, we

use the Monte-Carlo procedure [176]. Consider that the microcracks propa-

gate along the grain boundary after their nucleation at triple junctions. The

microcracks are arrested at the neighboring junctions because the adjacent

boundary facets are usually subject to internal compression [295]. The com-

puter algorithm of intergranular microcracking during cooling includes the

following steps:

(1) the deﬁnition of all triple points in model structure;

(2) the deﬁnition of sizes of all intergranular boundaries, crossing the triple

points;

(3) the re-consideration of the boundaries and their replacement by micro-

cracks in the case of fulﬁllment of the microcrackimg criterion (8.5) for

corresponding boundary.

8.1.3 Formation of Microcracks Around 211 Particles

in 123 Matrix

Thermal Stresses on 123/211 Interface

During cooling from the sintering temperature, inside 211 particles and around

them in 123 matrix, the thermal stresses form as a result of the thermal ex-

pansion factors diﬀerence between 123 and 211 phases. Consider the spherical

coordinate system {r, Θ, Φ} with origin coinciding with the 211 particle cen-

ter. Then, depending on hydrostatic stress P

applied to the particle of radius,

211

, the radial σ

123R

(r) and tangential stresses σ

123Θ

(r)=σ

123Φ

(r)inthe

matrix (for R

211

<r<∞) can be deﬁned as [957]

− 2σ

123Θ

(r)=−2σ

123Φ

(r)=σ

123R

(r)=P

211

/r)

; (8.6)

(α

211

− α

123

)ΔT

(1 + ν

123

)/2E

123

+2(1− 2ν

211

)/E

211

. (8.7)

For convenience, it is proposed that α

211

>α

123

;ΔT = T

− T

is the

temperature diﬀerence at cooling (where T

and T

are the sintering temper-

ature and room temperature, respectively). The stresses on 123/211 interface

are independent of the particle size R

211

. They decrease according to the low

of (R

211

/r)

and attain 12.5% from maximum value for r = R

211

. Select-

ing necessary values from Table 8.2 and taking into account the magnitude

of T

= 925

◦

C, we obtain stresses on the 123/211 interface, presented in

Table 8.3.

The obtained tangential stress in the c-direction (acting on equator of the

211 particle) is suﬃciently great and capable to form ab-microcracks around

the 211 particles. However, because the 211 particles are under compression,

ab-microcracks do not propagate through them, and the 123/211 interface

344 8 Computer Simulation of HTSC Microstructure and Toughening

Table 8.3. Stresses obtained on 123/211 interface

Stress σ

123R

(GPa) σ

123Θ

(GPa) σ

123Φ

(GPa)

c-Axis −2.01.0–

ab-Plane 0.43 −0.215 −0.215

does not fail unless the size diﬀerence between two neighboring particles is

too large. Because the radial stress in ab-plane is tensile, but both tangential

stresses are compressive, then taking into account a higher fracture toughness

in the planes, which are parallel to the c-axis, it may be expected that micro-

cracks, introduced by the 211 particles will not be parallel to the c-direction.

General picture of change of the value and sign for stresses σ

123R

and σ

123Θ

the 123 matrix on the 123/211 interface is presented schematically in Fig. 8.3.

The complicated picture of the stress state in the 123/211 composite leads

to alternative opinions about acting stresses and 123 lattice distortions near

211 particles. In particular, some authors neglect stress state around the 211

particles [748]. Other authors deﬁne perturbations of the 123 matrix far from

the 211 particles or higher density of crystalline defects near some 123/211

interfaces [37]. The considerable indirect experimental supporting of thermal

stresses, existing around 211 particles, is provided with the observed processes

of de-twinning or predomination of one twin variant along [100] directions in

the case of movement of the twin boundaries due to thermal stresses, caused

by 211 (422) particles [199, 200]. The stresses required for the motion of twin

boundaries and estimated by the value of 50 MPa [199] are well in agreement

with obtained magnitudes in ab-plane. The presented analysis of stress state

is also supported by observed distribution of microcracks around the 211 par-

ticles (see Fig. 8.4). The high radial compressive stresson poles of the 211

123R

a,b

(a)

