Gottstein G., Shvindlerman L.S. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications
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244 3 Grain Boundary Motion
T = 1320K
v [10
-6
m/s]
<221> Σ9 boundary
T = 1270K
1.0
1510520
0
0
0.5
1.5
2.5
2.0
P [kJ/m
3
]
01051520
P [10
-3
MPa]
T = 1320K
v [10
-6
m/s]
<221> Σ9 boundary
T = 1270K
1.0
1510520
0
0
0.5
1.5
2.5
2.0
P [kJ/m
3
]
01051520
P [10
-3
MPa]
FIGURE 3.70
Variation of boundary velocity v with driving force P for 221 tilt grain
boundaries in Fe-Si at 1270K and 1320K [300].
A
b
[m
2
/s]
1000/T [1/K]
10
-7
10
-10
10
-11
10
-8
10
-9
1.2 1.3 1.4 1.5 1.6 1.7
560
500
450 400 355
T [°C]
1.1
I
II
III
A
b
[m
2
/s]
1000/T [1/K]
10
-7
10
-10
10
-11
10
-8
10
-9
1.2 1.3 1.4 1.5 1.6 1.7
560
500
450 400 355
T [°C]
1.1
I
II
III
FIGURE 3.71
Temperature dependence of the reduced mobility of 38
◦
111 tilt grain bound-
aries in Al. (For the meaning of I, II, III see Table 3.5.)
© 2010 by Taylor and Francis Group, LLC
3.5 Experimental Results 245
1.1 1.2 1.3 1.4 1.5
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
1.1 1.2 1.3 1.4 1.5
1.1 1.2 1.3 1.4 1.5
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
0246810
v [m/s]
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
v [m/s]
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
1.1 1.2 1.3 1.4 1.5 1.1 1.2 1.3 1.4 1.5
400440500560 400440500560
1000/T [1/K] 1000/T [1/K]
T [°C]T [°C]
(a) (b)
I
II
III
IV
V
VI
1.1 1.2 1.3 1.4 1.5
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
1.1 1.2 1.3 1.4 1.5
1.1 1.2 1.3 1.4 1.5
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
0246810
v [m/s]
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
v [m/s]
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
1.1 1.2 1.3 1.4 1.5 1.1 1.2 1.3 1.4 1.5
400440500560 400440500560
1000/T [1/K] 1000/T [1/K]
T [°C]T [°C]
(a) (b)
I
II
III
IV
V
VI
FIGURE 3.72
Temperature dependence of the velocity of a 36.5
◦
111 tilt grain boundary
in iron-doped aluminum. (For meaning of I, II, III, IV, V, VI see Table 3.6.)
no effect was found on random (non-special) grain boundaries and in materials
with high impurity content. Apparently, it has been experimentally impossi-
ble to create a sufficiently large driving force to detach a grain boundary of
high adsorption capability from its impurity cloud. An analysis of the data
obtained in the framework of the L¨ucke, St¨uwe or Cahn approach showed that
the phenomena observed for 111 tilt boundaries in aluminum and 10
¯
10,
11
¯
20 tilt boundaries in zinc were related to the drag effect of adsorbed iron
atoms in aluminum and to adsorbed aluminum atoms in zinc [292, 301]. The
conceptual framework to determine the major parameters of interaction be-
tween grain boundary and impurities from experimental data of the breakaway
effect is given below.
Fig. 3.72 shows the temperature dependence of the mobility of 111 tilt
grain boundaries, ϕ =36.5 ± 0.5
◦
in high-purity aluminum samples (I–VI)
specially doped with iron. The iron concentration, though rather low, was sig-
nificantly larger than the concentration of any other dissolved element [297]
(Table 3.6). A constant driving force was provided by the surface tension of
a curved grain boundary (grain boundary quarter-loop technique). The mea-
surements were carried out by continuous boundary tracking (XICTD). The
effect was observed in the temperature range 460–550
◦
C; the specific temper-
ature depended on the driving force and impurity (iron) content (Fig. 3.72,
Table 3.6). The experimental data correlated well with predictions of the im-
purity drag theory. The energy of interaction between grain boundary and
impurity iron was found to be 0.134 ± 0.02 eV [297].
