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On the origin of cluster strengthening in aluminum alloys de Vaucorbeil, Alban 2015

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On the Origin of Cluster Strengthening in Aluminum AlloysbyAlban de VaucorbeilA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Materials Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2015c© Alban de Vaucorbeil, 2015AbstractUnderstanding the influence of atomic clusters formed during natural aging in aluminum alloys is a keyproblem to control more effectively the strength of alloys both during their processing and life. The yieldstrength of these alloys is controlled by the interaction between dislocations and solute atoms, and clusters.An empirical scaling law relating the dislocation-obstacle interaction force with the size of clusters hasbeen developed and successfully used to predict the yield stress of a cluster strengthened AA6111 industrialaluminum alloy. It is proposed that the strengthening effect captured by the scaling law could come from thegeometrical rearrangement of solute atoms from a random distribution to a clustered distribution, and/or fromthe change in strength of individual obstacles. A modified areal glide model was employed to investigate thestatistical problem of a dislocation moving through a set of clustered point obstacles in the glide plane. Theresults of these simulations suggest that the degree of clustering of solute atoms does not influence the criticalresolved shear stress. Then, molecular statics simulations were used to investigate the origin of the changein strength of individual clusters, in the simple case of Al-Mg alloys. A model based on elastic interactionbetween the solute atoms/clusters and an edge dislocation was developed and demonstrated to give goodpredictions for the maximum pinning force of single solutes, dimers and trimers. Using a detailed analysisof the model and the molecular statics simulations, it was shown that the strength of clusters principallycomes from the elastic interaction between dislocations and solute atoms forming the clusters. Further, thechange of topology of clusters was found to not significantly affect their strength at least in the case ofMg clusters in aluminum. Finally, this model was employed to determine the strengthening contribution ofdistributions of single solutes, dimers and trimers in binary Al-Mg alloys. The strength was found to roughlydepend linearly on the size of clusters, however, its slope is lower than in the case of the AA6111 alloywhich predominately contains a combination of Mg-Mg and Mg-Si clusters. The possible reasons for thisdiscrepancy are discussed.iiPrefaceAll the work in this thesis was completed by the author, Alban de Vaucorbeil, with the following exceptions.The atom probe experiments mentioned in Chapter 5 were done by Ross W. Marceau, and Warren J. Pooleestimated the yield stress at 0 K of the aged AA6111 alloy from the experimental yield stresses he obtainedat 293 K and 77.A version of Chapter 5 was published in Acta Materialia [R.K.W. Marceau, A. de Vaucorbeil, G. Sha,S.P. Ringer,W.J. Poole, Analysis of strengthening in AA6111 during the early stages of aging: Atom probetomography and yield stress modelling, Acta Materialia, vol. 61, no. 19, pp. 7285-7303, 2013].A version of Chapter 6 was published in Philosophical Magazine [A. de Vaucorbeil, C. W. Sinclair, andW. J. Poole, Dislocation glide through nonrandomly distributed point obstacles, Philosophical Magazine,vol. 93, no. 27, pp. 3664-3679, 2013].iiiTable of contentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Decomposition of super-saturated solid solution in aluminum alloys aged at room temperature 52.1.1 Solid solution decomposition sequence . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Guinier-Preston zones and clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Thermodynamics of clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7iv2.1.4 Experimental evidence of clustering in naturally aged aluminum alloys, and Al-Mgalloys in particular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.4.1 Electrical resistivity measurements . . . . . . . . . . . . . . . . . . . . . 82.1.4.2 Small angle scattering measurements . . . . . . . . . . . . . . . . . . . . 102.1.4.3 Differential scanning calorimetry measurements . . . . . . . . . . . . . . 112.1.4.4 Direct observation techniques: TEM, and Atom Probe Tomography . . . . 122.1.5 Origin of the rate of cluster formation . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Strengthening mechanisms associated with cluster strengthened aluminum alloys . . . . . . 132.2.1 Link between clustering and strengthening . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Physics of solute solution and cluster strengthening . . . . . . . . . . . . . . . . . . 142.2.2.1 Peierls stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2.2 Obstacles to dislocation motion . . . . . . . . . . . . . . . . . . . . . . . 152.2.2.3 Statistical effect of distributions of obstacles to dislocation motion . . . . 172.2.2.4 Atomistic simulations for studying the interaction between dislocationsand obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2.5 Areal glide model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Scope and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1 Areal glide model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Atomistic simulations: molecular statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.1 Creation and glide of an edge dislocation . . . . . . . . . . . . . . . . . . . . . . . 384.2.2 Peierls stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41v4.2.3 Interaction with a single solute on the glide plane . . . . . . . . . . . . . . . . . . . 425 Modelling the yield stress of a cluster strengthened AA6111 alloy . . . . . . . . . . . . . . . . 476 Clustering: Effect of spacing between solute atoms . . . . . . . . . . . . . . . . . . . . . . . . 566.1 Aggregation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Effect of the spacing inter-clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.3 Combined effect of the spacing inter-clusters and of the ‘effective’ strength of solutes . . . . 626.3.1 Behaviour of strong obstacles (Region I): . . . . . . . . . . . . . . . . . . . . . . . 646.3.2 Behaviour in the transition region (Region II): . . . . . . . . . . . . . . . . . . . . 666.3.3 Behaviour in the cluster-insensitive limit (Region III): . . . . . . . . . . . . . . . . 677 Intrinsic strength of individual obstacles: from single solutes to trimers . . . . . . . . . . . . 707.1 Strength of a single solute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.1.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.1.2 Dislocation-solute interaction: linear elasticity . . . . . . . . . . . . . . . . . . . . 777.1.3 Dislocation-Solute interaction: elasticity using the elastic strain field of the disloca-tion from MS simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.2 Strength of dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3 Strength of trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Relationship between cluster size and cluster strength in binary Al-Mg alloys . . . . . . . . . 1088.1 CRSS of random solid solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098.2 CRSS of randomly distributed dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138.3 CRSS of randomly distributed trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115vi8.4 CRSS as a function of cluster size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.1 Thesis summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A The value ofP for the case of a random distribution . . . . . . . . . . . . . . . . . . . . . . . 133B Detail of the derivation of P(L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134viiList of tablesTable 4.1 Elastic constants and intrinsic stacking fault energy (ESF ) of aluminum: comparison be-tween values obtained with Liu’s and Mendelev’s potentials and experimental ones. . . . 36Table 5.1 Experimental number densities (1015 cm3) of clusters tabulated within certain size rangesfor the AA6111 samples after natural aging at room temperature [57]. . . . . . . . . . . . 48Table 5.2 Experimental number densities (1015 cm3) of clusters tabulated within certain size rangesfor the AA6111 samples after artificial aging at for various times and temperatures [57]. . 49Table 5.3 Best-fit parameters of the average three-dimensional Guinier radius cumulative log-normalfunction at various aging conditions in the AA611 alloy with corresponding mean clusterradius and variance. Also given for each condition is the cluster volume fraction f clusterstotas well as the average breaking angle of the distribution of clusters Φ¯clustersc . Data takenfrom Ref. [57] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Table 5.4 Yield stress at 0 K extrapolated from experimental yield stresses at 293 K and 77 K [57]. 53Table 6.1 Comparison between the transition strengths βwc and β sc predicted and their possible rangeestimated from the simulation results, these correspond to the shaded regions in Figure 6.6 66Table 8.1 Critical resolved shear stress obtained from the areal glide simulations compared withselected simulation and experimental results. The values in parenthesis correspond to thecase where only solute atoms on the glide plane are considered. . . . . . . . . . . . . . . 113viiiList of figuresFigure 2.1 Schematic of a L12 crystal structure. Blue and green spheres represent Mg and Al atoms,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 2.2 transmission electron microscope (TEM) image of clusters in an Al-Cu alloy, from Etoet al. [32]. The thin dark lines are the one-atom thick Cu clusters. . . . . . . . . . . . . 6Figure 2.3 (a) TEM and (b) high resolution transition electron microscope (HRTEM) image of clus-ters in an Al-Mg-Si alloy taken from [30]. The arrows point at a few selected clusters. . 6Figure 2.4 Difference between the derivative of the change of free entropy and the derivative ofthe change of free enthalpy. When ∂∆Hmix∂χB >−TkBln(χB)1−χB ⇔ χB > χmin, clustering shouldoccurs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.5 Change of electrical resistivity of an Al-10%Mg alloy aged at 20◦C (from Sato et al. [16]). 9Figure 2.6 DSC curves of samples of an Al-16Mg alloy naturally aged during 2 years obtained atheating rates of 1.2, 2.5, 5, 10, 20, 40 and 80◦C/min.(from Starink and Zahra [54]) . . . 11Figure 2.7 Strengthening evolution during natural aging of (a) an AA6111 alloy [5] and (b) a binaryAl-10%Mg alloy [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.8 Schematic of one possible dislocation/obstacle interaction force (Fobs)/distance (a) curvewhich illustrates the activation energy ∆G (from Kocks et al. [75]). τbL is the externallyapplied force on a dislocation segment of length L. . . . . . . . . . . . . . . . . . . . . 16Figure 2.9 Experimental strength of solid solutions in high-purity binary aluminum alloys from [4] 18Figure 2.10 Schematic representation of a dislocation under stress bending between obstacles. . . . . 18ixFigure 2.11 Example of interaction force between two partial edge dislocations in Al and a substitu-tion Mg atom calculated by Patinet and Proville using molecular statics simulations [84]. 