Gottstein G., Shvindlerman L.S. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications

Подождите немного. Документ загружается.

194 3 Grain Boundary Motion

9 Energy of high-angle

0.15–0.2 Al B

grain boundaries

∼

2γ/a =

− 3 · 10

= const.

10 Energy of high-angle

0.1–0.2 Zn C

grain boundaries

∼

2γ/a =

5 · 10

− 10

= const.

11 Energy of high-angle

0.2 Al D

grain boundaries

∼

γ/a =

5 · 10

− 5 · 10

= const.

12 Energy of high-angle

0.2 Al D

grain boundaries

∼

γ/a =

5 · 10

− 5 · 10

= const.

13 Energy of high-angle

0.2 Fe-Si D

grain boundaries

∼

2γ/a =

− 5 · 10

= const.

14 Diﬀerence of

0.15–0.3 Fe A

surface tension

of free surfaces

of a foil

P =2Δγ

/δ

∼

− 10

= const.

15 Magnetic free

0.3 Bi B

energy in a

bicrystal with

anisotropic

magnetic

susceptibility

P = ω

− ω

(χ

− χ

)

Annotations

1. δ is the thickness of a sample.

2. Δt is the interval between two consecutive recordings of the grain boundary

position or displacement; the speed of recording of grain boundary motion.

3. ρ is a criterion describing the interaction of a moving grain boundary and

the free surface of a crystal. ρ

exp

is the magnitude of ρ in the series

of experiments discussed.

3.4 Measurement of Grain Boundary Mobility 195

30–300 V ∼ Pρ=

2γ

γa

[XI,

exp

= 10–12 XII]

speed of V ∼ Pρ=

2γ

γa

[XIII]

recording: ρ

exp

= 5–25

1 frame per

3–15s

––ρ =

γa

[XIV,

exp

=XVII]

exp

= 10–15

∼4measure- V ∼ Pρ

exp

= 10–30 [XVI,

ments per XVII]

second

∼4measure- V ∼ Pρ

exp

= 10–100 [X]

ments per

second

14 – 10

− 10

- ρ =

6Δγ

[XVIII]

exp

≈ 1

180 − 360 V ∼ Pρ=

Pδγ

=[XIX,

Pδ

0.09γ

exp

∼

2XX]

4. The ratio γ

/γ was taken as 3 [240, 263]. R is the radius of curvature

of a moving grain boundary. For the reversed-capillary technique

R =  sin α[1 − cos(Θ − α)] where  is the displacement of a boundary

along the specimen edge; values of α and Θ are given in the ﬁgures

relevant to the reversed-capillary technique. For the experimental

technique “15”; γ =0.3γ

; γ

=0.521 J/m

at 239

◦

C [254].

196 3 Grain Boundary Motion

5. “A” means that the grain boundary position was determined by chemical

etching; “B” by thermal grooving; “C” means that grain boundary motion

was recorded by observation with polarized light in an optical microscope,

the annealing of the sample was not interrupted; “D” means the grain

boundary motion was determined via X-ray interface continuous tracking

device (XICTD).

6. χ

and χ

are the susceptibilities of grains 1 and 2, respectively, along

the magnetic ﬁeld H; μ

is the absolute permeability.

I Aust KT, Rutter JW, Trans. AIME, 215119, 1959;

Aust KT, Rutter JW. Trans. AIME, 215:820, 1959;

Aust KT, Rutter JW. Acta Metall., 12:181, 1965;

Rutter JW, Aust KT. Trans. AIME, 218:682, 1960.

II Rath BB, Hu H. in Hu H, editor, TheNatureandBehaviour

of Grain Boundaries, Plenum Press, New York, 1972.

III Rath BB, Hu H. Trans. TMS− AIME, 236:1193, 1966;

Rath BB, Hu H. Trans. TMS− AIME, 245:1243, 1969;

IV Gr¨unwald W, Haessner F. Acta Metall., 18:217, 1970.

V Deminczuk DM, Aust KT. Acta Met., 21:1149, 1975.

VI Viswanathan R, Bauer CL. Acta Met., 21:1099, 1973.

VII Antonov AV, Kopetskii ChV, Shvindlerman LS, Sursaeva VG.