Of special interest are the nature and number of adsorption sites at a grain
boundary. The adsorption capacity of a special grain boundary is in the range
of ∼ 10
14
cm
−2
. This is distinctly lower than that measured by Auger spec-
© 2010 by Taylor and Francis Group, LLC
246 3 Grain Boundary Motion
TABLE 3.5
Grain Boundary Detachment I
Number of the sample Active impurity Fe
I II III
Concentration of the active 2 · 10
−5
1 · 10
−5
5 · 10
−6
impurities [at%]
Temperature of detachment 775 700 680
[T
0
,K]
Driving force P
i
[J/m
3
]6·10
2
6 · 10
2
6 · 10
2
D
m
[m
2
/s] at T
0
(experimental) 4.1 · 10
−10
1.7 · 10
−9
8.3 · 10
−9
D
im
[m
2
/s] at T
0
(experimental) 4 · 10
−17
1.4 · 10
−16
3.3 · 10
−16
D
m
0
[m
2
/s] (experimental) 6.1 · 10
−7
9.2 · 10
−6
6.7 · 10
−6
D
0
im
[m
2
/s] (experimental) 1.2 ·10
−13
2.5 · 10
−12
1.4 · 10
−11
H
D
m
[eV] 0.50 0.53 0.40
H
D
im
[eV] (experimental) 0.52 0.60 0.64
H
D
im
, diffusion 0.60 0.60 0.60
Fe in Al D
0
im
[m
2
/s]
∗)
4.1 · 10
−13
4.1 · 10
−13
4.1 · 10
−13
U
0
[eV] 0.17 0.24 0.24
Z [cm
−2
]2.3 · 10
14
7 · 10
13
2.4 · 10
14
*) Mirano K, Agarwala RP, Cohen M, Acta Metall., 10:857, 1962.
troscopy on random grain boundaries in polycrystals, which proves a much
higher adsorption capability of non-special grain boundaries.
The temperature dependency of the mobility of 111 tilt grain bound-
aries (ϕ =46.8
◦
, Σ19) in iron-doped aluminum samples (Figs. 3.73a,b)
exhibits two consecutive steps, followed by a dramatic decrease of the mi-
gration activation enthalpy. An analysis of the data obtained shows that
the observed mobility jumps are related to the breakaway of the migrating
grain boundary from adsorbed iron atoms, and the sequence of the break-
away is attributed to the existence of two types of adsorption centers in
the special grain boundaries mentioned, which differ in interaction energy
(U
01
=0.12 ± 0.01 eV, U
02
=0.3 ± 0.02 eV). The density of the adsorption
centers are z
1
=(1.5 ± 0.2) · 10
14
cm
−2
and z =(4.4 ± 0.3) · 10
12
cm
−2
,re-
spectively. It appears [294] that z
1
correlates with an average (with respect to
the inclination angle deviating from the symmetric position) surface density
of coincidence sites (∼ 1.8 · 10
14
cm
−2
). Hence, the active centers of type I
may be interpreted as most distorted regions between the coincidence sites. It
was suggested that the interaction centers of type II may be associated with
grain boundary dislocations.
It counts in favor of the hypothesis that the observed abrupt change in the
migration velocity is due to grain boundary detachment from its impurities
so that the value of the calculated interaction energy U
0
and the number of
adsorption sites z (within the limits of experimental uncertainty) remain con-
stant for a given grain boundary type but different impurity content (Table
© 2010 by Taylor and Francis Group, LLC
3.5 Experimental Results 247
1.1 1.2 1.3 1.4 1.5
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
A
b
[m
2
/s]
T [°C]
1000/T [1/K]
635
560 495 440 395
10
-7
10
-8
10
-9
10
-10
1.1 1.2 1.3 1.4 1.5
I
III
1.1 1.2 1.3 1.4 1.5 1.6
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
T [°C]
A
b
[m
2
/s]
1000/T [1/K]
1.1 1.2 1.3 1.4 1.5 1.6
10
-7
10
-8
10
-9
10
-10
350395635 560 495 440
II
IV
(a)
(b)
1.1 1.2 1.3 1.4 1.5
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
A
b
[m
2
/s]
T [°C]
1000/T [1/K]
635
560 495 440 395
10
-7
10
-8
10
-9
10
-10
1.1 1.2 1.3 1.4 1.5
I
III
1.1 1.2 1.3 1.4 1.5 1.6
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
T [°C]
A
b
[m
2
/s]
1000/T [1/K]
1.1 1.2 1.3 1.4 1.5 1.6
10
-7
10
-8
10
-9
10
-10
10
-7
10
-8
10
-9
10
-10
350395635 560 495 440
II
IV
(a)
(b)
FIGURE 3.73
Temperature dependence of the reduced mobility of a 46.5
◦
111 tilt boundary
in iron-doped aluminum. (For meaning of I, II, III, IV see Table 3.6.)
