Materials Science and Engineering A 400–401 (2005) 306–310

Dislocation processes in quasicrystals—Kink-pair formationcontrol or jog-pair formation control

Shin Takeuchi∗

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

Received 13 September 2004; received in revised form 24 November 2004; accepted 28 March 2005

Abstract

A computer simulation of dislocation in a model quasiperiodic lattice indicates that the dislocation feels a large Peierls potential whenoriented in particular directions. For a dislocation with a high Peierls potential, the glide velocity and the climb velocity of the dislocationcan be described almost in parallel in terms of the kink-pair formation followed by kink motion and the jog-pair formation followed by jogmotion, respectively. The activation enthalpy of the kink-pair formation is the sum of the kink-pair formation enthalpy and the atomic jumpactivation enthalpy, while the activation enthalpy of the jog-pair formation involves the jog-pair enthalpy and the self-diffusion enthalpy. Sincethe kink-pair energy can be considerably larger than the jog-pair energy, the climb velocity can be faster than the glide velocity, so that thep©

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lastic deformation of quasicrystals can be brought not by dislocation glide but by dislocation climb at high temperatures.2005 Elsevier B.V. All rights reserved.

eywords: Dislocation; Quasicrystal; Plasticity; Dislocation glide; Dislocation climb

. Introduction

A quasicrystalline lattice is described by the projectionf crystal lattice points in a high dimensional space (six-imensions for the icosahedral quasilattice) onto the realpace (or the parallel space) through a projection window inhe complementary space (or the perpendicular space). Sincehe Burgers vectorB of a perfect dislocation in a quasicrys-al can be defined by a lattice vector in the high dimensionalrystalline lattice, it is composed of the parallel space com-onentb‖ and the perpendicular space componentb⊥ [1],

.e.

= b‖ + b⊥. (1)

heb‖ component produces a phonon strain field around theislocation as in crystal dislocations, whereas theb⊥ com-onent produces a phason strain field which is specific toislocations in quasicrystals. For perfect dislocation motion,ither glide or climb, in a quasicrystal, both the phonon strain

∗

field and the phason strain field must accompany the moHowever, since the relaxation of the phason strain field natomic diffusion and can take place only at high temperatmotion of a perfect dislocation is possible only at high tperatures well above half the melting point. At lower tematures below half the melting point, dislocation motion mcreate a stacking fault with the fault vectorb⊥, often calledthe phason fault. Because the phason fault energy isthe dislocation is practically immobile without the pharelaxation.

Experimentally, since the first demonstration bypresent author’s group of the plastic deformation of Al–Cu icosahedral quasicrystal at high temperatures[2], ithas been shown that quasicrystals, both of icosahedradecagonal phases, are generally deformable at high teatures above 0.8Tm [3,4]. It has also been shown by electmicroscopy that the high temperature plasticity is brougmost cases by a dislocation process[5,6]. Until recent yearsit has been generally believed that the plasticity of quasitals is carried by a glide process of dislocations, and vamodels of the deformation mechanism have been propbased on the glide of dislocations[3,7–11]. However, in re

Tel.: +81 4 7124 1501x4305; fax: +81 4 7123 9362.

E-mail address:takeuchi@rs.noda.tus.ac.jp. cent years, Caillard and coworkers have shown by electron

921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2005.03.068

S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 306–310 307

microscopy observations, in situ as well as post mortem, ofdislocation motions in icosahedral Al–Pd–Mn that disloca-tions move not by a glide process but by a pure climb process[12–14].

The purpose of the present paper is to compare theoreti-cally the velocity of dislocation glide and that of dislocationclimb in a quasicrystal with a high Peierls potential, and toclarify the criterion of the controlling mode of dislocationprocess in high temperature plasticity of quasicrystals.

2. Peierls potential

In this section, we show, on the basis of both experimentand computer simulation, that the Peierls potential for dislo-cations in quasicrystals is high enough for the dislocations tobe confined in a Peierls potential valley.