Sov. Phys. − Dokl., 18:736, 1974.

VIII Sursaeva VG, Andreeva AV, Kopetskii ChV, Shvindlerman LS.

P hys. Met. Metall., 41:98, 1976.

IX Lejcek P, Paidar V, Adamek J, Kadeckova S. Interface Sci., 1:187,

1993.

X Furtkamp M, Gottstein G, Molodov DA, Semenov VN, Shvindlerman

LS. Acta Mater., 46:4103, 1998.

XI Fridman EM, Kopetskii ChV, Shvindlerman LS, Aristov VJu.

Zt. Metallk., 64:458, 1973.

XII Fridman EM, Kopetskii ChV, Shvindlerman LS. Zt. Metallk.,

66:533, 1975.

XIII Straumal BB, Sursaeva VG, Shvindlerman LS. Phys. Met.

Metalloved., 49:102, 1980.

XIV Aristov VYu, Kopetskii ChV, Molodov DA, Shvindlerman LS.

Sov. Phys. Solid State, 22:1900, 1980.

XV Molodov DA, Kopetskii ChV, Shvindlerman LS. Sov. Phys. Solid

State, 23:1718, 1981.

XVI Molodov DA, Czubayko U, Gottstein G, Shvindlerman LS.

Scripta Metall. Mater., 32:529, 1995; Molodov DA, Czubayko U,

Gottstein G, Shvindlerman LS. Mat. Sci. Forum, 207–209:133, 1998.

XVII Gottstein G, Molodov DA, Czubayko U, Shvindlerman LS.

J. P hysique IV , C3, 5:89, 1995.

XVIII Walter JL, Dunn CG. Trans. AIME, 215:46, 1959.

XIX Molodov DA, Gottstein G, Heringhaus F, Shvindlerman LS.

Scripta Metall. Mater., 37:1207, 1997.

XX Molodov DA, Gottstein G, Heringhaus F, Shvindlerman LS.

Acta Mater., 46:5627, 1998; Molodov DA, Gottstein G, Heringhaus F,

Shvindlerman LS. 1997 NHF ML Annual Report, 165, 1998.

3.4 Measurement of Grain Boundary Mobility 197

3.4.3 Shape of a Moving Grain Boundary

Grain boundary motion during grain growth is controlled by the curvature of

the grain boundary, which, in turn, is determined by the shape of the grain

boundary. When considering the complex grain boundary geometry of poly-

crystals, this problem is simpliﬁed by the assumption that all grain boundaries

possess equal mobilities and surface tensions irrespective of the misorientation

of the grains and the crystallographic orientation of the boundary planes, and

thus the shape of all grains is considered to be uniform and allows us to de-

scribe the grain boundary curvature by only one parameter — the mean grain

size. However, when the shape of individual grains cannot be neglected, like

in bicrystals — and truly also in polycrystals — it is necessary to determine

the shape of a moving grain boundary. Besides, the shape of a moving grain

boundary is a source of very useful data with regard to the orientation de-

pendence of grain boundary surface tension and grain boundary mobility or

the interaction between a moving grain boundary and diﬀerent kinds of ob-

stacles. In the following we will restrict our consideration predominantly to

the steady-state motion of a single grain boundary.

The problem of the shape of a moving grain boundary was ﬁrst considered

by Mullins [242], and his equations provide a basis for all studies in this area.

As mentioned, the general case of grain boundary steady-state motion under

the inﬂuence of its own surface tension was considered in [256]. Although that

study was dedicated to a U -shaped grain boundary (half-loop), all principal

results can be successfully applied to any other type of steady-state grain

boundary motion as to the quarter-loop geometry and to grain boundary

systems with triple junctions (see below). In [256] a grain boundary was con-

sidered that moved steadily at a velocity V along the x-axis (Fig. 3.32). It

was assumed that the system was homogeneous through the thickness of the

bicrystal, i.e. independent of the z-axis, which in Fig. 3.32 is perpendicular to

the plane of the diagram. Hence, there is a quasi-two-dimensional situation,

as for all known experimental techniques of grain boundary migration stud-

ies where the driving force is generated by the surface tension of the curved

boundary. The coordinate system was attached to the steadily moving bound-

ary, i.e. all boundary elements were at rest with regard to this system. A slight

