Friction mechanism of dislocation motion in icosahedral Al ... · Al± Pd± Mn quasicrystals...

PHILOSOPHICAL MAGAZINE A 1999 VOL 79 NO 9 2123plusmn 2135

Friction mechanism of dislocation motion in icosahedralAlplusmn Pdplusmn Mn quasicrystals

UMesserschmidty M Bartschy MFeuerbacherzBGeyery and KUrbanz

yMax Planck Institute of Microstructure PhysicsWeinberg 2 HalleS D-06120 Germany

zInstitute of Solid State Research Research Centre JuEgrave lichJuEgrave lich D-52425 Germany

[Received in revised form 9 September 1998 and accepted 27 October 1998]

AbstractResults from the literature and unpublished ones of the present authors are

summarized which are relevant to the mechanisms governing the mobility ofdislocations in icosahedral Alplusmn Pdplusmn Mn quasicrystals These results concernmacroscopic deformation tests conventional transmission electron microscopyin situ straining experiments in a transmission electron microscope and computersimulation experiments These experiments can best be interpreted by assumingthat the dislocation motion is controlled by the thermally activated overcoming ofMackay-type clusters The present paper gives an estimate of the activationvolume of this process It turns out that the activation volume of overcomingthe clusters individually is one order of magnitude smaller than the experimentallyobserved one The experimental observations can be interpreted in a consistentway in terms of the Labuschplusmn Schwarz theory of solution hardening in crystalswhich considers the interpenetrating clusters as extendedrsquo obstacles a number ofwhich are surmounted collectively Some implications of this new model arediscussed

1 IntroductionQuasicrystals are brittle at low temperatures but can be deformed plastically at

high temperatures This was shown for icosahedral Alplusmn Pdplusmn Mn single quasicrystalsby Wollgarten et al (1993) and by Inoue et al (1994) The intrinsic macroscopicdeformation parameters were determined by Feuerbacher et al (1995) for Alplusmn PdplusmnMn single quasicrystals During deformation the dislocation density increasesstrongly pointing to the operation of a dislocation mechanism (Wollgarten et al1993) This conclusion has been conreg rmed by Wollgarten et al (1995) by the directobservation of moving dislocations during in situ straining experiments in a high-voltage electron microscope These experiments showed that the dislocations movein a viscous way On the basis of this observation and taking into account themacroscopic deformation parameters Feuerbacher et al (1997) and Urban et al(1998) suggest that the interaction with the Mackay-type clusters occurring in thequasicrystal lattice provides the basic friction mechanism of mobile dislocationsthus controlling the rate of macroscopic plastic deformation The present paper

Philosophical Magazine A ISSN 0141plusmn 8610 printISSN 1460-6992 online 1999 Taylor amp Francis LtdhttpwwwtandfcoukJNLSphmhtm

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discusses this mechanism in more detail and shows that the theory by Labusch andSchwarz (1991) of overcoming extendedrsquo obstacles can be applied to the clusterfriction model yielding quantitative agreement with most of the experimental resultsThe basic experimental and theoretical observations relevant for a discussion ofdislocation friction in quasicrystals are briemacr y outlined in sect2 The model is describedin sect3 followed by a discussion in sect4

2 Experimental observationsandresults ofcomputer simulations

21 Macroscopic deformationMacroscopic deformation tests were performed in compression at strain rates of

10iexcl5 and 10iexcl4 siexcl1 along twofold quasicrystal axes by Feuerbacher et al (1995) andGeyer et al (1998) and along reg vefold axes by Brunner et al (1997) and Geyer et al(1998) According to the latter study the deformation parameters depend on thecompression direction only weakly

The results of the deformation tests on quasicrystals were discussed in terms ofthe models of dislocation motion in crystals where the strain rate _ is given by anArrhenius-type relationship (for a review see for example Evans and Rawlings(1969))

