e
a,⇑
RW
of
form
e 2
und
gr
n g
res
equ
was the analysis of a groove formed at the intersection of a
mine the triple junction line tension [5,6]. The method is
based on the equilibrium of four line tensions at their point
of intersection—the grain boundary triple line and three
and to design the grain microstructure of fine-grained
2. Measurement of triple line tension
The fundamental principle underlying the measurement
of triple line tension is the establishment of thermodynamic
equilibrium at the intersection of four triple junctions,
namely three groove root junctions (triple line between
⇑ Corresponding author.
E-mail address: gottstein@imm.rwth-aachen.de (G. Gottstein).
Available online at www.sciencedirect.com
Acta Materialia 59 (2011) 3510–35
grain boundary with a surface [1]. The fact that grain
boundary motion can be influenced by a thermal groove
[2] is established as textbook knowledge. High-resolution
transmission electron microscopy observations have shown
that the intersection of a grain boundary with a free surface
leads to a reconstruction [3]. This proves the existence of a
driving force, which is likely to be related to the existence
of a grain boundary–free surface line tension [4].
We have recently designed a method to correctly deter-
and nanocrystalline materials through their effect on recov-
ery, recrystallization and grain growth. Knowledge of the
magnitude of the grain boundary line tension can provide
a quantitative estimate of the contribution of grain bound-
ary triple junctions to the driving force for grain growth. In
this study we will show that the Zener force and the Gibbs–
Thompson relation and related phenomena are also
affected by the triple line energy.
1. Introduction
The term “triple line” identifies the intersection of three
interfaces, either external interfaces or internal interfaces of
bulk materials. Triple lines are differentiated based on the
interfaces that are intersecting, e.g. three phase boundaries,
three grain boundaries, or one grain boundary and two
phase boundaries.
One of the earliest studies of triple junctions in materials
triple lines at the bottom of the thermal grooves of the
merging boundaries. The line tension of the grain bound-
ary–free surface triple lines is determined by comparing
the dihedral angle at the root of a flat and a curved grain
boundary groove.
Grain boundary triple junctions have been recognized
recently to constitute another structural element of poly-
crystals, which can strongly impact microstructural evolu-
tion [7–9]. They open up new opportunities to control
Triple junction
B. Zhao a, G. Gottstein
a Institut fu¨r Metallkunde und Metallphysik,
b Institute of Solid State Physics, Russian Academy
Received 22 December 2010; received in revised
Available onlin
Abstract
The grain boundary–free surface triple line tension and grain bo
introduced novel approach. The effect of triple line tension on grain
results showed that the triple line tension has a considerable effect o
cially for nanocrystalline materials.
� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights
Keywords: Grain boundary triple junction; Zener drag; Gibbs–Thompson
1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All
doi:10.1016/j.actamat.2011.02.024
ffects in solids
, L.S. Shvindlerman a,b
TH Aachen University, Aachen, Germany
Sciences, Chernogolovka, Moscow District, Russia
10 February 2011; accepted 14 February 2011
1 March 2011
ary triple line tension were investigated in copper using a recently
owth, Zener drag and Gibbs–Thompson relation was studied. The
rain growth, particle–boundary interactions and void shape, espe-
erved.
ation; Triple line energy
www.elsevier.com/locate/actamat
18
rights reserved.
(b
eria
Groove root 1-2
Gro
ove
roo
t 2-
3
z
y
u2-3(r)
r
t
(a)
B. Zhao et al. / Acta Mat
two crystal surface and a grain boundary) and one triple
junction formed by the intersection of three grain bound-
aries (Fig. 1).
The equilibrium in the z-direction yields
clTP ¼ clS1�2 sin f1�2 þ clS1�3 sin f1�3 þ clS2�3 sin f2�3; ð1Þ
where clTP and c
lS
i�j are the grain boundary triple line tension
and the line tension of the triple lines at the bottom of each
thermal groove, respectively. fi�j are the angles at each
Groove root 1-3
x
Triple line
r
Fig. 1. (a) Schematic 3-D view of the line geometry at a triple junction. (b)
Fig. 2. (a) AFM topography measurement on a triple junction of a Cu tricr
γB
γSγS θ
γB
γSγS θ
(a) (b
Fig. 3. (a) Grain boundary groove formed at a straight, non-curved grain bou
grain boundary with a curved groove root [6].