displacement of a boundary element parallel to its tangential plane was as-

sumed to cause no physical changes in the system and, hence, would not cause

an energy change or dissipation. Therefore, each boundary element is solely

under the inﬂuence of a force f normal to the boundary element plane. This

force may be due to two reasons. First, it may stem from the diﬀerence of the

chemical potentials of the atoms in the adjacent grains, caused by a tensorial

external thermodynamic force (e.g. a magnetic ﬁeld in a magneto-anisotropic

material). Second, it may be due to the grain boundary energy dissipated

when the grain boundary area is reduced. The ﬁrst force component, f

,is

the same for all boundary elements, whereas the second component, f

,isre-

lated to the velocity v of boundary motion normal to the boundary element.

198 3 Grain Boundary Motion

a/2

y(x)

a/2

y(x)

a/2

FIGURE 3.32

Grain Boundary shape y(x) for the half-loop geometry. In steady state the

grain boundary is displaced as a whole with velocity V due to the action of

grain boundary curvature, which displaces each surface element of the bound-

ary with normal velocity v.

When the motion of the boundary is considered under the eﬀect of capillary

forces only, f

= 0. The relationship m = v/f

shows that the boundary

element mobility m is the physical property of the element.

In the following we will derive the shape of a grain boundary during steady-

state motion under the action of a dissipative force f

= f. For simplicity the

coordinate system is attached to the boundary, i.e. moves together with the

boundary. If the boundary runs straight through the thickness of a specimen,

its shape is given by the 2D shape function y = y(x) on the surface. In steady

state the free energy assumes a minimum and, therefore, an inﬁnitesimal vir-

tual displacement δn perpendicular to the boundary will not change the free

energy. Such displacement, however, will alter the area and thus the total en-

ergy of the boundary and dissipate the driving force in the swept area, i.e. in

δG = δ



γds +



fδnds (3.105)

where s is the line element of the shape curve y(x), which can be expressed

by r(t)= (x(t),y(t)). With ˙x = dx/dt,˙y = dy/dt

ds =

˙x

+˙y

dt; (3.106)

δnds = δxdy − δydx =(˙yδx − ˙xδy) dt;

3.4 Measurement of Grain Boundary Mobility 199

Diﬀerentiation yields the Lagrange equations

−f ˙y =

∂

∂ ˙x



˙x

+˙y



(3.107)

f ˙x =

∂

∂ ˙y



˙x

+˙y



Introducing the angle Θ between the normal to the boundary element and the

direction of motion (Fig. 3.32) (sinΘ = ˙x/

˙x

+˙y

, cosϕ =˙y/

˙x

+˙y

)we

obtain the boundary shape

x(Θ) =





γ +

dθ



cos Θ

dΘ (3.108)

y(Θ) =





γ +

dθ



sin Θ

dΘ

Since the displacement rate V of the boundary and the migration rate v of a

boundary element perpendicular to its plane (referred to as normal velocity

in the following) are related by the angle Θ (Fig. 3.32)

v = V cos Θ (3.109)

also

f = f

V cos Θ

(3.110)

Substituting Eq. (3.110) into Eq. (3.108), we obtain with the reduced mobility

= m



γ +

dΘ



(3.111)

x =



dΘ (3.112)

y =



tan ΘdΘ

The shape of a boundary will be a smooth curve as long as the boundary

surface tension varies continuously. If, however, it changes discontinuously

with the angle Θ, as is inherent in crystalline interfaces, facetting of the moving

In Sec. 3.5.2 we will introduce the simpliﬁed version A

= m

γ for the reduced mobil-

ity, which assumes that the grain boundary energy does not vary notably with changing

inclination. In the case of facetting, as discussed here, however, γ changes strongly with

inclination.

