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

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228 5 General Aspects of HTSC Modeling

considered materials the non-associated ﬂow rules. The function of plastic po-

tential, g, used to study plasticity of granular media, coincides nearly with the

yielding function, f, applied to separate plastic and elastic states. Diﬀerence

between these functions consists in that the internal friction angle, Φ,inthe

equation for f is replaced by the dilatancy angle, Ψ,inrelationtog.Thus,

in order to state a validity of the associated or non-associated ﬂow rules for

HTSC powder compaction, it is necessary to deﬁne the dilatancy angle, Ψ,

from (5.10) and to compare its value with the friction angle, Φ.

After a pick value in the “stress–strain” curve (see Fig. 5.4a) is reached in

the softening regime (III), there is unstable behavior of material, in particular

caused by thin shear bands, which separate the specimen in two more or less

rigid bodies. For such macroscopic non-uniform deformations, connected with

bulging of specimen in triaxial compression test, the strain increment is not

measured correctly. At the same time, the strain rate ratio is not so strongly

aﬀected by the localization into a shear band. The axial strain–volumetric

strain curve of Fig. 5.4c is much more informative, but the dilatancy angle Ψ

can be measured with acceptable accuracy.

Violation of the normality law, caused by actual resistance to deforma-

tion, comes from two sources: (ii) the dilatation of the bulk material during

shear enhances the yield stress under conditions of compression – due to this

it is necessary to do a work against the applied pressure and (ii) the fric-

tional dissipation of energy at the contact patches between the granules also

enhances the yield stress under applied loading. Below, we will not take into

account the work executed during material hardening and softening and also

the localization processes at deﬁnition of yield surface. Then, assuming that

the material is the strain rate independent of small deformation, the strain

rate is split up into elastic and plastic components. Each of these strain rates

can be also split up into volumetric, ˙e, and deviatoric,

, components. Con-

sider a typical dilatancy rule for rigid particles caused by the material volume

expansion at granule rearrangement [136, 137]:

˙e

= ν(

)

1/2

, (5.11)

where the constant of proportionality, ν, is a generalization of the dilatancy

angle, Ψ. For granules of ﬁnite strength, some deformation occurs at the con-

tact patches, so some of this expansion can be alleviated. Hence, it may be

assumed that energy is dissipated during this damage at the rate:

= l[ν(

)

1/2

− ˙e

] , (5.12)

where parameter l is proportional to the strength of the granules and the size

of the contact patches between granules. Then, the deformation of granular

material caused by the granules rolling and sliding deﬁnes the energy dissi-

pated due to friction at the contact patches. The rate of this energy can be

approximated by

= −μσ(

)

1/2

, (5.13)

5.1 Yield Criteria and Flow Rules for HTSC Powders Compaction 229

where μ is some measure related to the friction factor. Note that both energy

dissipation rates are obtained taking into account that (i) there can never be

negative dissipation function for all non-zero values of strain increment and

(ii) the resistance to deformation is rate independent.

During densiﬁcation, there are deformation and/or rearrangement of par-

ticles depending on the ratio of shear stress to compression. Bearing both

mechanisms in mind, a suitable dissipation function might be

D = {μ

+ l

[ν(

)

1/2

− ˙e

]

}

1/2

. (5.14)

The rate working per unit volume,

W , by the boundary tractions reduces

by Gauss’s theorem and the condition of equilibrium to

W = σ

˙e

. (5.15)

This energy input can be stored as an increase in elastic energy rate,

U, or dissipated,

D. Energy conservation gives

W =

U +

D. Splitting the

deformation rate into elastic and plastic components, we obtain

˙e

−

D =0. (5.16)

From Euler’s theorem on homogeneous functions, it is clear that

∂

∂ ˙e

. (5.17)

Equations (5.14) and (5.17) provide an implicit ﬂow rule, which would

satisfy the energy balance:



νl

(

)

1/2



ν(

)

1/2

− ˙e





; (5.18)

σ =



˙e

− ν(

)

1/2



. (5.19)

The yield surface can be found in this case by eliminating the strain in-

crements in the energy balance relation (5.16) using (5.18) and (5.19). We

obtain

[(s

)

1/2

+ σν]

− 1=0. (5.20)

Again, using (5.18) and (5.19), the ﬂow rule can be derived in terms of the

components of stress and the overall rate of energy dissipation:

[(s

)

1/2

+ νσ]

)

1/2

; (5.21)

˙e

Dσ



ν[(s

)