In situ electron microscopy of dislocation motion in qua-sicrystals has revealed that dislocations migrate steadilykeeping a straight form oriented in a symmetrical direction[6,15]. Such behavior is reminiscent of the dislocation glidein bcc metals at low temperatures and in covalent crystals athigh temperatures; in both types of crystals the dislocationmotion is established to be controlled by the Peierls mech-anism. The dislocation motion keeping a straight shape in acrystallographic direction indicates that the rate controllingp tioni mbp

dis-l avec om-p in ofs unc-tr nta-t ways ergy.T f 0.1( tionsi havea lidep

3f

ca-t rep justi ofq het esulto ff se

Fig. 1. A perfect dislocation in a Penrose lattice. After gliding the dislocationonly with b‖ component to the left, shaded tiles along the glide plane aredestroyed to produce intra-tile phason defects, whereas outside the glideplane tiling mismatch phason defects, examples of which are shown bydashed tiles and dotted tiles, respectively, before and after a displacementshown by an arrow of the dislocation position, are produced.

lattice. The former type of fault has a much higher energythan the latter type, and during kink-pair formation and kinkmigration the former type of phason defects should be relaxedby local rearrangements of atoms in the lattice planes facingthe glide plane. Thus, an activation energy of atomic jumpis indispensable for the migration of a kink to relax the highenergy intra-tile phason defects. As a result, the energy pro-file for the kink-pair formation process shown inFig. 2(a) asa function of the kinks separationl is like that schematicallydepicted inFig. 2(b) by a dashed curve, whereH′ correspondsto the activation enthalpy of atomic jump and the slopeΓ the

F rofileo dc

rocess of the dislocation motion is the kink-pair forman the glide process or the jog-pair formation in the clirocess.

In order to estimate the potential energy of a straightocation on a glide plane in a quasiperiodic lattice, we honstructed a realistic model quasiperiodic lattice in cuter and computed the potential energies at zero Kelvtraight dislocations oriented in various directions as a fion of their position on the quasiperiodic plane[16]. Theesults have shown that for dislocations in particular orieions their potential energy oscillates in a quasiperiodicuperimposing on almost constant phason production enhe Peierls stress component is as large as the order oGG: shear modulus), the same order as that for dislocan covalent crystals. Thus, dislocations in quasicrystals

strong tendency to lie along a Peierls valley on the glane.

. Dislocation glide motion controlled by kink-pairormation

With the migration in a quasiperiodic lattice of a disloion only with b‖ component, two kinks of phason fault aroduced, one is the intra-tile phason defects produced

n close vicinity of the glide plane as a result of cuttinguasiperiodic tiles by theb‖ dislocation, and the other is t

iling mismatch outside the glide plane produced as a rf the shift of the dislocation center[17]. These two kinds o

ault are illustrated inFig. 1 for a dislocation in the Penro

ig. 2. (a) Kink-pair formation process and (b) the enthalpy change pf the kink-pair as a function of the kink-pair spacingl without stress (dasheurve) and under an effective stressτ∗ (solid curve).

308 S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 306–310

energy produced by unrelaxed tiling mismatch outside theglide plane. Except for the slopeΓ , the potential profile isquite analogous to that of the kink-pair formation process fora dislocation in covalent crystals treated by Hirth and Lothe[18].

The thermally activated kink-pair formation is possible forstresses higher thanΓ /b‖. The enthalpy profile under stressis shown by a solid curve inFig. 2(b), which is composedof a kink-pair enthalpyHkp(τ∗) having a maximumH∗

kp(τ∗)at a kink separationl∗ and the superimposing atomic jumpenthalpyH′. The rate of kink-pair formation is given by theflow rate of kinks throughl∗ towards largerl value. Let theaverage distance between the potentials of atomic jump bed′ and the vibrational frequency of a kink beνk, the rateof the kink-pair formation per unit length of a dislocationlying in a Peierls potential is given, in perfect analogy to thekink diffusion theory of the kink-pair formation in covalentcrystals[18], as

νkp = νk

d′ exp

(−Hkp(τ∗) + H ′

kBT

)sinh

τ∗b||dd′

2kBT

≈ τ∗b||d2kBT

νk exp

(−Hkp(τ∗) + H ′

kBT

). (2)

In the equilibrium state of an infinite length of disloca-t ceb inksa

l

F1 thed l-lo gthLa

V

4f

eepP edb r ast thep ed inF thej owni or-m tion

Fig. 3. (a) Jog-pair formation process and (b) the enthalpy change profileof the jog-pair formation as a function of the jog-pair spacing without stress(dashed curve) and under an effective stressσ∗.

is governed by the rate of vacancy absorption or emission atthe jog site.