Grain3Grain2
u2-3(r)
lS
21−γ lS
31−γ
lS
32−γ Grain3Grain2
u2-3(r)
lS
21−γ lS
31−γ
lS
32−γ
)
lia 59 (2011) 3510–3518 3511
groove root of the corresponding grain boundary at the
center of the triple junction (Fig. 2).
Eq. (1) requires the determination of the triple line
tensions of the groove roots clSi�j. This information can be
retrieved from the curvature of the groove root triple
lines.
Fig. 3a depicts a typical thermal groove formed at a tilt
grain boundary which extends perpendicular to the free
surface. If the orientation of the surface on both sides of
Grain1
Triple line
lγ
Grain1
Triple line
lγ
AFM 3-D view of the line tension equilibrium at a triple junction [6].
ystal after annealing at 980 �C for 2 h. (b) Profiles along the lines in (a).
ξξ
)
ndary with no variation in height and (b) grain boundary formed at a flat
eria
3512 B. Zhao et al. / Acta Mat
the grain boundary is the same and the root of the groove is
straight, the specific grain boundary energy is given by:
cB ¼ 2cS cos
h
2
; ð2Þ
where h is the dihedral angle at the groove root under the
assumption that the grain boundary groove is symmetric,
and cB, cS are the grain boundary tension and the free sur-
face tension, respectively.
Fig. 4. Top view of AFM topography measurement of a grain boun
Fig. 5. (a) Top view of AFM topography measurement on a Cu wire after ann
and across the thin Cu wire.
lia 59 (2011) 3510–3518
The curved groove root in Fig. 3b gives:
cB �
clS
R
¼ 2cS cos
n
2
; ð3Þ
where R is the radius of curvature at a given point of the
groove root. The dihedral angle n has the same meaning
as h but may be different in magnitude owing to the curva-
ture. Combining Eqs. (1) and (2) yields the grain bound-
ary–free surface line tension clS:
dary groove in a Cu bicrystal after annealing at 980 �C for 2 h.
ealing at 300 �C for 2 h. AFM image step size 7 nm. (b) Profiles parallel to
eria
clS ¼ 2cS cos
h
2
� cos n
2
� �
� R: ð4Þ
The measurement of the angle h was performed on Cu
bicrystals with perfect surface quality [6] Fig. 4.
The dihedral angle h was determined from an atomic
force microsopy (AFM) scan perpendicular to the grain
boundary groove, and was found to be h ¼ 161:0� � 2:3�.
The frequency distribution of the measured dihedral angle
was Gaussian.
Measurement of the angle n (Eq. (3)) is more difficult.
For that, we determined the dihedral angle of the grain
boundary groove on very thin Cu wires, which were grown
by strain control within thin-film cracks [10] Fig. 5. Copper
wire grids were deposited on a silicon substrate, and
annealed at 300 �C in a vacuum furnace for 2 h.
To obtain maximum accuracy, the measurements
were performed with high-aspect-ratio tips (Olympus
AC11160BN-A2) in an atomic force microscope in non-
contact mode. The resolution of these tips was calibrated
with a tip-check sample, and the geometry of the tip was
derived by blind tip reconstruction. According to this cali-
bration, the tip radius was about 10 nm with a slope of 72�.
Therefore the step size of the AFM image of Cu wires was
also chosen to be610 nm for fast and accuratemeasurement.
The radius of the wire was in the range 80–150 nm. The
dihedral angles of the grooves were in the range 150–157�
depending on the groove curvature. This yielded on average
a line tension of the grain boundary groove root (triple line
grain boundary–free surfaces) clS = (2.5 ± 1.1) � 10�8 J/m.
Based on the assumptions that
1. the free surface energy is independent of orientation,
cS = 1.75 J/m
2,
2. the grain boundary–free surface line tension is constant
along the groove root, and
3. all grain boundary–free surface line tensions are the
same, the grain boundary triple line tension in Cu was
determined to be clTP ¼ ð6:0� 3:0Þ � 10�9 J=m.
