Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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422 10 Quasicrystals

Fig. 10.24. Schematic of the multiplication of dislocations in Al–Pd–Mn after a

change from the primary plane of motion PP onto a secondary one SP. After [711]

diﬀerent planes. After full bow-out, two dislocations move on the secondary

plane in opposite directions (stage 4). Thus, dislocation motion spreads from

one plane onto other planes as discussed in connection with Fig. 10.18, lead-

ing to the dislocation networks formed at high temperatures. However, at low

temperatures, the dislocations seem to return to planes parallel to the origi-

nal ones similar to the multiplication in crystals as indicated by the narrow

dislocation bands observed (Fig. 10.15b).

The following video clip illustrates dislocation generation, always con-

nected with some change of the plane of motion.

Video 10.5. Dislocation multiplication in i-Al–Pd–Mn single quasicrystals: The

three sequences were taken at 720, 685, and 675

◦

C. The parent dislocations are

marked by M. In the ﬁrst sequence, the dislocation changes its plane of motion to

a new one as described above. Two new dislocations emerge and move in oppo-

site directions on a plane with its trace perpendicular to the tensile direction. In the

second sequence, the multiplication event is connected with a small parallel displace-

ment of the plane of motion. One dislocation quickly moves backwards, followed by

two dislocations moving in forward direction. In the third sequence, the dislocation

undergoes a shift onto a parallel plane. At A, one dislocation moving back annihi-

lates with another one on the same plane. At R, two dislocations moving on planes

with parallel traces of diﬀerent inclination in the foil react with each other. As a

result, a dislocation with double contrast moves to the right.

The in situ study in [704] also revealed dislocation multiplication.

10.3.2 d-Al–Ni–Co

As classiﬁed in [716,717], the decagonal phase of Al–Ni–Co may have diﬀerent

modiﬁcations depending on the exact composition. In the electron diﬀraction

patterns, they show particular superstructure reﬂections, being ﬁrst observed

in [718]. The diﬀerent modiﬁcations correspond to diﬀerent states of structural

and chemical order. At the Co and Ni-rich ends of the decagonal phase ﬁeld,

the decagonal structures do not show superlattice reﬂections. These structures

are termed ‘basic Co-rich’ and ‘basic Ni-rich’ variants. The results following

below are conﬁned to the composition Al-15at%Ni-15at%Co with the type I

superstructure showing the so-called S1 and S2 superlattice reﬂections [718].

10.3 Microscopic Observations of Dislocations 423

Fig. 10.25. Parallel components of the Burgers vectors in d-Al–Ni–Co single qua-

sicrystals as determined in [720]. b

1

is directed perpendicular to the quasiperiodic

image plane

The Burgers vectors of dislocations in decagonal quasicrystals were ﬁrst

studied by Zhang and Urban [719] via contrast extinction, and later on by

other authors also by applying the CBED method (e.g., [719,720]). The paral-

lel components of the observed Burgers vectors are outlined in Fig. 10.25. The

grey arrows marked by 2 to 6 are the basis vectors in the quasicrystalline plane.

They are longer by a factor of τ than those introduced by Steurer et al. [667].

The Burgers vectors observed are the following: the periodic Burgers vector

directed along the periodic axis of the length of the c lattice constant in par-

allel space, in the full ﬁve-dimensional notation of Steurer et al. [667] written

as B

= [00001], the Burgers vectors in the quasiperiodic plane B

10010]

and B

1010] of the lengths of 0.29 nm and 0.47 nm in parallel space,

and a mixed Burgers vector B

1/5,

1/2] of the length of

0.25 nm in parallel space. The mixed Burgers vector is not a lattice vector

in hyperspace so that the respective dislocations are partial dislocations. The

strain accommodation parameters ζ for the quasiperiodic Burgers vectors are

−1

and τ

−3

. Compared to i-Al–Pd–Mn, in d-Al–Ni–Co these values indicate

a lower tendency of the dislocations to be accompanied with phason defects.

The dislocations induced by plastic deformation show very strong preferen-

tial orientations parallel or perpendicular to the periodic axis. The dislocations

with periodic Burgers vectors assume both these line directions whereas those

with quasiperiodic or mixed Burgers vectors are arranged only parallel to the

periodic axis.

