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

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184 3 Grain Boundary Motion

approximations should be used [236]. The driving force for the reversed-

capillary method is given by

P = γ/R =

f(α) (3.102)

where R is the radius of boundary curvature.

An example is the circular arc shape [236] (Fig. 3.26). To solve the equa-

tion which describes the shape of a moving boundary in a two-dimensional

sample, an assumption of shape invariance, i.e. a scaling behavior, must be

introduced [234, 235]. If the (2D) shape of the boundary is given in polar

coordinates (r, Θ)

f(α)=

dβ

dΘ

Θ=0

◦

(3.103)

where β is the inclination of the tangent at a given point of the grain boundary

(Fig. 3.26). As shown in [242] this approach does not take into account the

drag eﬀect of mobile obstacles, e.g. impurity atoms, mobile particles or surface

grooves [208, 210, 225]. (This problem will be addressed in Sec. 3.4.3.)

The important disadvantage of the reversed-capillary technique is the lack

of steady-state motion of a grain boundary. This disadvantage is avoided by

the half-loop technique, since the driving force remains constant (Fig. 3.25c,d).

The grain boundary half-loop (Fig.3.25d) or quarter-loop (Fig. 3.25c) moves

as a whole, and its shape remains self-similar during migration [227, 244]. The

average driving force on a half-loop (h.l.) [227, 243, 245] or on a quarter-loop

(q. l.) [244, 246] is

h.l.

2γ

(3.104a)

q.l.

(3.104b)

respectively.

The contention is that the driving force for constant driving force techniques

is substantially smaller than for the reversed-capillary techniques [229]. How-

ever, as shown in [247], it is possible to grow bicrystals with half-loop grain

boundary geometry, where the size of the shrinking crystals does not exceed

∼ 0.1mm. It is particularly emphasized that in all capillary techniques, except

for half-loop geometry, the moving grain boundary is exposed to the free side

surface of the bicrystal. In certain cases this may cause drag eﬀects by the

free surface, as will be discussed in Sec. 3.5.

3.4.2.2 Techniques to Monitor Grain Boundary Migration

As mentioned above, the observation of grain boundary migration is com-

monly performed by investigation of the motion of a macroscopic segment

3.4 Measurement of Grain Boundary Mobility 185

FIGURE 3.27

Traces of a moving boundary in a Zn bicrystal (reversed-capillary technique).

of the grain boundary. The velocity v of grain boundary motion is deﬁned

as the displacement Δx of this segment normal to itself in a time interval

Δt : v =Δx/Δt. For this purpose the position of the segment has to be

located. There are two essentially diﬀerent ways to determine the position of

a grain boundary in crystals and to measure the velocity of its motion: the

continuous method and the discontinuous method.

The discontinuous method is most common. The location of the boundary

is determined by its intersection with the crystal surface after discrete time

intervals. The position of a boundary can be revealed by the groove, which

forms on sample cooling or by chemical etching of the crystal surface. The

boundary groove can easily be observed under an optical microscope. The

principal advantage of this method is its simplicity; the major shortcoming

is the necessity to average the measured boundary displacement over a large

period of time between consecutive observations.

One variation of this method consists of periodic cooling of the sample by

20–30

◦

C for a short time in the hot stage of a microscope [227, 245]. Such

procedure causes the formation of a “vacancy” groove along the line of inter-

section of the grain boundary with the free surface of the crystal: an excess

of thermal vacancies moves from the bulk of the crystals to the grain bound-

ary and along the boundary to a free surface. Since the observation of grain

boundary displacement occurs in this case “under a microscope” it is more

“controlled,” which substantially reduces the principal shortcoming of this

method [227, 236] (Fig. 3.27).

