Effects of temperature and surface step on the incipient plasticity in strained aluminium studied by atomistic simulations
Pierre Hirel, Sandrine Brochard, Laurent Pizzagalli, Pierre Beauchamp
aa r X i v : . [ c ond - m a t . o t h e r] O c t Effects of temperature and surface step on theincipient plasticity in strained aluminiumstudied by atomistic simulations
P. Hirel a , ∗ S. Brochard a L. Pizzagalli a P. Beauchamp a a Laboratoire de M´etallurgie Physique, Bat. SP2MI, Bvd M. et P. Curie BP 3017986 962 Futuroscope Chasseneuil Cedex, FRANCE
Abstract
Atomistic simulations using an EAM potential are carried out to investigate the firststages of plasticity in aluminum slabs, in particular the effect of both temperatureand step geometry on the nucleation of dislocations from surface steps. Temperatureis shown to significantly reduce the elastic limit, and to activate the nucleation ofdislocation half-loops. Twinning occurs by successive nucleations in adjacent glideplanes. The presence of a kinked step is shown to have no influence on the nucleationmechanisms.
Key words: computer simulation, aluminium, surfaces & interfaces, dislocations,nucleation
The study of mechanical properties takes a new and more critical aspect whenapplied to nanostructured materials. While plasticity in bulk systems is relatedto dislocations multiplying from pre-existing defects, such as Franck-Readsources [1], nanostructured materials are too small for such sources to operate,and their plasticity is more likely initiated by dislocations nucleation from sur-faces and interfaces [2,3,4,5]. In particular, nucleation from grain boundaries isof great interest for the understanding of elementary mechanisms occuring inwork hardening of nano-grained materials [6,7,8,9]. The mechanisms involvingthe nucleation of dislocations from crack tips are also of great importance toaccount for brittle to ductile transition in semiconductors [10,11,12].In epitaxially-grown thin films, misfit induces a strain and can lead to theformation of dislocations at interfaces [13,14,15]. The presence of defects in asurface, such as steps, terraces or hillocks, can also initiate plasticity [16]. Inparticular, experimental and theoretical investigations have established that ∗ Corresponding author.
Preprint submitted to Elsevier tress concentration near surface steps facilitates the nucleation of disloca-tions from these sites [17,18]. Dislocations formation in such nanostructureschanges their mechanical, electrical, and optical properties, and then may havea dramatic effect on the behaviour of electronic devices [19]. Hence, the under-standing of the mechanisms initiating the formation of dislocations in thesenanostructures is of high importance.Since these mechanisms occur at small spatial and temporal scales, which aredifficult to reach experimentally, atomistic simulations are well suited for theirstudy. Face-centered cubic metals are first-choice model materials, because oftheir ductile behaviour at low temperatures, involving a low thermal activa-tion energies. In addition, the development of semi-empirical potentials formetals has made possible the modelling of large systems, and the accuratereproduction of defects energies and dislocation cores structures. Aluminiumis used here as a model material.In this study we investigate the first stages of plasticity in aluminum f.c.c.slabs by molecular dynamics simulations. Evidence of the role of temperaturein the elastic limit reduction and in the nucleation of dislocation half-loopsfrom surface steps is obtained. Steps in real crystals are rarely straight, and ithas been proposed that a notch or kinked-step would initiate the nucleationof a dislocation half-loop [20,21]. This is investigated here by comparing theplastic events obtained from either straight and non-straight steps.Our model consists of a f.c.c. monocrystal, with two { } free surfaces (Fig. 1).Periodic boundary conditions are applied along the two other directions, X=[0¯11] and Z= [011]. On one { } surface, two opposite, monoatomic steps arebuilt by removing atoms. They lie along Z, which is the intersection between a { } plane and the surface. Such a geometry is therefore well suited to studyglide events occuring in { } planes. We investigate tensile stress orthogonalto steps. In this case, Schmid analysis reveals that Shockley partials with aBurgers vector orthogonal to the surface step are predicted to be activatedin { } planes in which glide reduces the steps height [4]. In some calcula-tions, consecutive atoms have been removed in the step edge, forming a notch(Fig. 1), for investigating the effect of step irregularities on the plasticity. Var-ious crystal dimensions have been considered, from 24 × ×
10 (3680 atoms),up to 60 × ×
60 (142800 atoms). The latter crystal size was shown to belarge enough to have no influence on the results.Interactions between aluminum atoms are described by an embedded atommethod (EAM) potential, fitted on experimental values of cohesive energy,elastic moduli, vacancy formation and intrinsic stacking fault energies [22]. It iswell suited for our investigations since it correctly reproduces the dislocationscore structures. 2 =[100] X=[011]Z=[011]