123Θ

a,b

(b)

Fig. 8.3. Schematic representations of: (a)radialand(b) tangential stresses in 123

matrix at 123/211 interface. Sizes and direction of arrows demonstrate value and

also action regions of tensile and compressing stresses

8.1 YBCO Ceramic Sintering and Fracture 345

10 μm

Fig. 8.4. Microcrack pattern around 211 particles. The microcracks in ab-plane

avoid 211 particle poles and deﬂect in radial direction [196]

particle inhibits microcrack growth into ab-plane near the poles. Therefore,

ab-microcracks avoid the 211 particle poles and return to propagation along

radial direction in existence of tangential tensile stresses, according to Figs. 8.3

and 8.4.

Microcracking Criterion in ab-Plane

Formation of the ab-microcracks in the case when the particle demonstrates a

smaller thermal expansion compared to matrix demands existence of a microc-

rack nucleus on the 123/211 interface and supply by energy, which is necessary

for the defect growth. This is provided by elastic energies of the particle and

surrounding matrix. In isotropic case, total stored energy per unit volume can

be deﬁned as [177]

= P

πR

211



1+ν

123

2(1 − 2ν

211

)

211



. (8.8)

In this case, only part of the stored energy can be used to form ab-

microcracks. Using the 123 constants for c-direction, it may be assumed that

the energy approximately equals third part of the total energy, U

. Then,

the necessary, but insuﬃcient condition of microcrack formation is that the

energy of creation of new surface, U

= γ

A (where γ

is the eﬀective surface

energy of matrix and A is the square of the newly formed surface) must not

exceed U

/3, that is

/3 ≥ U

. (8.9)

Taking into account γ

=(K

)

(the upper indexes here and below

point the a, b and c directions) and selecting the corresponding data from

Table 8.2, we obtain γ

=7.16 × 10

−7

MPa · m. The square A is found from

346 8 Computer Simulation of HTSC Microstructure and Toughening

211

a, b

Fig. 8.5. Schematic representation of ab-microcrack formation due to release of

elastic energy introduced by 211 particle

Fig. 8.5 as A =2π(r

− R

211

). Considering the usual distance between 211

particle, which is equal to 4R

211

(for 20 mol% of the 211 particles) [196], it

is necessary to equate the value of r

to 2R

211

in order to create the ab-

microcrack, connecting two adjacent 211 particles. Then we have

=2π[(2R

211

)

− R

211

]γ

=6πR

211

. (8.10)

The existing energy is proportional to R

211

; at the same time the energy

absorbed during cracking is proportional to R

211

. Combining (8.8)–(8.10), we

deﬁne the critical size of the particle, R

c211

,as

c211

≥

18γ

)

[(1 + ν

123

)/E

123

+2(1− 2ν

211

)/E

211

]

. (8.11)

Substituting the obtained values of P

and γ

and also elastic constants

from Table 8.2 for the 123/211 composite, we deﬁne R

c211

=0.24 μm. Another

estimation of the critical particle size for microcracking in ab-plane (giving

the value < 1 μm [199]) agrees well with the value (R

c211

) obtained in the

present analysis. The strong proof that the ab-plane defects are not growth-

related defects is that they were not observed in the tetragonal 123 phase

[202, 535] as well as in the tetragonal Nd-123 system with small 422 particles

(see Fig. 8.6) [201].