The continuous method requires determination of the boundary position at

any moment in time and thus requires automation of the procedure to locate

the boundary position. Correspondingly, the boundary has to be identiﬁed by

the discontinuity of a physical property associated with the change in crystal

orientation. There are various techniques to distinguish diﬀerent crystal ori-

entations such as reﬂection of polarized light [228, 248], photoemission [223],

X-ray topography [249, 250], X-ray diﬀraction [251]–[253], or backscattered

electron density [254]. The observation of grain boundary motion by orienta-

tion contrast is convenient and reliable, and does not only yield the location

186 3 Grain Boundary Motion

35 s

1850 s70 s

65 s

55 s

50 s

40 s

530 s

80 s

75 s

515 s

505 s

35 s

1850 s70 s

65 s

55 s

50 s

40 s

530 s

80 s

75 s

515 s

505 s

FIGURE 3.28

Recorded grain boundary migration in a Zn bicrystal.

of a moving grain boundary, but also its shape at any moment. In Fig. 3.28

individual video frames of grain boundary migration in zinc bicrystals are

given [228] as observed by polarized light in an optical microscope. Evalua-

tion of this recording provides all parameters to determine the grain boundary

mobility: time dependence of grain boundary displacement and radius of cur-

vature of a moving grain boundary at any moment. The characteristic time

resolution is about 10

−2

s, the spatial resolution is a few μm, the thickness of

the samples is insigniﬁcant [228]. Unfortunately, this technique is applicable

only to optically anisotropic materials, i.e. of material with lower symmetry

crystal structure, for instance, hexagonal crystals.

X-ray topography permits us to obtain an image of a grain boundary as

the interface between the contiguous grains. Synchrotron white beam X-ray

topography (SWBXRT) oﬀers excellent time resolution and thus can be used

for “in situ” studies of grain growth, recrystallization and grain boundary mo-

3.4 Measurement of Grain Boundary Mobility 187

FIGURE 3.29

Shape of a moving 37.5

◦

111 grain boundary in Al at 480

◦

C [268].

tion [249, 250]. SWBXRT allows a spatial resolution in the range of several

μm, the time resolution is about 1 s, the ﬁeld of view extends to 2–3 cm

the sample thickness can range between 0.1–1 mm [250]. However, the study

of grain boundary motion by SWBXRT is restricted to samples of high per-

fection: the dislocation density should not exceed 10

−2

.Moreover,the

principal shortcoming of this technique is its limited availability and thus lim-

ited access.

Photoemission microscopy also can be utilized for continuous tracking of

grain boundary migration [223]. However, this technique is restricted to metals

with high melting points. A recently developed elegant method is the investi-

gation of crystal shape evolution by backscattered electrons in an SEM, since

the contrast by backscattered electrons is very sensitive to orientation. The

method also allows for orientation evaluation by EBSD with the same set-up.

The capabilities of the EBSD technique have been increased substantially in

the past years. In particular, the migration and shape of the grain boundaries

can be measured in situ in an SEM utilizing the orientation contrast revealed

by an electron backscatter detector [266]. With a specially designed laser

powered heating stage in situ investigations of grain boundary motion can be

conducted at temperatures up to 1000

◦

C [267]. The SEM is equipped with

a digital image scanning system, which records series of orientation contrast

images in predetermined time intervals. In Fig. 3.29 the shape of a moving

grain boundary quarter-loop in Al at 480

◦

C is presented [268].

A simple, precise and very versatile technique to identify the position of

a grain boundary utilizes X-ray diﬀraction. X-ray diﬀraction is very sensitive

to the structure and orientation of crystalline material and, therefore, it is an

188 3 Grain Boundary Motion

excellent probe for diﬀerences in crystal structure or crystal orientation. The

principle of the method is illustrated in Fig. 3.30 [251, 253]. The bicrystal

is placed in a goniometer of the X-ray Interface Continuous Tracking Device

(XICTD) in such a way that one grain (I) is in the Bragg position, while

the other is not. The maximum intensity I

of the reﬂected X-ray beam is

measured as long as the X-ray spot is located solely on the surface of crystal

I. If the spot is located on the grain boundary, the intensity of the reﬂected

beam should be intermediate between I

and I

. When the boundary moves

the sample can be displaced accordingly so that the reﬂected X-ray inten-

sity remains constant during grain boundary motion. Thus, the velocity of

the moving grain boundary (generally speaking, the interface) is equal to the

speed of sample movement at any moment in the course of the experiment.

