arXiv:cond-mat/9804279v1 [cond-mat.mtrl-sci] 24 Apr 1998 68.Tepaeo h uscytli ieigby line differing is the the by of phase bound ex- The plane being a [6,8]. to and confined from is it tending core, temperatures dislocation temperature low the melting at around out whereas the spread is to field close phason the that revealed have tal spectively. pc.Tedsoaini ecie ya“phonon”-like a by described physical is field in displacement dislocation also component The orthogonal dislocation complementary a the space. a in with component of a dimension, vector and space higher Burgers crys- of the periodic Hence is dimensional irra- higher fact [7]. as a the tal considered through to be cuts due can tional is structures exis- quasiperiodic in The that quasicrystals. mech- of in the tence deformation is plastic computer dislocations of of by anism motion and that [5] [6], simulations microscopy electron transmission 30% to up but deformed [3], [4]. be temperature, temperature can melting the then room of and 80% at about hard at ductile and become sam- brittle the that are disclosed quasicrystalline ples have AlCuFe icosahedral and Sev- standard AlMnPd the order. alloys the of mainly to studies is related it eral are which because defects, structure, by governed by particular, in influenced solid, strongly a is of behavior mechanical The re- structure–property quasicrystals lations. studying structure, millimeter for peculiar possibilities in new the open grown of Because be [2]. could sizes thermodynami- quasicrystals plas- high–quality, stable transport, since properties cally heat , and physical and mass measure years ticity, as to such few last quasicrystals, possible conven- of the become in In has as quasiperiodic. it periodic, [1]. but not axes , is tional order eightfold translational or Their five- noncrys- with e.g., but tallographic, ordered, orientationally are They ture: field en osqec fteqaieidcsrcue r a are structure, quasiperiodic the of walls, consequence phason a that – being processes fracture of simulations istic as denoted is therefore which wall, this “phason-wall”. of sides two the oeua yaissmltoso hae quasicrys- sheared a of simulations dynamics Molecular ocrigdciiyi a encnre ohby both confirmed been has it Concerning struc- unusual an by characterized are Quasicrystals nti ril edmntae–b mlyn atom- employing by – demonstrate we article this In w ( x ,woecnoritgasyield integrals contour whose ), mt ilcto olwdb hsnwl,aogwihth which along wall, phason crac a specific by a followed in cert dislocation result For a emits propagation. the of crack nature and quasiperiodic motion env dislocation atomic both coordinated highly for of role dominating the reveal rc rpgto ssuidi w iesoa decagonal dimensional two a in studied is propagation Crack u ( x 2 n pao”lk displacement “phason”-like a and ) a-lnkIsiu ¨rMtlfrcug esr 2 70 92, Seestr. f¨ur Metallforschung, Max-Planck-Institut 1 ntttfu hoeiceudAgwnt hskdrUnive der Physik Angewandte und f¨ur Theoretische Institut .Mikulla R. rc rpgto nquasicrystals in propagation Crack fffnadig5,D750Sutat Germany Stuttgart, D-70550 57, Pfaffenwaldring 1 .Stadler J. , ihu hardening without b || and 1 .Krul F. , b ⊥ b ⊥ re- , at 1 1 .R Trebin H.-R. , utn h ihl on e-ig.Tesraeener- surface The without ten-rings. lines bound these follow tightly the can connecting the crack lines cutting a parallel between the because in of clusters, those are separation fracture, large for the lines” “easy the i.e. tal, on tutrluiso h uscytlmdlfrall for model quasicrystal the of of choices tightest units the structural become clusters bound decagonal the coordination fprle ie,mtal oae y36 by families five rotated on mutually situated lines, are parallel clusters and of smallest clusters The super on. form so Further- clusters such quasicrystals. of clusters real assemblies Mackay many shell more, of for versions second significant 2D 1), a are as (Fig. that and considered atom large atoms be are central can small a These and of surrounding atoms shell clusters. large first of of a structure by hierarchical built a a (1) [11] (2) TTT are and the which from inherited quasicrystals, diffraction real decagonal for models structure we rep- which “atom direct tiling, the rhombic resentation”. of representation” a “bond type as obtains different denote one of (S) atoms connected, tiling. atom neighboring original are of small the centers of a triangle the and large When vertex each large of each A center at derived the into placed [11]. [9], is (TTT) (L) system tiling atom tiling T¨ubingen triangle the binary from the quasicystalline front. ordered of crack perfectly dimensional representative a three (2D) simulate a system we of dimensional Secondly simplifica- analysis two the requires a evade consider material to we complex First a tions. in fracture ical crack. obstacles brittle the as for structure, centers scattering quasicrystalline and clusters, the atomic low to identify at inherent we mechanism Furthermore, fracture brittle temperature. the in element crucial h odeege are energies bond The h emtia ns[] h nto eghi u sim- our in length of unit The ulation [9]. ones geometrical the rnae t2.5 at truncated a in arranged 2). family, (Fig. each sequence within Fibonacci separation large a and oaodpaesprto.Almse r e o1 The 1. to set are therefore masses is time All of unit separation. phase avoid to rfre laaepae fte2 oe quasicrys- model 2D the of planes cleavage Preferred ulttvl u oe hrscnrlpoete with properties central shares model our Qualitatively dynam- like process nonlinear strongly a of study The h tm neatwt enr-oe potentials Lennard-Jones with interact atoms The rpgto ehns:Tecaktip crack The mechanism: propagation k rnet ssrcueitiscobstacles intrinsic structure as ironments r i vrod,teeosalsadthe and obstacles these overloads, ain 0 aeiloesup. opens material e ε oe uscytl h simulations The quasicrystal. model = LL 1 r n .Gumbsch P. and 7 ttgr,Germany Stuttgart, 174 LS and r sgvnb h egho h Sbond. LS the of length the by given is 0 rsit¨at Stuttgart, ε h odlntsaecoe obe to chosen are lengths bond The . SS ε t mle hnone. than smaller 0 0 = = r ε 0 2 LS p m/ε and 0 ε u oterhigh their to Due . LL ◦ = ihasmall a with , ε SS 0 = . 