19Figure 2.12 Critical resolved shear stress of a square array of obstacles compared to the critical rre-solved shear stress (CRSS) of a random array due to Foreman and Makin [97] and Han-son and Morris [98]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 4.1 Schematic showing the different simulation tools used in this work as well as their link. . 30Figure 4.2 Schematic of the three possible evolution mechanisms for a dislocation segment: (a)meet an obstacle, (b) break an obstacle, and (c) by-pass an obstacle. From Nogaret andRodney [92]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 4.3 Simulated critical resolved shear stress of 10,000 randomly distributed obstacles as afunction of the obstacle strength. Each point represents the average of 10 simulations atthat strength level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 4.4 Under the same shear stress, the distance a dislocation glides for the same array of 10,000randomly distributed obstacles whose breaking angle isΦc= 0◦ when simulated with (a)Nogaret and Rodney’s method and (b) the ‘circle rolling’ method. All obstacles bypassedby the dislocation and dislocation loops are not represented. Clusters of obstacles belowthe dislocation are groups of looped obstacles (not represented in (b)). . . . . . . . . . . 34Figure 4.5 Enthalpy of mixing: of aluminum: comparison between values obtained with Liu’s andMendelev’s potentials and CALPHAD data [42]. . . . . . . . . . . . . . . . . . . . . . 37Figure 4.6 Schematics showing the different steps of the creation of an edge dislocation. . . . . . . 39Figure 4.7 Snapshot of the potential energy of atoms in and around the core of the dislocation. . . . 39Figure 4.8 Schematics showing the method used to make the dislocation glide. . . . . . . . . . . . 39Figure 4.9 Maximum pinning force of a single solute laying on the plane just above the glide planeas a function of the simulation box size. . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 4.10 Simulated stress-strain response of a split edge dislocation gliding in a perfect aluminumlattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42xFigure 4.11 Effect of the periodic boundary conditions on the MS simulations: the dislocation glidesthrough a rectangular array of obstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 4.12 Forces acting on a dislocation segment during glide. . . . . . . . . . . . . . . . . . . . 43Figure 4.13 (a) Disregistry projected on the x direction of a split edge dislocation. The symbolscorrespond to the simulation values, and the solid line corresponds to the Peierls-Nabarrofit. (b) The first derivative of the Peierls-Nabarro fit of the disregistry. . . . . . . . . . . 45Figure 5.1 Examples of best-fit of truncated cumulative density function for two aging conditions:(a) 3 days at 60◦C and (b) 1 week at room temperature. . . . . . . . . . . . . . . . . . . 52Figure 5.2 Schematic showing the interaction between the glide plane and a distribution of clustersmodelled by spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 5.3 Model results compared to the experimental yield stress (extrapolated to 0 K) for (a)artificial aging and (b) natural aging in the AA6111 alloy. . . . . . . . . . . . . . . . . . 55Figure 6.1 Schematic representing how distributions of obstacles with different levels of clusteringwere obtained. The 14 dashed circles are the virtual circles inside of which Nin = 5obstacles are randomly placed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 6.2 Partial representation (400 obstacles) of four different distributions of 40,000 obstacleswith different level of clustering with their respective CRSS. The level of clusteringincreases from (a), a quasi random distribution, to (d). For all distributions Nin = 5. . . . 58Figure 6.3 Processes involved in the calculation of the clustering parameter P . Given a set ofpoint obstacles (dots), a regular grid of ‘nodes’ (crosses) are superimposed such that theaverage square spacing is the same for both. A random node from the grid is selected(zoom in (b)) and the distances di j from these nodes to their neighbours are computed.ThenP is obtained as the average of the square of these distances (equation (6.1)) . . . 60Figure 6.4 Example of both neighbouring and non-neighbouring obstacles as defined by Kocks [135].Pj is a neighbour of Si because no points lies inside C1, the smallest circle going throughboth Pj and Si. However, Q is not a neighbour since an obstacle lies inside C2. . . . . . . 60xiFigure 6.5 Variation of the normalized CRSS with the degree of clustering as measured byP fromsimulations with 40,000 strong (β = 1) obstacles. These distributions were obtained byvarying r from r= 0.1 to r= 0.001 and the number of obstacles per clusters from Nin= 5to Nin = 20. The number of clusters remained unchanged as Nc = 4000. . . . . . . . . . 62Figure 6.6 Change in the relationship between normalized CRSS and obstacle strength (β ) with thedegree of obstacle clustering,P . The results forP = 1 corresponds to the data shownin Figure 4.3. The distributions of obstacles in the glide plane were generated using{r = 0.005, Nin = 5} for P = 2.5, {r = 0.001, Nin = 5} for P = 5.2 and {r = 0.005,Nin = 20} forP = 20. The critical values βwc and β sc fall within the shaded regions. . . 63Figure 6.7 An example of the data from Figure 6.6 for P = 5.2 plotted alongside the results ex-pected for a random distribution (equation 4.4) illustrating the three regimes of CRSSdependence on clustering. In region I, the CRSS is found to be independent of the obsta-cles’ strength, β . Region II represents a transition regime, while in region III it is foundthat the CRSS does not depend on the level of clustering. The Orowan stress is calcu-lated using equation (6.2) while the expression developed by Friedel [75] for randomlydistributed weak obstacles is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 6.8 Example of a 10-obstacle cluster and its envelope (dashed lines). . . . . . . . . . . . . . 65Figure 6.9 Definition of Aswept , the maximum area of a cluster the dislocation can sweep beforebreaking away from an obstacle lying at the cluster’s boundary. . . . . . . . . . . . . . . 67Figure 6.10 CRSS prediction at low obstacle strength. Two states of a dislocation are represented:one at zero stress (along the x-axis) and the second at the stress required to overcomeobstacle O . The dark grey shaded area is the swept area free of obstacles while the lightgrey shaded area contains at least one obstacle. The closest obstacle to O is thus foundbetween L and L+dL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 7.1 Schematic of the pressure field due to a split edge dislocation. Red corresponds to posi-tive pressure (compression) and blue corresponds to negative pressure (tension). . . . . . 72xiiFigure 7.2 Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides pasta single solute located 1/2 an atomic plane above the glide plane. . . . . . . . . . . . . . 73Figure 7.3 Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides pasta single solute located 1/2 an atomic plane below the glide plane. . . . . . . . . . . . . . 73Figure 7.4 Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides pasta single solute located 7.5 atomic planes above the glide plane. . . . . . . . . . . . . . . 75Figure 7.5 Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides pasta single solute located 7.5 atomic planes below the glide plane. . . . . . . . . . . . . . . 75Figure 7.6 Variation of the maximum pinning forceFmaxobs of a single Mg solute with its distance tothe glide plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 7.7 Variables used to model the interaction between dislocation and solute. . . . . . . . . . 79Figure 7.8 Generalized Stacking Fault energy surfaces for (a) pure aluminum and (b) aluminumwith one Mg solute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 7.9 Energy due to the change of the generalized-stacking-fault energy surface in the presenceof solute on the glide plane (d =±0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 7.10 Comparison between the maximum pinning force predicted by elasticity and the resultsof MS simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 7.11 Comparison between the predictions obtained from linear elasticity and the MS resultsfor an atom located 5 planes above the glide plane. . . . . . . . . . . . . . . . . . . . . 82Figure 7.12 Comparison between the predictions obtained from linear elasticity and the MS resultsfor an atom lying on the plane just above the glide plane. . . . . . . . . . . . . . . . . . 82Figure 7.13 Comparison between the displacement field ux of a split edge dislocation obtained from(a) MS simulations and (b) linear elasticity. . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 7.14 Comparison between the displacement field uy of a split edge dislocation obtained from(a) MS simulations and (b) linear elasticity. . . . . . . . . . . . . . . . . . . . . . . . . 83xiiiFigure 7.15 Comparison between the maximum pinning force predicted by the hybrid elastic modeland the results of MS simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 7.16 Disregistry: comparison between the case where the solute is just above the glide plane(d = +0.5) and the case where the solute is located 1.5 atomic planes above the glideplane (d =+1.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 7.17 Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom lying on the plane just above the glide plane. . . 88Figure 7.18 Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom located on the plane just below the glide plane. . 88Figure 7.19 Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom located 5.5 planes above the glide plane. . . . . 90Figure 7.20 Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom located 5.5 planes below the glide plane. . . . . 90Figure 7.21 Schematic showing how the dislocation bends when interacting with an obstacle (here asolute atom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 7.22 Schematic showing the position of the different close packs A (red), B (blue), and C(black) and the corresponding colour coding used in Figure 7.23. . . . . . . . . . . . . . 93Figure 7.23 Arrangement and maximum pinning forceFmaxobs of dimers. . . . . . . . . . . . . . . . . 93Figure 7.24 Distribution of maximum pinning force for all the different configurations of dimersdetermined from the results of MS simulations. . . . . . . . . . . . . . . . . . . . . . . 94Figure 7.25 Arrangements of Dimer I (strongest dimer) lying just above the glide plane before andafter passage of the dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 7.26 Arrangements of Dimer II (second strongest dimer) lying on the glide plane before andafter passage of the dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 7.27 Schematic of the compensative interaction that can exist between the two solutes of adimer when they are located just above, or just below the glide plane, respectively. Insuch configuration, the solute above attracts the dislocation, while the solute below repels it. 96xivFigure 7.28 Three body interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 7.29 The case of Dimer I lying just above the glide plane. Comparison between the predic-tions obtained from elasticity and the MS results. . . . . . . . . . . . . . . . . . . . . . 98Figure 7.30 The case of Dimer II on the glide plane. Comparison between the predictions obtainedfrom the hybrid elastic model and the MS results. . . . . . . . . . . . . . . . . . . . . . 