_hellipfrac12 T dagger ˆ _0 exp hellipiexcl Ghellipfrac12dagger=kT dagger hellip1daggerThe same approach is used in the present paper The pre-exponential factor _0depends on the mobile dislocation density and is considered constant during themeasuring procedures of the activation parameters G is the Gibbs free energyof activation which is a function of the shear stress frac12 k is Boltzmannrsquos constantand T is the absolute temperature The experimentalrsquo activation volumeVex ˆ kT =msr was calculated from the strain rate sensitivity r ˆ hellip frac14= ln _daggerT Here frac14 is the macr ow stress and ms is the orientation factor As shown belowms 04 The macroscopic data of Geyer et al (1998) cover the widest temperaturerange availableat present Therefore the following discussion is based on these dataThe strain rate sensitivity was determined mainly from stress relaxation testsBetween about 640 C and 830 C the macr ow stress at _ ˆ 10iexcl5 siexcl1 decreases fromvalues above 800MPa down to less than 100MPa At the same time the strainrate sensitivity r decreases from a rather constant value of about 140MPa between640 C and about 680 C down to about 25MPa at 830 C This corresponds to anincrease of the experimental activation volume from about 025 up to 15nm3

In equation (1) the strain rate is controlled by the Gibbs free energy of activationG Unfortunately this quantity is not directly accessible in the experiments

Instead the activation enthalpy H ˆ iexclkT 2hellip frac14= T dagger=r was calculated from thetemperature sensitivity of the macr ow stress hellip frac14= T dagger at a constant strain rate The H

values increase from about 2eV at 640 C up to about 44eV at 735 C Reliable dataof H characterizing the dislocation mobility cannot be determined for highertemperatures because of recovery as discussed below For crystals the Gibbs freeenergy of activation G ˆ H iexcl T S is usually calculated from H using anequation derived by SchoEgrave ck (1965) S is the activation entropy which is estimatedon the assumption that entropy terms arise solely from the temperature dependenceof the elastic moduli Applying this formalism also to quasicrystals and using thedata by Tanaka et al (1996) for the temperature dependence of the shear modulusyields G values increasing from about 1eV at 640 C up to 23eV at 735 C

2124 U Messerschmidt et al

A number of observations indicate that the long-range athermal component ofthe macr ow stress is small and that recovery plays an important role in the deformationbehaviour of quasicrystals particularly at high temperatures This concerns thefollowing observations

(a) During long-time stress relaxation tests the macr ow stress drops to very smallvalues

(b) Stress dip tests at 760 C show zero relaxation at about 20 of the macr owstress and back-macr ow at lower stresses (Feuerbacher 1996)

(c) The original relaxation curves and the curves obtained from repeating therelaxation tests after re-loading the specimens up to stresses smaller than thestarting stresses of the original relaxations do not coincide particularly athigh temperatures (Geyer et al 1998) The repeated relaxation curves showsmaller strain rates (which are proportional to the negative stress rates) atthe same stresses and therefore yield higher strain rate sensitivities than theoriginal relaxation curves The decrease of the strain rate can be explained byrecovery of the dislocation density during the reg rst relaxation After relaxa-tion the original dislocation density is only restored when some plastic strainhas evolved Then a yield point e ect appears the height of which is inagreement with the reduction of the mobile dislocation density during thereg rst relaxation

22 Microscopic observationsThe dislocations move on well-dereg ned slip planes which is concluded from

analysing the Burgers vector and dislocation line directions by conventional trans-mission electron microscopy (Rosenfeld et al 1995) and from in situ straining experi-ments in a high-voltage electron microscope (Wollgarten et al 1995 Messerschmidtet al 1998) According to these studies the planes perpendicular to reg vefold three-fold twofold and pseudo-twofold directions are the most frequent slip planes Theparallel components of the most frequent Burgers vectors lie along twofold direc-tions (Rosenfeld et al 1995) Considering the most frequent slip planes and direc-tions as well as the high degree of isotropy of the quasicrystals usually leading tomulti-slip ms 04 seems to be a good estimate of an average orientation factorWith increasing strain the ratio between the phason and phonon components of thesix-dimensional Burgers vectors of dislocations increases