While, because of the high symmetry of icosahedral Al–Pd–Mn, its plastic

behavior is very isotropic, decagonal Al–Ni–Co has a restricted number of

Burgers vectors of diﬀerent character and only a few dislocation line directions

resulting in a remarkable plastic anisotropy. It was ﬁrst revealed on single Al–

Ni–Co quasicrystals in [721]. Mostly, stress–strain curves were measured with

loading axes parallel to the periodic axis, perpendicular to it, and at an angle

of 45

◦

, which may be denoted A



, A

⊥

,andA

. The highest yield stress always

has the orientation A

⊥

, whereas the lowest one has A

. The amount of the

intermediate value of A



depends on the composition, it is slightly higher than

424 10 Quasicrystals

that of A

for Al-15at%Ni-15at%Co and slightly less than that of A

⊥

for the

basic Co-rich material.

A characteristic deformation microstructure produced during in situ defor-

mation along the A



axis is shown in Fig. 10.26a. The ﬁgure was taken with a

systematic row of reﬂections parallel to the periodic axis, i.e., containing the

G = (00002) reﬂection, with a higher reﬂection excited (high-order bright-ﬁeld

technique). This direction is also the tensile direction TD. Under these con-

ditions, dislocations with quasiperiodic Burgers vectors are extinguished. The

arrows in the ﬁgure point to dislocations with a periodic Burgers vector, which

appear as short dark lines that are attached to extended gray lines, which are

the traces of the dislocation motion. The periodic dislocations, which are seen

almost end-on, move along quasiperiodic planes perpendicular to the periodic

axis. Many other traces in the same direction as of dislocations running out

of the image area are visible as well. The dark double line in the center of the

ﬁgure is the trace of many dislocations. The gray lines parallel to the tensile

direction, marked by M, are dislocations which, according to [720], have mixed

Burgers vectors.

Figure 10.26b refers to a diﬀerent specimen region, taken with a system-

atic row containing the quasiperiodic reﬂection G =(

10010). Dislocations

with mixed and quasiperiodic Burgers vectors are visible but those with the

periodic Burgers vector are now extinguished. The entire image shows a back-

ground structure with a striation contrast parallel to the tenfold axis. This

contrast possibly originates from phason boundaries resulting from phason

disorder [722]. It appears under all diﬀraction vectors that exhibit any com-

ponent in the quasiperiodic direction. It makes the observation of dislocations

diﬃcult.

In macroscopic deformation experiments, all specimens were deformed

within the high-temperature range. In the A



orientation, the dislocations

form networks between dislocations with periodic and mixed Burgers vectors

as illustrated in Fig. 10.27a. Similar structures are observed in Al-15at%Ni-

15at%Co. Burgers vector analysis by the CBED method [720] showed that the

networks are formed by dislocation reactions with the periodic component of

the Burgers vector of the mixed dislocation being opposite to the direction

of the Burgers vector of the periodic dislocations, as outlined in Fig. 10.28.

The dislocations in Fig. 10.28a move across each other, forming a junction in

(b). The reaction is connected with a gain in energy. In (c), the junction has

rearranged. According to the deﬁnition in Sect. 10.3.1, the dislocations with

periodic Burgers vectors running horizontally in Fig. 10.27a are equivalent to

set A experiencing a high climb force but no glide force.

In the A

orientation, periodic dislocations arrange along the screw and

edge orientations with the screws dominating as illustrated in Fig. 10.27b.

The image plane is parallel to planes marked by slip steps on the specimen

surface. The ﬁgure also shows a dislocation loop. If it is assumed that the

loop plane is also the plane of motion, the loop has been formed by glide.

It may therefore be concluded that periodic dislocations in d-Al–Ni–Co can

10.3 Microscopic Observations of Dislocations 425

1 µm

(a)

1µm

(00004)

(b)

Fig. 10.26. Deformation microstructure during in situ straining of a d-Al–15at%Ni–

15at%Co single quasicrystal along the A



axis in an HVEM at 730

◦

C, taken at the

P2 twofold pole of the foil normal. (a) g vector in the direction of the periodic axis.

(b)Quasiperiodicg vector. From [666]. Copyright Mater. Res. Soc. (2005)

glide. In addition, also dislocations with mixed Burgers vectors contribute to

the deformation. At the A

⊥

deformation axis, dislocations with quasiperiodic

and mixed Burgers vectors dominate. They are arranged along the periodic

axis.

