Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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402 10 Quasicrystals

Fig. 10.8. Two-dimensional quasiperiodic structures generated by the density-wave

method. (a) Perfect pattern. (b) Pattern of (a) distorted by a spatial variation of

the phonon degree of freedom. (c) Pattern distorted by varying the phason degree

of freedom. (d) Dislocation. Reprinted with permission from Socolar et al. [679].

v34/p3345/y1986)

of the cut and projection procedure. The undisturbed arrangement of lat-

tice points is demonstrated in Fig. 10.8a, showing diﬀerent kinds of clusters

forming the structure. Looking at this ﬁgure at a glazing angle reveals the

diﬀerent atom rows. In Fig. 10.8b, the same structure is distorted by spatially

varying displacements of only phononic character. The originally straight rows

of atoms are now curved, corresponding to elastic strains. The eﬀect of varying

phasonic displacements is demonstrated in Fig. 10.8c. The rows of atoms are

straight again but they are jagged, i.e., they show lateral displacements. Sim-

ilar images were revealed by high-resolution electron microscopy (e.g., [681]).

Relaxation of the phason faults requires local diﬀusion of atoms. Therefore,

these processes occur far more slowly than phonon relaxations. The dynamics

of long-wavelength phason ﬂuctuations was studied by changes in the speckle

contrast in a coherent X-ray-scattering experiment [682]. According to that,

phason ﬂuctuations in i-Al–Pd–Mn are frozen-in below 500

◦

C. At 650

◦

C, the

10.2 Defects in Quasicrystals 403

corresponding diﬀusion coeﬃcient is 2.2 × 10

−18

−1

with an activation

energy of 2.3(±1)eV. This diﬀusion coeﬃcient is in the range of those of

Pd and Mn. Phason ﬂips were observed in situ at a high temperature in a

decagonal Al–Cu–Co quasicrystal by high-resolution TEM [683].

10.2.3 Dislocations

As in crystals, plastic deformation is realized by the motion of dislocations.

A recent review has been given by Bonneville, Caillard and Guyot [684].

Figure 10.9 shows the result of a computer simulation of the shearing and

introduction of an edge dislocation into a two-dimensional decagonal qua-

sicrystal from [685]. A Burgers circuit (see Sect. 3.1.1) is carried out around

the dislocation in B and the corresponding reference circuit is shown in A. It

has to be closed by the Burgers vector of the dislocation. Since the shearing

was carried out in physical space, there remain a number of places with vio-

lations of the matching rules marked by grey shading, i.e., of phason defects.

Therefore, the Volterra process to generate a dislocation should not be per-

formed in the physical space but in the higher-dimensional hyperspace so

that in icosahedral quasicrystals the Burgers vector is a six-dimensional vec-

tor [686]. The cutting procedure similar to Figs. 3.7 and 3.8 is carried out along

a ﬁve-dimensional half-plane comprising the whole perpendicular space. The

Fig. 10.9. Creation of phason strain by the introduction of a dislocation into a

two-dimensional decagonal quasicrystal. From [685]. Copyright Taylor & Francis

Ltd. (1995) (http://www.informaworld.com)

404 10 Quasicrystals

dislocation line is a four-dimensional manifold, a one-dimensional line in phys-

ical space plus the whole orthogonal space, since after (10.5), the strain ﬁeld

does not depend on the coordinates of the perpendicular space. The geomet-

ric properties of dislocations and their motion in the quasiperiodic lattice are

described in detail in [687].

In analogy to (3.1), the Burgers vector B in quasicrystals can be deﬁned

by a line integral over the displacements U along a closed curve C around the

dislocation in the physical space E



B =



dU =







+du

⊥



= b



+ b

⊥

. (10.6)

Thus, the integration over the phonon strain ﬁeld yields the parallel part b



of the Burgers vector. It corresponds to the usual Burgers vector in crystals

describing the continuous elastic strains around the dislocation line. For the

edge dislocation, b



is perpendicular to the dislocation line, whereas for the

screw dislocation it is parallel. In many processes, b



replaces the Burgers

vector known from crystals. The perpendicular part of the Burgers vector b

⊥

represents the phason strain ﬁeld around the dislocation line. Both phenom-

ena can be observed in the two-dimensional density wave image of an edge

dislocation in Fig. 10.8d. The lattice rows are bent around the inserted half-

row characterizing the elastic strain ﬁeld of the dislocation. In addition, there

appears a number of jags in the atom rows.