The sign of the grain boundary triple line tension
depends on the grain boundary–free surface line tension,
which is positive under equilibrium conditions. It was dem-
onstrated in Ref. [6] that a negative grain boundary–free
surface line tension would cause the grain boundary–free
surface interface to form a convex bulge which would be
higher than the surface. Therefore a negative grain bound-
ary–free surface line tension and the normal (classical)
shape of a thermal groove system cannot be in equilibrium.
Consequently, the grain boundary triple line tension must
be positive. This is different from the line tension for a spe-
cific system which can undergo a first-order wetting transi-
tion, where the line tension will change sign from negative
to positive values with increasing temperature [11–13].
Pompe et al. [14] found that the line tension of a three-
B. Zhao et al. / Acta Mat
phase system (solid–liquid–vapor) is of the order of
�2� 10�10 to þ8� 10�11 J/m. The experiments in Ref.
[14] were carried out on hexaethylene glycol, aqueous
CaCl2 solution and water with surface tensions in the range
of (4.5–7.2) � 10–2 J/m2. The measured grain boundary–
free surface line tension in our experiments on Cu
amounted to clS ¼ ð2:5� 1:1Þ � 10�8 J=m and the respec-
tive grain boundary triple line tension was determined to
be clTP ¼ ð6:0� 3:0Þ � 10�9 J=m with cS ¼ 1:75 J=m2 obvi-
ously in reasonable agreement with Ref. [14].
3. Impact of triple line tension on grain growth
3.1. Effect of triple junctions on the driving force for grain
growth
It was shown in Refs. [15,16] that the influence of grain
boundary junctions (triple line and quadruple junction in
3-D polycrystals) on grain growth kinetics can be described
in terms of a parameter K. The rate of 3-D grain growth
can be expressed as:
dhDi
dt
¼ mBP
1þ 1KQP þ 1KTP
� � ; ð5Þ
where D is the grain size, and KQP and KTP are the quadru-
ple point and triple junction parameters, respectively [17].
We note that 1K is a measure of the drag effect of a given
junction on grain growth. However, junctions do not only
exert a drag on grain boundary motion, but also contribute
to the driving force for grain growth due to the junction en-
ergy. The classical equation for the driving force P for con-
tinuous grain growth reads P ¼ 2cBR . Here, 2/R is the grain
boundary curvature. If, for simplicity, we assume R � hDi,
the total driving force from grain boundaries and triple
junctions reads:
P ¼ 2cBhDi þ
36clTP
phD2i : ð6Þ
The ratio PTPPB ¼
18clTP
pcB
¼ a defines the grain size for which
the grain boundary and triple junction contributions to
the driving force are equal. Assuming the measured value
of clTP � 6� 10�9 J=m2 for Cu, a constant grain boundary
surface tension of Cu cB ¼ 0:6 J=m2 and a constant ratio
clTP=cB, we arrive at a � 55 nm. In other words, up to a
mean grain size of about 55 nm the driving force stemming
from triple junctions is larger than that of the grain bound-
aries. As a consequence, a correct examination of grain
growth in nanocrystalline materials at least up to mean a
grain size a cannot be performed if the driving force of tri-
ple junctions is not taken into account [17].
3.2. Effect of triple line tension on grain boundary–particle
interaction
In 1948 Zener proposed a concept of how the interaction
lia 59 (2011) 3510–3518 3513
between a grain boundary and a particle can be estimated
quantitatively [18].
u < arccos 1g, the grain boundary would intersect the parti-
cle, and f* becomes a retarding force for grain boundary
motion.
In Cu, cB ¼ 0:6 J=m2, and if we assume cBP � clS ¼
ð2:5� 1:1Þ � 10�8 J=m, the critical size R* is about 40 nm;
if we take the value cBP � clTP ¼ ð6:0� 3:0Þ � 10�9 J=m,
the critical size R* is about 10 nm. According to Fig 7,
DG0I > 0 for R < R
* and increases with growing area of
intersection. The minimum value of DG0 is obtained at
ticle
0.0 0.5 1.0 1.5 2.0
-6
-5
-4 10nm
20nm
30nm
GB
h/R
Fig. 7. Dependency of DG0I on particle radius in Cu for c
BP ¼ clTP ; h is the
penetration depth of the boundary in the particle as defined in Fig. 6.
eria
When a spherical solid particle intersects a grain bound-
ary, it replaces part of the grain boundary area, and by
doing so reduces the free energy of the system:
DG ¼ �pr2 � cB; ð7Þ
where G is the Gibbs free energy, cB is the boundary energy,
and r is the radius of the intersected circular area (Fig. 6).