© 2010 by Taylor and Francis Group, LLC
248 3 Grain Boundary Motion
1.2 1.3 1.4
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.01
0.1
1
10
100
1000
1000/T [1/K]
T [°C]
A
b
[m
2
/s]
1.2 1.3 1.4 1.5
10
-6
10
-8
10
-9
10
-10
10
-11
10
-7
395440495560
I
II
1.2 1.3 1.4
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.01
0.1
1
10
100
1000
1000/T [1/K]
T [°C]
A
b
[m
2
/s]
1.2 1.3 1.4 1.5
10
-6
10
-8
10
-9
10
-10
10
-11
10
-7
10
-6
10
-8
10
-9
10
-10
10
-11
10
-7
395440495560
I
II
FIGURE 3.74
Temperature dependence of the reduced mobility of a 36.5
◦
111 tilt grain
boundary for samples I and II (Table 3.6).
3.6): the detachment temperature correctly shifts with increasing impurity
concentration and driving force. The temperature dependence of the velocity
of a 36.5
◦
111 grain boundary for two samples (I and II) with equal iron con-
tent but different driving force (P
I
/P
II
=1.6, the absolute value of the driving
force in this experiment being rather small, P/kT ∼ 10
−6
)ispresentedin
Fig. 3.74. The velocities and breakaway temperature intervals differ for the
two samples, but the mobilities in the low- and high-temperature regimes are
the same. This confirms that the driving force determines the temperature
interval of the effect. The good quantitative agreement also supports the pro-
posed idea of the nature of the effect [294]. It should be noted that the change
in the driving force by a factor of 1.5 was sufficient to shift the detachment
temperature by 25K.
The difference in the number of adsorption sites (adsorption centers),
especially obvious for grain boundaries in zinc (Table 3.6), gives rise to a
number of effects. So, at the same bulk impurity concentration the orien-
tation dependencies for 10
¯
10 and 11
¯
20 tilt grain boundaries in zinc are
quite different (Fig. 3.75). The difference in the adsorption capacity leads
to a faster adsorption saturation on 10
¯
10 grain boundaries than on 11
¯
20
boundaries [239, 276].
3.5.7 Athermal Motion of Grain Boundaries
Low adsorption capacity may result in athermal grain boundary migration
and was experimentally observed for both special and close to special 11
¯
20
© 2010 by Taylor and Francis Group, LLC
3.5 Experimental Results 249
<>
1010
2.5
2.0
1.5
0.5
1.0
0
H
m
[eV]
20 40 60 80
ϕ [deg]
(b)
0
H
m
[kJ/mol]
20 40 60 80
ϕ [deg]
<>
1120
(a)
1.5
1.0
0.5
0
H
m
[eV]
0
H
m
[kJ/mol]
50
150
100
0
50
150
100
200
250
0
<>
1010
2.5
2.0
1.5
0.5
1.0
0
2.5
2.0
1.5
0.5
1.0
0
H
m
[eV]
20 40 60 80
ϕ [deg]
(b)
0
H
m
[kJ/mol]
20 40 60 80
ϕ [deg]
<>
1120
(a)
1.5
1.0
0.5
0
H
m
[eV]
0
H
m
[kJ/mol]
50
150
100
0
20 40 60 80
ϕ [deg]
<>
1120
(a)
1.5
1.0
0.5
0
H
m
[eV]
0
H
m
[kJ/mol]
50
150
100
0
50
150
100
200
250
0
FIGURE 3.75
Dependence of activation enthalpy of migration on misorientation angle for
(a) 11
¯
20 and (b) 10
¯
10 tilt grain boundaries in Zn of different purity: ◦ –
99.995at%; • – 99.9995at%.
tilt grain boundaries in zinc (Figs. 3.76, 3.77) [298, 299]. The temperature
dependence of the reduced mobility A
b
(T ) is characteristic for breakaway of a
migrating boundary from its adsorbed impurities. An essential new feature is
that after the detachment from the impurity atmosphere the grain boundary
velocity becomes temperature independent
10
.
Experiments on 86
◦
11
¯
20 tilt grain boundaries under constant driving force
(grain boundary half-loop, technique N 10, Table 3.3) showed a strong de-
pendence of the mobility on grain boundary orientation (inclination) in the
athermal regime (Figs. 3.33, 3.76–3.79) [247]. The mobility maximum approxi-
mately coincided with the grain boundary plane {10
¯
12} of the shrinking grain.
It is interesting to analyze what mechanism may control the athermal mo-
tion of a grain boundary. It is conceivable that linear defects (grain boundary
dislocations or steps) move along the grain boundary in an athermal fashion.