200 3 Grain Boundary Motion

boundary will occur. If at a given value of the angle Θ = Θ

the derivative

changes by a step Δγ



, then at that point

dΘ

= −Δγ



δ (Θ − Θ

) (3.113)

Substituted into Eq. (3.112), it yields the following ﬁnite change of x and y

at a constant angle Θ = Θ

Δx =

(Θ

)Δγ



(Θ

)

(3.114)

Δy =

(Θ

)Δγ



(Θ

)tan Θ

This means a facet occurs, the length of which

ΔS

(Θ

)Δγ



(Θ

)

V cos Θ

(3.115)

depends on the given angle Θ

This facet length, however, is diﬀerent from the length of a facet on a

stationary (resting) boundary. Imagine a boundary that is pinned at its end

and subjected to a volume driving force f

. In analogy to the shape of a free

surface of a crystal, one obtains a facet if the grain boundary surface tension

changes discontinuously by Δγ



at inclination Θ

. The length of the facet

can be shown to be [257]

ΔS

=Δγ



(Θ

) /f

(3.116)

It is diﬀerent, however, from the facet length on a boundary moving under

the action of a driving force f

. For example, if Θ

and Θ

are two angles

atwhichanenergystepoccurs,thentherelation between the facet lengths

reads with Eqs.(3.115) and (3.116)

ΔS

(Θ

)cos Θ

(Θ

)cos Θ

(3.117)

Note that the length of the facet at steady-state motion depends on its mo-

bility and orientation relative to the direction of motion. The steady-state

velocity of a U -shaped grain boundary (grain boundary half-loop technique

(N9 in Table 3.3)) is

V =



π/2

−π/2

dΘ (3.118a)

In experiments with grain boundary half-loops the steady-state velocity and,

to a minor extent, the boundary shape depend on the half-loop width (driving

force), the grain misorientation angle, the orientation of the half-loop as a

3.4 Measurement of Grain Boundary Mobility 201

whole, i.e. on the direction of motion, on temperature, on material purity,

and so on

In general, the reduced mobility of a boundary element depends on its

orientation. To take that dependence into account, let us count the angle Θ in

Eq. (3.118a) from a given ﬁxed direction in the crystal, which may be deﬁned

by an angle β with regard to the x-axis of the coordinate system in Fig. 3.32.

V =



β+π/2

β−π/2

dΘ (3.118b)

i.e. the velocity of the half-loop motion may depend on its orientation. If the

reduced mobility of the boundary element does not depend upon the velocity

of normal motion, i.e. in the case of a linear relationship between the velocity

and the driving force

, the dependence of the steady-state velocity upon the

half-loop orientation is determined by the diﬀerence between the mobilities of

the straight (stationary) boundary sections.

dβ

(β + π/2) − A

(β − π/2)

(3.119)

If the crystal lattice has an inversion symmetry, then A

(β + π/2) = A

(β − π/2), and the velocity would not depend on the half-loop orientation.

In this case, experiments on half-loop migration determine the reduced mo-

bility, averaged over all directions present in the half-loop. It is stressed that

the averaged mobility thus determined basically diﬀers from the so-called av-

erage grain boundary mobility, obtained by studying the kinetics of grain

growth in a polycrystal, since in a polycrystal the average is taken over diﬀer-

ent boundaries, which diﬀer not only in orientation, but also in misorientation.

Averaging in a polycrystal is a statistical averaging; it is meaningful only for a

statistical grain boundary distribution, i.e. for a random texture in the sample,

a uniform distribution of the grain boundaries by misorientation parameters,

etc. These assumptions are not commonlysatisﬁedinexperiments,whichre-

sults in a relatively poor reproducibility of polycrystal properties. Contrary to

the averaging in polycrystalline experiments, the averaging by Eq. (3.118) is a

physical averaging, which remains unchanged from experiment to experiment.

The orientation-averaged boundary mobility turns out to be dependent on the

characteristics of the system, such as the misorientation between the grains,

temperature, pressure, purity of the materials used, etc. Consequently, despite

its average character, the evidence obtained by the techniques of single grain

Of course, all principal consequences pertaining to the grain boundary half-loop hold for

the steady-state migration of a grain boundary quarter-loop as well (technique N11, Table

3.3).

It should be noted that the anisotropy of the reduced mobility is usually neglected. A

nonlinear dependence of the velocity on the driving force can be due to an interaction of a

moving boundary with impurity atoms, as will be discussed below.