1/2

+ νσ]

. (5.22)

230 5 General Aspects of HTSC Modeling

Equations (5.21) and (5.22) predict that at small σ/l the granules behave

in an almost rigid manner, but as σ/l ≈ 1, compaction takes place. This is

in accordance with experiments for granular materials [591]. From (5.21), it

follows that these ﬂow rules give sensible results as long as

)

1/2

+ νσ > 0 . (5.23)

Successive development and application of constitutive models for HTSC

powder compaction should be based on the three-level hierarchical structure

of the models. At ﬁrst step, the non-hardening perfectly plastic model un-

der consideration provides an excellent introduction to the modeling of the

HTSC powders under compression, taking into account the particle cohesion

and friction. Second step includes consideration of the isotropic hardening

models together with study of isotropic softening and estimation of damage

variables, based on ideal plasticity, taking into account non-associated be-

havior. Finally, in the third step, anisotropic hardening and softening should

be studied together with modeling of material instability that leads today to

numerous diﬃculties in the application of ﬁnite element codes and other com-

putational methods for numerical solution of corresponding boundary-value

problems.

5.2 Void Transformations During Sintering of Sample

The experimental studies of the sintering duration eﬀect on superconducting

properties of monocore Bi-2223/Ag tapes [670, 671] have shown that critical

current, at ﬁrst, increases together with annealing time, reaches maximum

value and then decreases with reaction time (Fig. 5.6). Monotonous enhance-

ment of weight losses of the sample with simultaneous diminution of Pb con-

tent and increase of the Bi-2223 phase homogeneity (Figs. 5.7 and 5.8) at

remaining constant average size of superconducting grains about 25 μmhas

been observed, which has not depended on the annealing duration (in the

range of 40–200 h). This behavior of critical current has been related by the

authors to acting during calcining competing mechanisms, which initially im-

prove the quality of grain boundaries (or weak links),

and then decrease the

pinning properties of the superconductor. A decrease in critical current has

been explained by lead expelling during annealing and by the correspond-

ing degradation of the pinning strength. At the same time, as rule, during

BSCCO/Ag processing, there are signiﬁcant processes of the void formation

and other microstructure transformations due to CO

release, in particular

At relatively badly coupled structure of grains, magnetic ﬂux can penetrate

intergranular boundaries and trap pores or secondary phases. This non-

superconducting volume will be eﬀectively screened in the sample with better

intercrystalline properties, and corresponding fraction of superconducting volume

will be enhanced.

5.2 Void Transformations During Sintering of Sample 231

0 50 100 150 200 250

T = 4.2 K

B = 0 T

Annealing Time (h)

120

(A)

T = 77 K

Fig. 5.6. Annealing time dependence of the self-ﬁeld I

at 4.2 K and 77 K [670, 671]

leading to bubbles arising and the critical current decrease. As it has been

noted in Sect. 4.2, 200 ppm of carbon can cause about 36% porosity in the

core of Bi-2212/Ag tapes if all carbon forms CO

at high temperature [1196].

Though an average grain size observed in monocore Bi-2223/Ag tapes has

remained constant during various duration of annealing, the grain size dis-

tribution could be altered during reaction. This is supported by considerable

0.0

−

–0.2

−

–0.4

−

–0.6

−

–0.8

−

–1.0

0 50 100 150 200 250

Weight Loss of Tape (mg)

Pb Content (%)

2.00

−

1.75

−

1.50

−

1.25

−

1.00

−

0.75

0.50

Annealing Time (h)

Fig. 5.7. Weight loss of total sample, and the lead concentration (in atomic percent)

versus annealing time. The circulars sign values averaged on several single grains and

squares show measurements on the large region of about 400 grains [670, 671]

232 5 General Aspects of HTSC Modeling

10 μm

(a)

(b)

10 μm

Fig. 5.8. SEM micrographs of a typical transversal fracture surface after (a)40h

and (b) 240 h annealing time [671]

lead expelling monotonously increasing with annealing time. But then these

additions heterogeneously distributed into the material can inhibit only a

local grain growth in the Bi-2223 core. Moreover, as it has been shown in

the modeling of fracture processes in YBCO, the size distributions of mi-

crostructure elements play more important role to compare with average sizes

[821]. Hence, it may be assumed that the observed preservation of the aver-

age grain size in diﬀerent annealing times cannot assert an absence of grain

growth and other microstructure alterations. Therefore, another mechanism of

diminution of critical current during prolonged annealing is possible, namely

inevitable transformation of pores attached to intergranular boundaries, which

5.2 Void Transformations During Sintering of Sample 233

can lead to the formation of signiﬁcant closed porosity. Consider some regimes

of change of the void space: their displacement, shrinkage, coarsening, coales-

cence and possible separation from grain boundaries to the interior of the

grains that directly deﬁne useful properties of superconductor.

