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 365

TABLE 4.4

Crystallography and Kinetic Parameters of Grain Boundary and Triple

Junction Mobility in Al [437]

No. GB I GB II GB III ΔT,

◦

C H

temp. eV m

/s eV m

inter-

val

invest-

igated

121

◦

398–479 1.0 0.03 1.8 4.5 · 10

111111111

220

◦

380–420 2.0 1.8 · 10

111111111

320

◦

470–510 0.4 1.8 · 10

111111111

422

◦

460–495 3.3 1.8 ·10

100100100

512

◦

590–620 1.3 0.5

100100100

637

◦

500–550 3.6 6.8 ·10

100100100

712

◦

520–570 0.9 4.7 · 10

−4

4.4 1.8 · 10

100100100

827

◦

469–591 1.4 2.3 2.7 1.3 · 10

110110110

944

◦

530–591 1.3 0.4

110110110

TABLE 4.5

Transition Temperature (T

trans

) and Compensation Temperature (T

comp

)for

Al Bi- and Tricrystals [437]

Grain boundaries in Al bicrystals Triple junctions in Al tricrystals

comp.

melt.

comp.

melt.

trans.

melt.

100 1.08 [437] 0.82 [436] 0.84 0.85

111 0.77 [434] 0.75 [436] 0.75 0.76

110 0.92 [434] 0.93 [436] 0.85 0.85

triple junctions drag the motion of the system, whereas at temperatures

above T

the motion is governed by the grain boundary mobility.

366 4 Thermodynamics and Kinetics of Connected Grain Boundaries

-7

-3

=430

H, eV

, m

-7

-2

=510

H, eV

, m

-2

=520

H, eV

, m

(c)

(b)

(a)

-7

-3

=430

H, eV

, m

-7

-2

=510

H, eV

, m

-2

=520

H, eV

, m

(c)

(b)

(a)

FIGURE 4.26

Activation enthalpy H vs. pre-exponential factor A

for: a) 111: triple junc-

tions in Al tricrystals [437] —  ; grain boundaries in Al bicrystals —  [274],

 [435]; b) 100: triple junctions in Al tricrystals [437] — ; grain boundaries

in Al bicrystals —  [379],  [435]; c) 110: triple junctions in Al tricrystals

— ; grain boundaries in Al bicrystals —  [435],  [274].

4.5.1 Motion of a Grain Boundary System with Triple Junc-

tion in a Material with Impurities

At the beginning of this chapter we considered in general terms the inﬂuence of

impurity atoms on the motion of a grain boundary system with triple junction.

It was shown that the concurrent interaction between foreign atoms and the

two types of moving structural elements makes the process more complicated

and more intricate than for grain boundary motion. The ﬁrst experiments in

this area were reported in [433]. The experiments were conducted on tricrys-

tals manufactured from high pure Al (99.9999 wt.%) with diﬀerent contents

of Mg.

The total impurity concentration [433] includes a large number of chemical

elements of diﬀerent concentration. Since each element may interact diﬀer-

ently with grain boundary and junction it is important to select the element

which is most eﬀective for migration. This problem is hard to address exper-

imentally as it is impossible to produce a “pure” binary alloy with a rather

low concentration of the alloying element.

That is why in [433] alloys were produced where the concentration of one

species (Mg, “alloying element”) exceeded the total amount of other impu-

rities in the specimen. It was assumed that just this element in the given

concentration is responsible for the impurity inﬂuence on grain boundary mi-

gration (Table 4.6).

