What is the elementary process of dislocation glidein quasicrystals?

Shin Takeuchi *

Department of Materials Science and Technology, Tokyo University of Science, Yamazaki, Noda, Chiba 278-8510, Japan

Accepted 24 March 2003

Abstract

It is now well established that the plasticity of quasicrystals at high temperatures is governed largely by thermally

activated dislocation glide. Discussion is made of the elementary process of the thermally activated dislocation motion

overcoming a quasiperiodic Peierls potential, accompanied by a phason strain relaxation.

� 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Quasicrystals; Plastic deformation; Dislocation; Thermally activated processes; Phason relaxation

1. Dislocations in quasicrystal

Although the real atomic structures of quasi-

crystals are not completely understood yet, we

assume here that the structure is described basi-

cally by the projection of a high-dimensional

crystal lattice. On this basis, it was earlier estab-lished theoretically that perfect dislocations can

exist also in quasicrystals [1–3]. However, the es-

sential difference between dislocations in quasi-

crystals and in crystals is that the Burgers vector of

the dislocation in the former consists of two

components, i.e.,

bq ¼ bk þ b?; ð1Þ

where bk is the parallel component which corre-

sponds to the Burgers vector in crystals and pro-

duces a phonon elastic strain around the

dislocation, and b? is the perpendicular compo-

nent which produces a phason strain around the

dislocation. It should be noted that the phonon

strain field can instantaneously be relaxed with the

migration of the dislocation, whereas the relax-

ation time of the phason strain is quite slow since it

is governed by atomic diffusion. As a result thedislocation glide at room temperature cannot ac-

company the phason strain and must produce a

phason fault along the glide plane behind the

dislocation. Only at high enough temperature at

which atomic mobility by diffusion is faster than

the dislocation velocity the dislocation can migrate

accompanying both phonon and phason strains.

In an intermediate temperature region, disloca-tions migrate trailing a partially relaxed phason

strain field. The three situations are schematically

shown in Fig. 1. Thus, a phason drag stress, which

is a decreasing function of the temperature, should

oppose dislocation glide in quasicrystals. As a re-

sult, the stress to move a dislocation, sa, in a

quasicrystal can generally be written as

*Tel.: +81-471241501; fax: +81-47123962.

E-mail address: takeuchi@rs.noda.tus.ac.jp (S. Takeuchi).

1359-6462/03/$ - see front matter � 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

doi:10.1016/S1359-6462(03)00168-4

Scripta Materialia 49 (2003) 19–24

www.actamat-journals.com

sa ¼ seff þ sdrag þ sint; ð2Þwhere seff is the thermal stress component (the

effective stress necessary for the dislocation to

overcome short range barriers by thermal activa-

tion at a sufficient rate), sdrag is the phason drag

stress and sint is the internal stress due to longrange interaction with other dislocations.

In Eq. (2), we have a priori assumed the pres-

ence of an effective stress component, seff . In the

subsequent section, we show that the presence of

seff is experimentally verified and that seff is even

the dominant component of the applied stress.

2. Summary of experimental facts

Fundamental aspects of plasticity in quasicrys-

tals can be summarized as follows (see reviews

[4,5]).

(1) Any quasicrystals can be plastically deformed

at high temperatures above 0:75Tm (Tm: themelting point).

(2) After the yielding, the flow stress continuously

decreases with increasing plastic strain until it

levels off. The levelling-off stress is often as

small as 20% of the yield stress [6,7].

(3) The yield stress decreases rapidly with increas-

ing temperature and is strain-rate sensitive.

The same is true for the levelling-off stressbut the temperature dependence is much weaker

(e.g. [8]).

(4) Stress-relaxation experiments show that theflow stress consists dominantly of a strain-rate

dependent component, meaning that seff is the

main component of the flow stress.

(5) Analyzed activation volumes become as small

as 0.1 nm3 at relatively high stress level.

(6) Electron microscopy observations show that

plastic flow occurs in most cases by dislocation

glide. In situ observations reveal that disloca-tions move as rigid lines and in a steady and

continuous manner [9,10].

(7) The analysis of the Burgers vector of the dislo-

cations [5] shows that the jbkj is about 2 �AAwhich is the same as or even smaller than the

Burgers vectors of the dislocations in metallic

crystals.

The above facts (3), (4) and (6) indicate that the

plasticity is governed by a thermally activated

glide process of dislocations. Then the question

arises of the nature of the elementary process of

the thermally activated motion of the dislocations

in quasicrystals.