202 3 Grain Boundary Motion

-5 15 35 55 75 95

-07

-06

-05

-04

-03

1E-011

1E-010

1E-009

1E-008

1E-007

-5 15 35 55 75 95

015355575

Θ [deg]

-7

-8

-9

-11

-10

/s]

-5 15 35 55 75 95

-07

-06

-05

-04

-03

1E-011

1E-010

1E-009

1E-008

1E-007

-5 15 35 55 75 95

015355575

Θ [deg]

-7

-8

-9

-11

-10

/s]

FIGURE 3.33

Dependence of the reduced mobility of a half-loop 86

◦

11

20 tilt boundary in

a Zn bicrystal vs. the half-loop inclination angle.

boundary migration studies discussed conveys important physical information

on the kinetic properties of grain boundaries and reveals speciﬁc diﬀerences

between dissimilar boundaries.

In crystal lattices that do not possess an inversion symmetry, the half-

(or quarter-)loop velocity may depend on its orientation relative to speciﬁc

crystallographic directions. This dependence was indeed discovered for the

migration of 11

20 tilt boundaries in zinc with a misorientation angle of 86

◦

(Fig. 3.33). However, such a dependency may also be due to impurity drag,

when the dependence between velocity and driving force becomes nonlinear

due to breakaway of the boundary from adsorbed impurities. To answer the

question as to whether the observed dependence was due to the absence of an

inversion symmetry or to the nonlinearity due to breakaway eﬀects, one has

to consider the motion of a grain boundary half- (or quarter-) loop in a grain

boundary system with impurities [243]. In the case of the results presented in

Fig. 3.33, the half-loop velocity was high and the free boundary moved much

faster than the impurity-loaded boundary. In such a case Eq. (3.119) holds

and, therefore, the velocity was determined by the impurity free boundary seg-

ments, mutually related by a 180

◦

rotation, whereas the role of non-linearity

was insigniﬁcant [247].

In the following we consider the motion of a grain boundary half- (or

quarter-) loop in detail, using the approximation of “L¨ucke-Detert” [193]. Far

away from the vertex, the two branches of the boundary will be planar, and

their planes are parallel to each other and perpendicular to the plane of the

diagram. This renders the problem quasi-two-dimensional. The velocity of

3.4 Measurement of Grain Boundary Mobility 203

normal motion of a boundary element is [242]

v =

(3.120)

where R is the radius of curvature of the boundary, which changes along

the boundary and, therefore, the velocity v varies along the boundary. Under

steady-state conditions the shape of the half- (or quarter-) loop does not vary

with time and

v = V cos Θ = Vy





1+(y



)



−1/2

(3.121)

where the function y = y(x) describes the shape of the boundary in a co-

ordinate system carried along by the boundary, y



≡ dy/dx. We shall only

consider the top part of the half-loop, i.e. y ≥ 0 (for reasons of symmetry the

shape of the bottom part is the mirror image of the top part).

The shape of the boundary during steady-state motion is given by

Eqs. (3.120) and (3.121), and the well-known equation for the radius of cur-

vature R = −y





1+(y



)



3/2



= −

γm





1+(y



)



(3.122)

under three boundary conditions (Fig. 3.32)

y(0) = 0 (3.123)

y(∞)=a/2



(0) = ∞

The third boundary condition in Eq.(3.123) relates to a half-loop only. For a

quarter-loop the boundary condition (Fig. 3.34) should be substituted by



(0) = tan

Θ (3.124)

where the angle

Θ is deﬁned by the surface tensions of the grain boundary

and the free surfaces on the side of the sample [258]. Let us consider the shape

of the boundary in the quarter-loop technique (Fig. 3.34). The free energy of

the system in quasi-two-dimensional approximation can be represented as the

sum of boundary and surface energies

G =



b−a



γδ

1+(y



)

+2Δγ



dx+Δγ

δ+c ≡



b−a

Udx+W (3.125)

where Δγ

= γ

− γ

,Δγ

= γ

T 1

− γ

T 2

are the diﬀerences between the

speciﬁc free energies on the two sides of the boundary on the top and side

surface of the sample, respectively (Fig. 3.34); c is some constant. Taking into