However, this area has to be regenerated when the grain
boundary detaches from the particle. The respective retard-
ing force, i.e. the “Zener force”, is given by:
f ¼ dDG
dr
¼ �2pr � cB; ð8Þ
which retards the motion of the grain boundary.
For the past 60 years all considerations of grain bound-
ary motion and grain growth in solids with both immobile
and mobile particles have been based on this concept, i.e.
the triple line that forms at the intersection of sphere and
boundary was simply disregarded. In Section 2 we showed
that a triple line is a defect in its own right with specific
thermodynamic properties, and that the line tension of a
triple junction can be measured. Hence, such measure-
ments can be utilized to study its effect on particle bound-
ary interaction [19]. Then, Eq. (7) has to be rewritten as:
DG0I ¼ �pr2 � cB þ 2pr � cBP ; ð9Þ
where cBP is the line tension of the triple junction.
Triple line “grain boundary-particle”
Fig. 6. Schematic view of a par
3514 B. Zhao et al. / Acta Mat
Let us define the coefficient g ¼ cB�RcBP .
The derivative of Eq. (9) reads:
dDG0I
du
¼ 2pRcBP sinu � ðg cosu� 1Þ: ð10Þ
For g < 1, DG0I attains a maximum value at u ¼ 0. When
g > 1, the maximum shifts to cosu ¼ 1g, and DG0I assumes
a minimum at u ¼ 0. The sign of DG0I depends on the par-
ticle radius, as shown in Fig. 7.
From Eq. (9), we obtain the interaction force between
the grain boundary and the particle:
F � ¼ �2pr � cB þ 2pcBP ð11Þ
At the critical particle size R� ¼ cBPcB , f* changes sign.
Hence, for R < R* the particle will not attach to the grain
boundary. When the particle radius is larger than R*, and
ϕr
R
Grain boundary
Particle
h
r
R
r
R
r
R
h
intersecting a grain boundary.
-3
-2
-1
0
1
2
3 -60° 90°60°0°-90°
ΔG
0 I(1
0-
16
J)
GB
lia 59 (2011) 3510–3518
I
u ¼ �90�, therefore the particle will not attach to the grain
boundary at all. When R > R*, a further minimum of
DG0I ðuÞ appears at u ¼ 0�. On the other hand, the impact
of the triple junction on particle–boundary interaction
brings about the formation of an energy barrier. This bar-
rier can only be overcome with the help of an external driv-
ing force, as follows:
DGI ¼ �pr2 � cB þ 2pr � cBP � PV : ð12Þ
The external driving force P will expend a work propor-
tional to the swept volume V. The energy barrier still exists,
but the grain boundary will become curved under the driv-
ing force. If we assume that the radius of the curved grain
boundary is equal to the radius of the particle, the change
in the system energy DGC due to the curved grain boundary
reads:
eria
DGC ¼ �pr2 � cB þ 2pRh � cB � PV : ð13Þ
Fig. 8 gives an example of the dependencies DGI(u) and
DGC(u) for R = 30 nm, P = 25 MPa. Accordingly, the
grain boundary will first circumvent the particle (energy
curve DGC, broken line in Fig. 8) until the angle u� is
reached. Then it will intersect the particle (energy curve
DGI, thick solid profile in Fig. 8). The angle u� and the crit-
ical driving force for the process are:
u� ¼ arcsin 1� g
2
1þ g2 ð14Þ
Pcrit ¼ ð1þ g
2ÞcB
g2R
ð15Þ
A detailed derivation is given in the Appendix A.
When the driving force P > Pcrit, the grain boundary can
pass the particle, and the interaction force is given by Eq.