99Figure 7.31 Variation of the maximum pinning force of Dimer I as a function of its distance to theglide plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 7.32 Trimers could be seen as being composed of two ‘dimer’ segments. . . . . . . . . . . . 102Figure 7.33 Arrangements of Trimer I lying just above the glide plane before and after passage ofthe dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 7.34 Arrangements of Trimer II lying on the glide plane before and after passage of the dis-location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 7.35 Four body interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 7.36 The case of Trimer I lying just above the glide plane. Comparison between the predic-tions obtained from the hybrid elastic model and the MS results. . . . . . . . . . . . . . 105Figure 7.37 The case of Trimer II lying just across the glide plane. Comparison between the predic-tions obtained from the hybrid elastic model and the MS results. . . . . . . . . . . . . . 105Figure 7.38 Distribution of maximum pinning force for all the different configurations of trimersdetermined using the hybrid elastic model. . . . . . . . . . . . . . . . . . . . . . . . . 106Figure 7.39 Configuration of the strongest trimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 8.1 Normalized CRSS of randomly distributed solute atoms when Nplanes = 24 as a functionof the number of obstacles in the areal glide model. . . . . . . . . . . . . . . . . . . . . 110Figure 8.2 CRSS of randomly distributed single solutes as a function of the number of planes aroundthe glide plane considered Nplanes, and for a Mg concentration of 1 at%. . . . . . . . . . 111Figure 8.3 CRSS of randomly distributed Dimer I as a function of the number of planes around theglide plane considered Nplanes, and for a Mg concentration of 1 at%. . . . . . . . . . . . 113xvFigure 8.4 Critical resolved shear stress of randomly distributed solute atoms, dimers and trimers,for a total Mg concentration of 1 at%. The blue dots correspond to the stress of distribu-tions featuring all configurations of clusters. Whereas red dots correspond to the stressof distributions featuring only the strongest configuration. . . . . . . . . . . . . . . . . 116Figure 8.5 Comparison between the scaling law used in the study of a clustered strengthened AA6111and the strength of random distributions of solute atoms, dimers, or trimers as obtainedfrom the areal glide simulations.The blue dots correspond to the strength of distributionsfeaturing all configurations of clusters. Whereas red dots correspond to the strength ofdistributions featuring only the strongest configuration. . . . . . . . . . . . . . . . . . . 117Figure 8.6 When atoms on either side of the glide plane are both bigger than an Al atome.g. bothMg (a), their individual interactions with the dislocation nearly balance each other out.When one is bigger, and the other smaller, e.g. Mg and Si, respectively (b), their indi-vidual interactions with the dislocation reinforce each other. The shades of red representcompressive stresses, and shades of blue tensile stresses. . . . . . . . . . . . . . . . . . 118Figure B.1 CRSS prediction at low obstacle strength. Two states of a dislocation are represented:one at zero stress (along the x-axis) and the second at the stress required to overcomeobstacle O . The dark grey shaded area is the swept area free of obstacles while the lightgrey shaded area contains at least one obstacle. The closest obstacle to O is thus foundbetween L and L+dL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134xviList of variablesSymbol DescriptionA areaAa area of one atom on the glide planeAswept area swept by the dislocationb magnitude of the Burgers’ vectorCi concentration of non-clustered solute atoms of type i (Mg, Si or Cu) in atomic percentCi j anisotropic elastic constantsCi jkl stiffness matrixc concentration of solutescA concentration of species AcB concentration of species Bctot concentration of all species (A and B typically)C disregistry vectorC DimerII disregistry vector when the obstacle is Dimer IId distance between glide plane and solute or clusterd{111} distance between two {111} consecutive planes~dMg−Mg distance vector between the two Mg solutes forming the dimerdi j distance from the ith node to its jth neighbourdp distance between partial dislocationsdSF stacking fault length∆Eb energy barrier for each soluteEbind elastic binding energyxvii∆EDimerI change of total energy when the obstacle is Dimer I∆EDimerII change of total energy when the obstacle is Dimer II∆EchemicalDimerII change of chemical energy when the obstacle is Dimer II∆Etrimer change of total energy when the obstacle is a trimerEdipole change of elastic energy due to the presence of the elastic dipoleE int interaction energyE intsolute1 interaction energy of solute 1E intsolute2 interaction energy of solute 2E intsolute3 interaction energy of solute 3E intDimer interaction energy of a dimerESF intrinsic stacking fault energyEslip energy due to the change of generalized-stacking-fault energy surfaceEtot total energy of the system∆Etot change of total energy of the system∆G activation energy∆Hmix enthalpy of mixing of species A and B∆F total free energy to overcome one obstacleFobs drag force exerted by an obstacle on the dislocationFmaxobs maximum pinning force of an obstacleFmaxdimeri maximum pinning force of Dimer i lying on the glide planeFPK Peach-Kohler forceFP drag force due to the resistance of the latticef2D density probability function of the distribution of cluster radius on the glide planef3D density probability function of the distribution of cluster radiusf clusterstot total volume faction of clustersfPK Peach-Kohler force per unit lengthfP drag force due to the resistance of the lattice per unit lengthk spring constantki proportionality coefficient between the concentration of randomly distributed solute atoms of type ixviii(Mg,Si or Cu) to the power 2/3 and their critical resolved shear stresskB Boltzman constantl distance between two points forming the envelope of a clusterl¯ average distance between the points forming the envelope of a clusterL spacing between two obstacles on the planeL1 length of a dislocation segment connection obstacles from two different clustersL2 length of a dislocation segment connection obstacles from the same clusterLsolutei average square spacing between non-clustered solute atoms of type i (Mg, Si or Cu)Ls average square spacing between obstacles on the planeLclusterss average square spacing between clusters on the glide planeLx length of molecular statics simulation box along the x directionLy length of molecular statics simulation box along the y directionLz length of molecular statics simulation box along the z directionM Taylor factorMi number of neighbouring obstacles of the ith nodeN number of obstacles per unit areaNc number of clustersNin number of atoms per clusterNl p number of lattice planesNplanes number of planes consideredNS number of nodesP aggregation levelP(X) probability that event X happensP(L) probability to find obstacle between L and L+dLP dipole momentPi j elastic dipolep a numberq exponent of the addition lawr radius of a clusterxix< r¯ > average cluster radiusra radius of an atomrc critical radius at which a cluster becomes non-shearablermin minimum possible radius for a clusterR dislocation segment radiusR∗ normalized dislocation segment radiusSi jkl compliance matrixSi j anisotropic compliances∆Smix entropy of mixing of species A and BT temperatureV volumeVa atomic volumeVcell volume of a unit FCC cellxd position of the dislocation along the x directionα internal angle of the envelope of a clusterα¯ average of the internal angles of the envelope of a clusterβ obstacle strengthβ sc transition strength between Region III and II of an aggregated distribution of obstaclesβwc transition strength between Region II and I of an aggregated distribution of obstaclesΓ dislocation line tensionγAl generalized-stacking-fault energy surface of pure aluminumγAl−dimerII generalized-stacking-fault energy surface of aluminum with Dimer IIγAl−Mg generalized-stacking-fault energy surface of aluminum with one Mg soluteγxy engineering shear strain on xy planesδ magnitude of the displacement of the end of the springsεi j strain tensorε⊥ strain tensor of a split edge dislocationε 1/2⊥i j strain tensor of a half edge dislocationε˙ strain ratexxε˙0 strain rate of referenceζ half-width of the dislocation coreθ supplementary angle of Φcµ shear modulusµ0K shear modulus at zero KelvinµCDF mean of the cumulative density functionν Poisson’s ratioΦc obstacle critical breaking angleΦclusterc cluster critical breaking angleΦ¯clustersc average cluster critical breaking angleΦsolutei breaking angle of non-clustered solute atoms of type i (Mg, Si or Cu)σ0 contribution of the grain size to the strengthσclusters contribution of the clusters to the strengthσ2CDF variance of the cumulative density functionσi j stress tensorσ⊥i j virial stress tensor due to the split edge dislocationσSS contribution of the non-clustered solute atoms to the strengthσSSi contribution of the non-clustered solute atoms of type i (Mg, Si or Cu) to the strengthσy yield stressτ applied resolved shear stressτ∗ normalized resolved shear stressτ∗c normalized critical resolved shear stressτ∗clusters normalized critical resolved shear stress of a distribution of clustersτ∗break normalized resolved shear stress required to break or shear an obstacleτ∗meet normalized resolved shear stress required to meet an obstacleτ∗Orowan normalized resolved shear stress required to by-pass an obstacleτ∗SS normalized critical resolved shear stress of a distribution of randomly distributed solutesτc critical resolved shear stressτ∗c,random normalized critical resolved shear stress of an array of randomly distributed obstaclesxxiτ∗c,square normalized critical resolved shear stress of a square array of obstaclesτp Peierls stressτSS critical resolved shear stress of a distribution of randomly distributed solutesτdimers critical resolved shear stress of a distribution of randomly distributed dimersτtrimers critical resolved shear stress of a distribution of randomly distributed trimersτxy shear stress resolved on xy planesχB ratio of the concentration of B and A speciesxxiiGlossaryAPT Atom Probe TomorgraphyCRSS Critical Rresolved Shear StressDDD Discrete Dislocation DynamicsDFT Density Functional TheoryDSC Differential Scanning CalorimetryEAM Embedded Atom MethodFCC Face-centered CubicGP zones Guinier-Preston ZonesHRTEM High Resolution Transition Electron MicroscopeMS Molecular StaticsMD Molecular DynamicsSANS Small Angle Neutron ScatteringSAXS Small Angle X-ray ScatteringSSSS SuperSaturated Solid SolutionTEM Transmission Electron MicroscopexxiiiAcknowledgmentsI would likeTo express my gratitude to my supervisors Warren Poole and Chad Sinclair for their passion, their ethic,and for allowing me to restart my Ph.D. anew in a respectful, warm, and supporting environment that helpedme thrive,To thank my supervisor Warren Poole for the numerous opportunities he gave me to present my work tointernational conferences,To thank my other supervisor Chad Sinclair for all the intense technical discussions we had, and for hisdrive for excellence,To thank my dear friend Peter for the wonderful music sessions, his friendship and his hospitality,To thank my friends David, Nori, Michael, Trinley, Sebastian, Alec, Manami, David, Marı´a Alejandra,Roland, Kyle