All tricrystals studied had the same crystallographic parameters and dif-

fered only in Mg concentration. (The concentration of Mg in specimen Al II

is distinctly smaller than the total impurity concentration; however, the au-

thor of [433] used it as a starting point for the alloys studied.) The shape

and orientation of the grown tricrystals is shown in Fig. 4.27. The two curved

grain boundaries GB1 and GB2 were identical in terms of misorientation angle

4.5 Experimental Investigations of Triple Junction Motion 367

TABLE 4.6

Concentration of Mg in the Al Alloys

Studied [433]

Al Alloys Al II Al III Al IV Al V

Main solute

element Mg Mg Mg Mg

Concentration

wt. ppm 0.2 50 100 1000

=°

<111>

III

<112>

III

<110>

Grain I

Grain II

Grain III

GB1

GB2

GB1 (GB2)

40° <111> tilt

with 2° twist

angle around

<110>

°=Φ 2 °=Φ 2

321

≠==

GB3

4° <110> twist

<111>

<112>

<111>

<112>

<110>

GB3

°=Φ

=°

<111>

III

<112>

III

<110>

Grain I

Grain II

Grain III

GB1

GB2

GB1 (GB2)

40° <111> tilt

with 2° twist

angle around

<110>

GB1

GB2

GB1 (GB2)

40° <111> tilt

with 2° twist

angle around

<110>

GB1

GB2

GB1

GB2

GB1 (GB2)

40° <111> tilt

with 2° twist

angle around

<110>

°=Φ 2 °=Φ 2

321

≠==

GB3

4° <110> twist

<111>

<112>

<111>

<112>

<110>

GB3

°=Φ

GB3

4° <110> twist

<111>

<112>

<111>

<112>

<110>

GB3

°=Φ

<111>

<112>

<111>

<112>

<110>

GB3

°=Φ

FIGURE 4.27

Geometry and crystallography of grown tricrystals [433].

and rotation axis. The 40

◦

111 tilt grain boundary (GB1, GB2) was super-

imposed by a rotation around the axis perpendicular to the grain boundary

plane by an angle ψ = ±4

◦

. Thus, such a boundary can be described as a

◦

111 tilt boundary with a ±4

◦

twist component. GB3 is a low-angle twist

grain boundary with rotation axis 110 and misorientation angle 2ψ =4

◦

The width of the shrinking grain a (Grain III) was about 900–1000 μminall

specimens studied. The shape of the grain boundaries and the displacement

of the triple junction were recorded in situ using SEM. A sequence of video

frames of the recorded motion is presented in Fig. 4.28 [433].

For all specimens the velocity, the dihedral angle at the triple junction and

the criterion Λ were found to remain constant at constant temperature. This

conﬁrms the assumption of a steady-state motion of the grain boundary sys-

tem. In Fig. 4.29 the temperature dependence of the reduced grain boundary

and triple junction mobility, A

and A

, respectively, and the dihedral angle

2Θ at the triple junction for a specimen with 1000 wt ppm Mg are presented.

The kinetic parameters of the grain boundaries and triple junctions studied

368 4 Thermodynamics and Kinetics of Connected Grain Boundaries

FIGURE 4.28

Set of video frames of recorded SEM images for the motion of grain boundary

system with a triple junction in Al with 50 wt ppm of Mg [433].

FIGURE 4.29

Temperature dependence of the reduced mobility A

(solid symbols), A

(open symbols), and the dihedral angle 2Θ in an Al specimen with 1000 wt

% Mg [433].

4.5 Experimental Investigations of Triple Junction Motion 369

are given in Table 4.7.

TABLE 4.7

The Activation Enthalpy and Pre-Exponential Factor for Grain Boundary and Triple

Junction Migration

,eV A

gb0

/s H

,eV H

tj0

,eV

Al II 1.45±0.16 7.173 · 10

+1.152 · 10

1.60±0.24 1.098 · 10

+1.096 · 10

−6.752 · 10

−9.182 · 10

Al III 1.49±0.01 8.938 · 10

+2.269 · 10

2.01±0.14 7.005 · 10

+8.089 · 10

−1.809 · 10

−6.446 · 10

Al IV 1.47±0.08 3.867 · 10

+3.348 · 10

1.57±0.11 8.597 · 10

+5.969 · 10

−3.005 · 10

−7.514 · 10

Al V 1.32±0.02 3.589 · 10

+1.488 · 10

1.46±0.02 7.324 · 10

+2.211 · 10

−1.052 · 10

−1.702 · 10

As shown in [433] the dihedral angle 2Θ and the criterion Λ remain con-

stant at a given temperature, however, they increase with rising temperature

(Figs. 4.29–4.31).