10 μm

Orthorhombic

Phase

Tetragonal

Phase

Fig. 8.6. Dense ab-micxrocracks in the orthorhombic Nd-123 phase and no ab-

microcracking in the tetragonal region. The case of small 422 particles [201]

8.1 YBCO Ceramic Sintering and Fracture 347

8.1.4 Fracture Features at External Loading

The suﬃciently high thermo-mechanical and electromagnetic loading can lead

to fracture of material. In order to estimate critical fracture parameters, we

use a model of crack growth in anisotropic body [583]. Consider a polycrys-

talline aggregate of grains with hexagonal shape, containing an annular ﬂaw

(Fig. 8.7). The probabilistic condition of misorientation of the adjacent grains

leads to initiation of the greatest compression in the central grain, while the

surrounding grains will be subject to tension. It is assumed that the central

grain is forced into a cavity of diameter, D =2R, surrounded by the annular

ﬂaw with length of S. In this case, we shall take into account eﬀects of elas-

tic stress concentration from external tension, σ, and thermal strains on the

grain boundaries, ε =ΔαΔT . Then, the summary stress intensity factor (SIF)

due to concentration of the thermal stresses and applied external loading is

found by

tot

= K

+ K

. (8.12)

D = 2R

R + S

(a)

(b)

Fig. 8.7. Polycrystalline aggregate, containing annular ﬂaw: (a) hexagonal grain

under stress; (b) crack opening due to TEA (α

and α

are the thermal expansion

factors along a-andc-axes, respectively)

348 8 Computer Simulation of HTSC Microstructure and Toughening

The ﬁrst term is determined by the radial component of the thermal stress

concentration from the external side of spherical boundary [583, 814] as

2σ

1/2

(1 + S/R)

1/2



1 −



1 −

(1 + S/R)



1/2



, (8.13)

and the second term is found by the tangential component [583, 814] as

2σ

1/2



1 −

(1 + S/R)



1/2

(1 + S/R)

3/2

. (8.14)

Here,

ΔαΔT

3(1 − ν

)

is the thermal stress; E

, ν

are Young’s modulus and Poisson’s ratio for

porous cracked material, respectively. In order to deﬁne elastic constants of

porous superconductor, we use the modiﬁed cubic model [428]

= E

(1 − P

2/3

); ν

= ν

(1 − P

2/3

) , (8.15)

where P

is the volumetric fraction of closed porosity; E

, ν

are the elastic

constants of cracked no-porous ceramic. The elastic constants E

, ν

can be

calculated from the following relationships [892]:

=1/



16(1 − ν

)(10 − 3ν)

45(2 − ν)



; (8.16)

1+[(16/45)(1 − ν

)/(2 − ν)]β

1+[(16/45)(1 − ν

)(10 − 3ν)/(2 − ν)]β

, (8.17)

taking into account arbitrary-oriented microcracks. Here E, ν are the elastic

modules of non-defective material; β

= N

is the microcrack density

in the considered volume, V

; N

and a

are the number and characteristic

size of microcracks, respectively.

The SIF, caused by applied stress, K

, is deﬁned as [583, 814]

2σR

1/2

(1 + S/R)

1/2

, (8.18)

where



1 −

(1 + S/R)



1/2



(4 − 5ν

)

2(7 − 5ν

)

(1 + S/R)

2(7 − 5ν

)



(1 + S/R)

− 1



(1 + S/R)



. (8.19)

8.1 YBCO Ceramic Sintering and Fracture 349

We deﬁne strength of anisotropic body from the critical conditions of a

crack initiation: K

tot

= K

is the fracture toughness of grain boundary)

and σ = σ



2D(1 + S/R)



1/2

− σ

, (8.20)

where σ

is the critical stress;

=1−



1 −

(1 + S/R)



1/2



1 −

2(1 + S/R)



. (8.21)

Finally, we write a condition for critical displacement at crack opening (see

Fig. 8.7b): Δ

= Δ

(where Δ

is the linear thermal expansion of the central

grain) as [583]

DΔαΔT =

2(1 − ν



R + S



1/2

. (8.22)

Then, the critical grain size, D

, deﬁning further crack growth, can be

obtained, suggesting D = D

and deﬁning a positive root of quadratic equa-

tion (8.22) by [814]:

= b +(b

+4Sb)

1/2

; b =

(1 − ν

)

πE

(ΔαΔT )

. (8.23)

Then, we select average grain of model structure as grain size, R,and

deﬁne value of S (the length of nucleus microcrack) through mean microcrack

length, a

, formed during the ceramic cooling. Here and below, an absence

of the microcrack interaction is assumed.

