large separation between the decagons have a low surface energy of 1.9. Inspection of the binary tiling shows, that the tightly bound clusters are alignet along ...
NUMERICAL SIMULATIONS OF DISLOCATION MOTION AND CRACK PROPAGATION IN QUASICRYSTALS RALF MIKULLA and FELIX KRUL Institut fur Theoretische und Angewandte Physik, Universitat Stuttgart Pfa enwaldring 57, D-70550 Stuttgart PETER GUMBSCH Max-Planck-Institut fur Metallforschung, Institut fur Werksto wissenschaften Seestrae 75, D-70174 Stuttgart HANS-RAINER TREBIN Institut fur Theoretische und Angewandte Physik, Universitat Stuttgart Pfa enwaldring 57, D-70550 Stuttgart Crack propagationhas been studied in a two-dimensionalbinary model quasicrystal by two kinds of numerical simulation, the microconvergence method and the nite temperature molecular dynamics method. The computer experiments reveal two modes of crack motion: In the rst one the crack tip is emitting dislocations with straight phason walls in the wake, along which the quasicrystal breaks in a zig-zag shape. In the second one the crystal opens along a meandering line of minimal binding energy per unit length, which circumvents tightly bound clusters. In both cases the cleavage line is rough in accordance with scanning tunneling microscope observations.
1 Introduction Quasicrystalline structures can be constructed by irrational cuts through higher dimensional periodic crystals and thus possess a \hidden" translational symmetry. Therefore they may contain dislocations as defects, which, however, are accompanied not only by phonon-like strain elds, but also by phason elds 1 . The latter can be located on planes, which are bounded by the dislocation line. These planes are similar to the stacking faults of partial dislocations in periodic crystals. But the phason wall may also be broadened and dispersed. As a moving dislocation has to drag along its phason- eld, which is propagating by di usive atomic hopping motion, for a long time the dislocation mobilities had been considered as extremely low, and hence quasicrystals as brittle. Indeed, at room temperature, quasicrystals break under load. But at about 80% of the melting temperature, they become plastically deformable without hardening 2. By in-situ-observation in electron microscopy dislocation motion uniquely has been proven as mechanism of the plastic deformation 3 . Shearing of quasicrystalline model systems also was performed by molec1
Figure 1: Phason wall (stacking fault) obtained from low temperature dislocation glide 5 .
ular dynamics simulations of a binary model system, and dislocation motion was observed both at low 4 and at high temperatures 5 . After a dislocation pair had been formed and traversed the sample, a wall of phasonic defects indeed was left behind. On this wall secondary dislocation pairs were created. The wall was con ned to a sharp plane at low temperature ( gure 1), but broadened by shear-induced transverse di usion at high temperature to a large glide zone ( gure 2) enabling plastic deformation. It had been assumed that the low temperature phasonic wall is a region of high energy and weakened bonds and as such a good candidate for a cleavage plane.
Figure 2: Stacking fault at higher temperatures broadend by di usion 5
The (Lees-Edwards-) boundary conditions, however, were such that fracture could not occur. In the present article, therefore, we are going to simulate two-dimensionalquasicrystals at low temperature in con gurations corresponding to classical fracture experiments. We rst introduce the atomic model, then review basic notions of the theory of fracture and nally present two computer 2
experiments, where a crack is propagating through a dilated quasicrystal with a notch. In one of the experiments, where the microconvergence method is applied, a new mechanism for crack motion becomes apparent, namely dislocation emission by the crack tip and subsequent opening of the material along the phason wall in the wake of the dislocation. This mechanism is, however, not con rmed by a full- edged molecular dynamics simulation and thus up to further clari cation has to be considered as speculative. In both cases, the cleavage plane is not straight, but rough as observed in STM-experiments 6 .
2 The system
Figure 3: Binary tiling obtained from the Tubingen trangle tiling by decoration. Left: atoms are displayed as disks; right: bond representation
Our system is a two-dimensional binary tiling. It arises from the Tubingen triangle tiling, if large atoms are placed at the vertices of the tiling and small ones into the interior of the acute triangles. Apart from the direct atomic representation ( gure 3 left) we also visualize the con guration by the bond representation ( gure 3 right), where only bonds between di erent atomic species are drawn. A striking feature are clusters with a central large atom, which is surrounded by two ten-rings, the interior consisting of small atoms and the exterior of large atoms. In the bond-representation these clusters appear as ten-pronged stars. The atoms are made to interact by Lennard-Jones-potentials, of depth 1 for potentials between equal atoms, and of depth 2 between di erent atoms. The latter choice ensures, that no phase separation between the atomic species 3
is occurring and that the clusters of ten-rings are the tightest bound units due to their large number of inter-species bonds. The motion of the atoms is now simulated by two methods: either by temperature controlled molecular dynamics calculations (solution of Newton's equations of motion under constraints 7 ) or by the microconvergence method. Here the atoms move freely into their local potential minima, but are stopped as soon as they move out 8 . The microconvergence method is a very ecient relaxation algorithm corresponding to systems at zero temperature.
