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

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4.5 Experimental Investigations of Triple Junction Motion 345

In the case of steady-state motion of the entire boundary system the velocity

of the triple junction equals the velocity of the grain boundaries. Therefore,

the steady-state value of the angle Θ is determined by Eqs. (4.29) and (4.44)

2Θ

2cosΘ−

= Λ (4.45)

The dimensionless criterion Λ reﬂects the drag inﬂuence of the triple junction

on the migration of the system. One can distinguish two limiting cases:

a) Λ → 0: In this case the angle Θ tends to zero, i.e. the motion of the entire

boundary system is governed by the mobility of the triple junction and the

corresponding driving force. For the limit Θ = 0

◦

the velocity of the system

is given with Eq. (4.44) by:

v = m

(2γ − γ

) (4.46)

b) Λ →∞: In this case the angle Θ tends to assume its value at thermody-

namic equilibrium:

Θ=arccos



2γ



=Θ

(4.47)

The motion of the system is independent of the triple junction mobility and is

governed only by the grain boundary mobility and the corresponding driving

force. The velocity of the boundary system in this case with Eq. (4.29) and

(4.47) is given by

v =

2Θ

(4.48)

The two states of motion of the entire grain boundary system can be distin-

guished experimentally for a known ratio γ

/γ by measuring the contact angle

Θ.

Experimentally, this idea was ﬁrst realized in [183, 411]. The experiments

were carried out on zinc tricrystals with a grain boundary geometry as shown

in Fig. 4.5. The tricrystals were produced of high purity Zn (99.999 at%) by

a technique of directional crystallization in a high purity Argon atmosphere

(Fig. 4.7). The characteristics of the tricrystals studied are given in Table 4.1.

The orientations of the three adjacent grains of each sample were determined

by the Laue-technique, Electron Back Scatter Diﬀraction (EBSD) and by an

investigation of the fracture surface of a cracked sample. For the latter method

small cracks were induced by a sharp knife in the sample cooled by liquid ni-

trogen. The cracks propagated along the basal plane of each grain. Hence, the

misorientation could be determined by the orientation of the surface of the

cracks.

For measuring the migration rate and the geometry of the grain boundary

system during the motion temperatures a modiﬁed optical microscope operat-

ing with polarized light and a hot stage was used. An additional polarization

ﬁlter applied in the reﬂected beam allowed us to distinguish the diﬀerent

346 4 Thermodynamics and Kinetics of Connected Grain Boundaries

monocrystal

seed

bicrystal

seed

solidification

front

monocrystal

seed

solidification

front

bicrystal

seed

monocrystal

seed

bicrystal

seed

solidification

front

monocrystal

seed

bicrystal

seed

solidification

front

monocrystal

seed

solidification

front

bicrystal

seed

FIGURE 4.7

Method of tricrystal fabrication; the rectangular area represents the tricrystal

used for migration experiments.

orientations of the grains by the diﬀerent intensity of the reﬂected light. A

color video camera was attached to the microscope and connected to a video

cassette recorder to record the motion of the triple junction system during

the experiment. For each temperature the velocity of the triple junction sys-

tem, the angle Θ, and the width a of the vanishing grain were measured. For

this, single video frames were grabbed by a computer and in accordance with

Eq. (4.28) a computed shape of the grain boundary system was superimposed.

By ﬁtting the computed shape to the shape observed, the angle Θ and the

width a of the shrinking grain were determined.

TABLE 4.1

Misorientations of the Used Tricrystals

Sample GBI GBII GBIII

S1 82

◦

01

10 81

◦

01

10 4

◦

12

30

S2 61

◦

01

10 62

◦

01

10 3

◦

0001

S3 25

◦

10

10 23

◦

10

10 48

◦

10

10

S4 14

◦

10

10 16

◦

10

10 30

◦

10

10

Note: The numbering of the boundaries corresponds to Fig. 4.5.

4.5 Experimental Investigations of Triple Junction Motion 347

GBIII

GBII

GBI

(a)

(b)

GBIII

GBII

GBI

(a)

(b)

FIGURE 4.8

Sketch of an experiment to determine Θ

. (a) grain boundary system with

triple junction and notches to avoid a complete dissappearance of the grain

enclosed by GBI and GBII; (b) the same system after annealing at 655

◦

Unfortunately, this technique cannot be applied to tricrystals of Al. Early

in the development of this research the migration rate of a grain boundary sys-

tem with triple junction in Al was measured by the XICTD technique. (The

XICTD technique is discussed in greater detail in Chapter 3.) The main dis-

advantage of an X-ray-diﬀraction-based technique is the diﬃculty to visualize

the shape of a moving boundary. That is why in [436] the following procedure

was used. For each temperature the velocity V , the angle Θ and the width of

the shrinking grain α were determined. For this the sample was rapidly cooled

down after measurements of the migration rate at each temperature. The po-

sition of the grain boundary system and the angle Θ were recorded from the

grain boundary grooves. Before subsequent heating to measure the migration

rate the specimen was polished to remove the grooves. Measurements of Θ

during annealing close to the melting point were carried out by the following

experiment. A sample with grain boundaries was annealed at a temperature

close to the melting point (T = 655

◦

C) for 5 minutes. To avoid the complete

vanishing of the shrinking grain enclosed by GBI and GBII (4.8a) the sample

was notched. After annealing the sample was rapidly cooled down, and the

position of the grain boundaries and the angle 2Θ ≈ 2Θ

(Fig. 4.8b) were

measured by optical microscopy. To determine Λ for the samples SII and SIII

(Table 4.1) the grain boundary energy data calculated by Hasson and Goux

were used [184].