Under the action of the climbing forceσ∗b‖, there appearsan excess vacancy concentration at the dislocation, whoseosmotic force on the dislocation balances the climbing force.Considering that the vacancy diffusion along the dislocationcore is much faster than in the bulk, Hirth and Lothe obtainedthe jog velocityvj given by[18]

vj = 4πDsσ∗b||a

kBT ln(z/b||)exp

(�Hs

2kBT

), (5)

where a is the atomic distance along the dislocationline, z is the mean free life length along the disloca-tion line of a vacancy diffusing in the core given by ¯z =√

2a exp(�Hs/2kBT ), Ds the self-diffusion coefficient and�Hs is the difference between the activation enthalpy forself-diffusion in the bulk and that along the core.

In the climbing process by jog-pair formation followedby jog motion of a dislocation confined in Peierls valley, weagain assume, as in the case of dislocation glide controlled bythe kink-pair formation, that the jog mean free path is longerthan the dislocation lengthL, on the ground that jog-pairenthalpy is considerably larger than the activation enthalpyof self-diffusion. The climb velocity is written as[18]:

V

( )

w y

o(fa

V

ion, the mean kink separationl is determined by the balanetween the rate of kink-pair formation and the rate of knnihilation and is given by

= d′ exp

(H∗

kp

2kBT

). (3)

or a typical value of�H∗kp = 3 eV andd′ = 0.5 nm, l at

000 K is calculated to be 3 cm, which is much larger thanislocation segment length,L, lying in a Peierls potential va

ey. Therefore, the dislocation glide velocityVg is controllednly by the rate of kink-pair formation on the segment lenwhich is followed by the kinks motion over a distanceL,

nd is written as

g = τ∗b||d2L

2kBTνk exp

(−H∗

kp(τ∗) + H ′

kBT

). (4)

. Dislocation climb motion controlled by jog-pairormation

The climb process of a dislocation confined in a deierls potential valley occurs by jog-pair formation followy jog motion along the dislocation in a similar manne

he kink-pair formation and the kink motion mentioned inrevious section. The process is schematically illustratig. 3(a). The energy profile of a jog-pair as a function of

og spacing is analogous to the kink-pair formation, as shn Fig. 3(b). An essential difference from the kink-pair f

ation process is that the jog mobility along the disloca

c = 4πLDsvaσ∗

a2kBT ln(z/b||)exp −H∗

jp(σ∗) − �Hs/2

kBT(6)

hereva is the atomic volume,H∗jp is the activation enthalp

f the jog formation. WritingDs ≈ a2νD exp(−Hs/kBT )Hs: activation enthalpy of self-diffusion;νD: Debye-requency) and approximatingva ≈ a3, Eq. (6) is rewrittens

c = 4πLa3σ∗

kBT ln(z/b||)νD exp

(−H∗

jp(σ∗) + (Hs − �Hs/2)

kBT

)(7)

S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 306–310 309

By comparing Eqs.(4) and (7)for velocities of dislocationglide and dislocation climb, one sees that both equations havethe same functional form. Thus, it is impossible to discrimi-nate the two cases experimentally only from the analysis ofthe macroscopic plasticity.

5. Comparison ofVg andVc

In comparing Eqs.(4) and (7), we approximateνk ≈ νD,τ∗ ≈ σ∗ andd≈ b‖ ≈ a. Then, the ratioVg/Vc is written as

Vg

Vc≈ ln(z/a)

8πexp

[−{H∗

kp(τ∗) − H∗jp(σ∗)} + {H ′ − (Hs − �Hs/2)}

kBT

](8)