Some dislocation density data of deformed Al–Ni–Co single quasicrystals

are compiled in Table 10.1, revealing a slight dominance of periodic disloca-

tions in the A

orientation, where slip of periodic dislocations is possible,

with a comparatively very low value of the total dislocation density. All

426 10 Quasicrystals

(a)

(b)

Fig. 10.27. Dislocation structures in decagonal basic Co-rich Al–Ni–Co single qua-

sicrystals deformed in macroscopic compression. (a) A



compression axis, 890

◦

image plane containing the tenfold g vector g

.(b) A

compression axis, 790

◦

image plane parallel to slip plane. g vector parallel to tenfold direction. From [723],

courtesy of Peter Schall

(a) (b) (c)

Fig. 10.28. Schematic of the reaction after [720, 723] between dislocations with

periodic and mixed Burgers vectors in d-Al–Ni–Co to form a dislocation network.

The Burgers vectors are indicated by grey arrows

Table 10.1. Densities of dislocations with periodic 

,mixed

, and quasiperiodic

Burgers vectors 

in deformed Al–Ni–Co single quasicrystals in units of 10

−2



and 

are densities of periodic screw and edge dislocations. Data from [723]

material, axis T (

◦

C) ε (%) 



Basic Co, A



860 6 4.2 14.5

Basic Co, A

860 6 0.6 0.2 0.5

Al-15Ni-15Co, A

790 4.50.10.60.6

Basic Co, A

⊥

860 6 1.2 17 19

10.3 Microscopic Observations of Dislocations 427

dislocation densities are remarkably higher than those in undeformed but

annealed reference specimens. They decrease with increasing temperature.

In situ straining experiments in an HVEM [666] yielded some character-

istics of dislocation motion in the A



orientation at 730

◦

C. This temperature

is relatively low compared with the temperatures of the conventional TEM

experiments described above. Most dislocations observed moved on planes per-

pendicular to the periodic axis, like those indicated by arrows in Fig. 10.26a.

These dislocations consist of two straight segments with a sharp knee between

them. They move by pure climb since their Burgers vector is perpendicular to

the plane of motion. As the tensile direction TD is also perpendicular to the

plane of motion, the climb force acting on the dislocations from the applied

load is maximum, but there is no glide force. These dislocations may be called

periodic dislocations on the climb system and are equivalent to the dislocations

of set A in i-Al–Pd–Mn. The following video illustrates their motion.

Video 10.6. Motion of dislocations with periodic Burgers vectors in a d-Al–Ni–

Co single quasicrystal: The dominating feature is the continuous viscous motion of

the dislocations in d-Al–Ni–Co. In some cases, dislocations move on planes with

traces on the specimen surface parallel to those in Fig. 10.26a, but which have a

diﬀerent width, i.e., they are inclined with respect to the climb plane. In the ﬁrst

sequence, the plane of motion of the dislocation marked by a yellow circle changes

continuously accompanied with a reduction of the width of the trace. The motion

should then occur by a combination of climb and glide. In the second sequence, two

dislocations of opposite sign annihilate each other. In the third sequence, periodic

dislocations cross those of mixed Burgers vectors. The two types of dislocations do

not seem to interact with each other. As described above, the reaction occurs only if

the periodic Burgers vector component of the mixed dislocations is opposite to the

Burgers vector of the periodic dislocations.

Figure 10.29 demonstrates that dislocations with periodic Burgers vectors

may change between the climb plane, labeled c, and a plane with a trace

inclined by about 18

◦

with respect to the trace of the climb plane, labeled i.

The places of the transition are marked by black arrows. In Fig. 10.29, these

inclined planes are also imaged edge-on. With reference to the diﬀraction pat-

tern in Fig. 10.6 (Fig. 10.29 is slightly rotated against Fig. 10.6), these planes

may be indexed (0

1104). The mode of motion on these planes is a combination

of climb and glide. The inclined trace i in the middle of the ﬁgure changes

into the direction of the Burgers vector or the tensile direction, marked by g,

and back. The corresponding plane of motion is imaged edge-on, too. It may

be indexed (01

100). This plane is a glide plane of these dislocations.

Similar processes are illustrated in the ﬁrst sequence of the next video with

another dislocation iteratively changing its plane of motion.

Video 10.7. Climb and glide motion of a dislocation in d-Al–Ni–Co: In the ﬁrst

part, the dislocation D extends approximately perpendicular to the periodic axis.

