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

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274 3 Grain Boundary Motion

grain a: -34°[A]

grain d: -42°[D]

Al - rolled 30%

Matrix: (011)[100]

RDR D

(a)

(b)

grain a: -34°[A]

grain d: -42°[D]

Al - rolled 30%

Matrix: (011)[100]

RDR D

(a)

(b)

FIGURE 3.97

Examples of two growing grains in Al with equivalent orientation relationships

to the consumed grain but diﬀerent shapes (a); the orientation of the shovel

cut (b).

<hkl

<hkl>

<uwv

<uwv>

<hkl

<hkl>

<uwv

<uwv>

direction

of motion

direction of motion

ϕ<hkl>

tilt grain boundary

pure tilt

boundary

mixed tilt-twist

boundary

(a) (b)

<hkl

<hkl>

<uwv

<uwv>

<hkl

<hkl>

<uwv

<uwv>

direction

of motion

direction of motion

ϕ<hkl>

tilt grain boundary

pure tilt

boundary

mixed tilt-twist

boundary

<hkl

<hkl>

<uwv

<uwv>

<hkl

<hkl>

<uwv

<uwv>

direction

of motion

direction of motion

ϕ<hkl>

tilt grain boundary

pure tilt

boundary

mixed tilt-twist

boundary

(a) (b)

FIGURE 3.98

Sketch of a bicrystal with initially straight symmetrical ϕhkl tilt grain

boundary, which moves under a capillary driving force as (a) a pure tilt and

(b) a mixed tilt-twist boundary [311].

3.5 Experimental Results 275

grain I

grain II

<hkl>

<uvw>

directions of motion

<hkl>

FIGURE 3.99

Motion of a curved tilt boundary with diﬀerent sets of boundary planes. The

straight section of the boundary is asymmetrical with an inclination from a

symmetrical position [311].

Fig. 3.98 was investigated. The initial straight grain boundary in the investi-

gated bicrystals was mostly a symmetrical tilt boundary. The curved boundary

in the Fig. 3.98a conﬁguration keeps its tilt character, and only the inclination

of the boundary plane changes in the curved part. By contrast, the charac-

ter of the curved boundary in the conﬁguration of Fig. 3.98b changes along

the boundary and becomes mixed with increasing twist and decreasing tilt

components. The motion of such a mixed tilt-twist grain boundary was in-

vestigated. Since the moving curved part of the boundary consists of diﬀerent

boundary planes, it is impossible to study directly the eﬀect of grain bound-

ary inclination on its mobility by utilizing the grain boundary curvature as a

driving force. That is why in experiments [311] the orientation of the whole

moving curved boundary was changed by an inclination of the initial part of

the boundary from its symmetrical position (Ψ > 0, Fig. 3.99). Therefore, the

motion of diﬀerent sets of boundary planes for the same ϕhkl tilt bound-

ary can be measured and compared. The motion of 34

◦

111 and 36

◦

111

boundaries with asymmetrical straight tilt boundary having an inclination an-

gle Ψ up to 7

◦

(Fig. 3.99) was studied as well. The higher the Ψ, the greater

the diﬀerence in the orientations of the corresponding curved sections. If the

boundary mobility depends on boundary inclination, this would aﬀect the ve-

locity of steady-state motion of a curved boundary.

The results of measurements of the motion of 111 tilt boundaries in

opposite directions are shown in Figs. 3.100 and 3.101. As seen, practically

no diﬀerence has been found either in the case of an asymmetrical initially

straight boundary (Fig. 3.100), or in the case of an asymmetrical boundary

with inclination angles Ψ = 5.5

◦

orΨ=7.1

◦

(Fig. 3.101).

Another feature which reﬂects both the kinetic and thermodynamic prop-

erty of the grain boundary during steady-state motion is the shape of a mov-

ing grain boundary (see Sec. 3.4.3). Pure twist boundaries are planar bound-

aries. By deﬁnition, it is impossible to measure their motion by utilizing grain

276 3 Grain Boundary Motion

1.1 1.2 1.3 1.4 1.5

1/T (10

/K)

-10

-9

-8

-7

[

]

400

450

500550

T [°C]

36.8°<111>;

FIGURE 3.100

Reduced boundary mobility vs. temperature for the motion of a 36.8

◦

111

tilt boundary with a symmetrical (Ψ = 0) straight boundary section [311].

