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

Подождите немного. Документ загружается.

238 5 General Aspects of HTSC Modeling

will increasingly distort the boundary and eventually induce separation. It

is interesting to note that grain boundary coalescence is likely to result in

a dislocation or sub-grain boundary attached to the pore (especially in the

presence of some symmetry) [197, 530].

The speciﬁc condition that dictates the separation of pores from grain

boundaries depends on the grain conﬁguration to which the pore is attached.

Consider two conﬁgurations that occur during grain growth,

namely (i) a pore

located on one boundary of a three-sided grain (Fig. 5.9b) (the conﬁguration

grain that invariably precedes grain disappearance) and (ii) a pore located on

one boundary of a ﬁve-sided grain at the perimeter of a large grain subject

to exaggerated grain growth (Fig. 5.9a). Pore drag observations (Fig. 5.14)

indicate that the pore perturbs grain boundary motion over a certain inﬂuence

distance, z (see Fig. 5.11). Therefore, it is appropriate to examine the motion

of the grain boundary outside this inﬂuence distance relative to that of the

pore (obviously, it is assumed that the inﬂuence distance is less than the

grain radius). The rate of the grain boundary outside the inﬂuence zone in

the direction of pore (Fig. 5.11) is [445]

√

3γ

2/3

/R . (5.32)

By allowing this rate to exceed the peak steady-state pore rate, v

separation should be inevitable, provided a grain boundary displacement

Fig. 5.14. Pores in MgO subjected to motion with the grain boundary [445]

Other conﬁgurations, such as several pores on single grain boundary, will pro-

vide diﬀerent separation conditions. However, these separation events usually

occur after the ﬁrst separations have been induced and are, probably, less crit-

ical. For example, a modiﬁed phenomenological analysis with a pore ∝ R gives

the two limiting cases [445]: R ≤ 2M

kTγ

1/3

(17.9 − 6.2ψ)and

R ≥ δ

1/3

(17.9 −6.2ψ)/2γ

5.2 Void Transformations During Sintering of Sample 239

(i.e., grain size), suﬃcient to create boundary convergence at the dragging

pore, is available. This condition yields a pore size for separation given by

≥





1/3

kTM



(17.9 − 6.2ψ)

√

. (5.33)

For the ﬁve-sided grain, R remains essentially constant (Fig. 5.15). How-

ever, the most stringent condition for breakaway exists when R attains its

smallest value, which corresponds to the three-sided grain conﬁguration, pre-

ceding grain disappearance (Fig. 5.9b). In this case, separation is averted if

the pore converges onto the prospective three-grain junction (R ∼ 2a

)before

(5.33) can be satisﬁed. Thus, the critical condition becomes







1/3

kTM



2(17.9 − 6.2ψ)

√

, (5.34)

as shown in Fig. 5.15. This critical pore size represents a lower bound for pore

separation at all reasonable values of a

/R. Obviously, that obtained critical

value increases with increasing of surface diﬀusion or with increasing of grain

boundary mobility or dihedral angle.

HTSC materials demonstrate a spectrum of dihedral angles connected, in

the ﬁrst place, with various grain boundaries formed during grain growth.

Then, there is a range of the pore critical sizes that the very ﬁne pores can

localize at low angle or special boundaries, in the sites of coincidence of the

crystalline lattices. Similarly, a distribution of the surface diﬀusion parameters

and grain boundary mobility due to inhomogeneity of admixture distribution

should exist. Hence, diﬀerent spectrum of the pore critical sizes will corre-

spond to various superconducting systems and should be taken into account

in consideration with the pore breakaway. The microstructure changes, sup-

pressing the pore separation, should be estimated in comparison with the pore

critical size, with the trajectories of alteration of the pore and grain sizes at

the ﬁnal stage of sintering. These processes will be examined in detail in the

next section.