(11). The maximum interaction force f �max ¼ f �ðRÞ does
0.0 0.1 0.2 0.3 0.4 0.5
-6
-4
-2
0
2
Δ G
(1
0-
16
J)
h/R
ΔGI
ΔGI0
ΔGC
-90° -60°*
Δ
Fig. 8. The dependencies of DG0I , DGI and DGC on h/R for R = 30 nm,
cBP ¼ clTP ¼ ð6:0� 3:0Þ � 10�9 J=m.
B. Zhao et al. / Acta Mat
not depend on the driving force.
In other words, a particle cannot cross a grain boundary
spontaneously, but a certain driving force needs to be
applied to the grain boundary to overcome the barrier. In
the course of the process, the grain boundary first becomes
curved and eventually will pass the particle.
So far we have assumed that the particle is immobile. In
the following we will dwell on the interaction between mov-
ing grain boundary and mobile particles [15,20–23]. Since
the mobility of a second-phase particle strongly depends
on its size, the problem becomes particularly important
for fine-grained and nanocrystalline materials. In Ref.
[23] this problem was considered in the framework of the
Zener approach for steady-state motion of the boundary–
particle system. The latter assumes that the shape of a par-
ticle which moves together with a grain boundary changes
with the velocity of their joint motion. It is of interest to
evaluate the size of such a particle. For joint motion, the
velocity of a particle and grain boundary have to be equal:
vB ¼ vP ; ð16Þ
where vB ¼ 2mBcBhDi=2 , andmB, cB and hDi are the reduced bound-
arymobility, grain boundary surface tension andmean grain
size, respectively. The particle velocity vP ¼ mPpr ¼ 110 � DsbXakT �R4
pRcB [15], where Ds, b, Xa and R are the interface diffusion
coefficient, lattice constant, atomic volume and particle ra-
dius, respectively. For a Cu polycrystal with hDi � 10
lm ¼ 10�5 m at 937 K, cB ¼ 0:6 J=m2, Ds � 10�13 m2=s,
b ¼ 3 � 10�10 m and Xa ¼ 10�5 m3=mol, the radius R of a
particle moving together with a grain boundary is in the
range 1–2 nm. For R � 2 nm, g ¼ cB�RcBP < 1 and the critical
driving force P for the grain boundary motion which is nec-
essary to permit a second-phase particle to intersect the grain
boundary is given by Eq. (15), with parameters
cB ¼ 0:6 J=m2, cBP � clTP ¼ 6� 10�9 J=m, R � 2� 10�9 m
and Pcrit ¼ 7:7� 109 J=m3. This value is four orders of mag-
nitude larger than the capillary driving force! Even for a
mean grain size < D >� 20 nm the critical driving force
Pcrit is larger than the driving force for grain growth. In other
words, themotion of a 2–3 nm particle adsorbed to themov-
ing grain boundary according to the Zener model under a
capillary driving force is impossible [23].
3.3. Effect of triple line tension on the Gibbs–Thompson
relation
Let us consider the shape of a void at a grain boundary in a
polycrystal. For homogeneous materials the contact angle h
is constant, and the equilibrium surface will constitute a sur-
face of rotation [18,19] (Fig. 9).Owing to the symmetry of the
problem, the free energy of the system can be expressed as:
DG ¼
Z y1
y0
4px
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ x02
p� �
dy � cS � px20cB þ 2px0clS ; ð17Þ
where cS is the free surface energy, cB is the grain boundary
energy and clS is the grain boundary–free surface triple line
θ
γS
γB R
r
x
y
(x0,y0)
(x1,y1)
θ
R
Fig. 9. Schematic view of a void intersecting a grain boundary.
lia 59 (2011) 3510–3518 3515
tension.
Since the volume of the void V is constant:
V ¼ 2
Z y1
y0
px2dy ¼ const; ð18Þ
the problem can be reduced to the isoperimetric problem of
calculus of variations:
J ¼
Z y1
y0
4pxcS
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ x02
p
þ 2kpx2
� �
dy � �px20cB þ 2px0clS :
ð19Þ
The extrema of the function J correspond to the extrema
of the function xðyÞ, i.e. the shape of the void, in accor-
dance with the Euler equation:
U� x0Ux0 ¼ C1 ð20Þ
and the transverse conditions:
�Ux0 þ @U
@x0
����
����
y¼y0
¼ 0; ð21Þ
jU�
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