for all these wonderful moments spent together, ainsi que Lucie et Alain,Remercier mes parents, mes fre`res et sœur pour leurs soutiens et leur foi en moi,Agradecer a mi novia Marı´a Isabel por su amor, su apoyo, y la motivacio´n que me llevo´.xxivChapter 1IntroductionThe consumption of fossil fuels is partly responsible for making our planet warmer as it leads to the increaseof the atmospheric concentration of carbon dioxide, one of the most important greenhouse gases directlyemitted by humans [1]. In North America, about a quarter of these emissions emanate from the transportationsector [1]. Motivated by the rate at which the climate changes, governments in North America and aroundthe world are setting new taxes and standards in order to reduce the use of fossil fuels and the emissionsof carbon dioxide. In order to meet these new standards, vehicles need not only to be more efficient, butalso lighter. In fact, for a 10% weight reduction, greenhouse gases emissions of internal combustion enginevehicles would decrease by 6-8%.Today, the structure and parts of the vast majority of cars and trucks produced in the world are made outof steel. One possibility to reduce the weight of these vehicles is to replace some of the steel by aluminumalloys. This option has been used in particular by the Ford Motor company for its 2015 F-150 truck (the mostsold vehicle in North America), resulting in a weight reduction of around 300 kg (700 pounds) or 10% [2].Ford is not the only car maker increasing the amount of aluminum alloy used in their products. The totaluse of aluminum in light vehicles in North America is expected to increase by 28% by the end of this year(2015) compared to 2012 [3].Increasing the amount of aluminum alloys used in vehicles cannot be achieved without better understand-ing the strengthening of aluminum alloys during both production and service. Typically, automotive gradealuminum alloys are received by the car manufacturers in a T4 temper, meaning that the alloy was solution1heat treated, and then naturally aged (i.e. a period of time when the alloy is kept at ambient temperature).Further natural aging also occurs during storage, before the material is formed. During natural aging, soluteclusters form [4]–[6] and cause the alloy’s strength to increase [5]. This increase of strength could negativelyaffect the forming response, or positively affect the final strength of the material. Eventually, the materialis heat treated at high temperature (150-180◦C) during the so-called paint bake cycle, which sets the alloy’sfinal strength. The final strength does not only depend on the high temperature heat treatment, but also on theprevious aging treatments. More specifically, as natural aging progresses, the alloy strength after the paintbaking cycle decreases [7] due to the presence of clusters. In order to improve this, alloys are pre-aged atan intermediate temperature (50-150◦C) after solution treatment. However, during the following inevitablenatural aging, clusters may still form [8], thus affecting the strength.Therefore, in engineering aluminum alloys, understanding the influence of solute clusters that form dur-ing natural aging is key to controling the strength of alloys both during their processing and service. Thestrength of these alloys is well known to be controlled by the interaction between dislocations and obstacles.When these obstacles are solute clusters, however, the primary mechanisms responsible for strengtheningare not yet well known. The study of this problem is the main focus of the present work.In order to study the origin of cluster strengthening in aluminum alloys, this thesis is structured asfollows. First, a literature review (Chapter 2) on the state of the knowledge of the microsctructure of clus-ter strengthened alloys and more specifically their strengthening mechanisms is presented. Following that,Chapter 3 presents the specific scope and objective of this thesis. Then, in Chapter 4 the multiscale sim-ulation tools used in this work are introduced. In Chapter 5, a case study focused on predicting the yieldstress of a cluster strengthened AA6111 alloy is presented. In this chapter, the change of yield stress of acluster strengthened alloy with aging conditions will be modelled under the assumption of a linear scalinglaw for the strength of solute clusters as a function of their size. This scaling law is an extension of the usualhypothesis that the strength of precipitates scales with their radius. The relatively good predictions obtainedfrom this case study motivates the rest of this thesis, whose aim is to provide a physical basis for this lawand cluster strengthening in general.As will be seen in Chapter 2, the critical resolved shear stress depends on the local interaction betweendislocations and obstacles as well as the statistical distribution of these obstacles. In Chapters 6 and 7, eachof these effects will be studied one by one, in the case of binary Al-Mg alloys. In Chapter 6, the effect of2the statistical distribution of solute atoms is investigated using an areal glide model developed as part of thisthesis. Chapter 7 focuses on the change of intrinsic strength of individual clusters as a function of their size.The interaction between a single dislocation and a single solute cluster will be analyzed using molecularstatics. The number of possible arrangements of solutes within a cluster increases exponentially with itssize. The aim of this chapter is to develop a quasi-analytical model that can be parametrized and validatedby a limited number of computational expensive atomistic simulations.Finally, in Chapter 8, the models developed in Chapter 6 and 7 are coupled and used to compare theincrease of strength due to the clustering of solid solutions into dimers, and then trimers in binary Al-Mgalloys. This is compared with the analytical scaling law proposed in the case of the AA6111 alloy whereit is shown that the correct trends emerge despite the increased chemical complexity of the AA6111 alloycompared to the binary Al-Mg alloys, for which the model was parametrized.3Chapter 2Literature reviewHigh purity aluminum exhibits a very low yield stress [9], less than 10 MPa, but this can be changed dras-tically by adding a small amount of alloying elements. The discovery by Alfred Wilm in 1906 of an aged-hardened Al-Cu alloy (Duralumin) was a turning point, allowing aluminum alloys to be strengthened suffi-ciently for them to be used for structural applications. Such alloys eventually became the material of choicefor the aircraft industry [10]. Wilm discovered that after quenching, an Al-Cu(-Mg-Mn) alloy would slowlyharden (naturally age) when left at room temperature [11]. He reported that Duralumin (close to the modernAA2024) had a hardness of 125 HB in “its natural stage” [12], compared to 15 HB for pure aluminum. Thisdramatic increase of hardness was later found to be due to the presence of nanometre-sized precipitates orsolute atom clusters within the alloy [4].In a variety of aluminum alloys, during the early stages of aging, solute atoms form clusters (or Guinier-Preston zones) [13]. For some of these alloys, the formation of these clusters has been linked to an increaseof strength [5], [12], [14]–[16]. The first section of this chapter presents how clusters (or Guinier-Prestonzones) are known to form, in the literature, and how their existence has been shown experimentally, inaluminum alloys aged at room temperature. This section also addresses the difference between clusters andGuinier-Preston zones (GP zones). In the second section, experimental evidence from the literature of clusterstrengthening in aluminum alloys, and the state of the knowledge of the origin of the strength of clusters arepresented. Throughout this chapter, the example of binary Al-Mg alloys is often referred to since it is thealloy focused on in this thesis.42.1 Decomposition of super-saturated solid solution in aluminum alloysaged at room temperature2.1.1 Solid solution decomposition sequenceAge hardening alloys are typically heat treated at a temperature high enough to dissolve the alloying additionsinto a random solid solution. The material is then quenched in order to create a supersaturated solid solution(SSSS) having a random distribution of solutes [4]. Quenching is often followed by natural aging (i.e.aging at room temperature) and/or artificial aging (i.e. aging at higher temperatures). During aging thesupersaturated solid solution will decompose. This decomposition often follows a sequence such as [4]:SSSS→ Clusters→ Guinier-Preston zones→Metastable Precipitates→ Equilibrium PrecipitatesThis sequence is observed in particular in aluminum alloys such as 2xxx, 6xxx and 7xxx [6], [17], [18]. In theparticular case of binary Al-Mg alloys of interest in this thesis, the precipitation sequence is as follows [19]:SSSS→ Clusters→ Guinier-Preston zones→ β ′′→ β ′→ β ,where β ′′ (also called ordered Guinier-Preston zone) is a phase with a L12 crystal structure (see Figure2.1) of composition Al3Mg [16], β ′ is a semi-coherent hexagonal phase, and β is the equilibrium phase ofcomposition Al3Mg2 which has a complex face-centered cubic (FCC) structure [20], [21].Figure 2.1: Schematic of a L12 crystal structure. Blue and green spheres represent Mg and Al atoms,respectively.2.1.2 Guinier-Preston zones and clustersThe first step of the decomposition process is the formation of clusters, or Guinier-Preston zones (GP zones).The term ‘GP zones’ was historically applied to coherent aggregates of Cu atoms that were first indepen-5dently detected in Al-Cu alloys using X-Ray scattering techniques by Guinier [22] and Preston [23] in 1938.Decades later, transmission electron microscope (TEM) analyses have shown that GP zones are one atomiclayer thick aggregates rich in Cu [24]. Figure 2.2 shows an example of a TEM image of such monoatomiclayers. Guinier and Preston postulated that the zones had a structure coherent with that of the matrix. Thiswas later proved by transmission electron microscope (TEM) [24] and high resolution transition electronmicroscope (HRTEM) [25] work. Later, the term ‘GP zones’ was extended to also describe pre-precipitationphases in other systems such as 6000 and 7000 series aluminum alloys [26]–[29]. In such systems, GP zonesare usually spherical [30], [31]. An example of spherical clusters in Al-Mg-Si alloys observed by TEM andHRTEM is given in Figure 2.3 [30].Historically, the term ‘clusters’, or ‘clustering’, was used to describe the phenomenon responsible for anincrease followed by a decrease of resistivity [13], [33] measured in a number of alloys (for example Al-Zn)during aging. It was argued that this phenomenon was due to the clustering of solute atoms and quenched-in vacancies. The term ‘clusters’ is usually used to describe regions of the matrix richer in solute thanthe nominal composition which form aggregates that contain of the order of 100 atoms or less [34]. Untilthe development and use of atom probe tomorgraphy (APT), these aggregates were usually referred to asGuinier-Preston zones when they were large enough to be detected using X-Ray scattering techniques [13],[34], [35]. GP zones were found to have the same effect on resistivity as clusters, but due to limitationsin resolution they were only detected at later aging times [36], [37]. With the use of APT, the distinctionbetween clusters and GP zones blurred as it was possible to resolve smaller groups of solutes [28], [38].In the