The stressplusmn strain curves of icosahedral Alplusmn Pdplusmn Mn quasicrystals exhibit a pro-nounced yield drop phenomenon During loading up to the lower yield point thedislocation density increases by more than two orders of magnitude and decreases atlarger strains (Wollgarten et al 1993 Feuerbacher et al 1997) The dislocationdensity at constant plastic strain (13) measured on samples rapidly quenchedafter deformation decreases from 9 1013 miexcl2 at 695 C to 3 1012 miexcl2 at 820 C(Schall et al 1998) From the dislocation density raquo the long-range athermal compo-nent of the macr ow stress frac12i can be estimated using the well-known formula for Taylorhardening (Taylor 1934)

frac12i ˆ 05middotbraquo1=2 hellip2dagger

Friction mechanism of dislocation motion in quasicrystals 2125

where middot is the shear modulus and b is the modulus of the Burgers vector Withmiddot 50GPa (Tanaka et al 1996) b ˆ 0183nm (the most frequently observed lengthof the Burgers vector component in physical space) and the dislocation densitiesquoted above frac12i turns out to amount to only about 5 of the total macr ow stressFor dislocation densities even reg ve times larger during deformation (ie taking pos-sible recovery processes into consideration) the athermal component of the macr owstress would still be small agreeing well with the result of the dip tests mentionedabove Annealing a specimen at 730 C previously deformed at the same temperaturereduces the dislocation density (Schall et al 1998) The half-time of this process isonly about 10 min which emphasizes the importance of recovery even at this rela-tively low temperature

The in situ straining experiments in a high-voltage electron microscope at tem-peratures between 700 and 750 C have shown that the dislocations move in a steadyviscous way (Wollgarten et al 1995 Messerschmidt et al 1998) Dislocations ondi erent slip systems move at di erent velocities Figure 1 shows that under load thedislocations mostly are of angular shape with straight segments in preferential orien-tations The preferred orientations determined so far are parallel to twofold direc-tions on a threefold slip plane and to an orientation inclined about 60 to thetwofold direction on a pseudo-twofold plane (Messerschmidt et al 1998) The angu-


Figure 1 Dislocations moving during in situ straining in a high-voltage electron microscopeat about 750 C Tensile direction TD near to a twofold direction beam direction inthreefold direction The normal of the foil plane is tilted by about 15 with respect tothe beam direction g-vector g ˆ permil1=0 1=1 0=1Š d ˆ permil0=1 1=0 1=1Š Dashed linestraces of slip planes (1) plane perpendicular to a twofold direction imaged almostedge-on (2) di erent planes perpendicular to threefold directions the widths of thetraces of the slip planes are indicated All dislocations are arranged perpendicular totwofold directions The three twofold directions on a threefold plane are marked by adotted line (3)

lar shape of dislocations was not observed by post mortem electron microscopy ofdeformed material (Rosenfeld et al 1995) It should be pointed out that all results onthe dislocation structure originate from deformation tests at or above 730 C exceptthe in situ experiments performed at 700 C The dislocation arrangement observedunder these conditions may have been inmacr uenced by recovery and need not representthe situation at low deformation temperatures Nevertheless it seems quite clear thata great part of the macr ow stress of Alplusmn Pdplusmn Mn single quasicrystals is controlled by anintrinsic friction mechanism of moving dislocations the density of which stronglydepends on the deformation conditions

23 Results of computer simulation studiesThere is a number of studies in which the motion of dislocations in two- and

three-dimensional quasi-periodic structures is investigated by computer simulationThe results of molecular statics calculations indicate that the dislocations glide onwell-dereg ned planes producing a layer of a high concentration of phason defectsalong their glide plane (Dilger et al 1997 Takeuchi et al 1998) as well as isolatedphason defects (Takeuchi et al 1998) In a three-dimensional simulation the energyof the phason layers was estimated to be 58 of the surface energy (Dilger et al1995) In two-dimensional molecular dynamics simulations at 10iexcl6 of the meltingtemperature (Mikulla et al 1998) a dislocation placed on a plane between denselypacked planes was observed to dissociate into partial dislocations at a stress of 002middotIndividual partial dislocations have been observed experimentally (Rosenfeld et al1995) but no split dislocations of small width In the simulation both partialsstarted to move at a shear stress of frac120 ˆ 003middot In Alplusmn Pdplusmn Mn this corresponds toa shear stress of 15GPa which may be considered a theoretical estimate of thefriction stress close to 0K In the calculations the dislocations were observed tostop in front of local conreg gurations of high coordination the structure of which hasbeen identireg ed in the two-dimensional models Analogous structures for the three-dimensional case have not yet been specireg ed At low temperatures the layers ofphasons produced by the gliding dislocations remain macr at but at high temperaturesthe phasons di use away to form a wide damaged zone (Mikulla et al 1996)