The criterion Λ decreases dramatically toward lower temperatures, reveal-

ing that triple junction drag increases at low temperatures. However, the

absolute value of the criterion Λ even at the lowest temperature studied was

still rather large (Λ ≈ 12). Probably, the motion of the grain boundary system

was still controlled by grain boundary kinetics. It is of interest how the cri-

terion Λ changes with Mg concentration at constant temperature (Fig. 4.32).

Obviously, a rising impurity content markedly decreases the criterion Λ. Since

Λ is proportional to the ratio of grain boundary and triple junction mobility

such a behavior substantiates that foreign atoms aﬀect triple junction mo-

bility much more strongly than grain boundary mobility. Hence, there is a

signiﬁcant diﬀerence in interaction of solute atoms with grain boundaries and

triple junctions. This is evident also from the concentration dependency of

triple junction and grain boundary mobility (Fig. 4.33).

In essence, the two diﬀerent regimes of coupled triple junction and grain

boundary motion can be observed by a change in the angle Θ with temper-

ature. At low temperatures, where Λ is of the order of unity (Figs. 4.16 and

4.17), the motion of the boundaries is dragged by the hardly mobile triple

junction. Accordingly, the angle Θ is smaller than predicted by the equilib-

rium of grain boundary surface tensions (Figs. 4.11 and 4.12), and the motion

of the entire boundary system is controlled by the mobility of the triple junc-

tion. With increasing temperature the triple junction becomes more mobile

compared to the grain boundary as indicated by an increasing value of Λ

(Figs. 4.16 and 4.17). Therefore, triple junction drag decreases and at high

370 4 Thermodynamics and Kinetics of Connected Grain Boundaries

FIGURE 4.30

Dihedral angle 2Θ vs. temperature in diﬀerent Al alloys:  —AlII; —

Al III;  —AlIV; — Al V [433] (see Table 4.6 for composition of alloys).

FIGURE 4.31

Parameter Λ vs. temperature for diﬀerent Al alloys:  —AlII; — Al III;

 —AlIV; — Al V [433] (see Table 4.6 for composition of alloys).

4.5 Experimental Investigations of Triple Junction Motion 371

0.00 0.02 0.04 0.06 0.08 0.10

120

180

240

300

T=360°C

T=400°C

c [% wt.]

FIGURE 4.32

Criterion Λ vs. concentration of Mg atoms at constant temperature [433].

temperatures, close to the melting point, the motion of the entire boundary

system is governed by the grain boundary mobility only. As a consequence

of this transition of the state of motion the angle Θ tends to attain its ther-

modynamic equilibrium value with increasing temperature (Fig. 4.15). One

might argue that the grain boundary surface tension will also change with

temperature and may eﬀect a change in Θ. However, the observed tempera-

ture dependence of Θ cannot be explained in terms of a diﬀerent temperature

coeﬃcient of the grain boundary surface tension. First, the temperature co-

eﬃcient of the surface tension is much higher for high-angle grain boundaries

than for low-angle boundaries. Thus, the change of Θ for samples S1 and S2

should be opposite to the experimental observations, i.e. the angle Θ would

have to decrease with increasing temperature. Secondly, the order of change of

Θ is too large to be explained by thermodynamic reasons in terms of diﬀerent

temperature coeﬃcients of the surface tension, in particular for samples S3

and S4, which comprise nearly identical boundaries [183]. Consequently, the

temperature dependence of Θ for all samples must result from a change of the

kinetics of motion, reﬂected by the temperature dependence of Λ (Figs. 4.16,

4.17).

The reported investigations unambiguously prove the existence of a speciﬁc

mobility of triple junctions. The dimensionless criterion Λ speciﬁes the ratio

of triple junction mobility to grain boundary mobility (Eq. (4.45)). For low

temperatures Λ is on the order of unity and thus the mobility of triple junc-

tions is comparable to the grain boundary mobility (normalized by the width

372 4 Thermodynamics and Kinetics of Connected Grain Boundaries

(a)

(b)

, A

]

(a)

(b)

, A

]

FIGURE 4.33

Reduced mobility A

(solid symbols) and A

(open symbols) as a function

of Mg concentration c at a constant temperature of (a) 360

◦

and (b) 400

◦

[433].