However, electron backscatter diﬀraction (EBSD) is nowadays widely used

for the investigation of grain microstructure evolution. Since the contrast by

backscattered electrons (BSE) is sensitive to crystallographic orientation, the

348 4 Thermodynamics and Kinetics of Connected Grain Boundaries

FIGURE 4.9

Heating stage for in situ measurements of grain boundary motion in an SEM

[430, 431].

shape of a moving grain boundary can, in principle, be visualized in an SEM

(see Chapter 3). The major diﬃculty is obtaining good orientation contrast

at elevated temperatures. A hot stage was speciﬁcally designed and fabricated

at the Institute of Physical Metallurgy and Metal Physics of RWTH Aachen

University, for the study of grain boundary migration in the SEM [430, 431].

To reduce the heat radiation and to protect the BSE detector the hot stage

was surrounded by double reﬂecting walls (Fig. 4.9). An Everhart-Thornley

backscattered electron detector was used to obtain the orientation contrast.

The hot stage allowed to heat the specimens up to 640

◦

C; the temperature

could be held constant within ±2

◦

. A video frame grabber system recorded

a sequence of BSE images with a scanning rate between 1 to 30 seconds per

frame. The recorded video frames contained the whole information to deter-

mine triple junction and grain boundary mobility and the magnitude of the

criterion Λ, the time dependency of the triple junction and boundary displace-

ment and the shape of the moving grain boundaries (Fig. 4.10).

The motion of four diﬀerent triple junctions (Table 4.1) with a grain

boundary conﬁguration as in Fig. 4.8 was investigated in the temperature

range between 330

◦

C and 405

◦

C [183]. The triple junctions of samples S1 and

S2 consisted of two nearly identical high-angle tilt grain boundaries (curved

boundaries, GB I and II) and a low-angle tilt boundary (straight boundary,

GB III). Due to the diﬀerent properties of low-angle grain boundaries and

high-angle boundaries, the triple junction systems of samples S1 and S2 were

symmetrical junctions as described above. Under the assumption that the

properties of a high-angle boundary vary only little with misorientations, the

4.5 Experimental Investigations of Triple Junction Motion 349

360°C

340°C

400°C

420°C

380°C

360°C

340°C

400°C

420°C

380°C

FIGURE 4.10

Set of recorded SEM images of steady-state motion at diﬀerent temperatures

of grain boundary systems with triple junction in Al [433].

350 4 Thermodynamics and Kinetics of Connected Grain Boundaries

behavior of the triple junction system of samples S3 and S4 can be consid-

ered as an ideal triple junction as referred to in [403]. For both types of triple

junctions the shape of the moving grain boundary system was similar to the

shape predicted by theory. Fig. 4.11 shows a series of video frames of a moving

symmetrical triple junction in a Zn tricrystal. The straight grain boundary

(GB III) is invisible due to the small orientation diﬀerence (3

◦

) of the adjacent

grains. The solid line in the lower right picture was computed in accordance

with Eq. (4.28) and ﬁt the shape of the curved grain boundaries quite well.

The small deviation between the theoretical and observed shape of the bound-

aries at the transition from the straight part of GB I and II to the curved part

may have been caused by facetting, i.e. a dependency of γ on the inclination

of the boundary, which was neglected in the derivation of Eq. (4.28). For the

ideal triple junction the same behavior was observed (Fig. 4.12). Because of

the diﬀerent ratio γ

/γ the angle Θ for the ideal junction was smaller than for

a symmetrical one. As can be seen from this consideration, there is no reason

to take an impurity inﬂuence into account.

For all samples the velocities v (Fig. 4.13) and the angles Θ (Fig. 4.14)

were found to be constant for a given temperature over the entire temperature

range investigated. Evidently, the assumption of a steady-state motion of the

entire grain boundary system was justiﬁed.

As obvious from Figs. 4.11 and 4.12, Θ increased with increasing temper-

ature. In particular for the symmetrical triple junction the change of Θ was

dramatic (Fig. 4.15). In accordance with the temperature dependence of Θ, the

criterion Λ, determined by Eqs. (4.30) and (4.45), was found to be constant for

a given temperature, but increased with increasing temperature (Figs. 4.16,

4.17). At low temperatures Λ was on the order of unity and increased with

rising temperature up to 3 orders of magnitude. For the calculation of Λ for

symmetrical triple junctions (samples S1, S2) the ratio γ

/γ was determined

under the assumption that for temperatures near the melting point the value

of Θ reaches the thermodynamic equilibrium value.