literature, the term clusters is often used without being precisely defined. In this work, clusters areFigure 2.2: TEM image of clusters in an Al-Cu alloy, from Eto et al. [32]. The thindark lines are the one-atom thick Cuclusters.000200AlFigure 2.3: (a) TEM and (b) HRTEM imageof clusters in an Al-Mg-Si alloy takenfrom [30]. The arrows point at a fewselected clusters.6defined following the suggested definition of Harkness and Hren [39]: “a cluster is an aggregate, or group,of continuously connected solute atoms that have the same structure as that of the matrix”. In other words,a cluster is a set of solute atoms that are all a closest neighbour of another solute from the same cluster [31].Clearly, the smallest possible clusters are dimers (two neighbouring solute atoms). Even dimers induce anincrease of resistivity as shown by Be´al and Friedel [40].2.1.3 Thermodynamics of clusteringIn the absence of strain fields due to defects (such as dislocations) and at constant temperature, clusteringoccurs as a means to decrease the total free energy of the alloy. This could be analyzed analytically in thecase of a binary alloy made of two mixed species called A and B. If the ratio of the concentration of B andA species is χB (χB = cB/ctot), the system would cluster if the resulting change of free energy of mixing isnegative, thus when [41]:∂∆Hmix∂χB>−TkB ln(χB)1−χB , (2.1)where ∆Hmix is the enthalpy of mixing, T the temperature and kB Boltzman constant. Since 0 ≤ χB ≤ 1,ln(χB) is always negative or zero. Therefore, −TkB ln(χB)1−χB is positive, whatever the concentration χB. Asa result, if ∆Hmix is negative, the system tends to favour randomly distributed solid solution. When theenthalpy of mixing is positive, however, the system could cluster when the concentration of alloying atomsin solid solution is more than χmin (see Figure 2.4) and this depends on the alloy, the type of impurity andthe temperature. In the case of Al-Mg alloys, which are of particular interest in this thesis, CALPHADdata reveal that the enthalpy of mixing is positive for all concentrations of Mg [42]. Therefore, from athermodynamic point of view, the condition is met for the decomposition of SSSS into clusters to occur inbinary Al-Mg alloys.Aged hardened alloys at room temperature are not in their thermodynamic equilibrium state when thesolute atoms are all in supersaturated solid solution. Thus, there is a driving force for the precipitation ofthe equilibrium precipitate [41]. Though, the precipitation of the SSSS into the equilibrium precipitate isnot spontaneous as there exists an energy barrier to the nucleation of this precipitate [41]. This precipitationcould however happen through a nucleation mechanism [41]. But the decomposition of SSSS into clustersis more thermodynamically favourable since the associated energy barrier to their formation is lower owingto their lower interfacial energy [41].7-2-1.5-1-0.500.511.520 0.2 0.4 0.6 0.8 1χB = cB/ctotχmin−TpkB ln(χB)1−χB∂∆Hmix∂χBFigure 2.4: Difference between the derivative of the change of free entropy and the derivative of thechange of free enthalpy. When ∂∆Hmix∂χB >−TkBln(χB)1−χB ⇔ χB > χmin, clustering should occurs.It is believed that quenched-in vacancies play a crucial role in the nucleation process of clusters andGP zones [43], [44]. In some alloys, vacancies appear to help nucleation by lowering the misfit strain becauseof solutes bigger than Al [43], e.g. Mg or Zn. The binding energy between solutes and vacancies is alsobelieved to play a role in the nucleation of clusters [43].2.1.4 Experimental evidence of clustering in naturally aged aluminum alloys, and Al-Mgalloys in particular2.1.4.1 Electrical resistivity measurementsThe electrical resistivity of metallic alloys depends on the number and the distribution of point defects,dislocations, solute atoms, and vacancies [13]. In alloys, due to the low concentration of vacancies withrespect to that of solute atoms, the change of resistivity caused by the clustering of vacancies is negligiblecompared to the change resulting from the decomposition of the SSSS [13]. Therefore, changes in electricalresistivity of an alloy over time can be directly related to the formation of solute clusters [13].As the super-saturated solid solution decomposes and starts forming clusters, the electrical resistivity ofthe alloy increases, and then decreases [13], [33], [45], as shown in Figure 2.5. The increase of resistivitycomes from a higher scattering of electrons due to the non-random character of the distribution of soluteatoms [45]. The origin of the following decrease of resistivity, also owing to the continuing growth of clus-ters, is still being debated as two theories co-exist [45]. The first theory argues that the resistivity decreases8as a result of the typical cluster radius becoming large compared with the electron mean free path [46]. Theother theory claims that the decrease of the number of solutes between clusters means that fewer electronsinteract with obstacles [36]. Nonetheless, this observed decrease of resistivity is caused by the increase ofclustering of solute atoms [45].The change of resistivity associated with the decomposition of a solid solution into clusters at room tem-perature has been observed in Al-Mg binary alloys by Sato et al. [16], but also by Osamura and Ogura [19] .Similar resistivity changes have also been observed at room temperature in other binary alloys such as Al-Cualloys [13], Al-Zn alloys [29], and Al-Ag alloys [47]. In the case of binary Al-Si alloys, on the other hand,no resistivity change has been observed [13].The change in electrical resistivity has been used to follow the kinetics of cluster formation in a numberof Al alloys [13]. In Al-Mg alloys, even at high concentrations of Mg, the formation rate of clusters andthe change of resistivity with aging time are low compared to other alloys such as Al-Zn [16], [29]. Indeed,when aged at room temperature, the peak of resistivity of an Al-10%Zn alloy was found by Panseri andFederighi [29] to occur after 6 min, after an increase of resistivity of 250 nΩcm. In the case of an Al-10%Mg allow, however, Sato et al. [16] found the peak of resistivity to occur after ≈ 1000 min at 20◦C,(see Figure 2.5), and the change of resistivity at the peak was only 50 nΩcm. In contrast, Osamura andOgura [19] observed a continuous increase of resistivity in an Al-10.8%Mg alloy up to ≈ 6000 min duringaging at room temperature. This difference can be explained by the different solution heat treatment timeswhich could affect the ‘randomness’ of the SSSS of the ‘as-quenched’ alloys. While both samples weresolution heat treated at 430◦C, the samples of Sato et. al were treated for 1 hour [16], while those ofFigure 2.5: Change of electrical resistivity of an Al-10%Mg alloy aged at 20◦C (from Sato et al. [16]).9Osamura and Ogura were treated for 30 minutes [19]. Osamura and Ogura also measured the change ofresistivity of an Al-14.8%Mg alloy, aged at room temperature, and found the peak of resistivity to occuraround 1 hour [16]. However, in the case of an Al-Zn alloy of same solute concentration (Al-10%Zn), theresistivity peak was found by Panseri and Federighi to happen at between 2 and 3 minutes [29], also at roomtemperature. Therefore, Al-Mg alloys experience clustering, but the decomposition of SSSS into clusters isslow.The decomposition of the SSSS of Al-Mg alloys, also depends on the solute concentration. Sato etal. [16] also measured the change of resistivity of an Al-5%Mg alloy aged at room temperature. Over aperiod of 7 days, the resistivity increased slowly before plateauing [16]. For the same alloy, Osamura andOgura [19] reported the resistivity change over a longer period of time, i.e. 70 days. During the first week,they observed the same behaviour as Sato and coworkers. However, after a week, Osamura and Oguraobserved that the resistivity started to increase again [19]. Nonetheless, no peak of resistivity was observedover the observation time [19]. These observations would suggest that clustering happens in Al-5%Mgalloys, however, the kinetic of decomposition of the SSSS is extremely slow.As we have seen, for binary Al-Mg alloys, the formation of clusters is slow, and clusters do not form inbinary Al-Si alloys [13]. However, when both Mg and Si are used as alloying elements, i.e. in Al-Mg-Sialloys, the formation of clusters is faster than in Al-Mg alloys. In fact, Panseri and Federighi [48] measuredthe resistivity of an Al-2.8%Mg-1.4%Si aged at room temperature, and observed an increase of 100 nΩ.cmin resistivity after 100 minutes of natural aging. In contrast, Sato et al. [16] observed an increase of only40 nΩ.cm after the same aging in an Al-10%Mg alloy.2.1.4.2 Small angle scattering measurementsSmall angle X-ray scattering (SAXS) is a technique that uses the scattering of X-rays over small anglesaway from the incident X-ray beam. The resulting pattern is influenced by the presence of clusters andprecipitates [49], whose size, density, and shape can be determined from the variation of intensity profilewith scattering angle [49]. The size of the smallest detectable GP zones with this method is of the orderof 1 nm [39]. The decomposition of SSSSs has been intensively studied using this technique in naturallyaged Al-Cu alloys [13], but also in other systems such as Al-Zn alloys [13], [50], [51], Al-Ag alloys [51],Al-Zn-Mg alloys [49].10Al-Mg alloys are difficult to study using SAXS since the atomic scattering factors of aluminum andmagnesium are close [13]. In this case, small angle neutron scattering (SANS) has been used [52] sincethe scattering of neutrons does not systematically vary with atomic number [53]. SANS was used by Rothand Raynal [52] to study clusters of Al-7%Mg and Al-11.5%Mg alloys aged at room temperature for oneyear. From their observations, they concluded that GP zones exist in the Al-11.5%Mg alloy. However, thepresence of GP zones in the Al-7%Mg alloy could not be proven [52].2.1.4.3 Differential scanning calorimetry measurementsDifferential scanning calorimetry (DSC) is a technique that monitors the difference of heat flux requiredto increase the temperature of a sample, compared to a reference, as a function of the temperature. Thistechnique is used to monitor phase transitions, in particular in metals. In aged aluminum alloys, the differentexothermic peaks seen in the DSC results correspond to the formation of phases, and endothermic peakscorrespond to the dissolution of phases [54].Typically, the dissolution of clusters in Al-Mg alloys corresponds to the first endothermic peak whichoccurs at temperatures between 40 and 110◦C [19], [20], [54] for samples that have been naturally agedas shown in Figure 2.6. This peak has been identified as being induced by the dissolution of clusters aftercomparing the DSC results with other alloys for which the precipitation sequence is well known [54], andconfirmed by direct observation with TEM [16].Figure 2.6: DSC curves of samples of an Al-16Mg alloy naturally aged during 2 years obtained atheating rates of 1.2, 2.5, 5, 10, 20, 40 and 80◦C/min.