Fig. 1. System used in simulations, with periodic boundary conditions along X andZ, and free { } surfaces. The { } glide planes passing through the steps edgesare drawn (dashed lines). Here, a notch is built on the right-side step. Without temperature, the system energy is minimized using a conjugate-gradient algorithm. The relaxation is stopped when all forces fall below 6 . × − eV.˚A − . Then the crystal is elongated by 1% of its original length alongthe X direction, i.e. perpendicular to the step. The corresponding strain isapplied along the Z direction, according to the isotropic Poisson’s ratio of alu-minum (0.35). Use of isotropic elasticity theory is justified here by the verylow anisotropy coefficient of this material: A = 2 C / ( C − C ) = 1.07 (EAMpotential used here) ; 1.22 (experiments [23,24]). After deformation, a new en-ergy minimization is performed, and this process is repeated until a plasticevent, such as the nucleation of a dislocation, is observed. The occurence ofsuch an event defines the elastic limit of the material at 0K.At finite temperature, molecular dynamics simulations are performed with thexMD code [25], using the same EAM potential. Temperature is introduced byinitially assigning an appropriate Maxwell-Boltzmann distribution of atomicvelocities, and maintained by smooth rescaling at each dynamics step. Thetime step is 4 × − s, small enough to produce no energy drift during a300K run. After 5000 steps, ie. 20 ps, the crystal is deformed by 1%, similarlyto what is done at 0K, and then the simulation is continued. If a nucleationevent occurs, the simulation is restarted from a previously saved state andusing a lower 0.1% deformation increment.To visualize formed defects, atoms are colored as a function of a centrosym-metry criterion [26]: atoms not in a perfect f.c.c. environment, ie. atoms onsurfaces, in dislocation cores and stacking faults, can then be easily distin-guished. In case of dislocation formation, the core position and Burgers vectorare determined by computing the relative displacements of atoms in the glideplane. These displacements are then normalized to the edge and screw com-ponents of a perfect dislocation.At 0K, the deformation is found to be purely elastic up to an elongation of 10%.Then a significant decrease of the total energy suggests an important atomic3 ig. 2. Formation of two dislocations at 0K, after a 10% elongation of a 60 × × reorganisation. Crystal vizualisation reveals the presence of defects located in { } planes passing through step edges, and a step height reduction by 2/3(Fig. 2). Atomic displacements analysis in these planes shows that plasticityhas occured by the nucleation of dislocations, with Burgers vectors orthogo-nal to the steps and with a magnitude corresponding to a 90 ◦ partial. Thisis consistent with the 2/3 reduction of the steps height. The dislocations arestraight, the strain being homogeneous all along the steps, and intrinsic stack-ing faults are left behind. The formation of dislocations from a surface stephas already been investigated from 0K simulations, using quasi-bidimensionalaluminum crystals [4]. It has been shown that straight 90 ◦ Shockley partialsnucleate from the steps. However, the small dimension of the step line didnot allow the bending of dislocations. Here, although this restriction has beenremoved by considering larger crystals (up to 90 atomic planes along Z), onlystraight partial dislocations have been obtained.In order to bring the role of steps to light, calculations are performed with a30 × ×
30 crystal with two free surfaces, but without step. In that case,plasticity occurs for a much larger elongation, 20%, and leads to a complexdefects structure. It clearly shows the important role played by steps, by sig-nificantly reducing the energy barrier due to dislocation-surface interaction,and initiating the nucleation in specific glide planes.The effect of temperature has been first investigated at 300K. Plasticity occursfor a 6.6% elongation, showing that thermal activation significantly reducesthe elastic limit. Another important difference due to temperature is the ge-ometry of the formed defect. Instead of a straight dislocation, a dislocationhalf-loop forms and propagates throughout the crystal (Fig. 3). As expected,the nucleation of a half-loop dislocation is thermally activated. Contrary tothe 0K simulation, a dislocation has nucleated from only one step: no disloca-tion is emitted from the other surface step, which remains intact. Atomic dis-4 ig. 3. Evolution of the aluminum crystal after a 6.6% elongation at 300K. Samecolor convention as Fig. 2. The origin of time is when the applied strain is increasedto 6.6%. (a) At 12 ps, several dislocation embryos appeared on both steps (arrows).(b) At 20 ps, a faulted half-loop dislocation has nucleated on one step. (c) After76 ps, a stable twin was formed. The other step (on the right) remains intact.