4.5 Experimental Investigations of Triple Junction Motion 373

a of the shrinking grain). The mobility of the grain boundary and of the triple

junction was determined on the basis of the given approach (Eqs. (4.29) and

(4.44)). For convenience we use the reduced form (4.50, 4.51). For all triple

junctions investigated the activation enthalpy of triple junction migration H

was found to be higher than for grain boundary migration H

.Itispartic-

ularly stressed that in contrast to A

the reduced triple junction mobility

does not only depend on the intrinsic mobility but on the grain size a

as well (Eq. (4.51)). With increasing grain size the triple junction becomes

more mobile (in terms of its reduced mobility) in comparison to the grain

boundary mobility. As a consequence, the transition temperature between

the two kinetic regimes, where A

is comparable to A

, changes with grain

size. Fig. 4.34 [183] shows schematically the temperature dependencies of the

reduced mobility of a triple junction and the corresponding grain boundary

mobility for diﬀerent grain sizes. For a large grain size (a

), as characteristic

for grain growth experiments, the mobility of grain boundaries is comparable

to the mobility of triple junctions

at relatively low temperatures (Fig. 4.34).

The kinetics in the temperature regime, which is characteristic for conven-

tional recrystallization treatments, is controlled by the more slowly moving

grain boundary. For a very small grain size (a

), as characteristic for ultraﬁne-

grained material, e.g. nano-crystalline material, the situation is opposite. The

transition temperature is comparatively high, and the kinetics in the same

temperature regime is controlled by the triple junction mobility. Moreover, the

absolute value of the triple junction mobility and thus its migration rate at a

given temperature is distinctly lower for ﬁne-grained than for coarse-grained

microstructures, and there is circumstantial evidence that the surprising ther-

mal stability of ultraﬁne-grained materials is due to insuﬃcient triple junction

mobility [412, 413].

We feel that the results reported above are of particular importance for the

ﬁeld of grain growth. If during grain growth the motion of the grain boundary

system is controlled by the mobility of the grain boundaries, the velocity v is

proportional to the grain boundary curvature (∼ 1/a) (Eq. (4.29)). By con-

trast, in the case of triple junction controlled motion the velocity v is constant

(Eq. (4.27)). The kinetics of the evolution of the mean grain size in the former

case will be determined by the dependency

v =

∼

⇒<a>∼

√

t (4.52)

i.e. the mean grain size increases in proportion to the square root of the

annealing time. By contrast, in the latter case

v =

=const.⇒<a>∼ t (4.53)

In other words, m

∼

· a

374 4 Thermodynamics and Kinetics of Connected Grain Boundaries

1/T

log A

, lo

g A

FIGURE 4.34

Sketch of the temperature dependence of the reduced grain boundary mobility

and reduced triple junction mobility A

for diﬀerent grain size a

; a

; the shaded area indicates the temperature regime characteristic for

recrystallization experiments.

i.e., the mean grain size increases in proportion to the time of annealing.

4.6 Triple Junction Drag and Grain Growth in 2D Poly-

crystals

4.6.1 Introduction

The grain microstructure in a polycrystal is a mosaic. A 2D mosaic is a sur-

face, in particular, a plane, which is closely packed by 2D shapes. Similarly,

if a space is closely ﬁlled by 3D bodies, we can call this construction a 3D

mosaic. Grain growth in 2D and 3D polycrystals is a living mosaic. During

grain growth the grains are growing, shrinking, or disappearing. The grain

boundaries are moving as if they are trying to escape the sample, without

much success, though.

The observation and study of 2D grain growth in polycrystals — in real

polycrystals or in computer simulations — aﬀord, beside pure scientiﬁc plea-

sure, a great aesthetic enjoyment. That is why we admire the mosaics of Seville