5.2.1 Void Separation from Intergranular Boundary

In discussion of pore breakaway processes, ﬁrst, it should be noted that pores

attached to grain boundaries decrease due to boundary-grain diﬀusion. How-

ever, when a pore separates from grain boundary and locates into the grain,

one can shrink only, thanks to much more slow diﬀusion process of the crys-

talline lattice. Total separation of pores occurs after their displacement at the

boundaries between two grains (Fig. 5.9). Therefore, a preliminary displace-

ment of pores from triple junctions of grain boundaries to interfaces of two

grains must precede to total pore breakaway.

Pores

Large

Grain

(a)

(b)

Grain

Boundaries

Fig. 5.9. (a) A schematic of the grain disappearance process involved in grain

growth of large grain at the expense of ﬁve-sided small grains located in its periphery;

(b) the three-sided conﬁguration associated with ultimate grain disappearance

234 5 General Aspects of HTSC Modeling

The phenomenological analysis is essentially based on solutions for the in-

teraction of a boundary with a rigid second-phase particle. Speciﬁcally, the

interaction between a pore and a moving grain boundary assumes a spherical

pore moving in isotropic, homogeneous material with a rate determined by

the surface diﬀusion coeﬃcient (into the framework of surface diﬀusion mech-

anism, appropriating to small pores). Then, unique pore mobility is derived,

retaining the spherical symmetry of the pore (i.e., neglecting the changes in

pore shape needed to maintain the atom ﬂux over the pore surface). The

corresponding pore mobility is given by [973]

kTπa

, (5.24)

where D

is the surface diﬀusion parameter, a

is the pore radius and Ω is

the atomic volume. The force F exerted by the grain boundary on the pore

(which eventually dictates separation) is derived from the assumption that

the contact line between the boundary and the pore can move freely over the

pore surface. The force has a maximum value:

max

= πa

. (5.25)

Equations (5.24) and (5.25) yield a peak pore rate as

ΩD

kTa

. (5.26)

The pore breakaway is considered to occur when the grain boundary rate

exceeds this peak pore rate.

Following [445], consider the motion of pores with grain boundaries, taking

into account a ﬂux of atoms from the leading to the trailing surface of the pore

(see Fig. 5.10). The driving force for the atom ﬂux is caused by the existence

of a gradient in the curvature of the pore surface, that is, pore distortion is

a necessary consequence of pore motion. The conﬁguration selected is an axi-

symmetric pore, moving due to surface diﬀusion. Initially, steady-state motion

(all locations on the surface, moving at the same rate) is considered. The axi-

symmetric pore exhibits two curvatures (see Fig. 5.11): an in-plane curvature,

, and an axi-symmetric curvature, k

. These curvatures are related to the

coordinates of the problem (x, y)by

=(d

y/dx

)[1 + (dy/dx)

]

−3/2

; (5.27)

=(1/x)(dy/dx)[1 + (dy/dx)

]

−1/2

. (5.28)

The ﬂux J

of atoms in the presence of these curvatures is given by

= −





d(k

+ k

)

, (5.29)

5.2 Void Transformations During Sintering of Sample 235

Trailing

Surface

Atom

Flux

Leading

Surface

Fig. 5.10. A schematic of a moving pore, indicating the atom ﬂux and the inclina-

tion of the grain boundary, θ

In-plane Radius of

Curvature, R

Axi-symmetric

Radius of Curvature,

Axis of Symmetry

Pore

Influence

Distance, z

Grain

Boundary

Fig. 5.11. The axi-symmetric conﬁguration associated with pore drag, illustrating

the important curvatures, the inﬂuence distance and the pore and grain boundary

rates

where ds is an element of pore surface in the ﬂow direction (see Fig. 5.10), γ

is the surface energy (assumed isotropic). For a pore moving with a rate, v

in the y-direction, conservation of matter requires that

2πxJ

= ±πx

/Ω , (5.30)

where the positive sign refers to the leading surface and the negative sign to

the trailing surface.