The traces directly connected with the dislocation are parallel to the periodic direc-

tion. Afterwards, the dislocation shifts to a plane with inclined traces, which have the

428 10 Quasicrystals

1μm

Fig. 10.29. Micrograph showing traces of periodic dislocations moving on diﬀerent

planes. From [666]. Copyright Mater. Res. Soc. (2005)

same orientation on the surface as the traces i in Fig. 10.29. Later on, it moves again

on the same plane as at the beginning of the recording. In contrast to Fig. 10.29,

the planes with the inclined trace and with the trace in the direction of the periodic

axis are not imaged edge-on. Considering the ratio between the widths of the two

traces, the corresponding planes of motion may be indexed (0

1002) for the inclined

trace and (01000) for the trace in the direction of the periodic axis. In principle,

the planes of motion of a dislocation can change by glide and cross glide. The Burg-

ers vector would then be parallel to the intersection line between the two planes,

which is parallel to a quasiperiodic direction. These Burgers vectors are, however,

extinguished under the present imaging conditions with a g vector in the periodic

direction. Therefore, it is supposed that the dislocation also has a periodic Burg-

ers vector, since, according to [720], only these dislocations assume line directions

perpendicular to the periodic axis. Thus, the dislocation is a pure edge dislocation.

Consequently, the mode of motion is a combination of climb and glide for the inclined

traces, and it is a pure glide motion for the other one. The occurrence of the glide

motion is surprising, since in the current geometry, there should be no glide force in

a specimen with a homogeneous cross section. A glide force acting on the dislocation

could be due to the inhomogeneous shape of the microtensile specimen. The second

part of the video shows essentially the same. Now the dislocation changes between a

plane which is probably inclined with respect to the tenfold axis and the one normal

to it.

The dislocation velocities in the diﬀerent modes of motion are quite diﬀerent.

Measurements from the video recording yield 1.1and7nms

−1

for glide, and

325 and 285 nm s

−1

for the combination of climb and glide. In the same speci-

men area, a dislocation moved on the climb plane at a velocity of 460 nm s

−1

This shows that even for the periodic dislocations the glide velocity may be

much lower than the climb velocity. However, because of the very diﬀerent

forces acting, the relation between the velocities does certainly not describe

the relation of the dislocation mobilities on the diﬀerent planes.

10.3 Microscopic Observations of Dislocations 429

Both the macroscopic experiments in the A



orientation and the in situ

experiments show another set of dislocations with mixed Burgers vectors

marked by M in Fig. 10.26a. They move on planes containing the periodic

axis. The dislocation MCM expands by the curved segment C moving upwards

thus trailing the long straight dislocations labeled M. Accordingly, the long

straight dislocations parallel to the periodic axis do not represent the mobile

parts of the mixed dislocations. The mode of motion can be determined only

by indexing both the plane of motion and the Burgers vector. The determi-

nation of the Burgers vector was not possible during the in situ experiments

because of the unsuitable imaging conditions with most g vectors (see above,

Fig. 10.26b). As the plane of motion contains the tensile direction, neither

a glide nor a climb force from the applied load acts on the dislocations in

a macroscopic sense. Thus, these dislocations are equivalent to set B in i-

Al–Pd–Mn. Dislocation velocities were measured from a video recording of

a moving mixed dislocation segment as well as periodic dislocations on the

climb system. At the same applied stress, the dislocations on the climb system

are 20 times faster than the mixed dislocation.

As described above (Fig. 10.28), dislocations with periodic Burgers vec-

tors can interact with dislocations with mixed Burgers vectors. In most cases,

however, the moving periodic dislocations cut the mixed ones without a vis-

ible sign of reaction (third sequence in Video 10.6). However, sometimes the

reaction takes place as in the following video.

Video 10.8. Reaction between a dislocation with a periodic Burgers vector and

onewithamixedBurgersvectorinad-Al–Ni–Co single quasicrystal: The event

is marked by a yellow dot. The periodic dislocation exhibits a sharp knee, like

the dislocations marked by arrows in Fig. 10.26a. It moves to the left towards the

mixed dislocation. The latter shows a similar curved shape as that in Fig. 10.26a.

Both dislocations attract each other and join to form a segment of the product

dislocation. At the same time, the climbing periodic dislocation segment is shifted

onto a parallel climb plane (downwards in the video) due to the mutual elastic

interaction. The latter segment is ﬁxed at the junction with the product dislocation

and starts to extend on the new climb plane (E). Later on, the segment indicated by

the straight yellow line moves back (upwards in the ﬁgure) and removes the product

dislocation. As a result, the mixed dislocation is restored and moves slowly upwards

while the periodic dislocation resumes its motion on the new climb plane. The dark

contrasts mark the traces of the dislocation motion.

In other cases, the same dislocation reaction results in the formation of a

dislocation source, as demonstrated in the video.