10 20 30 40
Atom label in the plane N o r m a li ze d r e l a ti v e d i s p l ace m e n t ( *b E DG E ( ° ) ) t=16ps 22ps 26ps Fig. 4. Calculated edge component of the relative displacements of atoms in theactivated glide plane and in the Z-layer corresponding to the dislocation front line,at 300K and for different times (triangles). They are fitted with an arctan function(solid lines) according to elasticity theory, for monitoring the dislocation core posi-tion during the simulation. The abcissa labels the depth of the atoms from the topsurface : 1 corresponds to atoms at the edge of the initial step, and 40 correspondsto the opposite surface, at the bottom of the system. placements at different simulation times (Fig. 4) indicates that this half-loopdislocation has a Burgers vector orthogonal to the step, the screw componentbeing almost zero. The formed dislocation is then a Shockley partial, leavinga stacking fault in its path. Atomic displacements have been fitted with anarctan function, according to elasticity theory. This allows to monitor the po-sition of the dislocation core defined as the maximum of the derivative, duringthe simulation.Before the complete propagation of a dislocation, several half-loop embryosstarting from both steps have been observed, appearing and disappearing(Fig. 3). Only one of them will eventually become large enough and prop-agate into the crystal (Fig. 3). This is related to the existence of a criticalsize for the dislocation formation, due to attractive interaction with the freesurface. As the dislocation moves through the crystal and reaches the oppo-site surface, a trailing partial does not nucleate. Though it would significantlyreduce the total energy of the system, especially in aluminum which have a5 ig. 5. Dislocations nucleated in a crystal with one straight step, and one irregular,elongated by 10% at 0K. high stacking-fault energy, this would require the crossing of a high energybarrier. On the contrary, the successive nucleation of dislocations in adjacent { } can be achieved with a much lower energy barrier. So, although it re-laxes less energy than a trailing partial would, this mechanism is more likelyto be activated. This is what we obtained in most simulations, similar to thetwinning mechanism proposed by Pirouz [20]. The remaining smaller step onthe top surface, as well as the step created by the emergent dislocation on thebottom surface, become privileged sites for the nucleation of other dislocationsin adjacent { } planes, leading to the formation of a twin. While sufficientstress remains, successive faulted half-loops will be formed in adjacent planes,increasing the thickness of the twin. After 76 ps, the crystal structure does notevolve anymore. The plastic deformation is then characterized by a micro-twin(Fig. 3), located around the previous position of the step, with an extension ofeight atomic planes, and delimited by two twin boundaries whose total energyequals the energy of an intrinsic stacking fault.We have also investigated how the dislocation formation process is modifiedin the case of irregular steps. We used a crystal with the same geometry, ex-cept that 10 consecutive atoms have been removed from one surface step edge(see Fig. 1), creating two step kinks between which lies a notch. The otherstep remains straight. First, at 0K, no defect is obtained up to 10% elonga-tion, beyond which plasticity occurs. This elastic limit is similar to the oneobtained for the system with perfect steps. Moreover, nucleated dislocationsare also Shockley partials with a Burgers vector orthogonal to the step, andare emitted from both surface steps, despite the system asymmetry. However,two dislocations have been formed from the irregular step. In fact, a secondpartial nucleates and propagates in the { } plane passing through the notch(Fig. 5). Both dislocations remain in their respective glide plane, leaving twostacking faults. This suggests that kinks are strong anchors for dislocations.Nevertheless, at 0K, it seems they have a negligible effect on the elastic limit,or regarding the nature of the nucleation event.6 ig. 6. Evolution of the aluminum crystal with an irregular step, under a 6.6%elongation at 300K. Same color and time conventions