Once a step (or dislocation) moves it will sweep along the grain boundary at
the speed of sound (c ≈ 3·10
5
cm/s). For a step height h
∼
=
10
−8
cm, the grain
boundary mobility can be estimated as A
∼
=
hc ≈ 10
−3
cm
2
/s, a magnitude
which is close to the maximum mobility observed in experiment (Fig. 3.33).
The macroscopic observations presented above are closely connected with the
motion of faceted grain boundaries.
10
The term athermal does not mean that the activation energy of grain boundary is exactly
equal to zero. It reflects only the experimentally established fact that the motion does not
depend on temperature, which also can occur for E kT, i.e. every trial to overcome the
activation barrier will be successful.
© 2010 by Taylor and Francis Group, LLC
250 3 Grain Boundary Motion
1.4 1.5 1.6 1.7 1.8 1.9 2
1E-006
1E-005
0.0001
0.01
0.1
1
1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
100
1000/T [1/K]
1.4 1.5 1.6 1.7 1.8 1.9 2.0
350 280 225
T [°C]
1000/T [1/K]
1.5 1.6 1.7 1.8 1.9
395 350 315 220 250
T [°C]
1000/T [1/K]
1.7 1.9 2.1 2.3
315 250 200 160
T [°C]
10
-7
A
b
[m
2
/s]
10
-8
10
-9
10
-10
10
-11
(a)
1.5
395
A
b
[m
2
/s]
10
-8
10
-9
10
-10
(b)
440
A
b
[m
2
/s]
10
-7
10
-8
10
-9
10
-10
(c)
1.4
440
1.4 1.5 1.6 1.7 1.8 1.9 2
1E-006
1E-005
0.0001
0.01
0.1
1
1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
100
1000/T [1/K]
1.4 1.5 1.6 1.7 1.8 1.9 2.0
350 280 225
T [°C]
1000/T [1/K]
1.5 1.6 1.7 1.8 1.9
395 350 315 220 250
T [°C]
1000/T [1/K]
1.7 1.9 2.1 2.3
315 250 200 160
T [°C]
10
-7
A
b
[m
2
/s]
10
-8
10
-9
10
-10
10
-11
(a)
1.5
395
A
b
[m
2
/s]
10
-8
10
-9
10
-10
(b)
440
A
b
[m
2
/s]
10
-7
10
-8
10
-9
10
-10
(c)
1.4
440
FIGURE 3.76
Temperature dependence of the reduced mobility of 11
¯
20 tilt grain bound-
aries in Zn: (a) ϕ =34
◦
(Σ23); (b) ϕ =56
◦
(Σ36); (c) ϕ =85
◦
(Σ11).
© 2010 by Taylor and Francis Group, LLC
3.5 Experimental Results 251
1.4 1.5 1.6 1.7 1.8 1.9 2
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
A
b
[m
2
/s]
1000/T [1/K]
10
-7
10
-9
10
-8
10
-10
1.4 1.5 1.71.6 1.8 1.9 2.0
1
2
1
2
250315
395
T [°C]
1.4 1.5 1.6 1.7 1.8 1.9 2
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
A
b
[m
2
/s]
1000/T [1/K]
10
-7
10
-9
10
-8
10
-10
1.4 1.5 1.71.6 1.8 1.9 2.0
1
2
1
2
250315
395
T [°C]
FIGURE 3.77
Temperature dependence of the reduced mobility of 56
◦
11
¯
20 tilt grain
boundaries in Zn with 1 · 10
−5
at%Al (1) and 3 · 10
−5
at%Al (2).
(0001) (0001) (0001)
Θ = 0° Θ = 86°Θ = 45°
Θ
Θ
ΙΙΙΙΙΙΙ Ι Ι
(0001) (0001) (0001)
Θ = 0° Θ = 86°Θ = 45°
Θ
Θ
ΙΙΙΙΙΙΙ Ι Ι
FIGURE 3.78
Bicrystals (half-loop geometry) with different inclination angle Θ of a tilt
grain boundary 86
◦
11
¯
20.
© 2010 by Taylor and Francis Group, LLC
252 3 Grain Boundary Motion
1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85
1E-008
1E-007
1E-006
1E-005
0.0001
1
10
100
1000
10000
1000/T [1/K]
1.5 1.6 1.7 1.8
A
b
[m
2
/s]
10
-8
10
-9
10
-10
10
-11
10
-12
1.4
440 395 280315350
T [°C]
1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85
1E-008
1E-007
1E-006
1E-005
0.0001
1
10
100
1000
10000
1000/T [1/K]
1.5 1.6 1.7 1.8
A
b
[m
2
/s]
10
-8
10
-9
10
-10
10
-11
10
-12
1.4
440 395 280315350
T [°C]
FIGURE 3.79
Temperature dependence of the grain boundary reduced mobility: tilt bound-
ary 11
¯
20 in Zn: ϕ =86
◦
; θ =4
◦
. The solid circles pertain to heating exper-
iments.