At microcrack development in brittle materials, there are processes caused

by the material structure properties, which lead to the material toughening.

The eﬀects of microcracking, crack branching and bridging and so on render

signiﬁcant, but no simple eﬀect on change of strength properties and frac-

ture resistance. The considerable correction of eﬀects of diﬀerent toughening

mechanisms may occur, taking into account their joint action. A domination

of one from the mechanisms is determined by microcrack distribution, and

their density causes the toughening value.

8.1.5 Microcracking Process Zone near Macrocrack

In the vicinity of growing macrocrack, a microcracking process zone initiates,

which leads to the macrocrack shielding and change of the ceramic fracture

toughness [572, 264, 294]. Both phenomena are caused by microcrack distri-

bution. The compliance of the process zone increases the fracture toughness;

however, the microcracks, directly neighboring the macrocrack tip, decrease

fracture resistance and toughness of material [264, 294]. The critical number

350 8 Computer Simulation of HTSC Microstructure and Toughening

of microcracks per volume unit, N

∗

, deﬁning initiation of coalescence in the

vicinity of the crack, depends on proper microcrack size, a

[1125]:

∗

=9/(64a

) . (8.24)

Deﬁne the width of the process zone, 2h

, as [107]

= a

, (8.25)

where I

= β

/(E − E

) is the parameter of elastic interaction of the

microcracks; β

= N

is the microcracking density in the considered

sample volume, V

; N

is the number of microcracks [892].

The ﬁnite-element analysis [572] shows that the stress state of macrocrack

to a greater degree promotes growth of microcracks, which are parallel to the

macrocrack propagation, and to a lesser degree promotes microcracks of any

orientation ahead in the direction of the macrocrack propagation. The stress

state of macrocrack also impedes microcracks, which are perpendicular to the

macrocrack and disposed at the side from direction of its growth. Moreover,

the process zone size in the crack plane is approximately two times smaller

than in perpendicular direction. These features are taken into account in com-

puter simulation of microcracking in the process zone and in deﬁnition of its

extent. The simulation of macrocrack growth along intergranular boundaries

(see also a preliminary discussion in Sect. 5.7) is carried out on the basis of

the graph theory [514, 515], using the Viterbi’s algorithm [72]. During tran-

sition from one sequence to other, the process zone width from (8.25) and all

triple points in the zone inside are estimated. Then, we model microcracks

on proper boundaries with length of l, using the critical facet size: l

=0.4l

[294]. The procedure of the microcrack modeling repeats the algorithm used

in Sect. 8.1.2 at the stage of spontaneous cracking. In this case, the following

are taken into account: the microcracks do not cross the macrocrack and the

condition, deﬁning an absence of the microcrack coalescence: N

∗

Obviously, the microcracking density, β

, increases together with the grain

size that guaranties increasing of the process zone size, h

As has been shown

by calculation of I

in dependence on the Young’s modulus ratio, E

/E [109],

at increasing density of microcracking, the function, I

, attains a maximum

value, corresponding to the critical density of microcracks, β

= β

.This

value divides the zones of the material toughening and the crack ampliﬁcation.

The fracture toughness change due to the alternative trends, caused by

the spontaneous cracking and by the process zone of microcracking (Fig. 8.8),

However, in contrast to inﬁnite growth of the process zone, stated in [572], even

for grain sizes, which are smaller than the critical one, in reality the process zone

must be ﬁnite. This contradiction is the consequence of the selected model in [572]

for array of hexagonal grains, in which all grain boundaries have the same length,

l, and on each of them the triple point (microcrack nucleus) exists. Therefore, for

l → l

the condition, h

→∞is carried out. Our model is free from the above

shortcoming.