3 The Grith theory of fracture ∆ y
2b
x L ∆
Figure 4: Geometry of the simulations with constant energy release rate.
Due to Grith 9 the critical load for crack propagation can be evaluated as the load at which the surface energy of the two fracture surfaces is exactly balanced by the elastic energy released upon crack advance. One starts from a strip of matter ( gure 4 ) of width 2b and in nite length with a central crack at x=0 and dilated on both sides by an amount . At x 0 the nonzero components of the stress tensor are (1) = 1 ?E 2 b = (2) and the energy density is 1 E ( )2 : (3) w= 2 1 ? 2 b E is the Young modulus, the Poisson number. If the crack moves on, the energy release per unit length is G = w 2 b. The crack can only proceed, when 4 yy
xx
yy
this energy surmounts twice the crack surface energy , i.e. if G 2 . From this estimate we obtain the critical strain necessary for crack formation: p (4) "crit = 2 (1 ? 2)= bE To determine "crit we need rst the elastic constants. These have been measured by compression and shearing the system and simultaneous monitoring of the potential energy 10 . More attention is necessary for determining the surface energies of di erent fracture planes.
4 Candidates for cleavage planes
3514 2 Figure 5: Line structures in a twodimensional binary quasicrystal. Lines cutting through the decagonal clusters (1), (5) or traversing the small separation between the decagons (2) possess the highest surface energy of 2.3 Lennard Jones units. Those passing through the large separation between the decagons have a low surface energy of 1.9.
Inspection of the binary tiling shows, that the tightly bound clusters are alignet along ve families of planes, rotated mutually by multiples of 36 and 5
separated by a large and a small distance ( gure 5). Now the binding energy per unit length was measured for several straight lines. It turned out maximal and of about the same value (2.3 in Lennard-Jones units) for lines traversing directly the clusters, the small separation, or being placed arbitrarily. Good candidates for cleavage planes are the lines within the large separation , of surface energy 1.9. These lines were also identi ed as glide planes of dislocations in low temperature simulations of gure 1 4 5. ;
5 Fracture experiment with the microconvergence method
Figure 6: Crack obtained from the quasistatic simulation (micrconvergence method)
A strip has been taken of 11836 atoms with a half width of b = 30 in LJ units (which is the edge length of a bond). A notch was inserted in one of the easy planes, i.e. within the large separation of the lines connecting the clusters. From the pGrith relation and the surface energy we obtain a critical load " = 0:164= b = 0:0299. We applied a supercritical load of " = 1:3 "crit and started the microconvergence algorithm. The crack moved on straight, but after a few time steps turned to the left by 36 into an equivalent easy plane ( gure 6). Proceeding a few bond length the crack tip rst stopped, then deviated into an easy plane inclined to the vertical direction by 36 to the right. Thus the crack tip zig-zagged about the vertical line until it came to a halt short of the upper, open boundary. 6 crit
A close investigation shows that after initial propagation the crack tip is being stopped at an incomplete ten ring on its path. It then emits a dislocation which is running in an easy plane closest to the direction of maximum shear strain. This direction is at 45 to the vertical direction, the next easy plane at 36 . The dislocation is leaving in its wake a phason wall, where the crystal is weakened and { according to our assumptions from low temperature shear simulations 5 { is breaking. In detail, the crack tip is building up strain, then emits the dislocation, follows up, shoots over the dislocation core, and then stops. It is then being de ected in the radial strain eld of a cluster. Thus the fracture occurs along an irregular, rough zig-zag line and provides a fractal like appearence. Further experiments with larger samples, lower dilation or di erently placed notches revealed, however, a less sweeping gesture of the crack motion. The crack is meandering along the previously determined curved line of lowest surface energy. To check whether the behaviour is dependent on the method, we also performed molecular dynamics simulations in a similar geometry.
6 Fracture experiment with the molecular dynamics method
Figure 7: Crack obtained form the molecular dynamics simulation. Left: The atoms are displayed as disks. Right: x{component of the velocities on a logarithmic scale. The waves traveling around the crack become clearly visible.
7
Here the sample was furnished with the initial dilation eld of the previous experiment. The applied uniaxial load was 1.1 "crit. In the course of crack formation a large amount of binding energy is released in the form of phonons ( gure 7 right). These are re ected from the boundaries and can heat the sample up to melting. To avoid such a situation, we damped the phonons as soon as they traversed an elliptical boundary around the crack tip. Along the cleavage planes these phonons also are re ected. At instances they evaporate atoms which can form bridges in the crack channel. In total, the crack tip now is not propagating according to the dislocation-emission mechanism, but follows the easy plane. The fracture mechanism is now totally brittle ( gure 7). The energy of the straight surface generated by the crack is reduced to a value of 1.8. Thus the dislocation{emission mechanism at rst appears to be an artefact of the microconvergence method or the chosen geometry. Yet it seems to be feasible if in quasicrystalline systems local accretion of strain is possible as realised by the microconvergence method. Detailed investigations are in progress.
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