as in Fig. 3. The positionof the notch is highlited in red. (a) After 7.4 ps, a faulted half-loop dislocationnucleates in the original { } plane. (b) At 10 ps, another dislocation is emittedin the adjacent { } plane, passing through the notch. At 300K and for the same geometry, the elastic limit is reached for a 6.6%elongation, i.e. similar to the crystal with straight steps. Again, it suggeststhat irregular steps have no effect on the elastic limit. The dislocation half-loopdoes not nucleate from a step kink, but about 15 atomic planes away from it(Fig. 6a). It propagates into the crystal, but stays anchored to the kink, whichacts like an obstacle to the movement. Then, another dislocation nucleatesin the adjacent { } plane, within the notch (Fig. 6b). Another simulationon a similar system leads to a dislocation nucleation from the straight step,despite the presence of a kinked step. These results show that kinks are notpreferential sites for nucleation. It can be explained because step kinks are 0-Ddefects, contrary to straight steps, what prevents to initiate 1-D defects suchas dislocations. After the first nucleation, the twinning mechanism, alreadydescribed above, is observed. At about 70 ps, the formed twin cannot bedistinguished from the one obtained in the crystal with straight steps. Finally,there is no indication left whether the step was initially irregular or not.Molecular dynamics simulations have been used to investigate the influenceof temperature and of step geometry on the first stages of plasticity in f.c.c.aluminum slabs. Surface steps were shown to be privileged sites for the nu-cleation of dislocations, significantly reducing the elastic limit compared to aperfect surface. Simulations with straight surface steps have revealed that onlystraight 90 ◦ dislocations could nucleate at 0K. Temperature reduces the elas-tic limit, and makes possible the nucleation of faulted dislocation half-loops.Due to the system geometry and the strain orientation, only Shockley par-tials were obtained. Successive nucleation of partials in adjacent { } planesare observed, similar to the twinning mechanism described by Pirouz in semi-conductors. Simulations with an irregular step have shown that a kink is nota systematic site for nucleation. Instead, half-loops have been obtained froma straight portion of the step. The kinks introduced along a step seem to bestrong anchor points for dislocations, making their motion more difficult along7he step.During all simulations including temperature, several dislocation half-loop em-bryos were observed before one eventually becomes large enough and propa-gates into the crystal. Calculations are in progress to determine the criticalsize a half-loop must reach to fully propagate. To determine the activation en-ergy of the nucleation from surface steps, two methods may be used. First, thenudged elastic band method [27,28,29], applied to the nucleation and propa-gation of a half-loop, would provide the minimum energy path for this event.Second, by performing several simulations at a given strain, one would obtainthe average nucleation time as a function of temperature, thus allowing de-termination of the activation energy from Arrhenius plots. The dislocationsspeeds, as well as the size and shape of the dislocation half-loops, can beexpected to depend on temperature, which will also be investigated throughsimulations. As a sequel to the nucleation event, several scenarii were ob-served. The twinning mechanism is supposed to be in competition with thenucleation of a trailing partial, which requires the crossing of a higher energybarrier. However this last mechanism was obtained during a simulation, show-ing it is still possible. More investigations would allow to determine the exactdependancy on temperature, strain, or other parameters.P. Hirel’s PhD work is supported by the R´egion Poitou-Charentes. We greatlyaknowledge the Agence Nationale de la Recherche for financing the project(number ANR-06-blan-0250). References [1] J. P. Hirth, J. Lothe, Theory of dislocations (Second Edition), KriegerPublishing Company, 1982.[2] M. Albrecht, S. Christiansen, J. Michler, W. Dorsch, H. Strunk, P. Hansson,E. 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