5 ε 0 gies [12] of different crystallographically equivalent, but left a phason wall in its wake. This phason wall carries structurally distinct lines (1-4 in Fig. 2) are lowest for some interface energy, and hence the cohesive strength of those between the widely spaced ten-rings (3,4 in Fig. 2), the material is reduced at this wall. After some time of namely 1.9 ε0/r0 as compared to 2.3 ε0/r0 for the other strain build-up the crack begins to open along the phason surfaces studied here (1,2,5 in Fig. 2). Furthermore, they wall (Fig. 4, bottom). It immediately regains its initial also have the lowest “unstable stacking energy”, i.e. propagation velocity until the next obstacle is met. Due increase in energy when the two halfs are shifted relative to the tightly bound complete ten-rings, the cleavage sur- to each other. Hence these lines are also preferred for face roughness is about 10 r0. dislocation motion [13]. When a dislocation is gliding At intermediate loads, between 1.3% and 1.5% strain, through a quasicrystal, leaving a phason wall behind, it a regime of steady crack propagation is observed. The creates an even more preferred cleavage line, since the crack tip proceeds with almost uniform velocity, but the phason wall is characterized by a positive defect energy. appearance of the cleavage plane is unchanged. The For the shortest Burgers vector the cleavage energy along dislocation-emission–phason-wall cleavage mechanism is the phason wall can be reduced by an additional 0.3 ε0/r0 still operates but without stopping the propagating [13]. crack. The fracture simulations are performed on a rectan- At even higher overloads (≥ 1.5% strain) we observe gular slab of approximately 250 000 atoms, 250 r0 high an onset of crack tip instabilities, which create defects and 1.000 r0wide. The slab is uniformly strained by an that are not bound to the crack tip. At a strain of amount ∆ normal to the long axis, and in the outer- 1.5% (Fig. 5) we observe dislocation emission at an an- most layer of width 3r0 all atoms are held fixed at these gle of 72◦, which reduces (due to shielding) the load on positions. A precrack of length 200r0 is inserted from the crack tip. Arrested by an obstacle, the dislocation one side along an easy line by cutting the bonds and moves back and annihilates at the fracture surface af- relaxing the sample. For the relaxation the externally ter the crack tip has propagated further. Such a pro- applied homogeneous strain is fixed at the Griffith value, cess, so–called “virtual dislocation emission”, was first at which the strain energy per unit length just equals proposed by Brede and Haasen to explain their fracture twice the surface energy. The resulting displacement field experiments in silicon [16]. At a load of 1.7% the crack then is scaled linearly above the critical dilation where bifurcates and stops. the crack tip starts moving. The evolution of the sys- In quasicrystals as in periodic crystals, we find a min- tem is followed by standard molecular dynamics (MD) imal velocity for brittle crack propagation of 14% of the technique where the atoms obey Newtonian equations of shear velocity vT at low loads. This suggests that motion. The system was started at an initial temperature there exists a lower band of forbidden crack velocities −6 of T=10 Tmelt. The IMD molecular dynamics package [17,18]. With rising load, the crack tip speed increases was used [14]. During crack propagation, the crack tip to about 35% of vT . This value is again similar to the bonds are broken and the strain energy above the surface maximum crack tip speed in periodic crystals [19]. The energy (the overload) is released in the form of acoustic lower than expected upper crack velocity, which linear emission or dislocation generation. To avoid heating of elasticity theory would place at the Rayleigh wave veloc- the slab and phonon reflections from the boundaries an ity [21] (vRayleigh ≈ 0.9 vT ), may be explained with the elliptical stadium is created outside which the are non-linearity of the atomic interaction, which has been damped gradually by a ramped friction term in the equa- shown to drastically reduce crack tip velocities in peri- tions of motion [15]. odic crystals [19,20]. At high loads the upper limit on For the phason-free precracked sample the critical the crack velocity is determined by the onset of crack strain is ǫc=1.12%. Straight brittle cleavage along the tip instabilities: (virtual) dislocation emission and crack easy line occurred only very close to this value and for branching. It seems likely that the instability is driven short segments. In all other cases a rough surface is cre- by a critical overload or a critical accumulation of specific ated. The propagating crack is monitored and its current shear waves. position is plotted versus time for three different loads in While the crack tip velocity is of similar magnitude in Fig. 3. Depending on the load level one can distinguish quasicrystals and periodic crystals, the crack propaga- three different fracture regimes. tion mode differs significantly. At small overloads an in- An intermittent propagation regime is found at low termittent crack propagation mode occurs, during which load levels. For example, at a strain of 1.131% the crack the average crack velocity can be low but microscopically proceeds with constant velocity along an easy fracture separated into distinct propagation and arrest periods. line until it hits an obstacle (Fig. 4, top), which can be The fracture surfaces are rough, governed by cluster in- a complete or an incomplete ten-ring. It then stops and homogeneities. Intermittency and roughness can both be emits a dislocation along one of the easy planes inclined attributed to the novel dislocation-emission–phason-wall by 36◦, along which the shear is high. The disloca- fracture mechanism. tion moves away from the crack tip until it is stopped it- At comparable external loads the brittle fracture of self by an obstacle(Fig. 4, middle), whereupon the entire perfect periodic crystals is characterized by steady crack configuration rests for some time. The dislocation has propagation along well defined flat and smooth cleav-