Video 10.9. Dislocation reaction and formation of a dislocation source in a d-Al–

Ni–Co single quasicrystal: Several dislocations with a periodic Burgers vector move

on the climb plane from right to left close to the bowed part of the dislocation

M with a mixed Burgers vector. After the mixed dislocation has slowly proceeded

upwards, the periodic dislocation following P reacts with the mixed one to form a

junction. Now, however, the segment of the periodic dislocation remains attached

to the junction and draws a long dislocation towards the left side. After the new

430 10 Quasicrystals

1 µm

(a)

(b)

Fig. 10.30. Video frame of a dislocation climb source in d-Al–Ni–Co (a)and

schematic of several stages of the source (b). From [666]. Copyright Mater. Res.

Soc. (2005)

long segment of the periodic dislocation parallel to the specimen surface escaped

through the surface, the remaining new dislocation moves out of the image area

and the segment attached to the junction rotates to the right side of the mixed

dislocation and expands there. Finally, the source has emitted a dislocation moving

to the left whereas the rotating arm of the source has already turned to the right side

(arrows), with the source being restored, as illustrated in Fig. 10.30a. Some stages

of the operation of the source are schematically drawn in Fig. 10.30b. Afterwards,

the source continues its revolutions. After each revolution, the rotating segment

shifts slightly down onto a parallel plane. In the second part of the video, this spiral

character of the source is clearly visible.

Dislocation sources operating with climbing dislocations are called Bardeen-

Herring sources [724]. Several sources of this kind have been observed,

emitting tens of dislocations each. Owing to the spiral character, the generated

deformation bands grow in width.

During the observation of one source, the load was changed to study

the dynamic dislocation behavior. From the video recordings, the frequency

of the revolutions of the source f

source

was determined as a function of

the load F . In Fig. 10.31a, the results are plotted in a double-logarithmic

scale. Each symbol corresponds to one revolution of the source. During the

ﬁrst part of the experiment, about 30 dislocations were emitted from the

source. The frequency of the source decreased only slightly while the load

remained almost constant (downward open triangles). Afterwards the speci-

men was partly unloaded until the source almost stopped to revolve. In the

plot of Fig. 10.31a, this phase of the experiment is characterized by a lin-

ear decrease of the frequency of the source with a load or stress exponent



sc ul

=dln(f

source

)/dlnF =dln(f

source

)/dlnσ =7.1. Afterwards, the spec-

imen was reloaded until fracture (full upward triangles). The ﬁgure shows that

unloading and reloading are reversible over the main part of the load range.

The reloading curve is linear, too, yielding a stress exponent of m



sc l

=4.4.

10.4 Macroscopic Deformation Parameters 431

In[F (N)]

In [f

source

−1

)]

1.8 2.0 2.2 2.4

−5

−4

−3

−2

−1

sc ul

= 7.1

sc l

= 4.4

(a)

In[F (N)]

In [v

(nm s

−1

)]

1.8 2.0 2.2 2.4

m' = 3.9

(b)

Fig. 10.31. Load dependence of the frequency of the operation of a dislocation

source (a) and velocity of dislocations with periodic Burgers vectors moving away

from the source (b). Open downward triangles: unloading; full upward triangles:

re-loading. Data from [666]

For higher loads, the unloading and reloading curves do not coincide. In addi-

tion, the velocity v

of the dislocations emitted to the left was determined

from digitized video frames and plotted in Fig. 10.31b also as a function of

the load. This curve is linear, too, yielding a load or stress exponent of the

dislocation velocity of m



=dlnv

/dlnF =dlnv

/dlnσ =3.9.

In summarizing this section, the following may be stated.

• At high temperatures, dislocations in icosahedral Al–Pd–Mn quasicrystals

are mostly straight and oriented along crystallographic directions. They

move on crystallographic planes.

• The motion is smooth in a viscous way with a stress exponent of about

4–6.

• Many dislocations move on planes where they experience a climb force

but no glide force (set A), or even no force at all from the external load

(set B). The determination of both the Burgers vectors and the plane of

motion showed that climb is the dominating mode of dislocation motion

in icosahedral quasicrystals.

• The dislocations drag phason walls which disappear at high temperatures

by retiling.

• At low temperatures, the dislocations are curved.

• In decagonal quasicrystals, periodic dislocations may climb and glide. The

mode of motion of the quasiperiodic and mixed dislocations is not yet

clear.

10.4 Macroscopic Deformation Parameters

The whole methodology, which for more than 50 years had been designed to

study the plastic deformation of crystals, was applied to quasicrystals within

a few years. As diﬀusion is involved in the plastic deformation of quasicrys-

tals, respective data, usual deformation experiments at ambient pressure are