This procedure does not interfere with the process of grain boundary migra-

tion. The measurable velocity ranges from 1 to 1000 μms

−1

with a temporal

resolution of about ﬁve measurements per second. In Fig. 3.31 the displace-

ment vs. time diagrams are shown for a grain boundary in Al, which moves

under a constant driving force (Fig. 3.30) [255]. The hot stage of the XICTD

allows the sample to be heated up to 1200

◦

C in nitrogen or an inert gas atmo-

sphere to suppress oxidation and grooving; the temperature is kept constant

within ±0.3

◦

C. The accuracy of the velocity measurements of grain boundary

motion is better than 2%.

It is stressed again that any reliable information on grain boundary mobility

can be obtained only from a physically proper and reproducible experiment,

carried out on speciﬁc single grain boundaries with given crystallography and

deﬁned chemistry. Respective experiments have to comply with several re-

quirements: controlled or constant driving force of grain boundary migration,

its reproducibility and potential to change it over a wide range free of in-

terference with the process of grain boundary migration itself, feasibility of

manufacturing bicrystals free of fabrication defects. Only under these condi-

tions is it possible to observe “the free motion of a grain boundary,” which we

deﬁne as the motion of a single grain boundary, when the inﬂuence of drag

factors can either be neglected or can be accurately taken into account. Actu-

ally, the interaction of a moving grain boundary with point defects (impurity

atoms, vacancies), dislocations or bulk defects (second-phase particles, voids)

usually is taken into account, or such defects are eliminated by appropriate

procedures.

Generally speaking there are two groups of basically diﬀerent experimental

techniques of grain boundary migration studies (Table 3.3). The ﬁrst group

includes the experimental techniques, where the driving force is determined

by the diﬀerence of the free energy of adjacent grains but not aﬀected by the

geometry of the sample or by the shape and energy of the grain boundary.

The second group comprises the experimental techniques in which the driving

force is provided by the surface tension (free energy) of the grain boundary.

The techniques of the latter group are advantageous due to the high stability

of the driving force both with respect to time and under a change of tem-

3.4 Measurement of Grain Boundary Mobility 189

direction of motion

X-ray

source

intensity

detector

grain I

grain II

sample

boundary

direction of motion

X-ray

source

intensity

detector

grain I

grain II

sample

boundary

FIGURE 3.30

Principle of operation of the XICTD. The diﬀraction conditions generate an X-

ray intensity gradient across the boundary. The specimen is moved to maintain

a constant recorded intensity.

190 3 Grain Boundary Motion

0 20 40 60 80 100 120

time [s]

13650

13700

13750

13800

13850

13900

[

]

T=484°C

v=1.9 µm/s

0 20 40 60 80 100 120

time [s]

12850

12950

13050

13150

13250

13350

13450

[

]

T=504°C

v=4.1 µm/s

0 10 20 30 40 50 60 70 80 90

time [s]

12300

12400

12500

12600

12700

12800

12900

13000

13100

[

]

T=525°C

v=7.9 µm/s

0 10 20 30 40 50 60

time [s]

16700

16900

17100

17300

17500

17700

[

]

T=547°C

v=14.5 µm/s

0 5 10 15 20 25 30 35

time [s]

11600

12000

12400

12800

13200

13600

[

]

T=558°C

v=51.7 µm/s

0 5 10 15 20 25 30 35

time [s]

5000

5700

6400

7100

7800

8500

[

]

T=579°C

v=101.9 µm/s

T = 504°C

v = 4.1

m/s

100

200

300

400

500

0 20 12060 10080

T = 484°C

v = 1.9

m/s

T = 547°C

v = 14.5

m/s

60503020100

400

200

600

800

1000

time [s]

T = 579°C

v = 101.9

m/s

2250

750

3000

0 101520253035

time [s]

1200

800

400

2000

1600

0 5 10 15 20 25 30 35

T = 558°C

v = 51.7

m/s

time [s]

T = 525°C

v = 7.9

m/s

800

600

400

200

150

200

250

100

02040 4060 80 100 120

time [s]

01020

40 4050 60 70 80 90

position [

m] position [

m]position [

600

1500

3750

FIGURE 3.31

Typical XICTD recordings of grain boundary position vs. annealing time from

low (≈ 1 μm/s) to high (≈ 100 μm/s) migration rates.

3.4 Measurement of Grain Boundary Mobility 191

perature, its reproducibility and the feasibility to control the magnitude of

the driving force. One shortcoming of the techniques in this group is that

the inﬂuence of the free surface may interfere with the migration process (see

groove dragging, Sec. 3.3.6).