3 Model ofdislocationfrictioninquasicrystalsAs shown by Boudard et al (1992) the structure of icosahedral Alplusmn Pdplusmn Mn

quasicrystals can be described either as a quasi-periodic arrangement of two typesof Mackay-type clusters or as the stacking of densely and less densely packed planesBoth views are important for the understanding of the mechanism of dislocationmotion The Mackay-type clusters are strongly bound structural units with a dia-meter of about 09nm consisting of a central Mn atom a core consisting of 8 Alatoms an inner icosahedron of either Mn and Al or Mn and Pd atoms and an outericosidodecahedron of Pd and Al atoms or Al atoms only which Boudard et al(1992) termed pseudo-Mackay clusters of type 1 and 2 respectively PerfectMackay-type clusters comprise about 60 of the atoms while the rest of theatoms between the clusters are arranged in patterns which can be considered tobe related to incomplete clusters The investigation of cleavage surfaces of Alplusmn PdplusmnMn quasicrystals by means of scanning tunnelling microscopy demonstrates that theclusters are mechanically highly stable Complete clusters are preserved at the sur-face which shows that cracks circumvent the Mackay-type clusters which therefore


have to be considered rigid obstacles capable of demacr ecting propagating cracks (Ebertet al 1996)

The centres of the clusters are arranged along crystallographic directions oncrystallographic planes This is demonstrated in reg gure 2 which is plotted after thedata of Boudard et al (1992) for the twofold directions on planes perpendicular to areg vefold axis Between the densely packed planes containing the centres of clustersthere are planes of low packing density which are not shown in the work byBoudard et al but between which the dislocations most probably glide (Yang etal 1998) This assumption was made also by Mikulla et al (1998) The spacingbetween the planes packed with cluster centres certainly never exceeds 06nmThis means that the slip planes cut at least the outer shells of the Mackay-typeclusters

It was suggested by Feuerbacher et al (1997) and Urban et al (1998) that theseMackay-type clusters act as obstacles to the dislocation motion and that the result-ing friction controls the plastic deformation This mechanism was termed a clusterfriction model by the authors The main argument supporting this view is that theexperimental activation volume Vex ranging from 025 to 15nm3 depending on thedeformation conditions applied which corresponds to 40b3 to 245b3 is considerablylarger than a few b3 which would be expected for the lattice friction mechanism(Peierls mechanism) in crystals Thus the dislocation mobility should be controlledby items of a larger length scale than double kinks forming along a dislocation line incrystals The mutual distance as well as the size of the Mackay-type clusters reg t this


Figure 2 Atomic arrangement in a plane perpendicular to a reg vefold axis (from a data setgenerated by Boudard et al (1992)) Al atoms Ccedil Mn atoms Rings of 10 Al atomsaround a Mn atom arranged along twofold directions characterize the Mackay-typeclusters

length scale It is the aim of the present paper to work out the cluster friction modelin more detail

According to the above discussion it is most probable that the Mackay-typeclusters are not cut near their equators but along planes of low packing densitybetween the planes of cluster centres Owing to the relation between the diametersand the distances between the clusters their outer shells have to be cut in any caseand it is probably these events that control the dislocation motion This implies thatthe dislocations will arrange parallel to orientations of a high cluster density Aspointed out above in situ experiments revealed preferred orientations of the disloca-tions As described in sect23 there may also be hard spots along the slip planes otherthan some features of the Mackay-type clusters like those identireg ed by Mikulla et al(1998) in the two-dimensional simulations To the authorsrsquo knowledge however todate no corresponding result of calculations of a three-dimensional structure hasbeen obtained It is expected that the distances between these other hard spots will beof the same order of magnitude as those between the centres of the Mackay-typeclusters as they are on the same level of the structure hierarchy

According to the model of the thermally activated overcoming of individuallocalized obstacles in crystals the activation volume is given by (see for exampleEvans and Rawlings (1969))