Using the symbols, p and q (≡ dy/dx) for the slopes of the trailing and the

leading surfaces, respectively, (5.27)–(5.30) lead to two diﬀerential equations

of second order. The motion of the pore is subject to the requirement that

the total dihedral angle, ψ, between the grain boundary and the pore surfaces

(Fig. 5.10) be invariant. Moreover, it is necessary that the chemical potential

236 5 General Aspects of HTSC Modeling

be continuous at the intersection of the leading and trailing surfaces. Finally,

the atom ﬂux and surface slope must be zero at the axis of symmetry. The

solution of the diﬀerential equations at the pointed conditions is a non-linear

problem. A solution is obtained by linearizing a trial solution and then using

a ﬁnite diﬀerence scheme [445].

The inclination, θ, of the grain boundary tangent to the plane of con-

tact between the grain boundary and the pore emerges from the analy-

sis as a unique function of the dihedral angle, ψ, and normalized pore:

=(v

)(γ

/γ

). The radius of contact, a, between the boundary and the

pore can also be deduced by requiring that the pore volume be independent

of pore rate in order to permit a unique comparison between the dimensions

of the stationary and moving pores.

Convergent pore shape solutions are found to exist over a limited range

of pore rates, implying the existence of a steady-state rate maximum, v

.Its

existence is associated with an ability to simultaneously satisfy the require-

ments that (i) the dihedral angle be speciﬁed, (ii) the curvature be continuous

and ﬁnite and (iii) the pore rate be uniform. The consequences of increasing

the rate above v

, leading to violation of one of these imposed conditions,

can be presented when the dihedral angle ψ ≈ π (Fig. 5.12). Shape changes,

which induce a continuous atom ﬂux in the requisite direction for pore motion

(i.e., a continuous gradient in surface curvature), cannot be constructed due

to an inevitable counter-ﬂux of atoms. Thus, steady-state motion by surface

diﬀusion of a pore with ψ ∼ π is impossible.

The steady-state rate maximum

may be required as the equivalent of the peak rate, v

, given by (5.26), and

can be expressed as [445]

= v

(17.9 − 6.2ψ)/2cos(ψ/2) . (5.31)

Counter-Flux

Stationary Pore

Grain

Boundary

Moving

Pore

Fig. 5.12. A schematic illustrating the development of a counter ﬂux with ψ = π is

distorted to achieve a net atom ﬂux and hence pore motion with the grain boundary

The inability of a pore with ψ ∼ π to exhibit steady-state motion implies that

such pores will always detach from grain boundaries. This is intuitively clear

because when ψ = π, the grain boundary energy is zero (surface energies are

always ﬁnite) and there is no preference for pores to locate on grain boundaries.

5.2 Void Transformations During Sintering of Sample 237

In this case, for all reasonable choices of the dihedral angle, the peak

steady-state rate exceeds the value anticipated by the phenomenological

model.

Non-steady-state solutions, in which the rate varies over the pore surface

(with a maximum at the axis of the leading surface), can be found for net

ratesinexcessofv

. These solutions coincide with a marked change in pore

shape and a decreasing in the grain boundary contact radius, a,asshownby

the shape depicted in Fig. 5.13. It is presumed that, in this case, the contact

radius a will rapidly diminish to zero, causing the grain boundary to converge

onto the pore axis and to initiate breakaway. Hence, the upper bound steady-

state pore rate can be used as a rate, which, if exceeded, will inevitably result

in non-steady-state pore motion and breakaway.

It may be shown that the grain boundary, subjecting to pore drag, exhibits

a rate component normal to the axis of symmetry, indicative of a tendency

toward instability [445]. Therefore, when the grain boundary rate, v

, exceeds

, the axi-symmetric radius of curvature, R

(see Fig. 5.11), decreases during

the motion of the pore–grain boundary conﬁguration, especially within the

immediate vicinity of the pore. The decrease in R

exceeds the change in R

and therefore, the conﬁguration is intrinsically metastable, but steady state

of the complete pore/grain boundary ensemble is impossible. The pore drag

Pore

Pore Shape at Peak

Steady-State Rate

Non-Steady-State

Pore Configurations

Grain Boundary

= V

> v

Fig. 5.13. Schematic illustration of the pore shapes under non-steady-state condi-

tions. For the pore volume to remain constant, the matter removed from the leading

surface, V

, must equal the matter deposited on the trailing surface, V

.Moreover,

under non-steady state conditions, the average rate of the leading surface,

,must

exceed that for the trailing surface,

. In order to satisfy these requirements, the

contact radius a must decrease rapidly with increase in net pore rate