The techniques N11-13 and N9-10 (Table 3.3) are most advantageous. They

stand out because of the constant driving force, no eﬀect of the free sur-

face on grain boundary motion and the feasibility to describe analytically the

steady-state shape of the moving grain boundary. This permits us to ﬁnd a

relationship between the microscopic kinetic properties of the grain boundary

and the macroscopic properties of a grain boundary half- or quarter-loop (the

techniques N9 and N11, respectively), as will be considered below.

Regrettably, the techniques of grain boundary migration studies based on

a reduction of the boundary area have the disadvantage that it is impossi-

ble to investigate the motion of a planar boundary. A curved grain boundary

implies that its structure changes along the boundary, since it is composed

of diﬀerent grain boundary planes. The mobilities obtained can, therefore, be

related to a speciﬁc misorientation and boundary character (tilt), but not to

a speciﬁc grain boundary structure. A planar boundary can be forced to move

in a bicrystal comprised of grains with some orientation-dependent properties

like elastic constants or magnetic susceptibility. Unfortunately, only few ma-

terials show such strong anisotropy. An example is Bi, which exhibits a strong

anisotropy of its magnetic susceptibility [189], or some other hexagonal met-

als.

This raises the fundamental question whether it is at all possible to study

the mobility of grain boundaries in experiments with a curved boundary. The

general solution of grain boundary motion under the eﬀect of its own sur-

face tension was given in [256]. Among other issues, it was comprehensively

discussed what can be measured in experiments with a curved grain bound-

ary. As shown in [256], experiments on steady-state motion of a curved grain

boundary for half-loop or quarter-loop geometry, or in the case of the reversed-

capillary technique, determine the average mobility only, i.e. averages over all

boundary planes present. But, as stressed in [256], contrary to experiments on

polycrystals, this averaging is a physical averaging, the same from experiment

to experiment. Such averaged boundary mobility proves to be dependent on

misorientation between grains, on purity of the material used and on other

characteristics of the system, as will be presented in Sec. 3.5.

192 3 Grain Boundary Motion

TABLE 3.3

Bicrystal Techniques of Grain Boundary Migration

N Source of Experimental δ [cm] Metal Grain

driving force technique boundary

[Pa] displace-

ment

recording

1 Energy of low-angle

0.25 Pb A

grain boundaries

∼

4 · 10

= const.

2 Energy of high-angle

0.25–0.5 Al B

grain boundary

P = γ/R =

5 · 10

− 10

3Energyofdeformed 0.01 Au B

matrix

∼

− 10

4 Energy of high-angle

0.5 Al B

grain boundary

P = γ/R

∼

− 10

5 Energy of high-angle

0.1 Cu B

grain boundary

P = γ/R

∼

− 10

6 Energy of high-angle

0.1–0.2 Zn C

grain boundary

P = γ/R

∼

5 · 10

− 10

7 Energy of high-angle

0.2 Fe-Si A

grain boundary

P = γ/R

∼

8 · 10

− 2 · 10

8 Energy of high-angle

0.2 Fe-Si A

grain boundary

P = γ/R

∼

8 · 10

− 2 · 10

3.4 Measurement of Grain Boundary Mobility 193

N Time Time interval Depend- Criterion Ref.

dependence Δt(s) between ence

of grain consecutive V (P ) ρ

boundary measurements

displace- or speed of

ment recording

0.6 · 10

− – ρ =

Pδγ

[I]

1 · 10

2– ∼ 10

V ∼ Pρ= [II,III]

Rγ

∼

3δ

exp

=0.5 − 4

6 · 10

− – ρ =

Pγ

[IV]

1 · 10

exp

5 · 10

− 5 · 10

0.15 · 10

V ∼ Pρ=

Rγ

=[V]

−2 · 10

exp

∼

5– –V ∼ P –[VI]

speed of for the ρ

exp

∼

10 [VII,

recording: initial VIII]

1 frame jumps

per 3-15s V ∼ P

∼ 5 · 10

− V ∼ Pρ

exp

∼

5[IX]

2 · 10

∼ 2 · 10

V ∼ Pρ

exp

∼

[X]

10 − 15