Available online at www.sciencedirect.com
Acta Materialia 59 (2011) 19–29
www.elsevier.com/locate/actamat
The formation of stacking fault tetrahedra in Al and Cu
III. Growth by expanding ledges
H. Wang a,⇑, D.S. Xu a, R. Yang a, P. Veyssie`re b
a
Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China
b
LEM, CNRS-ONERA, BP 72, 92322 Chaˆtillon, France
Received 10 February 2010; received in revised form 9 July 2010; accepted 27 July 2010
Available online 27 September 2010
Abstract
Ledge expansion and the concomitant growth of a stacking fault tetrahedron (SFT) are investigated in Al and Cu by molecular
dynamics (MD) simulations by addition of vacancy rods with selected lengths. Ledge expansion is largely governed by the site preference
of vacancies on the SFT edges resulting in distinct stable ledged SFTs. Both edge- and corner-facing ledge conﬁgurations may be
adopted. The growth of SFTs, especially large ones, is controlled by thermal agitation. The mobile part of the ledges consists of a dipole
of Shockley partials generally oriented in the 60° mixed orientation that move in a thermally activated manner, reﬂecting a certain core
reorganization of the Shockley dipole.
Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Stacking fault tetrahedron; Growth; Ledge mechanism; Molecular dynamics simulations; Shockley dipole
1. Introduction
In Au essentially but also in other metals such as Ag, Ni
and Cu and certain of their alloys stacking fault tetrahedron
(SFT) edges amount to a few tens of nanometers [1–8]. After
quenching and annealing edge lengths of the order of
100 nm, and thus containing at least 105 atoms, have been
observed [9,10]. Whereas small sized tetrahedra may grow
by successive absorption of vacancies at jogs on the tetrahedron faces [11–16] (see also Wang et al. Part II [26]), one may
wonder if this mechanism is still operative for SFTs containing one to two orders of magnitude more vacancies than the
several nanometer large SFTs generated by irradiation or
plastic deformation [17,18]. As an alternative to the Silcox
and Hirsch mechanism [1], SFT growth (and shrinkage) by
expansion of ledges on SFT faces has been proposed
[11,19], although little investigated. Experimental evidence
is scarce [20]. Because the distance between certain segments
of the ledges is of the order of the inner cut-oﬀ radius, SFT
growth by ledge expansion cannot be studied by linear elas⇑ Corresponding author. Tel.: +86 24 23971946.
E-mail address: [email protected] (H. Wang).
ticity calculations of interactions between piecewise dislocations. Recently, ledges were observed by MD simulations of
the growth of an initially defective triangular Frank loop
[16,21] and upon SFT intersection by a moving dislocation
[22–24].
An SFT, which is in general not perfect, may evolve in
signiﬁcantly distinct ways depending on where the next
vacancy or vacancy cluster comes from. Several local conﬁgurational minima may be attained, such as the double
SFT discussed in Wang et al. Parts I [25] and II [26]. Furthermore, a few additional vacancies may have segregated
in the time needed for a vacancy to ﬁnd its lowest energy
position, which makes a systematic MD simulation coverage of SFT growth by vacancy absorption intractable in
the absence of simplifying assumptions. Here we investigate
the alternative ledge propagation process in relation to the
site preferences determined in Wang et al. Part II [26].
2. Method
Because the simulations have already revealed an unexpected bulging upon the addition of the third vacancy on
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2010.07.045
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H. Wang et al. / Acta Materialia 59 (2011) 19–29
the SFT edge (insets in Fig. 4a and b in Wang et al. Part II
[26]), we have forced a perfect SFT to grow by addition of
rigid rods comprising a given number of vacancies along
h1 1 0i, named full and partial rods depending upon whether
or not the preset number of vacancies is enough to yield the
next perfect SFT. For this purpose, rods containing 7 and 20
vacancies (dubbed 7-rod and 20-rod, respectively) were
added to an edge of a 21-SFT and a 190-SFT, respectively
(see Fig. 1a and b). In addition, partial growth of the 190SFT was studied by adding three partial rods, 3-, 8- and
13-vacancies long. In the cross-section of the perfect SFT
corner (Fig. 1c) the site labeled V is where the rod is placed
(Fig. 1d). For a ledge to expand along the (1 1 1) SFT face
whose trace on the cross section is embodied by the dashed
Fig. 1. (a) A 190-SFT with atoms colored according to their centro-symmetry parameters. (b) As (a) with an extra vacancy row added to the bottom
facing one SFT edge. (c and d) Cross-sections of the stair-rod partial at the edge center (dashed lines in (a and b), respectively). Atoms with fcc centrosymmetry parameter (12) are not displayed in (a and b). All atomic conﬁgurations are visualized using AtomEye [43].