The drag eﬀect of a grain boundary triple junction was readily illustrated

in [434]. The authors undertook the attempt to compare “directly” the mo-

bility of a “free” grain boundary and a grain boundary system with triple

junction (Fig. 4.18). Actually, the boundary conﬁguration in Fig. 4.1 without

triple junction constitutes a grain boundary half-loop (Fig. 4.19).

Essentially, the procedure was as follows: the motion of a grain boundary

half-loop in Zn of 99.995% purity was studied. Independently, the motion of

practically the same half-loop attached to a ﬂat low-angle boundary (Fig. 4.18)

was investigated.“Practically the same” means that the incorporation of a low-

angle grain boundary changed the misorientation of curved grain boundaries

only negligibly. Hence, one can conduct a direct comparison between the mo-

tion of a grain boundary half-loop and the motion of practically the same

half-loop with a triple junction. In other words, we can determine the mo-

bility of a moving grain boundary system with and without triple junction.

It was mentioned above that the approach used for an analysis of a uniform

4.5 Experimental Investigations of Triple Junction Motion 351

T = 300°C

a = 135 µm

θ = 45°

a = 115 µm

θ = 40°

100 µm

T = 370°C

a = 130 µm

θ = 65°

a = 115 µm

θ = 60°

100 µm

T = 380°C

a = 140 µm

θ = 67°

a = 110 µm

θ = 60°

100 µm

T = 385°C

a = 150 µm

θ = 80°

a = 140 µm

θ = 75°

100 µm

T = 390°C

a = 155 µm

θ = 85°

a = 145 µm

θ = 75°

100 µm

T = 395°C

a = 168 µm

θ = 86°

a = 155 µm

θ = 75°

100 µm

FIGURE 4.11

Video frames for diﬀerent temperatures of a moving symmetrical triple junc-

tion (sample S2). GB III (see Fig. 4.5) is invisible, located at the tip of the two

visible boundaries. At T = 300

◦

C the grain boundary system did not move at

all. The solid line generated according to Eq. (4.28) in the lower right frame

ﬁts the boundary shape.

352 4 Thermodynamics and Kinetics of Connected Grain Boundaries

T = RT

a = 310 µm

θ = 50°

a = 220 µm

θ = 50°

500 µm

T = 389°C

a = 390 µm

θ = 48°

a = 290 µm

θ = 50°

500 µm

T = 395°C

a = 400 µm

θ = 50°

a = 260 µm

θ = 52°

500 µm

T = 401°C

a = 330 µm

θ = 53°

a = 280 µm

θ = 55°

500 µm

T = 404°C

a = 410 µm

θ = 60°

a = 340 µm

θ = 55°

500 µm

T = 407°C

a = 210 µm

θ = 63°

a = 340 µm

θ = 60°

500 µm

FIGURE 4.12

Video frames for diﬀerent temperatures of an ideal triple junction of sample

S3. The room temperature (RT) frame shows the initial position. The solid

line was computed according to Eq. (4.28). It ﬁts the shape of the boundaries.

4.5 Experimental Investigations of Triple Junction Motion 353

0 50 100 150 200 250 300

time [s]

v = 0.6 µm/s

T = 370°C

0 20 40 60 80 100 120 140 160 180

time [s]

100

200

300

400

500

600

700

[

]

v = 3.9 µm/s

T = 380°C

0 40 50 60 70 80 90 100 110 120

time [s]

T = 385°C

v = 5.1 µm/s

0 10 20 30 40 50 60

time [s]

100

200

300

400

500

600

[

]

v = 9.8 µm/s

T = 390°C

T = 395°C

v = 20.4 µm/s

160

140

120

100

450

400

350

300

250

200

150

100

T = 395°C

v = 20.4

m/s

T = 370°C

v = 0.6

m/s

T = 390°C

v = 9.8

m/s

T = 385°C

v = 5.1

m/s

T = 380°C

v = 3.9

m/s

700

600

500

400

300

200

100

position of triple junction [

700

600

500

400

300

200

100

position of triple junction [

600

500

400

300

200

100

position of triple junction [

0 50 100 150 200 250 300 0 40 60 80 100 120 140 160 18020

30 40 50 60 70 80 90 100 110 120 0 10 20 30 40 50 60

0 5 10 15 20 25 30 35

time [s]

FIGURE 4.13

Triple junction position vs. time for diﬀerent temperatures for sample S2.

354 4 Thermodynamics and Kinetics of Connected Grain Boundaries

60,625°

63°

71,625° 78,75° 79,5°

# measurement

370°C 380°C 385°C 390°C 395°C

79.5°78.75°71.625°63°60.625°

55 656060 65 65 70 75 80 75 80 80

[°]

FIGURE 4.14

Reproducibility of measurements of the angle Θ at diﬀerent temperatures for

sample S2.

380°C

370°C

385°C

390°C

395°C

300°C

395°C

385°C

370°C

390°C380°C

300°C

FIGURE 4.15

Evolution of the shape of the grain boundary system of sample S2 with in-

creasing temperature.