1.1 1.2 1.3 1.4 1.5

1/T (10

/K)

-10

-9

-8

-7

[

]

400

450

500550

T [°C]

36.3°<111>;

=5.5°

1.1 1.2 1.3 1.4 1.5

1/T (10

/K)

-10

-9

-8

-7

[

]

400

450

500550

T [°C]

34.3°<111>;

=7.1°

1.1 1.2 1.3 1.4 1.5

1/T (10

/K)

-10

-9

-8

-7

[

]

400

450

500550

T [°C]

36.3°<111>;

=5.5°

1.1 1.2 1.3 1.4 1.5

1/T (10

/K)

-10

-9

-8

-7

[

]

400

450

500550

T [°C]

34.3°<111>;

=7.1°

FIGURE 3.101

Temperature dependence of the mobility of two curved tilt grain boundaries

with asymmetrical straight boundary sections. Open and ﬁlled symbols indi-

cate the boundary motion in opposite directions [311].

3.5 Experimental Results 277

100 µm

0 200 400 600 800 1000

100

200

300

400

500

600

Y [µm]

X [µm]

(a) (b)

100 µm

0 200 400 600 800 1000

100

200

300

400

500

600

Y [µm]

X [µm]

(a) (b)

FIGURE 3.102

(a) Measured and (b) calculated shape of a moving 40.6

◦

111 mixed tilt-twist

boundary in pure Al (99.999%) at T = 602

◦

C (solid line is measured shape,

dotted line is calculated shape) [311].

boundary curvature as a driving force. However, it is possible to study the

motion of mixed boundaries which comprise both tilt and twist components

(Fig. 3.98b). In such a conﬁguration the boundary character changes from

pure tilt to almost pure twist along its curved part, although the boundary

retains the same misorientation angle ϕ and axis hkl of rotation. In [311] the

shape of a “mixed” grain boundary which comprises both tilt and twist com-

ponents has been studied (Fig. 3.98b). In such a conﬁguration the boundary

character changes from pure tilt to almost pure twist along its curved part,

although the boundary retains the same misorientation angle ϕ and axis hkl

of rotation (Fig. 3.102). The theoretical basis of such examination is given by

Eqs. (3.122), (3.148)–(3.150).

The consistency of measured and calculated grain boundary shape (Fig.

3.102) allows us to conclude that diﬀerent elements of the investigated bound-

ary have the same reduced mobility, irrespective of grain boundary character,

i.e. their composition of tilt and twist components. That means an increase in

the twist component along a curved mixed boundary in such geometrical con-

ﬁguration does not aﬀect its steady-state motion. The motion of 111 mixed

boundaries was measured in [311] in the angular misorientations interval be-

tween 37

◦

and 42

◦

in the vicinity of the special Σ7 orientation. It was shown

that both activation parameters of grain boundary migration — the enthalpy

of activation and the pre-exponential factor — changed non-monotonically

with a minimum at the Σ7 misorientation and very similar to the respective

dependencies for a 111 tilt grain boundary.

278 3 Grain Boundary Motion

3.5.9.3 Motion of Flat Boundaries

In the previous sections we discussed the dependency of grain boundary mobil-

ity on crystallography and impurity content on results obtained in experiments

where the motion of curved grain boundaries was studied. The only excep-

tions were the experiments of Aust and Rutter, where planar grain boundaries

moved under driving forces provided by the striated microstructure (sub-

boundaries) [180]–[182]; however, the boundaries were not perfectly planar,

and the striations may have caused other diﬃculties (Chapter 4).

A curved grain boundary implies that its structure changes the boundary,

since it is composed of diﬀerent grain boundary planes. The mobilities ob-

tained cannot, therefore, be related to a speciﬁc grain boundary structure.

However, a planar boundary can be forced to move, if a bicrystal with grains

that have some orientation-dependent property, like elastic constants or mag-

netic susceptibility, are utilized. Under the impact of a respective directed

external ﬁeld the property anisotropy will generate a free energy diﬀerence

between adjacent grains that creates a driving force for grain boundary dis-

placement. This driving force does not depend on boundary properties and

moves a boundary from the grain with lower free energy toward the one with

higher free energy. In particular, such conditions can be obtained by the ac-

tion of a magnetic ﬁeld on a bicrystal of a material with anisotropic magnetic

susceptibility.