Fig. 2. The various growth stages at 300 K over 10 ps of a 21-SFT/7-rod combination in (a) Cu and (b) Al.
H. Wang et al. / Acta Materialia 59 (2011) 19–29
arrow in Fig. 1d the location of V must be unique. The simulations were conducted at 0, 300 and 600 K in Cu and Al
and at 900 K in Al. Initially, one of the partial rod’s extremities was either forced to the corner or centered on the SFT
edge (see Fig. 4a0 and b0, respectively).
3. Results
There are two possible conﬁgurations for a ledge formed
by vacancy absorption. The ledge referred to as obtuse, Iledge or 109.5° moves towards the SFT corners, whereas
the other termed acute, V-ledge or 70.5° moves towards
the SFT edge [13].
3.1. The growth of ledges by addition of full vacancy rods
Growth is ﬁrst investigated in conﬁgurations containing
a 21-SFT and a 7-rod aged for 10 ps at 300 K in both Cu
21
and Al, together with a 190-SFT and a 20-rod aged for
100 ps at 600 and 900 K in Cu and Al, respectively. Both
the 21-SFT and 190-SFT grow by expansion of ledges
(Figs. 2 and 3, respectively). In a given metal growth is largely independent of SFT size (compare Figs. 2a and 3a for
Cu and Figs. 2b and 3b for Al). Perceivable diﬀerences
between Cu and Al are found in the initial ledge forming
stage. In Cu ledges nucleated at the two rod extremities
(Figs. 2a1, a2, 3a1 and a2). In Al incipient ledge expansion
is more homogeneous along the SFT edge than in Cu (see
Fig. 3b2). SFT Growth behavior, however, rapidly
becomes similar in the two metals (compare Figs. 2a4
and b4 or Fig. 3a3 and b3).
The above material-dependent properties are consistent
with the vacancy site preferences in Al and Cu reported in
Wang et al. Part II (Figs. 3b and 6a in [26]). In Cu the binding of a vacancy is strongest in the middle of the SFT edge
and weakest at the SFT corner, thus encouraging sideways
Fig. 3. Growth of a 190-SFT/20-rod ensemble at 900 K over 100 ps in (a) Cu and (b) Al.
22
H. Wang et al. / Acta Materialia 59 (2011) 19–29
ledge expansion. Conversely, site preference in Al favors
SFT apexes, but only slightly, and hence the rather homogeneous ledge motion.
3.2. Ledge expansion from partial rods
At 0 K in Cu a rod placed at the edge center is stable
(not shown), whereas, depending on the number of vacancies in the rod, the same forced to the corner undergoes a
perceivable structural relaxation. Below 3 vacancies the
rod is stable. The 3-rod relaxes into the same ledge as in
Fig. 4a1. Beyond 3 vacancies the relaxed conﬁgurations
exhibit a half grown ledge connected to the rod (see the
inset between Fig. 4a2 and a3). The barrier that opposes
either growth from the central part of the SFT edge or further ledge expansion is, however, overcome upon aging at
600 K (Fig. 4b1–b3), again consistent with the relative stabilities of the corner and central conﬁgurations in Cu
reported in Wang et al. Part II [26].
The snapshots selected in Fig. 5 reﬂect the growth processes that yield the ﬁnal conﬁgurations in Fig. 4a3 and b3
that are adopted in Cu by a 190-SFT combined with a 13vacancy rod. Fig. 5a1–a8 shows that, when placed at the
SFT corner the rod tends to generate ledges that expand
horizontally to the left, while vacancies located in the central part of the SFT edge will not expand until they are
met by the ledge front, at which stage they proceed downward forming a symmetrical, ledged 193-SFT. The ﬁnal
conﬁguration is entirely consistent with the “V-ledge” analyzed by Kuhlmann-Wilsdorf [13]. Vacancy absorption
causes the ledge to move towards an SFT edge. The situation diﬀers signiﬁcantly when the vacancy rod is initially
forced to its central, minimum energy position, thus pre-
venting ledges from nucleating at corners (Fig. 5b1–b8).