(from Starink and Zahra [54])112.1.4.4 Direct observation techniques: TEM, and Atom Probe TomographyDirect observation of cluster formation is difficult because of their small size. With TEM, clusters smallerthan about 100 atoms can be observed but can be difficult to analyze if the atomic number difference betweenmatrix and solute is low, as in the case of Al, Mg, and Si [17], [30], [55]. It is thus often difficult toexperimentally determine the size, morphology and number density of clusters in aluminum alloys fromTEM observation [30].Sato et al. [16] reported TEM observations of the microstructure in Al-5%Mg alloys and Al-10%Mgalloys both aged at room temperature. In the case of the Al-5%Mg alloy, they did not observe any decom-position of the solid solution over a period of 816 hours [16]. However, in the case of the Al-10%Mg alloy,at the beginning of aging, they observed a fluctuation in concentration that was attributed to the formation ofclusters [16].Recent advances in 3D APT have started to improve our ability to experimentally observe and studysmall atomic scale clusters. 3D atom-probe tomography uses an electric field applied between the tip of thespecimen and the detector [56]. Under the forces due to the electric field or a laser, atoms are evaporatedfrom the specimen and collected on a screen [56]. The type of atom is determined by its kinetic energywhen it hits the detector, and its 3D position in the sample is determined from the location of its impactwith the detector [56]. 3D APT allows the position and type of atoms to be identified within a small volume(∼ 105 nm3) [17]. This technique has mostly been used to study the decomposition of SSSS at early stagesof aging of commercial alloys such as Al-Mg-Si(-Cu) alloys [6], [57] and Al-Zn-Mg(-Cu) alloys [58]. Fromsuch data, 3D maps of solute atoms can be reconstructed [59]. The technique suffers from a detectionefficiency that is only approximately 60% [57], [59]. These maps provide evidence that during natural aging,the distribution of solute atoms in certain aluminum alloys is not totally random anymore as atom densitycan be found to be higher than the bulk density, and thus that clusters form [6], [57]. Quantification ofthese clusters from atom probe data sets can be problematic as well since the type of undetected atomsis not known. Nevertheless, many cluster finding algorithms exist that are all based on the principle thatatoms within a cluster are at a certain distance from each other [60]. Such a definition does not guaranteethat clusters found with these algorithms fulfill the previously stated unambiguous definition of clusters (seeSection 2.1.2) as the maximum allowed distance between atoms of a same cluster is an adjustable parameter.Typical values used in the literature for this distance vary between 0.5 to 0.8 nm [60]. Useful data can,12however, be gathered from atom probe data such as the ratio of species in clusters, their approximate size, andeven their spatial distribution [57]. The data can also be used to analyze short range ordering (randomness ata small scale) of solid solutions [8].2.1.5 Origin of the rate of cluster formationThe rate of cluster formation has been argued to be dictated by the concentration of vacancies that controlthe rate of transport of solute atoms from the matrix to the clusters [61]. This hypothesis was tested byadding low concentrations of large radius alloying elements that trap vacancies, for example Sn in Al-Cualloys [62]. It was found experimentally that the growth rate of Cu rich clusters in Al-Cu-Sn alloys was muchslower than in Al-Cu alloys [62]. This result was also confirmed by computer simulations by Hirosawa andcoworkers [63]. Also, Zurob and Seyedrezai [64] suggested that the growth of clusters would be limited bythe rate at which trapped vacancies are released. They predicted that the number of solute atoms per clusterwould increase proportionally with the logarithm of time [64].2.2 Strengthening mechanisms associated with cluster strengthenedaluminum alloys2.2.1 Link between clustering and strengtheningMany experimental studies have linked the presence of clusters to an increase of strength in Al-Cu [12],Al-Mg-Si [5], Al-Mg-Zn [14], and Al-Cu-Mg alloys [15]. In the case of Al-Mg alloys, Sato et al. [16]monitored the change of hardness and the change of resistivity (see section 2.1.4) of an Al-5%Mg alloy andan Al-10%Mg alloy aged at room temperature. While the hardness of the Al-5%Mg alloy remained constantover the duration of the experiment, i.e. 125 days, the hardness of the Al-10%Mg alloy was equal to thehardness of the sample right after quenching until the peak of resistivity was achieved after 1000 hours [16].Then, as the resistivity decreased, the hardness of the sample increased. At the end of the experiment, Satoet al. observed an increase of just 8% of the hardness of the Al-10%Mg alloy [16].Surprisingly, not all alloys featuring solute clustering exhibit strengthening with natural aging, for ex-ample binary Al-Zn alloys [65]. These alloys cluster when left at room temperature as shown by smallangle neutron scattering and electrical resistivity [66], [67], however, there is no accompanying increase in13hardness [65]. In fact, a decrease of hardness is seen at long aging times [65].Chemical composition seems to play an important role in cluster strengthening. A good example of asystem with strong cluster strengthenin is Al-Mg-Si, which, as shown in Figure 2.7a, strengthens rapidly atroom temperature. Indeed, Esmaeili and coworkers showed that the strength of an AA6111 nearly doublesafter 24h of natural aging and almost triples after about 2 weeks of natural aging [5]. Compared to theAA6111, the strengthening of Al-Mg binary alloys is slow. Sato and coworkers reported the yield stress ofan Al-10%Mg alloy aged at room temperature for up to 13 years [16] (see Figure 2.7b). They showed, thatthe yield stress of the alloy increased by only 10% after 3 months and finally plateaued at a stress twice ashigh as the as-quenched stress, but only after more than 8.5 years [16]. This could be caused by the slowformation of clusters in binary Al-Mg, and/or the low strengthening effect of Mg clusters. On the other hand,binary Al-Si alloys do not seem to experience any clustering during natural aging as reported by Kelly andNicholson [13]. This raises the question as to the origin of cluster strengthening and the large strengtheningdifference exhibited by naturally aged Al alloys, for example Al-Zn and Al-Zn-Mg, or Al-Mg, Al-Si andAl-Mg-Si.050100150200AQ 24h3 days2 weeksYieldstress(MPa)Ageing time(a)1001502002503003501 10 100Yieldstress(MPa)Ageing time (months)(b)Figure 2.7: Strengthening evolution during natural aging of (a) an AA6111 alloy [5] and (b) a binaryAl-10%Mg alloy [16]2.2.2 Physics of solute solution and cluster strengtheningThe yield stress of solution treated aluminum alloys containing solute clusters derives from the superpositionof the stress required to move dislocations through the perfect crystal, and to overcome obstacles, i.e soluteatoms and clusters.142.2.2.1 Peierls stressIn FCC metals, owing to the lack of directional bonding, the minimum stress required to move a dislocationat 0 K in the perfect crystal is low [68]. In aluminum, this stress known as ‘Peierls stress’ is of the order of afew MPa. Using molecular statics (MS), Patinet [69] found the Peierls stress to be 2 MPa, while Olmsted andcoworkers [70] found it to be in a range of approximately 1.5 to 2.5 MPa for an edge dislocation. In the caseof a screw dislocation researchers have reported, using empirical embedded atom method (EAM) potentials,the Peierls stress to be between 1 MPa [71] and 88 MPa [72], [73]. The majority of researchers have foundthe Peierls stress of a screw dislocation to be around 20 MPa. Patinet [69], Shenoy and Phillips [74], andOlmsted and coworkers [70] found with values of 20, 22, and 16 to 18 MPa, respectively, using molecularstatics methods, but different boundary conditions. Indeed, Patinet [69] used periodic boundary conditionsin all directions, while Shenoy and Phillips [74] and Olmsted et al. [70] used fixed displacement conditionsoutside a cylinder of radius 70 A˚, and 50 or 70 or 90 A˚, respectively. The difference between the values foundby Shenoy and Phillips [74], and Olmsted and coworkers [70] can be explained by the fact that they useddifferent interatomic potentials. These researchers constantly found the Peierls stress of a screw dislocationto be higher than that of an edge dislocation. This difference was argued to come from the differenceof distance between two consecutive rows of atoms perpendicular to the gliding direction of each type ofdislocation [69], [70]: b/2 for an edge dislocation, and√3b/2 for a screw dislocation [69].2.2.2.2 Obstacles to dislocation motionAt the scale of individual obstacles, the athermal (i.e. 0 K) interaction between dislocations and obstaclesis classically described by the phenomenological approach described by Kocks et al. [75]. Each obstacleis characterized by the force, Fobs, required to push the dislocation past the obstacle (see Figure 2.8). Forexample, assuming that the obstacle is repulsive, when the dislocation is far away from the obstacle, nointeraction exists, i.e. Fobs = 0. As the dislocation approaches the obstacle the force increases, owing tothe repulsion between dislocation and obstacle. In this case, the maximum force is reached just before thedislocation reaches the centre of the obstacle. The force drops to zero when the dislocation is right in thecentre where it is in a metastable equilibrium. When the dislocation has passed the obstacle, the stresschanges sign as the dislocation is pushed away by the obstacle until it is out of the interaction range. Whenthe obstacle is attractive, however, the sign of the stress changes as shown in Figure 2.8.15aFobsτbL∆GFmaxobsRepulsive obstacleAttractive obstacleFigure 2.8: Schematic of one possible dislocation/obstacle interaction force (Fobs)/distance (a) curvewhich illustrates the activation energy ∆G (from Kocks et al. [75]). τbL is the externally appliedforce on a dislocation segment of length L.At T 6= 0 K, the dislocation can overcome the obstacle if the applied force τbL (where τ is the externallyapplied resolved shear stress and L the length of a dislocation segment) is lower than the maximum pinningforce Fmaxobs by means of ‘thermal activation’ [75]. The thermal energy required by the system to overcomethe obstacle corresponds to the Gibbs free energy of the obstacle ∆G (shaded area in Figure 2.8). Thermalactivation being a thermal process [75], the strain rate as a function of temperature can be written usingtransition state theory [75] as [76]:ε˙ = ε˙0exp(− ∆GkBT), (2.2)where ε˙0 is a constant which acts as a scaling parameter for the strain rate and is the hypothetical strain rateat zero temperature. The free energy can be approximated following Kocks et al. [75] by:∆G= ∆F[1−(ττˆ)p]q, (2.3)where ∆F is the total free energy to overcome one obstacle or in other words ∆F = ∆G when τbL = 0, τˆ =Fmaxobs /bL is the flow stress to cut the obstacle at 0 K, and p and q are two parameters describing the shape ofthe dislocation/obstacle force-distance profile with 0 < p≤ 1 and 1≤ q≤ 2 [75]. If the dislocation/obstacleinteraction profile is as shown in Figure 2.8, for example, p = 1 and q = 3/2. Serizawa et al. [76] solvedEquations (2.2) and (2.3) for τ and obtained a relationship between the contribution to the strength of clusters16at finite temperature, and their contribution at 0 K:σclusters(TP) = σclusters(0 K)µµOK[1−(kBT∆Fln(ε˙0ε˙))1/q]1/p, (2.4)where µ and µ0K are the shear modulus of aluminum at room temperature, and 0 K, respectively.The phenomenological description of the obstacle-dislocation interaction proposed by Kocks et. al [75]represents a simple and useful approach to examine this