The origin of the driving force for grain boundary migration in a magneti-

cally anisotropic material was considered by Mullins [185]. The expression for

the driving force, as applied to bismuth, reads

P = μ

Δχ



cos

− cos



(3.178)

where H is the magnetic ﬁeld strength, Δχ the diﬀerence of the susceptibili-

ties parallel and perpendicular to the trigonal axis, Θ

and Θ

are the angles

between the magnetic ﬁeld and the trigonal axes in both grains of the Bi

bicrystal.

Several eﬀorts were made in the past to utilize a magnetic ﬁeld for the

study of grain boundary kinetics in Bi. The orienting eﬀect of a magnetic

ﬁeld on the solidiﬁcation and grain growth of Bi was investigated by Goetz

[326] and Mullins [185], respectively. Fraser et al. [327] showed that a grain,

which was generated by recrystallization following a local deformation of a

Bi single crystal, grew and consumed the initial single crystal under the ac-

tion of a magnetic driving force. However, no speciﬁc boundary motion was

investigated. Since the available magnetic driving force was rather small, the

magnetic ﬁeld in the experiments of Mullins and Fraser et al. was superim-

posed on growing grains, i.e. a magnetic force was applied to boundaries that

were already moving under the force exerted by their surface tension and cur-

vature.

In a recent investigation grain boundary motion was measured by apply-

ing a magnetic ﬁeld to a specially prepared bicrystal, i.e. to a system that

3.5 Experimental Results 279

was completely in equilibrium without an applied magnetic ﬁeld [189]. The

experiments were carried out on bicrystals of high purity (99.999%) bismuth.

The 90

◦

112 boundary was examined. The misorientation angle between the

trigonal axes in both crystals of the bicrystal was chosen to be 90

◦

in order

to gain the maximum possible magnetic driving force (Eq. (3.178)). The in-

vestigated boundary was a mixed 90

◦

112 boundary with a deviation of the

boundary plane from the ideal tilt position of 15

◦

±2

◦

(twist component) due

to the systematical deviation from the pure tilt position by 13 − 17

◦

during

crystal growth [189, 328]. Prior to their exposure to a magnetic ﬁeld (the ﬁeld

strength used was 1.63 · 10

A/m) the samples were annealed for 10 hours at

230

◦

C in vacuum. To ensure suﬃcient boundary mobility the magnetic ﬁeld

was imposed on the samples at a temperature of 255

◦

C (0.97T

The experiments unambiguously conﬁrmed that grain boundaries in Bi

bicrystals actually moved under the action of a magnetic driving force (Fig.

3.103). The observed linear dependence of boundary displacement on anneal-

ing time proves the free character of its motion (Fig. 3.104). It was found that

after the application of the magnetic ﬁeld the initially inclined grain bound-

ary plane changed its position and inclination as illustrated in Fig. 3.105 to

become a planar grain boundary perpendicular to the free surfaces, which

minimized the boundary area. This planar boundary was an asymmetrical

(near tilt) grain boundary, inclined 45

◦

to the symmetrical boundary plane.

The rotation of the boundary plane during the initial stage of its movement

may be explained by the action of an additional driving force for boundary

motion, provided by the boundary surface tension. Evidently, this additional

driving force was not suﬃcient to move the grain boundary during annealing

without a magnetic ﬁeld, but it forced the boundary into a position with min-

imum surface area once it started moving under the action of the magnetic

ﬁeld.

To prove that boundary motion was caused exclusively by the magnetic

driving force, the experiment was carried out in two diﬀerent ways. First, a

specimen was mounted in a holder such that the c-axis (111)ofcrystal1

was directed parallel to the ﬁeld (Fig. 3.105a). The 111 axis in crystal 2

in this case was perpendicular to the ﬁeld, and the grain boundary moved

in the direction of the latter crystal due to its higher magnetic free energy.

Second, a specimen was mounted in a position where the axis 111 in crystal

2 was close to the ﬁeld direction, and the corresponding axis in crystal 1 was

perpendicular to the ﬁeld. The direction of boundary motion in this case was

opposite, from crystal 2 toward crystal 1 (Fig. 3.105b). This result provides

unambiguous evidence that the grain boundaries in the bicrystals were forced

to move by the magnetic driving force only. In addition, some bicrystals were

annealed in a magnetic ﬁeld in both positions, and boundary motion in the

opposite direction was observed in the same specimen depending on its posi-

tion with regard to the magnetic ﬁeld.