In this case the 3-rod and 8-rod emit one wedged dipole of
Shockley partials (Fig. 4b1 and b2), probably because both
rods are too short to relax otherwise (the dipolar Shockley
partial conﬁguration is shown schematically in (c) and discussed in Section 4.3). In contrast, the 13-rod emits one
wedged Shockley partial at each of its extremities, while
its median part remains unaﬀected. This behavior again
reﬂects the property shown in Fig. 6a of Wang et al. Part
II [26], that the further a vacancy lies away from the SFT
corner, the more stable it is. Because of diﬀerences in binding energy along the rod, thermal activation at 600 K
encourages the nucleation of a Shockley dipole at the rod
extremities. The asymmetrical ledge development in
Fig. 5b originates from the fact that the rod extremities
are closer to the upper than to the bottom SFT corner, so
that the lateral edge of the upper Shockley partial reacts
with the neighboring SFT edge before the lower Shockley
partial can do so (Fig. 5b3 and b5). The property that ledge
expansion is inﬂuenced by vacancy site preference is manifest in both cases by the strong central pinning in
Fig. 5a3–a5 and by the central trailing cusp in Fig. 5b4–b7).
The simulations were performed in Al for the same rod
sizes on a 190-SFT at 600 K over 50 ps. The ﬁnal conﬁgurations after quenching consist of an acute ledge when the
rod is forced at the corner. They show a pronounced propensity to forming a fragmented ledge when the rod is
forced to the middle of the SFT edge (Fig. 6). It is worth
emphasizing that within the same aging period of 50 ps
the ledge expands to the maximum possible surface in Cu
(Fig. 4) but not, however, in Al (Fig. 6), indicating that
the motion of the Shockley dipole is relatively more diﬃcult in Al than Cu.
Fig. 4. A vacancy rod is added to the corner (a0) and the center (b0) of a 190-SFT edge in Cu and relaxed at 600 K during 50 ps. The atoms in (a0 and b0)
are colored according to their coordination number. (a1–a3) The relaxed conﬁgurations of a 3-, 8- and 13-rod laid at the SFT corner. The inset between (a2
and a3) is obtained after relaxation at 0 K of the same initial conﬁguration as for (a3). (b1–b3) As (a1–a3), respectively, with the rods centered on the SFT
edge.
H. Wang et al. / Acta Materialia 59 (2011) 19–29
3.3. The eﬀects of additional vacancies
Following the work of Osetsky et al. on Cu, conducted,
however, under a diﬀerent potential [27], the ledge properties were investigated by addition of individual vacancies.
After the introduction of vacancies from the apex of a
190-SFT at 0 K, no perceivable changes were found in
Cu until the third vacancy was added, forming a ledge
analogous to that shown in Fig. 4a1. The following vacancies provoke little relaxation of the ledge–rod conﬁgurations, which are all similar to the inset in Fig. 4. It is
only with the last one of the full vacancy rod, the 20th
vacancy, that another localized ledge comparable with that
located at the other apex is created (e.g. the two corners in
Fig. 7a). In Al no such relaxation is found after vacancy
addition at 0 K. On the other hand, after the introduction
of vacancies to a fully grown ledge on a 203-SFT (Fig. 7b),
23
imperfect SFTs behave similarly in Cu and Al. Relaxation
at 0 K does not encourage ledge growth and conﬁgurations
such as that exempliﬁed in Fig. 7b are stable. Subsequent
aging at 600 K favors ledge growth, which is complete in
Cu within 20 ps (Fig. 7c), while it is incomplete in Al (similar to, for example, Fig. 5a7).
In agreement with previous results (Section 3.4 in Martinez et al. [27]), the present MD simulations at 0 K show
that adding vacancies to a pre-existing SFT in Cu is not
enough to stimulate the growth of a perfect SFT to the next
perfect SFT because of various intermediate metastable
states. We also conﬁrm that SFT growth is strongly size
dependent. A diﬀerence lies in the fact that SFT reconstruction is observed in Martinez et al. [27] at 0 K upon addition
of 12 and 14 vacancies to a 91-SFT and 136-SFT, respectively, compared with the addition of three vacancies to
the 190-SFT in the present investigation (Fig. 4a1).