3. Glide resistance

As mentioned in Section 1, dislocation glide in a

quasicrystal produces an unrelaxed phason strain,

which exerts a force on the dislocation. In addition

to this, since the lattice is discrete in a quasiperi-odic manner on the glide plane, the core energy of

Fig. 1. Three possible cases of the structure of a dislocation gliding from left to right in a quasicrystal: (a) the glide of a partial

dislocation with b ¼ bk leaving behind a phason fault with fault vector b?, (b) the glide with a partial relaxation of the phason strain

field and (c) the glide of a perfect dislocation both with the phonon strain and phason strain fields.

20 S. Takeuchi / Scripta Materialia 49 (2003) 19–24

the dislocation should change with the position of

the dislocation, reflecting the lattice discreteness.

For crystal dislocations, the energy variation re-

flecting the lattice discreetness is known as thePeierls potential, and the stress necessary for the

dislocation to overcome the potential without

thermal activation is called the Peierls stress. The

Peierls stress is shown to be an exponentially de-

creasing function of the h=b value, where h is the

lattice spacing of the glide plane and b is the

strength of the Burgers vector [11]. A difference of

the Peierls potential in quasicrystals from that incrystals is that the former is quasiperiodic while

the latter is periodic.

Thus, the energy variation during dislocation

migration without any phason relaxation (migra-

tion of a partial dislocation with bk) consists of a

monotonically increasing phason energy to which

is superimposed the quasiperiodic Peierls poten-

tial. In Fig. 2 is shown an example of the potentialprofile for a straight dislocation migration com-

puted for a realistic quasiperiodic lattice [12].

For the Peierls potential of a crystalline dislo-

cation, the potential height along the dislocation

lying in the Peierls potential valley is regarded as

constant provided that the crystal does not contain

defects. However, for the potential in quasicrys-

tals, the potential height should be considerablyfluctuating along the dislocation line due to a

heterogeneous structure with a wavelength of a

few nanometer, which is typically due to a quasi-

periodic packing structure of atomic clusters with

a diameter of 2 nm. As a result, the two-dimen-

sional view of the potential for a dislocation lying

in a particular direction is like the one depicted in

Fig. 3. Thus, the potential profile of Fig. 2 is the

averaged one along a length of the dislocation line.We should note that due to such variation of

the potential along the dislocation line, the Peierls

potential picture becomes no more valid at very

high stress level where the potential is no more

trough-like but point barrier-like. In the range of

the experimental stress level, however, the point

barrier-like motion in such a short wavelength

(typically 2 nm) potential variation cannot takeplace due to the effect of the line tension of the

dislocation. Thus, in the actual stress level of de-

formation, the Peierls potential picture should be

valid. There are two regimes for the Peierls

mechanism, the smooth kink regime and the

abrupt kink regime, depending on the kink mo-

bility [13]. Due to the presence of a fluctuating

potential for the kink migration in quasicrystals,there is a high possibility that the Peierls mecha-

nism in quasicrystals is in the abrupt kink regime.

4. Thermally activated motion

An elementary glide process of a dislocation

blocked by a potential barrier takes place by

overcoming the potential barrier with the assis-

tance of a thermal activation from one mechani-

cally stable configuration to another mechanically

stable one, sweeping an area s0. The activation enth-

alpy profile for a dislocation to proceed througha saddle point configuration is schematically

Fig. 2. An example of computed potential profile for a glide of a

straight dislocation without phason strain relaxation in a real-

istic model quasicrystal [12]. A quasiperiodic Peierls potential is

superimposed on the phason fault production energy.

Fig. 3. A schematic illustration of a two-dimensional potential

profile in a quasicrystal.

S. Takeuchi / Scripta Materialia 49 (2003) 19–24 21

presented in Fig. 4. The enthalpy difference be-

tween the saddle point and the stable point, DH �,

is the activation enthalpy and the area difference

between them, s�, is the activation area. Since

dDH � ¼ dsbs�, the activation volume, v�, definedby the stress-derivative of the activation enthalpy

is given as

v� oDHos

¼ bs�: ð3Þ

Let the activation site density along the dislocation

line be n. The dislocation velocity is then written as

v ¼ ns�m exp

� DH �

kBT

�; ð4Þ

where m is the frequency factor and kBT has the

usual meaning.