The grain boundary mobility m is given by the ratio of velocity v and

driving force P , m = v/P. The measurement of boundary motion under a

280 3 Grain Boundary Motion

FIGURE 3.103

Displacement of a ﬂat grain boundary in a Bi bicrystal during annealing in a

magnetic ﬁeld. A and B are the initial and ﬁnal positions of the boundary.

0 100 200 300 400

t [s]

[

]

0 100 200 300 400

<111> in growing grain perpendicular

<111> in growing grain close

to direction of motion

<111> in growing grain perpendicular

to direction of motion

<111> in growing grain close

to direction of motion

0 400100 200 300

t [s]

Δl/P [10

-4

/J]

0 100 200 300 400

t [s]

[

]

0 100 200 300 400

<111> in growing grain perpendicular

<111> in growing grain close

to direction of motion

<111> in growing grain perpendicular

to direction of motion

<111> in growing grain close

to direction of motion

0 400100 200 300

t [s]

Δl/P [10

-4

/J]

FIGURE 3.104

Normalized displacement of a grain boundary vs. annealing time for the same

grain boundary moving in opposite directions.

3.5 Experimental Results 281

<111>

crystal 1

crystal 1 crystal 2

crystal 2

grain boundary

(a)

(b)

<111>

crystal 1

crystal 1 crystal 2

crystal 2

grain boundary

(a)

(b)

FIGURE 3.105

Geometry of investigated bicrystals and sense of driving force P with regard

to direction of the magnetic ﬁeld H.(a)Theaxis111 in the crystal 1 is

directed parallel to the ﬁeld. (b) The axis 111 in the crystal 2 is directed

parallel to the ﬁeld, while the axis 111 in the crystal 1 is perpendicular to

the ﬁeld.

constant magnetic driving force provides a unique opportunity to determine

the absolute value of grain boundary mobility. In experiments with curved

grain boundaries only the reduced mobility A

= m

γ,whereγ is the not

exactly known boundary surface tension, can be obtained. For the driving

force provided by a striation microstructure, both the surface tension of the

low-angle grain boundaries, which form the substructure, and their density

are unknown. Also, the average surface tension of the subgrain boundaries

and their density may vary along the way of the moving high-angle grain

boundary in the sample. The mobility of the boundaries investigated in Bi

bicrystals was found to be rather high: on average m

≈ 1.1 · 10

−7

/J·sat

250

◦

C(=0.97T

), which results in a very high velocity v ≈ 40 μ m/s at a

comparably small driving force P ≈ 350 J/m

. For comparison, the driving

force acting on a curved boundary with a radius of curvature of ≈ 0.5 mm,

which is typical for experiments on curvature driven boundaries in bicrystals

of high purity aluminum, is ≈ 10

J/m

(assuming γ ≈ 0.5J/m

), and the

boundary mobility at 480

◦

Cism

480

◦

≈ 5 ·10

−7

/J ·s [190]. The boundary

mobility in high purity aluminum has not been measured at a homologous

temperature T/T

=0.97, as used in the current experiment with Bi but an

extrapolation of experimental data to this temperature (which is T = 630

◦

for Al) would yield a boundary mobility m

630

◦

≈ 5 · 10

−5

/J · s.

In contrast to the symmetric tilt boundary, for the asymmetric tilt boundary

the measured boundary mobilities were found to be distinctly diﬀerent for mo-

282 3 Grain Boundary Motion

tion in opposite directions (Fig. 3.106). For the chosen crystallography of the

bicrystals the boundary was less mobile when the c (111)-axis in the growing

grain was perpendicular to the direction of motion (m

⊥

=8.0 · 10

−8

/J·s)

and moved faster when the trigonal c-axis in the growing grain was close to

the direction of motion (m

=1.3 · 10

−7

/J·s). There are several potential

reasons for this anisotropy. First, there is an essential diﬀerence in the dis-

tance between the crystallographic planes on each side of the boundary. An

estimation shows that this factor may change the velocity of grain bound-

ary motion; however, this factor is unlikely to change the velocity by more

than 20%, which is distinctly less than the observed eﬀect. Second, because

grain boundary motion in Bi bicrystals may be inﬂuenced by impurity drag,

the diﬀerence in the diﬀusivity of impurities in two opposite direction in the

anisotropic structure of Bi should be taken into consideration. In this respect

it is interesting that the symmetric tilt boundary exhibited a much higher

mobility than the asymmetric tilt boundary and did not show a dependence

of boundary mobility on the sense of motion (Fig. 3.106).