Fig. 5. The growth of the relaxed conﬁgurations in Fig. 4a3 and b3 (13-rod at 600 K in Cu) showing two distinct growth directions. Whereas the surface of
the untransformed stacking fault diﬀers between (a8 and b8), the number of vacancies (i.e. 7) needed to generate the next perfect 210-SFT is the same.
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H. Wang et al. / Acta Materialia 59 (2011) 19–29
4. Discussion
The true role played by the ledge mechanism in SFT
growth is relatively diﬃcult to apprehend because the initial conﬁguration is preset, which biases its own evolution.
Several general features are, however, worth discussing.
4.1. The operative growth mechanism
Numerous studies have been devoted to the stability of
SFTs versus that of other vacancy clusters utilizing either
energy calculations under linear elasticity [28–31] or MD
approaches under a variety of atomic potentials [32–35].
Because the detailed growth mechanism is experimentally
unknown and in view of the variety of diﬀusion routes
towards a cluster, one is led to making use of simplifying
assumptions. As far as elasticity calculations are concerned, jog nucleation and subsequent ledge expansion
are not tractable under the approximation of linear elasticity. On the other hand, the study of the relative stability of
SFTs versus faulted or perfect loops has been restricted to
the Silcox and Hirsch mechanism (see, for example, section
10–4 in Hirth and Lothe [36]), while this mechanism is conﬁrmed neither experimentally nor numerically. The wellaccepted result that beyond a critical size perfect SFTs
should form depending on the metal properties (stacking
fault energy and elastic constants) relies on a comparison
between the energies of various conﬁgurations. This comparison in turn postulates that vacancy segregation engenders perfectly triangular Frank loops [36], which have
never been observed experimentally while hexagonal Frank
loops are profuse. Deviations from the perfectly triangular
Frank loop are suﬃcient to explain the well-documented
presence of truncated SFTs [37] as is clearly illustrated by
the numerical simulations of Kadoyoshi et al. (Fig. 3 in
Kadoyoshi et al. [21]), Martinez et al. ([27], Fig. 3) and
Poletaev and Starostenkov [16]. On the other hand, the differences between the present work on ledge expansion and
the analysis of SFT growth by vacancy addition in Osetsky
et al. [38] mostly stem from how the vacancies are forced to
the SFT. In Osetsky et al. [38] Cu vacancies were added one
by one from one vertex to the next, then statically relaxed
and forced to stay in metastable positions. Similarly in Part
II of this series [26] vacancies were initially placed at their
most favored sites and relaxed at various temperatures in
order to reveal their possible path join the SFT.
In other words, and as far as numerical approaches are
concerned, the ﬁnal SFT and associated growth properties
are to a large extent determined by the postulated initial
conﬁguration. In spite of the considerable eﬀort devoted
Fig. 6. As Fig. 4 in Al. The ledges in (a2, b2, a3 and b3) are not fully expanded after 50 ps relaxation at 600 K.
Fig. 7. (a) The conﬁguration obtained by successive addition of 20 vacancies onto the edge of a 190-SFT in Cu and relaxation at 0 K. (b) The
conﬁguration relaxed at 0 K after introduction of one additional vacancy next to the fully grown ledge of a 203-SFT in Cu. (c) As (b) after 600 K aging for
20 ps resulting in full ledge zipping.
H. Wang et al. / Acta Materialia 59 (2011) 19–29
so far to this complex matter, a suﬃciently reliable description of the actual growth process is not available. In this
context, some early experimental investigations of SFTs
in quenched and annealed gold [20,39] are worth keeping
in mind. They indeed show that nucleation of vacancy clusters is completed in the very early stages of annealing and
that growth occurs by successive absorption of vacancies
thus supporting the ledge rather than the Silcox and Hirsch
mechanism.
4.2. 70.5° versus 109.5° ledges
In spite of a pronounced growth asymmetry, the ledge
engendered by rods forced to an SFT apex (Fig. 5a) moves
towards an edge (acute ledge), whereas the ledge resulting
from either a rod placed at the middle of an edge
(Fig. 5b) or a full rod (Figs. 2 and 3) moves towards an
SFT corner (obtuse ledge). The origin of this property
can be understood by inspection of the dislocation-equivalent conﬁguration of a vacancy absorbed on an SFT edge.