5.2.2 Size Trajectories in the Pore/Grain Boundary System

During Sintering

The sintering process is generally accompanied by grain growth, pore shrink-

age and pore coalescence [161, 1165]. These concomitant processes result in

pore/grain size trajectories that typically entail pore size enlargement dur-

ing the intermediate stage and pore size reduction prior to ﬁnal densiﬁcation

Moreover, exaggerated grain growth may initiate as a result of the pore sep-

arations from grain boundaries [126, 127]. Therefore, it is very important to

study the pore/grain size trajectory, especially with reference to the critical

size for pore breakaway. In this section, following [1005], concurrent grain

growth, pore shrinkage and coalescence are analyzed and compared to the

240 5 General Aspects of HTSC Modeling

Critical Pore

Size Three-

Sided Grain, a

100

Drag

Initiation

Separation Condition

Five-Sided Grain

max

Inadmissible Region

ψ = 80°

= 100°

= 120°

10 100

Normalized Pore Size

Normalized Grain Radius

Fig. 5.15. The various separation and admissibility conditions, identiﬁed in the

analysis of [445], are plotted for three values of the dihedral angle. The abscissa axis

presents normalized pore size (kT M

/Ω

1/3

)

1/2

; the ordinate axis shows

normalized grain size (kTM

/Ω

1/3

)

1/2

5.2 Void Transformations During Sintering of Sample 241

maximum pore size achieved during densiﬁcation, with the critical pore size

at pore–grain boundary separation. Mutual dependencies between the grain

and pore sizes, caused by concomitant grain growth, pore coalescence and

pore shrinkage, can be used together with expressions for the grain boundary

rate and the pore shrinkage rate to establish features of the pore/grain size

trajectories during sintering.

It is assumed that pores exist at each three-grain junction (see Fig. 5.16).

Subject to this condition, pore coalescence during the disappearance of small

grains results in a simple magniﬁcation of the microstructure, while concomi-

tant pore shrinkage reduces the size of the pores relative to the grain size.

Thus, the relative change in pore volume comprises an increase due to coars-

ening and a decrease due to shrinkage given directly by

−

, (5.35)

where dV

is the resultant volume change of the average pore in a small time

interval, dt;dV

is the corresponding grain volume change; dV

is the absolute

pore volume change due to shrinkage; and V

and V

are the average pore and

grain volume, respectively. Dividing (5.35) throughout by the time interval

dt, and rearranging, we obtain

−

, (5.36)

Small

Grain

Grain Boundaries

Large Grain

Pore

Fig. 5.16. A schematic indicating the grain and pore conﬁgurations used to calculate

pore drag eﬀects

242 5 General Aspects of HTSC Modeling

where a dot above a symbol signs the time derivation. Herein, it will be as-

sumed that the pore coarsening will be balanced by the shrinkage, that is,

/dV

= 0. In particular, this condition deﬁnes the pore volume, V

,at

which a transition from pore coarsening to pore shrinkage occurs during nor-

mal sintering (i.e., when the average number of pores per grain remains). This

is given by

. (5.37)

Taking into account the relations:

∝ l

;

∝ 3l

l ; V

∝ a

l ;

∝ a

l +2l ˙aa ;

l = λv

, (5.38)

where l is the grain facet length, a is the pore size, v

is the grain boundary

rate and λ is a grain shape coeﬃcient (1/3

√

3 <λ<1/

√

3) [421, 1165].

Incorporating (5.38) into (5.36) and (5.37) yields a coarsening pore rate as

da/dl = a/l −|˙a

|/λv

, (5.39)

and a transition pore size as

/l = |˙a

|/λv

. (5.40)

Pore Rate

A moving pore, located at a three-grain junction, must exhibit a distortion

with general characteristics (depicted in Fig. 5.17) contingent upon the next

requirements. The surface curvature must increase continuously from the lead-

ing to the trailing surfaces of the pore in order to establish the net atom ﬂux

that constitutes pore motion. Moreover, the equilibrium dihedral angle must

be maintained at each grain boundary intersection. Hence, the triangular zone

that connects the grain boundary intersections must exhibit an apex angle, 2θ,

that diminishes as the pore rate increases. This angular change inﬂuences the

orientation of the grain boundary tangent at each pore intersection, increasing

its curvature and reducing the permissible grain boundary rate (Fig. 5.18).