Vind ˆ ldb hellip3daggerwhere l is the obstacle distance along the dislocations and d is the so-called activa-tion distance The latter is the distance shown in reg gure 3 in forward direction alongthe forceplusmn distance proreg le of the dislocation interaction between the stable equili-brium conreg guration ye in front of the obstacle and the respective labile one yl wherethe dislocation leaves the obstacle d ˆ yl iexcl ye Both l and d decrease with increasingshear stress The area A ˆ ld is also called the activation area This is the area sweptby the dislocation during the activation event The situation is schematically shownin reg gure 4 for cutting the clusters near their equators An upper limit of Vind can beestimated in the following way According to Boudardrsquos structure model the dis-tances between the centres of the Mackay-type clusters are of the order of 1nm Forexample in the reg vefold planes the shortest distances are l1 ˆ 078nm and


Figure 3 Forceplusmn distance curve of the interaction between a dislocation and a localizedobstacle

l2 ˆ frac12 l1 ˆ 126nm Here frac12 ˆ hellip51=2 Dagger 1dagger=2 denotes the number of the golden meanThus assuming l 1nm for the obstacle distance should represent a reasonableestimate l would be smaller by a factor of 21=2 if it were considered that the clustersfrom both sides intersect the slip plane The value of d at zero stress d0 depends onthe shape of the interaction proreg le and is usually a fraction of the diameter of theobstacle Figure 4 shows a dislocation in the equilibrium position waiting for thermalactivation as a full line and after having overcome the central obstacle as a dottedline The hatched area between both conreg gurations is the corresponding activationarea A The cluster diameter is certainly an upper bound of the activation distanceso that d0 ˆ 09nm With these data the maximum value of the activation volume ofovercoming individual clusters should be of the order of magnitude ofVind0 ˆ 1 09 0183nm3 016nm3 In principle d0 might also be much largerthan the obstacle diameter However the area per obstacle which is equal to l2 is anultimate upper limit of the activation area In reg gure 4 this area is limited by the fulland dashed lines and it leads to an almost equal estimate of Vind0 If the clusters arenot cut at their equators which is supposed here d0 as well as Vind0 should beconsiderably smaller Vind0 has to be compared with the largest experimental valueof Vex ˆ 15nm3 This value corresponds to small stresses or high temperatures Asdiscussed in sect21 it may be inmacr uenced by recovery but not by an order of magni-tude Thus the experimental activation volume Vex determined from the strain ratesensitivity of the macr ow stress is at least one order of magnitude larger than the


Figure 4 Schematic drawing of a dislocation bowing out between Mackay-type clusters Thearrangement of atoms is a section of reg gure 2 In the reg gure it is assumed that theclusters are cut near their centres The hatched area is the activation area of over-coming a single obstacle

activation volume Vind0 of overcoming the Mackay-type clusters individually Atincreasing stress (or decreasing temperature) the experimental activation volumedecreases but the mismatch between the model and the experimental value remainsunchanged

The model of overcoming the Mackay-type clusters as individual localized obsta-cles treated so far does not remacr ect the nature of the quasicrystal structure where theindividual clusters penetrate each other A better description of the dislocationmotion may be obtained by the theory of Labusch and Schwarz (1991) for so-calledextendedrsquo obstacles This theory was derived for solution hardening in crystalswhere a relatively rigid dislocation moves in the reg eld of weak obstacles In thiscase the dislocation changes its position not only at and near an obstacle which isovercome by an event of thermal activation but also at a number of neighbouringobstacles The theory has to be applied if the width of the obstacle proreg les is notsmall compared to the bow-out of the dislocation segments between the obstaclesAccording to Labusch and Schwarz (1991) this criterion is fulreg lled if the normalizedobstacle width dereg ned by

sup20 ˆ d0hellip2Gc=F0dagger1=2 hellip4dagger

is not small with respect to unity c is the concentration of obstacles on the slip planewhich is given by c liexcl2 F0 is the maximum force the obstacles can sustain and G isthe dislocation line tension It is approximated here by G ˆ middotb2=2 ˆ 84 10iexcl10 Nwith the shear modulus middot 50GPa at high temperature as quoted above F0 isunknown According to the theory of Labusch and Schwarz (1991) there is a rela-tion between F0 the obstacle concentration c and the athermal macr ow stress of theobstacle array