We regard jog nucleation along the SFT edge (Fig. 8) as
the ﬁrst move of the fault in plane d to plane d0 . The jog
25
may adopt two alternative conﬁgurations. In one (Fig. 8a
or, equivalently, Fig. 8b) it subsequently expands sideways
by glide of two Shockley partials with distinct Burgers vectors (dA and Cd) to reach both extremities, forming a full
V-ledge (Fig. 8c). This obtuse (109.5°) conﬁguration often
found in models [11,12,36] never forms in MD simulations
(Figs. 4 and 5), except of course when an SFT is cut by a
moving dislocation [40,41]. The detailed description of
the obtuse ledge is nevertheless useful to the understanding
of the growth processes described hereafter. Once the ﬁrst
ledge has fully expanded to the SFT edges (not shown)
the upper stacking fault ribbon is bordered by a dipole of
±db sessile stair-rod partial dislocations. A modiﬁed form
of this ledge is shown in Figs. 17–20 of Hirth and Lothe
[36] after the dA Shockley partial has merged with edge
AB or, alternatively, as a conﬁguration possibly adopted
by a vacancy nucleated at an apex (see Fig. 10). The mobile
parts are comprised of two pairs of Shockley dipoles ±Cd
and ±Cb bordering an undefected surface. These partials
are glissile parallel to edge AC in planes d and b, respectively, where they should not show an orientation preference and, hence, the equivalence between Fig. 8a and b.
Fig. 8. Dislocation-equivalent SFT conﬁgurations resulting from the absorption of one vacancy (the proportions are not respected on purpose). The
triangle denoted (d0 ) embodies the SFT face {1 1 1} plane of the next perfect SFT. In this ﬁgure, as well as in the following schemes, grey ﬁgures are
stacking fault free parallelograms. (a) Centered jog with two pairs of glissile Shockley partial dipoles glissile sidewise to form a ledge with Burgers vector db
along edge AC. (b) A variant of (a) illustrating that the Shockley partials can adopt any orientation in their own slip plane. (c) The conﬁguration resulting
from expansion of the jog up to the SFT edges and the absorption of a second vacancy. (d) A jog again centered on edge AC, however, not glissile along
the direction of AC. This jog, which would generate a V-ledge if formed on edge AB instead of edge AC, is actually that shown in (a). (e) The conﬁguration
after this jog has fully expanded up to the other SFT edge. (f) Ledge growth by absorption of a second vacancy.
26
H. Wang et al. / Acta Materialia 59 (2011) 19–29
After the ledge generated by the ﬁrst vacancy has fully
expanded along the SFT edge a second vacancy is added
onto this ledge, resulting in transport of the upper ±bd
stair-rod dipole one step above (Fig. 8c). An alternative
jog conﬁguration is presented in Fig. 8d, where, at variance
with Fig. 8a, the Shockley partials ±Bd and ±Bc are now
glissile parallel to AB in planes parallel to d and c in Fig. 8e
trailing a stair-rod quadrupole (±cd) (there is an equivalent
conﬁguration glissile parallel to CB). Fig. 8f shows the
ledge fully extended between two SFT edges, a conﬁguration clearly unfavorable compared, for example, with the
same assuming it were nucleated on AB (see the bottom
ledge along AC in Figs. 17–19 of Hirth and Lothe [36]).
Similarly to Fig. 8c, Fig. 8f represents the next growth step
of this hypothetical ledge after absorption of a second
vacancy. It is worth noting that if operative up to apex
C, the mechanism yields an acute ledge. Conversely, the
nucleation of jogs onto the other side until the step merges
with edge AB yields an obtuse ledge. In so far as jog nucleation at the middle part of an edge is concerned, a variant
of Fig. 8d is shown in Fig. 9a where the lateral faces of the
jog are not mutually parallel but parallel to the SFT faces a
and c. In this case the glide motion of the upper loop results
in a sessile triangular jog (Fig. 9b). Restricted to one
vacancy, this conﬁguration makes little sense, but the same
formed on a rod several vacancies long is remarkably similar to the triangles generated by the simulations (see, for
example, Figs. 4b1, b2, 5a3, b2 and b5).