It is ﬁnally noted that a ﬂat grain boundary can also be moved by the

application of other external ﬁelds, e.g. an electrical ﬁeld in anisotropic di-

electric materials or an elastic stress in elastically anisotropic materials. The

latter has been studied extensively by experiments [341]–[357] and computer

simulations [358]. In this context it was proposed that the motion of grain

boundaries is always associated with a shear deformation. This is the subject

of current research

3.5.9.4 Generation of electical currents and magnetic ﬁelds by

grain boundary motion

The motion of a grain boundary in a magnetic ﬁeld can be considered as a

motion of a conductor in a magnetic ﬁeld, which causes an electromotive force

in accordance with the Faraday eﬀect. Such approach was put forward and

has been considered comprehensively in [340]. It is evident that the struc-

ture of a grain boundary including the electronic structure is diﬀerent from

the structure of the interior of the adjacent crystals. Thus, in principle, the

migration of a grain boundary can also be considered as the motion of an

electrical conductor with electromagnetic properties diﬀerent from the bulk.

The motion of a conductor in a magnetic ﬁeld causes an electromotive force

(Faraday, 1831), i.e. dissipates energy.

In a microscopic model a grain boundary is assumed to migrate by the

exchange of lattice sites across the boundary. In the mesoscopic model the

grain boundary is associated with a volume of diﬀerent magnetic suscepti-

bility. The atomic exchange will take a small, but nevertheless ﬁnite time τ.

In the course of the time τ the magnetic induction changes in the volume

where the atomic rearrangement takes place. This leads to the generation

of an electric ﬁeld in accordance with Faraday’s law. The induced electrical

current generates its own magnetic ﬁeld H

(Biot-Savart-Laplace law). The

3.5 Experimental Results 283

directions of the induced current and the magnetic ﬁeld H

are determined

by the Lenz rule in such a way that the generated magnetic ﬂux counteracts

the imposed magnetic ﬁeld, which corresponds to an electromotive force. As

a result the magnetic driving force decreases. This can be expressed as a re-

duction of grain boundary mobility m.

The migration process can be described as a displacement of a grain bound-

ary area S ≈ r

by a distance r with the velocity v ≈

. The change of the

magnetic ﬂux ΔΦ which occurs in a time τ through the area r

normal to

the direction of the magnetic ﬁeld with the strength H can be estimated as

ΔΦ ≈ ΔχHr

,whereΔχ is the diﬀerence of the magnetic susceptibilities of

adjacent grains along the magnetic ﬁeld direction.

This induces an electromotive force

ind

≈

ΔΦ

cτ

ΔχHr

cτ

(3.179)

(c — speed of light) and, in turn, the magnetic ﬁeld

ΔχH ≡

ΔχH (3.180)

where σ is some eﬀective conductivity of a nano-region in close proximity to

the grain boundary for the time of its rearrangement. It is conceivable that

this conductivity is diﬀerent from the bulk conductivity. The characteristic

time τ

σr

has a physical meaning. It is the time for attenuation of elec-

tromagnetic perturbations in the volume V ≈ r

. The eﬀect can be considered

in terms of a reduction in the driving force of grain boundary motion, or as

a reduction in grain boundary mobility m. This conclusion as well as the

derivation is quite general and may be applied to any moving interface. In

particular, it can be applied to the asymmetry of grain boundary mobility for

grain boundary motion in opposite directions in a magnetic ﬁeld as experi-

mentally observed in Bi. So, for the motion of an asymmetrical grain boundary

in opposite directions the parameter β =

Δm

should be essentially diﬀerent

where m is the mobility at zero ﬁeld. The only exception is a symmetrical

grain boundary where β must be exactly the same. The induced perturbation

will eventually be converted to Joule heat Q which contributes additionally

to the grain boundary slowdown. However, the calculations show that this

contribution is negligible.

As an alternative approach let us consider the motion of a grain boundary

with constant velocity v. The induced magnetic ﬁeld H

and the dragging

force F can be expressed as

 =

σv

ΔχH (3.181)

F = H

H