frac120 ˆ 094hellip1 Dagger 25sup20dagger1=3F 3=2

0 hellipc=2Gdagger1=2=b hellip5dagger

As described in sect2 a theoretical estimate of the athermal macr ow stress isfrac120 ˆ 003middot ˆ 15GPa This seems to be a realistic estimate as it is about reg ve timesthe maximum macr ow stress (frac14 ˆ 800MPa corresponding to frac12 ˆ 320MPa) at the lowesttemperature of the experiments Using this value and the above estimates of c G andd0 equations (4) and (5) can be solved simultaneously to yield sup20 ˆ 197 andF0 ˆ 352 10iexcl10 N d0 ˆ 09nm is certainly too large for cutting the clusters o -centre A smaller and more realistic value of d0 ˆ b ˆ 0183nm results in sup20 ˆ 035and F0 ˆ 454 10iexcl10 N In both cases however sup20 is well in the range of applica-tion of the theory of Labusch and Schwarz (1991) so that the situation of cuttingMackay-type clusters should be described by the theory of overcoming extendedrsquoobstacles

The theory implies that the activation volume Vex determined macroscopicallyfrom the strain rate sensitivity can be larger than that of surmounting individualobstacles by several orders of magnitude This is shown in reg gure 5 where thenormalized experimental activation volume Vex=V0 is plotted versus a normalizedtemperature T =T0 The normalizing constant of the activation volume is given by

V0 ˆ bgsup20=c hellip6daggerand that of the temperature by T0 ˆ G0=hellipk ln hellip _0= _daggerdagger g is a factor depending onthe shape of the interaction potential which is close to unity Using the data dis-cussed above yields a value for V0 of the order of magnitude of 01nm3 ThusVex=V0 is about 10 and T =T0 25 follows from reg gure 5 The ratio of T =T0 equals


the ratio between the experimental activation energy Gex and the activation energyof surmounting individual obstacles Gind

T =T0 ˆ Gex= Gind hellip7daggerThus the energy barrier to overcome an individual cluster is certainly smaller thanthe respective values observed in the experiments

4 DiscussionThe present paper gives an estimation of the activation volume of the cluster

friction model established by Feuerbacher et al (1997) and Urban et al (1998) Themicroprocesses during the thermally activated events are still not clear As describedabove the clusters are most probably cut o -centre along planes of large distancesMolecular dynamics simulations in three dimensions similar to the two-dimensionalones by Mikulla et al (1998) may help to identify the atomic conreg gurations acting asobstacles Nevertheless these conreg gurations will be of the size and spacing of theMackay-type clusters in the structural hierarchy of quasicrystals whatever theirexact atomic structure

The estimations of sect3 show that the maximum value of the activation volume ofovercoming individual clusters is about one order of magnitude smaller than themaximum volume measured in macroscopic deformation tests This discrepancy canbe overcome if the obstacles are considered to be extended obstacles in the sense of


Figure 5 Dependence of the normalized experimental activation volume V ex=V 0 on thenormalized temperature T=T0 after the theory of overcoming extended obstaclesRe-plotted from the paper of Labusch and Schwarz (1991)

the theory by Labusch and Schwarz (1991) The theory predicts that at high tem-peratures the obstacles are not surmounted individually but that in a thermallyactivated event the dislocation is shifted at a number of obstaclesConsequently the activation energy controlling dislocation glide is larger thanthat of an individual obstacle and at the same time the dislocation sweeps a largerarea during the activation Thus a combination of the cluster friction and theextended obstacle models can explain the large values of the experimental activationvolume

According to equation (7) and reg gure 5 the fact that Vex=V0 is about 10 impliesthat the experimental Gibbs free energy of activation Gex should be about 25 timeslarger than the value Gind of the individual obstacles The same conclusion can bedrawn from reg gure 4 of the paper by Labusch and Schwarz (1991) considering thatthe experimental stress range is below one reg fth of the maximum stress frac120 Asdescribed in sect21 in the experiments quite large activation enthalpies are observedUsing the data of Geyer et al (1998) the formalismfor crystals of Sch ock (1965) andthe temperature dependence of the shear modulus according to Tanaka et al (1996)yielded G values of 1eV at 640 C and of 23eV at 735 C Reasonable values of