3.5.7.1 Effect of Faceting on Grain Boundary Motion
The surface of minerals is often composed of planar faces which are referred
to as facets. They are due to the crystalline nature of the materials and reflect
an anisotropy of the surface energy. However, facets are not only observed on
crystal surfaces but can also occur on internal interfaces, like grain bound-
aries, and appear as straight grain boundary segments in an optical micro-
graph. Prominent examples are coherent twin boundaries in low stacking fault
energy materials [310]. The relation between grain boundary (GB) faceting
and grain boundary behavior, in particular, grain growth and grain bound-
ary migration, was the subject of investigation in the past [141], [312]–[314].
However, only recently have quantitative methods been put forward to tackle
this problem [315]–[317]. The migration of incoherent twin grain boundaries
with facets was studied by Straumal et al. [316]. It was demonstrated that
the rather complicated kinetics of grain boundary motion, in particular the
strong non-Arrhenius temperature dependence, could be explained by consid-
ering the motion of two competing facets. Rabkin considered the influence
of grain boundary faceting on grain growth and the motion of a 2D grain
boundary half-loop [317]. Using several simplifying assumptions he showed
that faceting might modify the Von Neumann-Mullins relationship for 2D
grain growth; it was also found that the length of a facet on a half-loop de-
pended on its mobility, i.e. the length of a facet increased with rising facet
mobility.
© 2010 by Taylor and Francis Group, LLC
3.5 Experimental Results 253
TABLE 3.6
Grain Boundary Detachment II
Metal Tilt Drag- Concen- Energy Number of Adsorption
boundary ging tration of of adsorption Γ[at/cm
2
)]
atoms dragging inter- sites at
impurity action grain
c
1
U
0
[eV] boundary
10
−5
at% Z
1
[cm
−2
]
A1 111 Fe 2 0.17 2.3 · 10
14
8 · 10
8
ϕ =38
◦
10.247· 10
13
-
10.232.4 · 10
14
-
Zn 11
¯
20 Al 1 0.36 9.0 · 10
13
4.8 · 10
9
ϕ =81
◦
11
¯
20 10.348.0 · 10
13
4.8 · 10
9
ϕ =56
◦
Zn 11
¯
20 Al 3 0.35 4.8 · 10
13
1.5 · 10
10
ϕ =45
◦
30.335.9 · 10
13
1.5 · 10
10
11
¯
20 30.358.8 · 10
13
1.5 · 10
10
ϕ =88
◦
10
¯
10 30.191.4 · 10
15
1.5 · 10
10
ϕ =57
◦
10
¯
10 30.181.4 · 10
15
1.3 · 10
10
ϕ =30
◦
A1 111 Fe 7* 0.15 1.0 · 10
14
7.1 · 10
8
ϕ =36.5
◦
sample I
7* 0.12 1.4 · 10
14
6.0 · 10
8
sample II
80.121.4 · 10
14
6.2 · 10
8
sample III
1.15 0.12 1.4 · 10
14
8.9 · 10
8
sample III
Al 111 Fe 1.6 0.12 1.1 · 10
14
11.3 · 10
8
ϕ =36.5
◦
sample V
1.65 0.12 1.3 · 10
14
15.9 · 10
8
sample VI
111 8** 0.12 1.4 · 10
14
6.1 · 10
8
ϕ =46.5
◦
sample I 0.30 4.9 · 10
12
2.1 · 10
8
111 6** 0.12 1.7 · 10
14
7.4 · 10
8
ϕ =46.5
◦
*** sample II 0.32 4.2 · 10
12
2.6 · 10
8
5** 0.14 1.3 · 10
14
5.6 · 10
8
sample III 0.32 3.8 · 10
12
1.9 · 10
8
* P
I
/P
II
= 1.6 (Fig. 3.74).
** The experimental data were analyzed in the framework of the model with
two kinds of adsorption sites.
*** Sample IV corresponds to the initial aluminum.
The impact of faceting on the motion of a high-angle grain boundary
in Zn was investigated in [320, 321]. The study was conducted [320, 321] for
© 2010 by Taylor and Francis Group, LLC