A linearized analysis of surface diﬀusion (see Fig. 5.17b) provides an ade-

quate solution as the deviations of the surface tangent from the ordinate are

suﬃciently small that the following approximations apply:

ds ≈ dx, 1/r ≈ d

y/dx

, (5.41)

where ds is an element of surface and r is the radius of curvature of the

surface. Subject to this linearization, the atom ﬂux can be expressed as [154]

= −



(γ

/r)



= −





, (5.42)

5.2 Void Transformations During Sintering of Sample 243

(a)

(b)

2θ

Trailing Pore

Surface

Leading Pore

Surface

Atom Flux

Grain

Boundary

ψ/2

Fig. 5.17. Schematics of the geometry of moving pores at the grain junctions:

(a) the pore distortion, (b) the coordinate system (x, y) used for analysis of the

distortion of the leading and trailing surfaces

244 5 General Aspects of HTSC Modeling

Radius of

Curvature, R

Pore

Small

Grain

Fig. 5.18. The grain boundary parameters used to determine the grain bound-

ary rate

and the atom ﬂux gradient (based on matter conservation) becomes [154]

, (5.43)

where D

is the surface diﬀusivity, γ

is the surface energy, Ω is the atomic

volume and v

is the component of the pore rate in the y-direction. Combining

(5.42) and (5.43), the governing diﬀerential equation is given by

= −

. (5.44)

This equation can be solved for the leading and trailing pore surfaces

subject to the next boundary conditions (see Fig. 5.17b):

The leading surface: j

=0anddy/dx =0,atx

= d, whereupon



1 −







−

(d/a

)







− 2







, (5.45)

where ψ

is the pore surface inclination at the boundary intersection, subscript

l refers to the leading surface and ν

= v

kTa

Ω is the dimensionless

pore rate identiﬁed in Sect. 5.2.1 (a

is the initial, zero rate, pore dimension).

5.2 Void Transformations During Sintering of Sample 245

The trailing surface: j

=0anddy/dx = θ − ψ/2, at x

= b, giving

sin θ(b/a

)







− 4









− 2









− θ







−







. (5.46)

Continuity of chemical potential requires that the curvature of the leading

and trailing pore surfaces be the same at the boundary intersection. Incor-

poration of this condition permits the apex angle θ to be related to the pore

rate and to the pore dimension, b:

3[2 sin θ(ψ/2 − θ) − 2θ − (ψ − π)/2](a

/b)

sin

θ +(5/4) sin

θ +(1/4) sin θ

. (5.47)

Finally, requiring that the volume V of the pore be independent of the

rate:

V/l =



dx +



dx +(b

sin θ cos θ)/2 , (5.48)

the dimensionless pore rate can be expressed in terms of the apex angle θ and

the dihedral angle ψ. Simple functional relations that adequately describe

the dependence of θ, α and d on the normalized pore rate in the important

application angular range (π/3 <ψ<2π/3andπ/12 <θ<π/6) are given

by [1005]

θ =(π/6) exp[−ν

/(0.07 + 0.4ψ)] ; α ≈ π/6+0.013ν

(1 + 3ψ);

d/a

≈ 1/2 − 0.013ν

(1 + ψ) . (5.49)

Grain Boundary Rate

The grain boundary rate, v

, is found by the grain boundary radii of curva-

ture R

and R

and also by the boundary mobility, M

, that is regarded as

independent of the driving force:

= γ

2/3





. (5.50)

The dependence of the radii of curvature on the extent of pore distortion,

through the pore drag angle, α (Fig. 5.18), establishes a unique link between

the boundary rate and the pore rate (with the intermediate determination

of α). For a boundary of uniform curvature, the in-plane radius of curvature,

,isgivenby

√

3l − d

√

3cosα − sin α

. (5.51)