G ie values corresponding to a reasonable pre-exponential factor _0 inequation (1) amount to about 25kT which is about 2eV at 640 C and 22eV at735 C This means that the experimental values are of a reasonable order of magni-tude According to reg gure 5 the Gibbs free energy Gind to overcome the individualobstacles should then amount to fractions of 1eV Thus the clusters need not bevery strong obstacles to explain the activation energy observed in macroscopicexperiments at low temperatures The present model does not explain the strongdecrease of the macr ow stress with increasing temperature However as described insect21 recovery plays an important role in the deformation of Alplusmn Pdplusmn Mn quasicrys-tals at high temperatures resulting in a reduction of the macr ow stresses

So far it was assumed that the dislocations cut the clusters certainly o -centreand not near their equators At high temperatures it may be easier for the disloca-tions to circumvent the strongly bound clusters rather than to cut themCircumvention is a motion of the dislocations out of their glide plane which shouldbe connected with climb The occurrence of recovery above about 700 C shows thatdi usion-controlled processes take place at most deformation temperaturesHowever if the climbing process controlled the mobility of the dislocations theactivation volume should be small ie of the order of magnitude of the atomicvolume This clearly contradicts the experimental results

In quasicrystals the dislocation motion produces a layer of phasons at lowtemperatures This process is not considered in the present model As described insect23 the computer simulations by Dilger et al (1997) suggest an energy for the layersof phasons of 58of the surface energy reg With the experimental value after Dubois(1998) of reg ˆ 17mJmiexcl2 the generation of the phason layer would cause a frictionstress of frac12f ˆ 058reg=b 54MPa This value corresponds to a considerable fractionof the macr ow stress particularly for the deformation at high temperatures In analogywith the production of stacking faults in crystals the formation of the phason layershould be an athermal process which does not inmacr uence the activation volume If theextension of the phason layer by thermally activated jumps of individual atomscontrolled the dislocation mobility the activation volume should again be verysmall in contradiction with the experimental observations The same holds for thesituation where the temperature is high enough so that the phasons di use away as


soon as they are created It may therefore be concluded that the formation ofphasons should not be neglected but that this is certainly not the rate controllingmechanism of dislocation motion

In the theory of Labusch and Schwarz (1991) it is assumed that the obstacles aredistributed randomly on their slip plane Although the distribution of the Mackay-type clusters is not periodic it is also not random As described above the clustersare aligned along crystallographic directions and the distances between the clustersalong these directions follow a Fibonacci sequence The random distributionassumed in the theory is certainly not a bad approximation of this non-periodicarray On the other hand the regular character of the arrangement of the clustersis expressed in the crystallographic orientation of moving dislocations observed inthe in situ experiments Thus a natural alternative description of the motion ofdislocations could be a Peierls model adapted to the larger length scale in quasicrys-tals Such a model does not exist up to now Nevertheless very simple estimates canbe made by formally applying the elastic theory of the Peierls stress as described egin the textbook of Hirth and Lothe (1982) According to this theory the Peierlsenergy is given by

UP ˆ frac12Pab=ordm hellip8daggerwhere frac12P is the Peierls stress and a 1nm is the width of the Peierls valleys ThePeierls stress should be equal to the athermal stress dereg ned above iefrac12P ˆ frac120 ˆ 15GPa It follows that UP 87 10iexcl11 N This can be used to estimatethe kink energy

Gk ˆ hellip2a=ordmdaggerhellip2UPGdagger1=2 hellip9dagger

It turns out that Gk 15eV using the above value of the line tension G Theactivation energy of dislocation motion is then approximately equal to 2 Gk whichis slightly too large but not in a wrong order of magnitude However the Peierlsmodel in its formulation for crystals predicts an activation volume of the order of theatomic volume ie it fails to describe the correct order of magnitude of the experi-mental activation volume Thus at present the cluster friction mechanism seems tobe the best model to explain the dislocation mobility in quasicrystals To establishagreement with the macroscopic deformation data the clusters should be consideredextended obstacles which are overcome in a collective way