problem, but it does not explicitly treat the detailsof the interaction at the atomic level. Outside the core of a dislocation, the interaction is well describedby classical elasticity [77]. It is however not valid inside the core of the dislocation as the solution for thestrain field of a dislocation given by classical elasticity diverges at the center of the dislocation [77]. Thetotal dislocation/obstacle profile interaction profile, as well as τˆ and ∆F can be obtained using atomisticsimulations, as it will be shown shortly.2.2.2.3 Statistical effect of distributions of obstacles to dislocation motionAt the next larger length scale, dislocations bend and interact with multiple obstacles simultaneously, thisbeing equally important for the prediction of stength. Estimating the strengthening from the different con-tributions is non-trivial [78]. At this scale, obstacles can be characterized by the maximum pinning forceFmaxobs (see Figure 2.8), rather than the entire force-displacement profile. This force can either be calculatedfrom experimental results on binary alloys such as those of Sanders et al. (see Figure 2.9) [4], using a setof models accounting for the statistical behaviour of the sample, or directly calculated from force-distancecurves presented above. As the applied resolved shear stress increases, the dislocation will bend betweenthe pinning obstacles. It will eventually break away from an obstacle when its critical breaking angle Φc isreached (see Figure 2.10). The critical breaking angle is a function of the maximum pinning force of theobstacle and is related to it as [79],Fmaxobs2Γ= cosΦc2(2.5)where Γ is the dislocation line tension.The effect of the concentration of obstacles that are randomly distributed in the glide plane has beenmodelled using either of two theories [80]. In the strong-pinning theory proposed by Friedel [77], the short-range interaction between dislocations is considered as the principal strengthening mechanism. The critical17Figure 2.9: Experimental strength of solid solutions in high-purity binary aluminum alloys from [4]ΦcTTfFigure 2.10: Schematic representation of a dislocation under stress bending between obstacles.rresolved shear stress (CRSS), τc, then varies as c1/2. In the statistical theory proposed by Labusch [81],the long-range interaction is considered and τc is proportional to c2/3. It has been suggested that Friedelstatistics should apply for dilute solid solutions whereas for concentrated solutions, Labusch’s model wouldbe more appropriate [82], [83]. Overall both models give reasonable estimates of the CRSS due to solidsolutions (i.e. the experimental data typically cannot distinguish between c1/2 and c2/3), however their rangeof validity is still being discussed today [75], [80]. It is worth noting that both theories approximate the totalstrengthening effect to only the strengthening induced by solutes in the glide plane of gliding dislocations,the influence of out of plane solutes being presumed to be negligible.2.2.2.4 Atomistic simulations for studying the interaction between dislocations and obstaclesRecent progress in computational physics has now provided atomistic tools that can resolve the core ofdislocations and compute the dislocation glide resistance due to a single obstacle as, for example, shown18in Figure 2.11. It can be seen that the classical phenomenological glide resistance description is in goodagreement with the atomistic simulations when the obstacle is a repulsive single solute in the glide plane‘Mg above’), noting that there are now two peaks: one for each dislocation. In the case of an attractivesolute (‘Mg below’) there is divergence as the interaction force presents now four peaks instead of two, asexpected, caused by the interaction of the solute atom with the stacking fault [84]. Examples of numericaltools for the simulation of materials at the atomistic scale include density functional theory (DFT), molecularstatics (MS), and molecular dynamics (MD). DFT is a method that determines the total energy of a systemof atoms from quantum mechanical models. High precision comes at the expense of computation timemeaning that this method is reserved for systems containing < 103 atoms. Dislocation cores being a fewBurgers vectors wide [72], DFT can thus be used to calculate the energy of the core of dislocations [85], andtheir interaction energy with obstacles like single solutes, but only if careful attention is paid to correctlycapture the long range strain field of the dislocation with the help of other techniques [86]. Leyson andcoworkers [86], [87] and Yasi and coworkers [88] used DFT to determine the interaction energies betweendislocations and different kinds of solute atoms, in Al and Mg alloys. From these DFT results, the stressrequired to bypass a single solute atom were determined. Then, to determine the CRSS of a distribution ofthese solute atoms, i.e. a solid solution, a statistical model needs to be used. The classic statistical modelsused in the literature are the model of Labusch [81], and that of Friedel and Fleischer [79]. The detailsof these two models are presented later in this section. Leyson and coworkers determined the CRSS ofsolid solutions using Labusch’s model [86] whereas Yasi and coworkers used that of Friedel [88]. AlthoughFigure 2.11: Example of interaction force between two partial edge dislocations in Al and a substitutionMg atom calculated by Patinet and Proville using molecular statics simulations [84].19Leyson et. al and Yasi et. al used different statistical theories to determine the CRSS, they used the sametechnique to calculate the interaction energies between dislocations and solutes. This technique starts withthe creation of a dislocation in a lattice free of solute. Then one of the atoms is replaced by a solute atom [86],[88], thus imposing the relative position between the solute and the dislocation to take only discrete valuesseparated by the magnitude of a Burgers’ vector. The fact that the distance between the dislocation andthe solute cannot be continuously varied could have a potentially non-negligible influence on the resultinginteraction energy profiles.This problem can be resolved using MS or MD simulations, at the expense of the accuracy of certainpredicted properties. Indeed, in MS simulations, the empirical interaction between atoms is approximated byinter-atomic potentials and the equilibrium position at 0 K of the system is found when the potential energy,calculated by summing the interatomic potential energy of the atoms minus the work of forces appliedexternaly to the system, is minimized [89]. Like MS, MD uses empirical inter-atomic potentials, however, itis used to study the time evolution of systems at finite temperature [89]. In MD, the trajectory of each atomis computed using Newton’s laws [89]. Thanks to the use of simpler equations compared to DFT, MS andMD can be used to simulate systems up to ∼ 106 atoms [69]. The accuracy of these simulations is limitedby the potentials as they are only an approximation of the interaction energy between atoms.Using MS or MD simulations, there are two ways to determine the CRSS of a solid solution:1. Simulate the interaction of a dislocation with a single solute to determine its strength, and then use astatistical model to determine the CRSS of a random distribution of solutes, as in the case of DFT.2. Simulate directly the interaction with a dislocation with a random distribution of solute atoms in thewhole simulation box that has to be large enough to capture the effects of the solute distribution andthe long range elastic field.Patinet and Proville used molecular statistics to determine the glide resistance due to a single soluteatom, in binary Al-Mg [69], [84], and Ni-Al alloys [69]. The method they used forces the solute atom tobe fixed and the dislocation to glide and interact with it by changing an imposed macroscopic shear stress.Using this technique, the entire interaction energy curve can be determined. The result for a split edgedislocation interacting with a Mg atom in aluminum as shown in Figure 2.11. It can be seen that the classicalphenomenological glide resistance description is in good agreement with the simulations when the obstacle20is a repulsive single solute in the glide plane (‘Mg above’ in Figure 2.11), noting that there are now twopeaks: one for each partial dislocation. In the case of an attractive solute (‘Mg below’ in Figure 2.11) thereis divergence as the interaction force presents four peaks instead of two. Patinet and Proville also studiedthe interaction between dislocations and different configurations of dimers located just above or across theglide plane. While single solutes have one degree of freedom (owing to the symmetry of the dislocation-solute interaction), i.e. their distance to the glide plane, dimers have three; namely the distance betweenthe pair of solutes and the glide plane, the distance between the solutes and their orientation with respect tothe gliding dislocation. For each of the configurations identified, they reported the maximum pinning force,however force-displacement curves were not extracted from the simulations [84]. Their simulations showthat the strength of dimers significantly depends on their orientation with respect to the gliding dislocation.The strength of the strongest configuration was reported to be 70% stronger than that of a single solute [84].However, some configurations were found to be similar in strength to a single solute, and some others evenweaker [84]. This result raises the question of what the strengthening effect of a distribution of dimers is.Would the strength of the strong dimers overcome the weakness of the other dimers?Most work on strengthening induced by obstacles to dislocation motion, such as solid solution atomsand clusters of solute atoms, focus on their influence when the obstacle is located just above, just below oracross the glide plane. It is thus assumed that the influence of out-of-glide plane obstacles have only a weakcontribution to strengthening. As the stress field of a dislocation decays (i.e. as the inverse of the distance toits core [77]), the elastic interaction can still be high even when the obstacles are a few lattice spacings awayfrom the glide plane as shown by Patinet [69]. On the other hand, when obstacles are randomly distributed inthe material, the number of out-of-glide plane obstacles is substantially higher that the number of obstaclesin the glide plane. Patinet looked at the relative influence of randomly distributed solute atoms out-of-glideplane, and showed that they contribute for only 20% of the total strength of a random distribution of soluteatoms [69]. Therefore, one could argue about whether or not it is reasonnable to neglect the contributionof solute atoms out-of-glide plane. This also raises the question of how this result changes when obstaclesbecome clusters such as dimers or trimers.Thanks to the improvement in the computational power of supercomputers, researchers have recentlytried to directly simulate the strength of solid solutions using MS or MD simulations [69], [89], [90]. Anumber of studies have been published in the case of Al-Mg alloys [89]. Patinet [69] and Olmsted et al. [90]21used MS simulations to determine the strength of a random solid solution at 0 K in the case of both Al-Mg alloys and Ni-Al alloys, and only Al-Mg alloys, respectively. For the case of Al-Mg alloys, Patinet andOlmsted et al. used Liu’s EAM