246 5 General Aspects of HTSC Modeling

We have from (5.49) and (5.51):

/l ≈

√

3+[0.013(1 + ψ)ν

− 1/2]a/l

[2.5 − 0.023ν

(1 + 3ψ)]

. (5.52)

Thus, the radius of curvature is dictated by the relative pore size, a/l,

the dihedral angle, ψ, and the dimensionless pore rate, ν

. The out-of-plane

curvature is assumed to be of secondary signiﬁcance because one is indepen-

dent of the pore movement. Hence, the radius R

may be neglected. Then,

the grain boundary rate can be obtained from (5.50) and (5.52) as

2/3

≈

2.5 − 0.023ν

(1 + 3ψ)

√

3+[0.013(1 + ψ)ν

− 1/2]a

. (5.53)

Since the pore and boundary remain attached, it is required that

cos α = v

. (5.54)

A dimensionless grain boundary rate ν

= v

kTa

Ω can be de-

termined directly from (5.49), (5.53) and (5.54) in terms of a dimensionless

parameter, ξ = M

kTa

1/3

, that reﬂects the relative mobility of

pores and grain boundaries. The boundary rate can be characterized by two

regimes (see Fig. 5.19). For small values of ξ, the motion of the pore/grain

boundary ensemble is limited by the motion of the grain boundaries. Within

this boundary mobility limited regime, grain boundary motion is not cognizant

Pore Drag Regime

(ν

is independent of ξ)

Boundary Drag Regime

(ν

is independent ofD

)

Relative Mobility, ξ

Dimensionless

Boundary Rate, ν

Fig. 5.19. A schematic indicating the pore drag and boundary drag regimes [1005]

5.2 Void Transformations During Sintering of Sample 247

of the presence of the pores. Therefore, the boundary rate increases linearly

with ξ and has form:

= M

2/3

/l . (5.55)

For large values of ξ, the motion of the ensemble is caused by the mobility

of the pores (i.e., by the rate of surface diﬀusion) and the rate of the boundary

becomes independent of ξ (Fig. 5.19), causing α → π/3, and the boundary

rate approaches the pore drag limit:

=8/(1 + 3ψ) . (5.56)

The transition between these two regimes occurs over a range of ξ,as-

sociated with a critical value, ξ

(Fig. 5.19). The grain boundary mobil-

ity dominated grain growth is considered to obtain when ξ<ξ

,that

is, when

/l) < 8D

1/3

kTγ

(1 + 3ψ) . (5.57)

Pore Shrinkage

The shrinkage of pores at three-grain junctions in a material, consisting of

grains of equal size, is given for grain boundary diﬀusion dominance by [442]

˙aa

(1 − a/l)

= −(8/3)F (ψ)(ΩD

/kT l) , (5.58)

where

F (ψ)=sin(ψ/2 − π/6)



√

3[ψ − π/3 − sin(ψ − π/3)]

2sin

(ψ/2 − π/6)



−1

, (5.59)

is the boundary diﬀusivity. Evidently, the pore shrinkage rate, ˙a,in-

creases as the grain facet length, l, diminishes. It might be anticipated that

the shrinkage rate in the three-sided grain conﬁguration would increase as

the grain boundaries converge. However, the tendency toward enhanced pore

shrinkage results in the development of residual tensile stresses. These residual

stresses constrain the local shrinkage, such that the shrinkage rate is approx-

imately that of circumventing material, as given by (5.56), in which l is the

mean facet length. Similarly, a tendency toward enhanced shrinkage, induced

by the change in curvature at the distorted moving pore, is suppressed by the

matrix constraint.

Coarsening Trajectories

The pore and grain coarsening that occur during normal sintering may be

deduced, inserting (5.58) and (5.59) into (5.39) and (5.40), in the case of the

constrained pore shrinkage rate, and inserting (5.55) and (5.56) into (5.39) and