2 age planes [18]. In comparison to periodic crystals, two [1] Ch. Janot, “Quasicrystals: A Primer”, Clarendon Press specific structural components of the quasicrystal are im- 1994, Oxford. portant to explain the observed behaviour. Firstly, the [2] A.P. Tsai in New Horizons in Quasicrystals: Research role of the ten-rings as tightly bound clusters which act and Applications, edited by A.I. Goldman, D.J. Sordelet, as structure intrinsic obstacles to both dislocations and P.A. Thiel and J.M. Dubois (World Scientific 1996) p. 1. cracks is obvious. At low overloads the clusters may tem- [3] S. Takeuchi, H. Iwanaga and T. Shibuya, Jpn. J. Appl. porarily arrest the crack, and even at high overloads and Phys. 30 (1991) 561. 32 high crack tip speeds, the clusters act as scattering cen- [4] S. Takeuchi and T. Hashimoto, Jpn. J. Appl. Phys. ters for the cracks. (1993) 2063. [5] M. Wollgarten, M. Beyes, K. Urban, H. Liebertz and U. K¨oster, Phys. Rev. Lett. 71 (1993) 549. In perfect agreement with a previous study of dislo- [6] R. Mikulla, J. Roth and H.-R. Trebin, Phil. Mag. B 71 cation mobility in quasicrystals [13], we find here that (1995) 981. the clusters also act as obstacles to dislocation motion. [7] J. Bohsung and H.-R. Trebin, in: “Aperiodicity and Or- As a consequence, dislocations that have been nucleated der” Vol. 2, ed. M.J. Jari´c, Academic Press Inc., (1989) at the crack tip cannot get farther away than the next 183 sufficiently strong obstacle. Since the interaction of the [8] R. Mikulla, J. Roth and H.-R. Trebin, in: Proc. of the crack tip and the dislocation produces a driving force on 5th Int. Conference on Quasicrystals, eds. C. Janot and the dislocation which decreases with distance from the R. Mosseri, World Scientific 1995, p. 297. crack tip [22], the strength of a cluster, measured as the [9] Deconal binary tilings(BT) [10] are mixtures of two capability to arrest the dislocation, increases with dis- species of particles where the bond lengths fulfill the re- tance. Strong local structural variations, which are a quirement that five particles of type A surround one B– characteristic of the quasicrystalline structure, are there- particle and ten B–particles sourround one A–particle. fore very efficient in blocking dislocation motion and tend Provided with Lennard–Jones type potentials glassy, to keep dislocations very close to the crack tip. Usually crystalline, perfect quasicrystalline and random tiling this would be considered beneficial, since a dislocation quasicrystalline phases are possible. The perfectly orderd emitted from the crack tip also shields the crack tip from quasicystalline phases like the BT derived from the TTT the applied load, with the shielding decreasing with dis- is strictly meta stable but at modest temperatures me- tance to the crack tip [22]. As a consequence, one would chanicly stable over time scales accessible with MD sim- ulations. expect a high fracture if dislocation emission [10] F. Lan¸con and L. Billard in Lectures on Quasicrystals ed. occurs, while dislocations are kept close to the crack tip. by F. Hippert and D. Gratias, les editions de physique, However, this is not what happens, due to the second Les Ulis (1994) 265. structural peculiarity of quasicrystals: the phason wall [11] M. Baake, P. Kramer, M. Schlottmann