We now consider jog nucleation at an SFT apex
(Fig. 10). In the dislocation-equivalent description absorption of the ﬁrst vacancy at A again makes no sense, as
shown by Fig. 10a. It is only after the second vacancy is
absorbed that the conﬁguration resembles the expected
70°5 ledge (Fig. 10a), similar to Figs. 17–19 in Hirth and
Lothe [36]. The growth mechanism is shown in Fig. 10c
with the absorption of a third vacancy. The conﬁguration
contains a loop whose edges consist of two pairs of dipolar
Shockley partials (±Ca and ±Cd) glissile in distinct planes
parallel to a and d, respectively, transporting the ledge
stair-rod da by a zipping mechanism analogous to
Fig. 8c. The numerical simulations presented in Wang
et al. Part II [26] have shown that the most favored sites
for the ﬁrst vacancy is located at the SFT apex, however,
not at site 1 but at site 2 (Fig. 3a of Wang et al. Part II
[26]). The most likely site for the second vacancy is A
(Fig. 4b of Wang et al. Part II [26]), fairly consistent with
Fig. 8c.
Some structural transformations resulting from vacancy
absorption by a relatively small 66-SFT have recently been
investigated by MD simulations using a model potential
for fcc crystals [16]. They show several signiﬁcant diﬀerences from the present results. In particular, (i) the ﬁnal
ledge conﬁguration, facing either a corner or an edge, is
concluded to depend on the number of added vacancies relative to the initial number of vacancies, a property dubbed
‘step sign change’ in Poletaev and Starostenkov [16], and
(ii) in the course of the simulations the same ledge may
reorient itself, changing the corner that it faces with an
apparently moderate activation energy. In Poletaev and
Starostenkov [16] the SFT is engendered from a platelet
of vacancies with a jogged triangular shape which upon
relaxation ﬁrst collapses into a Frank loop, then into an
SFT by a mechanism a` la Silcox and Hirsch. We believe
that the initial set-up is largely responsible for the step sign
change. The jogged triangular platelet is actually equivalent to a vacancy rod placed along one edge of the triangle,
in a corner position, before the SFT is formed. The origin
of property (ii) is unclear as it involves a considerable structural rearrangement under no clear driving force and may
originate from the atomic potentials utilized in Poletaev
and Starostenkov [16].
4.3. The core structure of the mobile dislocations bordering
the ledge
Common to Al and Cu is the property that the ledges
show a preferred orientation along h1 1 0i close packed
directions at 60° from the initial vacancy rod. A possible
explanation can be drawn from the evolution of an asymmetrical jog (Fig. 9). However, the sequences shown in
Fig. 5a3–a5 and Fig. 5b2–b4, where triangular ledges move
back and forth, together with the blocked conﬁgurations in
Fig. 9. (a) An asymmetrical variant of Fig. 8d. (b) The same fully expanded by glide. (c) Dislocation-equivalent sketch of segmentation of the Shockley
partial dipoles observed during MD simulations of SFT growth (see, for example, Fig. 3). (d) A schematic representation of the structure of wedged
Shockley dipoles consistent with the simulated conﬁgurations of, for example Fig. 3a5 and b6.
H. Wang et al. / Acta Materialia 59 (2011) 19–29
Fig. 6, suggest a core controlled behavior. Fig. 9c shows
that whereas the lateral edges of the dislocations bordering
the surface step are stair-rod in nature, the inner portion
comprises a glissile dipole of Shockley partials (i.e. ±Cd,
see Fig. 9d). It is interesting to compare the kinetics of
SFT growth observed in Al and Cu after a full vacancy
rod is added to a 190-SFT in order to form the next perfect
210-SFT. In both cases growth takes place by a ledge-like
process with a markedly trailing central portion. Our
results on site preference in Cu would therefore oﬀer a reasonable explanation for the preferred 60° orientation.
However, the fact that a similar trailing behavior is
observed in Al (Fig. 3b6), where site preference should
encourage the propagation of the central part at the
expense of the lateral segments, rules out this
interpretation.