The full six-dimensional Burgers vector of (three-dimensional) quasicrys-

tals can be determined by several methods. As shown by Wollgarten et

al. [688], the contrast extinction criterion (2.9) of diﬀraction contrast TEM

can be extended to yield

G · U = g



· u



+ g

⊥

· u

⊥

=0, (10.7)

with G = g



+ g

⊥

. Since the cut in six-dimensional space has an irrational

orientation, each g



belongs to a single G in the six-dimensional reciprocal

lattice, and accordingly also to a single g

⊥

in reciprocal space [654,689]. The

criterion (10.7) can be fulﬁlled in two ways. The ﬁrst one is by making both

terms zero, i.e.,



· u



= g

⊥

· u

⊥

=0. (10.8)

This is called a “strong” contrast extinction condition. Then, the dislocation

contrast disappears for a whole systematic row of reﬂections through the origin

of the diﬀraction pattern. Two such conditions have to be found to determine

the direction of the Burgers vector in both the physical and orthogonal space.

In the second case,



· u



= −g

⊥

· u

⊥

. (10.9)

This is the “weak” contrast extinction condition, which is fulﬁlled only for

periodically spaced reﬂections within a systematic row but not for other reﬂec-

tions of the same row. The determination of the length of the Burgers vector

requires a quantitative analysis of the dislocation contrasts as in crystals.

10.2 Defects in Quasicrystals 405

Wang and Dai [690] were the ﬁrst to use convergent beam electron diﬀrac-

tion (CBED) to determine the six components of the Burgers vector in

icosahedral quasicrystals. In this technique, the dislocation image is super-

imposed with the defocus CBED pattern where zero-order Laue zone (ZOLZ)

lines split n times when a dislocation line with |G · B| = n is crossed. B

is determined completely by six independent equations G · B = n.Afurther

method employs high-resolution TEM by observing lattice fringe images taken

in two diﬀerent orientations of the specimen [691].

The elastic properties of dislocations in quasicrystals are treated in the

framework of an extended elasticity theory (e.g., [692], for an introduction

see, e.g., [693, 694]). Considering only short-time elastic responses, icosahe-

dral quasicrystals behave isotropically so that their elastic properties can be

described by two elastic constants (e.g., the Lam´e constants λ and μ). They

can be measured in the usual way by ultrasound transmission. In the gen-

eralized Hooke’s law, there appear also two phason elastic constants K

and

, which are determined indirectly by diﬀuse X-ray or neutron scattering

(e.g., [695]), and a phonon–phason coupling term R. Like for strains, there

occur stress components in physical space and orthogonal space. The stresses

in real space depend on the elastic strains via the Lam´e constants, and on

the phason strains via the coupling constant. On the other hand, the stresses

in perpendicular space depend on the phason strain via the phason elastic

constants, and on the phonon strains via the coupling constant. A set of elas-

tic constants for i-Al–Pd–Mn used in another terminology in [692] comprises

λ = 85, μ = 65, K

=0.084, K

=0.036, and R =0.41 (all in GPa). The elas-

tic energy of a quasicrystal is composed of three terms, one from the phonon

part, one from the phason part, and a coupling term.

Not only the phononic strains of dislocations but also the phasonic ones

decrease with 1/r,wherer is the distance from the dislocation line. Only pure

screw and edge dislocations along a twofold orientation have simple displace-

ment ﬁelds with displacements only parallel to the line direction for the screws,

and perpendicular to it for edges. In all the other cases, the properties are not

so simple. The equilibrium positions of parallel dislocations with respect to

glide were calculated in [696]. This is based on a generalized Peach–Koehler

force, in analogy to (3.22),





+ b

⊥



× ξ, (10.10)

where σ

⊥

is the stress tensor in perpendicular space. In [697], the formula is

quoted for one-dimensional hexagonal quasicrystals.

In terms of the extended elasticity theory, the energy of a screw dislocation

in the same quasicrystal is given by [697]



3

+ K

⊥3

+2R

3

⊥3



ln (R/r

)

4π

with R and r

being the outer and inner cut-oﬀ radii as introduced in

Sect. 3.2.2. Thus, the dislocation energy depends on the squares of both

406 10 Quasicrystals

Burgers vector components b



and b

⊥

. Assuming that the phason part of the

dislocation energy is proportional to the number of matching rule violations,

there should arise a linear dependence on b

⊥

[679,698]. Though the necessity

of the extension of the elasticity theory for quasicrystals is quite obvious, the

consequences have widely been ignored so far in interpreting the experimental

results on dislocation motion and plastic deformation.