and D. Zeitler, left behind a moving dislocation. Int. J. Mod. Phys. B 4(15/16) (1990) 2217. [12] Because of the lack of periodicity, the surface energy of Since the phason wall carries excess energy, it provides the quasicrystal must be calculated on finite samples. To a new low–energy path for the crack and thereby effec- minimize sampling errors, the surface energy is deter- tively allows the crack to bypass the structure-intrinsic mined as the average energy difference per unit length obstacles. The existence of the phason wall, trailing dis- between an intact and an (artificially) cleaved quasicrys- locations nucleated at the crack tip, is ultimately respon- tal for several parts of the structure with different length. sible for the observed of the quasicrystals. [13] R. Mikulla, P. Gumbsch and H.-R. Trebin, cond- mat/9708102. [14] J. Stadler, R. Mikulla and H.-R. Trebin, Int. J. Mod. Although the binary model quasicrystal used in this Phys. C 8 (1997) 1131. study is very simplistic, it reflects the most important [15] B.L. Holian and R. Ravelo, Phys. Rev. B 51 (1995) features of real quasicrystals, i.e. quasiperiodicity and 11275. clusters. It permits conclusions relevant to real qua- [16] M. Brede and P. Haasen, Acta metall. 36 (1988) 2003. sicrystals. Our simulations reveal the dominating role [17] M. Marder and S. Gross, J. Mech. Phys. Solids 43 (1995) of highly coordinated atomic environments as structure 1. intrinsic obstacles for both crack and dislocation motion. [18] P. Gumbsch, S.J. Zhou and B.L. Holian, Phys. Rev. B Possible toughening effects, however, are over compen- 55 (1997) 3445. sated by the embrittling nature of the phason wall. The [19] P. Gumbsch, Z. Metallkd. 87 (1996) 341. phason wall modifies the Griffith criterion and leads to [20] B.L. Holian, R. Blumenfeld and P. Gumbsch, Phys. Rev. an immediate crack opening of the solid in the path of a Lett. 78 (1997) 78. single dislocation. The existence of the phason degree of [21] L.B. Freund, “Dynamical ”, Cam- freedom gives raise to this special mechanism of brittle bridge University Press 1990, New York. fracture in quasicrystals. [22] R. Thomson, in Solid State Vol. 39, edited by H. Ehrenreich and D. Turnbull, Academic Press 1986, New York, pp. 1–129.

3 FIG. 1. Ten-rings in atom (middle) and bond representa- tion (right) and their origin in the triangle tiling (left).

FIG. 4. Crack propagation mechanism in the intermittent regime. Top: The crack tip has stopped. Middle: It has emit- ted a dislocation which is followed by a phason wall. Bottom: The quasicrystal has opened along the wall. 3 5 1 4 2

FIG. 2. Binary tiling in the bond representation and spe- cial planes, whose surface energy is explained in the text.

300.0 1.13% 1.131% 1.14 % 1.3 % 1.5 % 1.6 % ]

0 200.0

100.0 propagated distance [r

0.0 0.0 35.0 70.0 105.0 140.0

time [t0 ] FIG. 3. Propagated distance of crack tip versus time. De- FIG. 5. Virtual dislocation emission observed at a strain of pending on the dilation three different regimes are found: An 1.5% in the bond representation. Top: Snapshot of a simula- intermittent (ε=1.13%, 1.131% and 1.14%), smooth (ε=1.3%) tion after 73.5t0; a dislocation has been emitted at an angle ◦ and unstable regime (ε=1.5 and 1.6%). of 72 . Bottom: After 77t0 the dislocation has returned.

4