The most probable pathway of a dislocation

during thermally activated glide depends on the

type of obstacles. For point-like obstacles, the

pathway is of a zigzag shape, while for a linear

obstacle like the Peierls potential the pathway is akink-pair formation followed by kind motion. The

feature (6) in Section 2 indicates that the obstacle

for thermally activated glide of dislocation in

quasicrystals is a Peierls potential-like linear ob-

stacle.For a thermally activated process to occur, the

enthalpy value at the final state should be lower

than that at the initial state. Hence, for a potential

such as shown in Fig. 3, the thermal activation can

occur only above the critical stress given by

sc ¼ C=bk; ð5Þ

where C is the slope of the potential below the

Peierls potential or the phason fault energy. sc is

considered to be quite high, of the order of GPa[2]. In actual high temperature experiments, the

thermally activated deformation occurs at quite

low stress levels. This indicates that phason re-

laxation occurs simultaneously to dislocation mi-

gration.

A successful thermally activated event takes

place in a short time of a quarter cycle of dislo-

cation oscillation; for a dislocation segment of ‘,the duration of an oscillation cycle is of the order

of ð‘=bÞmD (mD: the Debye frequency). For a pha-

son relaxation to take place in such a short period

of time, the driving force for the relaxation should

be large and the temperature be high enough.

Thus, the phason relaxation during a thermally

activated dislocation jump occurs only at the very

vicinity of the dislocation core at high temperaturewell above Tm=2. The modification of the potential

accompanying the phason relaxation under an

applied stress is schematically shown in Fig. 5. DUs

and DUp indicate the amount of the relaxation

energy at the saddle point and that at the next

Peierls potential valley, respectively. Both DUs and

Up increase with increasing temperature and hence

the critical stress sc for a thermal activation to takeplace, which corresponds to the stress component

sdrag in Eq. (2), decreases with increasing temper-

ature. We should note that DUs and Up may not be

the same, which results in a modification of the

Peierls potential depending on the temperature.

It should be noted that during the waiting time

between successful thermally activated events the

phason strain relaxation occurs around the dislo-cation to reduce the phason drag stress; the degree

Fig. 4. A schematic illustration of the thermally activated glide

of a dislocation. DH is the enthalpy change, DH � the activation

enthalpy, s the area swept by the dislocation and s� the acti-

vation area.

22 S. Takeuchi / Scripta Materialia 49 (2003) 19–24

of the relaxation is an increasing function of

temperature.

Incidentally, dislocation theory indicates that

DH � in Eq. (4) is given, in the smooth kink regime,by [14]

DH � ¼ 2ffiffiffiffiffiffiffiffi2E0

p Z xm

x0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUðsÞ U0

pdx; ð6Þ

where E0 is the self-energy of the dislocation and

other symbols are given in Fig. 5. Equation (6)

expresses the relation between the Peierls potential

and the activation enthalpy. As the plastic strain

applied to a quasicrystal increases, the phason

defect density increases, which modifies the Po-

tential profile of Fig. 5, leading to a change in theactivation enthalpy given by Eq. (6). A simple in-

terpretation of the observed pronounced work

softening phenomenon has been presented by

the present authors based on a Peierls mecha-

nism modified by introduction of phason strains

[4,15].

The value of the activation volume is closely

related to the nature of the corresponding ther-mally activated mechanism and signifies the

deformation mechanism. The observed small

activation volume of 0.1 nm3, which corresponds

to the activation area of �50 �AA2, is consistent with

the Peierls mechanism.In summary, the elementary process of the

thermally activated dislocation glide in quasicrys-

tals is special, different from that in crystals, in that

a partial relaxation of the phason strain, which

involves atomic diffusion, accompanies the pro-

cess. As a result, the potential barrier determining

the dislocation mobility is variable depending not

only on the applied stress but also on the tem-perature and on the phason defect density in the

specimen. This makes the plasticity of quasicrys-

tals special and complex, and the analysis of the

experimental results difficult.

For deeper understanding of the dislocation

glide process in quasicrystals, the following studies

are important: (1) computer simulation studies

of dislocation process in realistic model quasi-crystals, as already done in some numerical models

[16,17], (2) dislocation mobility measurements as a

function of stress and temperature, as has been

done for crystal dislocations, and (3) detailed,

high-resolution observation of the nature of

the glide dislocations by transmission electron

microscopy.

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