The microprocesses of dislocation motion depend on the temperature. If

the temperature is high enough for a rapid phason rearrangement, the equilib-

rium state of the phason stress ﬁeld around the dislocation can move together

with the phonon ﬁeld. The phason ﬂips accompanied with the motion occur

also far from the dislocation itself as illustrated by a simulation of the dislo-

cation motion in a decagonal quasicrystal [699]. However, if the temperature

is not so high, the dislocation may decompose [700] and the phononic part

of it will move alone as it is outlined highly schematically in Fig. 10.10. In

Fig. 10.10a, the dislocation of Burgers vector b



+ b

⊥

sketched as a full cir-

cle is surrounded by its phason cloud marked by dashed circles. Caillard and

coworkers called such a dislocation a perfect dislocation (e.g., in [684,701]). In

Fig. 10.10b, a dislocation with only the parallel part of the Burgers vector b



has moved from A to B owing to an external stress. It may be called an imper-

fect dislocation. In its wake, the imperfect dislocation created a layer of high

b+b

⊥

=0 du

⊥

= b

⊥

(a)

(b)

(d)

(c)

Fig. 10.10. Schematic of the motion of a perfect dislocation. (a) Starting dislocation

with phonon and phason stress ﬁelds. (b) Intermediate state with separation of

phonon and phason stress ﬁelds. (c) Burgers circuits around dislocations A and B.

(d) Final state

10.2 Defects in Quasicrystals 407

phason disorder called a phason wall and indicated by the thick dashed line.

At the original site A, there remains a dislocation with the Burgers vector b

⊥

representing the phasonic strain ﬁeld of the original dislocation. The situation

is illuminated by the two grey Burgers circuits around A and B in Fig. 10.10c.

While the dislocation in real space at B has only the Burgers vector b



,the

Burgers circuit in orthogonal space around B yields the Burgers vector b

⊥

when crossing the concentrated phason strain of the phason wall. The Burg-

ers circuit around A crosses the phason wall in opposite direction resulting in

a contribution of −b

⊥

from the phason wall, which cancels the contribution

of the phason strain from the original dislocation. The whole system is not in

thermal equilibrium. The equilibrium conﬁguration with a perfect dislocation

at B as in Fig. 10.10d can only be attained by diﬀusional retiling, where the

phason ﬁeld at A and the phason wall dissolve. Computer simulations on a

two-dimensional decagonal quasicrystal estimate the stress necessary to cre-

ate the phason wall by glide at zero temperature to amount to τ

≈ (1/27) μ

[685]. This would be a very high stress. Therefore, quasicrystals were consid-

ered brittle at low temperatures. On the other hand, measurements from the

width of pairs of dislocations moving together by climb yield very low values

[702].

Since the imperfect dislocations have only a phononic Burgers vector b



they fulﬁll only the strong extinction condition (10.8) with g



· b



=0.In

the orientation where the corresponding perfect dislocations obey the weak

extinction condition, a residual contrast remains. The phason walls have a

displacement vector which can be deﬁned either by r



= b



or by r

⊥

= b

⊥

The shift vectors of phason walls are determined in a way similar to that of

stacking faults in crystals. Since g



·r



is irrational in quasicrystals, extinction

occurs only for g



· r



= 0, but not for g



· r



= integer. A TEM diﬀraction

contrast analysis of phason faults is published in [703].

If the temperature is not too low, the distribution of the phason defects

in the solid can be rearranged by spreading of the phasons of the wall and by

annealing to form the equilibrium phason distribution around A and the new

cloud around the dislocation at B. The process is called phason dispersion

or retiling. All the existing phasons certainly do not diﬀuse to the dislo-

cation at B. Instead, the retiling occurs by local diﬀusion processes which,

however, include chemical diﬀusion over distances larger than the atomic dis-

tances. Thus, the atomic mobility has to be high enough. At the end, the

new equilibrium conﬁguration of phonon and phason strain ﬁelds may be re-

established (Fig. 10.10d). During rapid deformation at lower temperatures,

dislocations are certainly generated as imperfect dislocations. If the Burgers

vector in hyperspace is a fraction of a lattice vector, the dislocation is a partial

dislocation.