As a typical case, we assumeT≈ 1000 K, Hs≈ 2 eV and�Hs≈ Hs/2. Then, ¯z = √

2a exp(�Hs/2kBT ) ≈ 570a andhence the pre-exponential factor of Eq.(8) is∼0.25. It seemsreasonable to assume that the phason jump enthalpy at dis-location coreH′ is approximately equal toHs/2 (the same asthe activation energy of the core diffusion). It follows then[ ∗ ∗ ∗ ∗ ]

ofq tionc e oft rft ro-d ya d byt -eV

d ac-c ion,t

E

wl s,G

E

T

E

The ratio is then written as

Ek

Ej= 5√

βd

h. (13)

The kink heighth is always of an atomic spacing, whereasthe Peierls valley distanced is determined by the periodicityof the lattice on the glide plane and can be larger than anatomic spacing in a complex lattice. The value ofβ(=τp/G)of a dislocation in a complex structure like quasicrystal is ofthe order of 10−1 to 10−2, and henceEk/Ej = (0.5–1.5)d/h.In conclusion, since generallyd>h, Ek can be larger thanEjand consequently the climb velocity of a dislocation can behigher than the glide velocity in quasicrystal.

6. Work-softening mechanism

The specific feature of macroscopic plasticity in qua-sicrystals is a pronounced work-softening up to high strains;the flow stress often becomes only one-fifth of the yield stress[19,20]. We have shown from the activation analysis that thedecrease of the flow stress is due to a decrease of the activationenthalpy with increasing plastic strain[20,21]. Assuming thatthe kink-pair formation controls the deformation, the presentauthor and his colleagues have earlier proposed a mechanismwhich explains the decrease of kink-pair formation enthalpyc ofp mo-t o-t . Ins tiallyb ten-t e by

F lidep tionc

Vg

Vc≈ 0.25 exp −Hkp(τ ) − Hjp(σ )

kBT(9)

Thus, it is found that whether the plastic deformationuasicrystals is controlled by dislocation glide or dislocalimb is essentially determined by the relative magnitudhe kink-pair formation enthalpyH∗

kp(τ∗) and the jog-paiormation enthalpyH∗

jp(σ∗). Sinceσ∗ ≈ τ∗, the work-doneerms−τ∗b‖l and−σ∗b‖l are comparable. The phason puction energyΓ for dislocation glide and that for climb malso be similar. Thus, the relative magnitude is determine

he relative magnitude of the kink energyEk and the jog enrgyEj . If Ek >Ej , Vc can be larger thanVg, and if Ek <Ej ,g is larger thanVc.

The kink energy depends on the Peierls potential anording to the line tension approximation of the dislocathe kink energy is written as[18]:

k ≈ 2d

π

√2EpE0, (10)

hereEp is the height of the Peierls potential andE0 is theine energy of the dislocation. Forτp =βG (τp: Peierls stres: shear modulus) and forE0 ≈ Gb2, one obtains:

k ≈ 2√

2β1/2

π3/2 ab2G ≈ 0.5√

βGdb2. (11)

he jog energy is written as[18]

j = Gb2h

4π(1 − ν)≈ 0.1Gb2h (12)

ontrolling the dislocation velocity with the introductionhason defects on the glide plane by forest dislocation

ion [3,11]. Fig. 4 illustrates the quasiperiodic Peierls pential valleys on a glide plane cut by forest dislocationsuch a situation, dislocation glide is determined esseny the kink-pair formation at narrowly spaced Peierls po

ials; widely spaced Peierls potentials can be overcom

ig. 4. The figure illustrates either the Peierls valley distribution in a glane for dislocation glide or glide plane spacing distribution for dislocalimb, after passage of a forest dislocation.

310 S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 306–310

side motion of kinks produced at narrowly spaced Peierlspotential.

In the jog-pair formation controlled deformation, work-softening can occur by essentially the same mechanism asthe kink-pair formation controlled deformation. In icosahe-dral quasicrystals, the lattice spacings of the glide plane onwhich edge dislocations lie are not periodic but quasiperi-odic, consisting of wide and narrow ones. In phason freestate, the dislocation climb velocity is determined essentiallyby jog-pair formation between wide lattice planes, and in aphason-defected state, it is determined by the jog-pair forma-tion between narrow lattice planes; this is because the sidemotion of a jog from a narrowly spaced part to a widelyspaced part (e.g., position 2 to position 3 inFig. 4) can occurwith a smaller activation enthalpy.

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