We believe that ledge expansion is controlled by the
length of the Shockley partial dipole. For simplicity, we
consider the growth of an SFT with edge length L via (i)
two equilateral triangles with edge length, LL = aL, (ii) a
central equilateral triangle and (iii) a trapezium with equal
surfaces all (Fig. 11). We consider symmetrical conﬁgurations, i.e. the Burgers vector of the Shockley dipole is Bd
and the ledges propagate in plane d. In conﬁgurations (i)
and (ii) the bordering Shockley dipoles are 60° in character
(self-energy proportional to 4 m=4ð1 mÞ), but pure edge
in conﬁguration (iii) (self-energy proportional to
1=ð1 mÞ). Hence, growth via a central equilateral triangle
p
necessitates a length of dipolar Shockley partials 2 as
large as that required for the two lateral triangles, consistent with the behavior observed for Al and Cu. However,
growth
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ via the trapezium necessitates a length equal to
4ð 14a2 1Þ
LL so that, with m ½ and for a 6 ½, the trapeað4mÞ
zoidal ledge should always be favored, clearly at variance
with Fig. 3 (a similar conclusion is reached with a Burgers
vector that makes the conﬁguration asymmetrical, e.g. Cd
in plane d).
27
In order to explain the observed segmentation one is led
to the hypothesis that the ±60° Shockley dipoles reorganize
themselves at the atomic scale to form a stable core relative
to all other dipole orientations. Formally, instead of the
dipolar ±Bd Shockley partials bordering a non-faulted
strip, there is a possibility in the ±60° orientations that, driven by strong attractive forces, each ±Bd partial splits into
±(Bc + cd) or ±(Ba + ad) in planes parallel to c and a,
respectively, and that the Shockley partials mutually annihilate by glide in the appropriate {1 1 1} plane. MD simulations indicate that ledge expansion is signiﬁcantly
impeded by lattice friction, suggesting in turn some complex rearrangement similar to what was found for edge
dipoles in fcc metals [42].
The eﬀects of lattice friction are best viewed in dynamic
sequences where the ledges do not move continuously but
stop at times. Motion takes place by kink nucleation and
propagation along the 60° portions (Fig. 3a4 and b5) with
occasional backward motion. Fig. 12 shows snapshots
taken every 2.5 ps in Al annealed at 900 K (see Fig. 3b4)
Fig. 11. Three possible ledge conﬁgurations segmented along the h1 1 0i
crystallographic directions.
Fig. 10. Conﬁgurations resulting from the absorption of vacancies at the SFT apex A. (a) One vacancy. (b) Two vacancies (an alternative view is shown at
site B). (c) The growth mechanism of an acute ledge.
28
H. Wang et al. / Acta Materialia 59 (2011) 19–29
Fig. 12. Snapshots of the ledge growth in Al at 900 K selected around Fig. 3b4, showing the back and forth motion of ledge kinks in places. Arrows
indicate growth directions.
where kink motion occurs backwards in Fig. 12b and forward in Fig. 12c on the right-hand side Shockley dipole
portion, then on the left-hand side (Fig. 12c–e).
5. Conclusions
Highly dependent on the method chosen for its investigation, the actual process of SFT growth is largely
unknown. Here rods containing selected numbers of vacancies were forced on the edge of SFTs of various sizes in Al
and Cu, chosen for their signiﬁcantly distinct stacking fault
energies. At the atomic level the ledge mechanism exhibits
the following properties.
1. Ledge nucleation and the early stage of ledge expansion
are both material and temperature dependent. In Cu
vacancy rods prefer to aggregate at the middle of SFT
edges, while in Al they aggregate at SFT vertexes.
2. Ledge transformation depends critically on the material
and, within a given material, on rod location.
3. The ledge growth rate increases with temperature. It is
faster in Cu than in Al at the same temperature.
4. Depending on temperature and on the number of vacancies in the rod relative to that stored in the SFTs, ledge
expansion yields diﬀerent conﬁgurations, including
edge- and corner-facing ledges
5. The Shockley partial dipole that borders a ledge is
highly segmented along h1 1 0i directions, reﬂecting several kinds of core reorganization, consistent with the
observed dependence of ledge expansion upon
temperature.
Acknowledgements
The support of the Ministry of Science and Technology
of China under Grant No. 2006CB605104 and the Natural
Science Foundation of China under Grant No.
50911130367 and No. 50631030 is gratefully acknowledged. Patrick Veyssie`re should like to thank the Institute
of Metal Research, Chinese Academy of Sciences for ﬁnancial support and outstanding hospitality.
Appendix A. Supplementary material
Supplementary data associated with this article can be
found, in the online version, at doi:10.1016/j.actamat.2010.
07.045.
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