An important issue of dislocation motion in quasicrystals was the ques-

tion whether the dislocations move by glide or climb. Although the ﬁrst in

situ straining experiments in a TEM on i-Al–Pd–Mn single quasicrystals were

performed at a relatively high temperature (at 0.88 T

,withT

being the

408 10 Quasicrystals

melting temperature) [647], they immediately showed that dislocation motion

in this material is of crystallographic character, i.e., many dislocations showed

straight crystallographically oriented segments, and they moved on crystal-

lographic high-symmetry planes. Therefore, for several years it had been

assumed that the dislocations move by glide without questioning this hypothe-

sis. To prove the mode of dislocation motion, it is necessary to determine both

the direction of the parallel component of the Burgers vector and the plane of

motion of always the same dislocations. This is only possible by in situ strain-

ing experiments in a TEM. Only recently, Caillard and coworkers succeeded in

performing such experiments, at ﬁrst during in situ heating where the disloca-

tions moved under internal stresses [704], and then also during in situ straining

[705]. Both experiments showed dislocations moving by pure climb. Thus,

climb is certainly a main mode of dislocation motion in quasicrystals. The

following microstructural results have to be considered keeping this in mind.

10.3 Microscopic Observations of Dislocations

While information on dislocation mechanisms of quasicrystal deformation

has been gained by TEM on a number of materials of either icosahedral

or decagonal symmetry, for other materials like Al–Cu–Fe the evidence of

dislocation motion is still missing, in spite of a wealth of macroscopic defor-

mation data available. The following sections are restricted to two single

quasicrystalline materials, icosahedral Al-21at%Pd-8.5at%Mn and decago-

nal Al-15at%Ni-15at%Co. These materials may be considered prototypes of

quasicrystals. Their dislocation microstructures have most frequently been

studied, also by the author of this book and his coworkers.

10.3.1 i-Al–Pd–Mn

As it was mentioned earlier, the dislocation properties strongly depend on the

temperature. This inﬂuences also the macroscopic deformation parameters,

which will be described in detail in Sect. 10.4. At high temperatures where

diﬀusion is rapid, quasicrystals can be deformed up to high strains. After a

yield drop eﬀect, there develops a range of almost steady state deformation.

Most deformation experiments and microstructure studies were carried out

in this temperature range. Only later, experiments succeeded at lower tem-

peratures where deformation at low rates is accompanied with very strong

work-hardening. The following results consider both ranges. In situ strain-

ing experiments, however, were successful only in the high-temperature range

where the ﬂow stresses are not so high.

Transmission Electron Microscopy of Deformed Specimens

The ﬁrst experiment which proved a dislocation mechanism of the plastic

deformation of quasicrystals was the observation by Wollgarten et al. [646] of

10.3 Microscopic Observations of Dislocations 409

an increase of the dislocation density after deformation. Later on, the Burgers

vectors of deformation-induced dislocations were analyzed by the CBED tech-

nique [706]. 60 dislocations were chosen from specimens deformed at 800

◦

up to diﬀerent plastic strains, and at diﬀerent temperatures down to 732

◦

at a ﬁxed strain, all in the high-temperature range. Fifty-two of the 60 dis-

locations had parallel parts b



of the Burgers vectors in twofold directions,

32 of which with a length of |b



| =0.183 nm. Five dislocations had parallel

Burgers vectors in the ﬁvefold direction. They are super-partial dislocations.

To characterize the contribution of phason strain, the so-called strain accom-

modation parameter ζ = |b

⊥

|/|b



|, i.e., the ratio between the lengths of the

perpendicular and parallel parts of the Burgers vectors, was calculated. In

deformed samples, the values varied between τ

≈ 29.0andτ

≈ 4.2witha

maximum of τ

≈ 11.1. Higher values of the ratio occurred more frequently

with increasing plastic strain. In the in situ and post-mortem TEM studies

of the group of Caillard [702,704, 705] performed at lower temperatures, only

dislocations with longer parallel parts of the Burgers vectors b



were analyzed,

i.e., of 0.257, 0.296, 0.348, 0.456, 0.513, and 0.563 nm.

In addition to the Burgers vectors, also the line directions were deter-

mined of 25 dislocations with twofold Burgers vectors [706]. Forty percent

of the lines were oriented along twofold directions. The character β of the

dislocations ranged between 7 and 90

◦

. Thus, pure screw dislocations have

not been observed. The parallel parts of the Burgers vectors and the line

directions span the slip planes of the dislocations, and the dislocations were

believed to glide on these planes, although there were already hints that this

may be wrong.