ACKNOWLEDGEMENTS

The authors wish to thank the Deutsche Forschungsgemeinschaft for reg nancialsupport

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Rev L ett 77 3827Evans A G and Rawlings R D 1969 Phys status solidi A 34 9


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Wollgarten M Bartsch M Messerschmidt U Feuerbacher M Rosenfeld RBeyss M and Urban K 1995 Phil Mag L ett 71 99

Wollgarten MBeyss MUrban KLiebertz H and Koiumlster U 1993 Phys RevL ett 71 549

Yang W GFeuerbacher MBoudard M and Urban K 1998 to be published


discusses this mechanism in more detail and shows that the theory by Labusch andSchwarz (1991) of overcoming extendedrsquo obstacles can be applied to the clusterfriction model yielding quantitative agreement with most of the experimental resultsThe basic experimental and theoretical observations relevant for a discussion ofdislocation friction in quasicrystals are briemacr y outlined in sect2 The model is describedin sect3 followed by a discussion in sect4

2 Experimental observationsandresults ofcomputer simulations

21 Macroscopic deformationMacroscopic deformation tests were performed in compression at strain rates of

10iexcl5 and 10iexcl4 siexcl1 along twofold quasicrystal axes by Feuerbacher et al (1995) andGeyer et al (1998) and along reg vefold axes by Brunner et al (1997) and Geyer et al(1998) According to the latter study the deformation parameters depend on thecompression direction only weakly

The results of the deformation tests on quasicrystals were discussed in terms ofthe models of dislocation motion in crystals where the strain rate _ is given by anArrhenius-type relationship (for a review see for example Evans and Rawlings(1969))

_hellipfrac12 T dagger ˆ _0 exp hellipiexcl Ghellipfrac12dagger=kT dagger hellip1daggerThe same approach is used in the present paper The pre-exponential factor _0depends on the mobile dislocation density and is considered constant during themeasuring procedures of the activation parameters G is the Gibbs free energyof activation which is a function of the shear stress frac12 k is Boltzmannrsquos constantand T is the absolute temperature The experimentalrsquo activation volumeVex ˆ kT =msr was calculated from the strain rate sensitivity r ˆ hellip frac14= ln _daggerT Here frac14 is the macr ow stress and ms is the orientation factor As shown belowms 04 The macroscopic data of Geyer et al (1998) cover the widest temperaturerange availableat present Therefore the following discussion is based on these dataThe strain rate sensitivity was determined mainly from stress relaxation testsBetween about 640 C and 830 C the macr ow stress at _ ˆ 10iexcl5 siexcl1 decreases fromvalues above 800MPa down to less than 100MPa At the same time the strainrate sensitivity r decreases from a rather constant value of about 140MPa between640 C and about 680 C down to about 25MPa at 830 C This corresponds to anincrease of the experimental activation volume from about 025 up to 15nm3

In equation (1) the strain rate is controlled by the Gibbs free energy of activationG Unfortunately this quantity is not directly accessible in the experiments

Instead the activation enthalpy H ˆ iexclkT 2hellip frac14= T dagger=r was calculated from thetemperature sensitivity of the macr ow stress hellip frac14= T dagger at a constant strain rate The H

values increase from about 2eV at 640 C up to about 44eV at 735 C Reliable dataof H characterizing the dislocation mobility cannot be determined for highertemperatures because of recovery as discussed below For crystals the Gibbs freeenergy of activation G ˆ H iexcl T S is usually calculated from H using anequation derived by SchoEgrave ck (1965) S is the activation entropy which is estimatedon the assumption that entropy terms arise solely from the temperature dependenceof the elastic moduli Applying this formalism also to quasicrystals and using thedata by Tanaka et al (1996) for the temperature dependence of the shear modulusyields G values increasing from about 1eV at 640 C up to 23eV at 735 C































































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Friction mechanism of dislocation motion in icosahedral Al ... · Al± Pd± Mn quasicrystals...

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Transcript of Friction mechanism of dislocation motion in icosahedral Al ... · Al± Pd± Mn quasicrystals...