The dislocation microstructure of deformed samples was also imaged by

diﬀraction contrast in the HVEM allowing thick specimens to be investigated

so that the three-dimensional character of the dislocation structures could be

studied [707]. After deformation, some of the samples were cooled under load

to better preserve the microstructure. Some micrographs reveal dislocations

being arranged in bands, but inside the bands, they form a homogeneous three-

dimensional network at high temperatures, as shown in Fig. 10.11. Diﬀraction

contrast analysis using strong contrast extinctions at two diﬀerent g vectors

indicates that dislocations with diﬀerent Burgers vectors form the nodes of

the network. There exist groups of dislocations with their parallel component

of the Burgers vector being parallel to the compression axis, i.e., these dis-

locations experience a climb force but no glide force. At 800

◦

C, the average

link length in the network is of the order of magnitude of L =0.5 μm. Many

dislocation segments are quite straight and oriented along crystallographic

directions. This is remarkable for the high temperature.

At the lowest temperature with steady state deformation, many disloca-

tions form narrow bands. In Fig. 10.12, taken near a twofold pole parallel to the

compression axis, two bands cut each other. Dislocations are arranged on two

steeply inclined planes, most probably planes perpendicular to threefold direc-

tions. The bands include elongated dislocations as well as strongly bowed-out

410 10 Quasicrystals

Fig. 10.11. Dislocation structure in an i-Al–Pd–Mn single quasicrystal deformed

at 770, 800, and 818

◦

C along a ﬁvefold axis. Image taken near this pole. From [707].

dislocation segments and many small dislocation loops, as shown in Fig. 10.13.

The radii of curvature of the bowings and the loops are as small as 70 nm.

The dislocation density inside the bands is approximately 1.2 × 10

−2

Figures 10.12b to d exemplify the analysis of the parallel components of the

Burgers vector of the band running downwards from the upper right corner

of the ﬁgure. The hole in the specimen may serve as a marker. Because of

the bending of the specimen, a ﬁeld-limiting aperture of 2 μmdiameterwas

placed near the center for taking the diﬀraction patterns shown in the insets.

Thus, the respective diﬀraction conditions are fulﬁlled only near the centers

of the images. The discussed dislocation band is extinguished in Figs. 10.12c

and d. Accordingly, the parallel component of the Burgers vector is perpen-

dicular to the extension of the band and to the compression axis. Thus, these

dislocations are near the edge orientation and experience neither a glide force

nor a climb force from the applied stress.

At a lower temperature of 532

◦

C, where plastic deformation is achieved in

stress relaxation tests in the strong work-hardening range, many dislocation

segments are straight and oriented crystallographically but other segments

very strongly bow out between cusps indicating pinning agents, as marked by

an arrow in Fig. 10.14. The distances between the cusps (obstacle distances)

10.3 Microscopic Observations of Dislocations 411

1µm

(a)

1µm

(d)

1µm

(c)

1µm

(b)

Fig. 10.12. Slip bands after steady state deformation by ε = 2% at 580

◦

Cat

−6

−1

along a twofold axis with analysis of the parallel component of the Burgers

vector of the band running approximately vertically. The deformation curve is shown

in Fig. 10.32b. Compression axis and image normal [0/00/00/2] in (a, b), g vectors

(

1/00/10/0) (a, b), (1/00/10/0) (c) and (

1/0

10/1) (d). The insets show the

respective diﬀraction patterns. From [708]

are of the order of magnitude of 100 nm and the radii of curvature of the

bowed-out segments are about δ = 50 nm. At the lowest temperatures of 487

and 482

◦

C, where plastic deformation was obtained in conventional compres-

sion tests, diﬀerent dislocation structures are observed. Some specimen regions

again exhibit the straight crystallographically oriented dislocation segments.

However, Fig. 10.15a presents a totally diﬀerent dislocation structure. The

dislocations are arranged in strongly curved loops, but the cusps indicating

pinning agents at 532

◦

C are missing now. Frequently, the area inside the loops

exhibits a homogeneous area contrast, depending on the diﬀraction conditions

applied. Stereo pairs show that the loops belong to diﬀerent families of planes

but that the individual loops have segments which bow out onto planes that

diﬀer from those of the main parts of the loops. In another specimen shown

in Fig. 10.15b, two ordinary dislocation bands are crossing. In addition to the