potential [91]. In this case, Patinet found a CRSS of 56, 148, and 254 MPa forsolute concentrations of 2, 6, and 10 at%, respectively [69]. Olmsted et al. however simulated concentrationsbetween 1% and 8% and found lower values: ≈ 50 and ≈ 80 MPa for 2 and 6 at%, respectively [90]. Onemust be careful with these results as they depend on the length of the simulated dislocation and the numberof solutes present in the simulation box [90], [92]. For example, Patinet used a box size of 149× 11.5×4.6 nm3 [69], and thus, the number of obstacles to the dislocation motion in his box were approximately1,000, 3,000, and 10,000 for the respective concentrations. Nogaret and Rodney [92] showed using linetension simulations that one needs at least 10,000 obstacles of the same kind in order for the CRSS toconverge. Based on this, the number of obstacles simulated by Patinet may have been too low to captureaccurately the statistics of the problem.To solve this problem, Olmsted and coworkers used a semi-analytic model to predict the CRSS of arandom solid solution [90]. First, using MS, they calculated the interaction energy between a edge dislocationand a Mg solute as a function of the site occupied by the solute atom. The dislocation was kept from movingby freezing the position of the atoms in the plane perpendicular to the dislocation line and containing thesolute. This allows for the distance between the dislocation and solute to be fixed to a pre-determinedvalue while the dislocation bends in order to minimize the energy of the system. This was argued to allowthe determination of the interaction energy when a gliding dislocation is at the same distance from thesolute [90]. In the model, they assumed the dislocation line to be perfectly straight and summed the differentinteraction energies due to the presence of randomly distributed solutes. The position of the dislocation wasvaried to obtain the difference of total energy as a function of its position, and finally the CRSS was derivedfrom it. Using this model, Olmsted and coworkers found a value of 57 MPa for the CRSS of a 2 at% randomsolid solution [90].2.2.2.5 Areal glide modelAreal glide models (also called line tension models [92]) are simple 2D models that allow for the statisticaltreatment of the interaction between a dislocation and many obstacles. These models assume the line tensionof the dislocation as constant and the obstacles as points. Each obstacle is characterized by a breaking angle22Φc function of its strength β such that [92]:β = cos(Φc2). (2.6)Recently, these dislocation-obstacle models have been extended in more sophisticated discrete dislocationdynamics (DDD) studies that include non-constant line tension, finite size obstacles, kinetics (and thereforetemperature effects), see for example References [93]–[96]. However, the increase of detail of this toolcomes at the expense of significant computational cost.Nogaret and Rodney used an areal glide model to study the effect of the finite size of simulationboxes [92]. They concluded that the difference between estimated CRSS from finite box size simulationsand its equivalent infinite size limit could be up to 50% in the case of a square box [92], however, this valuedepended on the box geometry and size.The spatial distribution of obstacles also has an effect on the strengthening that one cannot realisticallycontrol and quantify through experimental methods. If the obstacles are arranged following a perfect squarearray, a simple analytic solution exist. In this case, under the constant line tension approximation, the CRSSis [75], [97]:τc,square = β2ΓbLs(2.7)where b is the magnitude of the Burgers’ vector and Ls is the average spacing of obstacles [79]. If Friedel’sspacing is used, Ls = 1/√N [79] with N being the number of obstacles per unit area, however, if that ofLabusch is used, Ls = 1/N3/2. When the obstacles are randomly distributed however, the mathematicsbecome far more complex. Using branching probabilities, Hanson and Morris [98] derived the CRSS ofarrays of randomly distributed obstacles. They found, in the limit of very weak obstacles (i.e. β → 0) that:τc,random = 0.8871β 3/22ΓbLs. (2.8)The effect of the spatial distribution of obstacles has also been studied using computer simulations. Linetension simulations are particularly appropriate to simulate such problems. Foreman and Makin [97] werethe first to use computational models to estimate the CRSS of a randomly distributed array of obstacles. Theirmethod, the 2D circle rolling method [79], [98], computes the equilibrium position of a dislocation under23the assumption of a constant line tension. In this method, the dislocation samples the distribution of pointobstacles at a given applied stress starting from the bottom of the array until it finds a stable configuration.At this point the dislocation is returned to its origin and the stress increased. This is repeated until nostable configuration can be found. The lowest stress level for which no stable configuration can be foundcorresponds to the CRSS. The simulations of Foreman and Makin all featured arrays of 10,000 obstacles,and their results can be adequately represented by the empirical equation:τc,random = 0.956β 3/2(1− β28)2ΓbLs. (2.9)Figure 2.12 shows a comparison between the CRSS of a random distribution of obstacles (given by themodels of Hanson and Morris, and Foreman and Makin) and the CRSS of a square array of obstacles. Atlow strengths, both models converge towards the same CRSS. However, as the strength increases, the effectof distribution becomes non-negligible. Indeed, when obstacles are non-shearable, i.e. β = 1, the CRSS of arandom distribution is almost 20% lower than that of a square array.In a recent study, Leyson and Curtin [80] used analytical models with interaction energies calculatedfrom DFT to argue about the domain of validity of the models of Friedel and Labusch for solid solutionhardening. They concluded that Labusch’s model is valid over a wide range of dislocation core structures00.20.40.60.810 0.2 0.4 0.6 0.8 1τ cbLs/(2Γ)βForeman and MakinHanson and MorrisSquare arrayFigure 2.12: Critical resolved shear stress of a square array of obstacles compared to the CRSS of arandom array due to Foreman and Makin [97] and Hanson and Morris [98].24and temperatures (i.e. thermal activation is taken into account) when solute concentrations are higher than∼ 10−4. However, the results from Patinet’s simulations of solid solutions are closer to Friedel’s model thanLabusch’s and contradict the conclusions of Leyson and Curtin [69], [80]. But as noted before, the validityof Patinet’s results concerning solid solution strengthening can be questioned because of the low number ofobstacles present in his simulations. This study is of great interest as it is one of the few studies to account,not only for obstacles in the glide plane, but also for obstacles away from it. Patinet showed that in thespecific studied case, the solutes out of the glide plane contribute to about 1/3 of the strengthening [69].However, MS or MD are costly methods to analyze the effect of the concentration and the distribution ofobstacles. In fact, simulations like the ones performed by Patinet can take weeks on supercomputers [69].The question remains as to what mechanisms are responsible for solid solution and cluster strengthening.Clouet et al. [99] argued in the case of Fe-C that the strength arising from solute atoms can be explained byelasticity theory. Solute atoms are point defects in the matrix and the strain field associated with these defectsinteracts elastically with dislocations [100]. This interaction gives rise to the dislocation/obstacle interactionforce-distance profile [99] (Figure 2.8). In addition to the elastic interaction, Ma et al. [101] argued that thechange of stacking fault energy caused by the presence of solute atoms in the stacking fault between partialdislocations also participate to the strength of solute atoms. In the case of clusters, Starink et al. [102] havehypothesized that the strength of clusters primarily rise from nearest neighbour atomic interactions. Withinthe limits of such a hypothesis, Starink et al. have used classic thermodynamics to derive the strength ofclusters [102]. They affirmed that the strengthening mechanisms related to obstacle shearing results from: (a)order strengthening and (b) modulus hardening. Order strengthening which also includes the stacking faulthardening comes from the change in conformation energies when clusters are sheared by a dislocation [15].The resulting strengthening was claimed to be equal to the corresponding change of energy per unit area ofslip planes [15]. This clearly requires further consideration as no mention is made of the elastic interactionbetween dislocations and clusters.Proville et al. [103] as well as Patinet [69] studied the interaction between a single split dislocation anda single dimer using MS. The dimer was located in or across the glide plane with different orientationswith respect to the gliding dislocation. They examined the influence of the change in topology of the dimerresulting from the shearing of a split dislocation. Their work suggests that this change, in the case of Mgdimers, has little influence on the CRSS [103], which is in contradiction with the assumptions made by25Starink et al. [102]. Indeed, following Starink et al., order strengthening should be the dominant mechanismas the modulus strengthening for a dimer would be low as a result of the low concentration of Mg atoms andthe low difference between the shear modulus of Al and Mg [102].The spatial distribution of obstacles can also have a significant effect on the CRSS. On the one hand, theCRSS of a glide plane populated by obstacles arranged in a square fashion is higher than that of the sameplane populated by the same obstacles but randomly distributed as shown by Foreman and Makin [104],i.e. the CRSS increases as the ordering of the obstacle distribution increases [104]. On the other hand, itis difficult to predict how the CRSS will change as the distribution becomes more clustered. There are twopossible competing consequences of obstacle clustering. First, in areas where obstacle density is higher thanaverage like in clusters, the decrease of obstacle spacing increases the resistance to dislocation motion. If theobstacles within a cluster are closely spaced then the dislocation will not pass through the cluster. Insteadthe dislocation will loop around the group. In this scenario, it is as if the clustering increases the ‘effective’strength of an individual obstacle. In contrast, elsewhere in the glide plane regions of low obstacle densitywill appear as the degree of clustering increases. These less populated areas will be easier for the dislocationto bow into, thereby leading to a reduction in glide resistance. It is hard to predict which mechanism is themost important.2.3 SummaryMany aluminum alloys experience decomposition of the super saturated solid solution into clusters, e.g Al-Cu, Al-Zn, Al-Mg, Al-Mg-Si, Al-Mg-Zn, Al-Mg-Cu alloys. In most of them, the decomposition of theSSSS into clusters lead to an increase of strength whose magnitude depends on the chemistry of the alloy.In the literature, this increase of strength is argued to have various origins. At the mesoscopic level, clusterstrengthening could originate from the way solute solutions are distributed and how a dislocation interactswith the distribution. At the atomistic scale, cluster strengthening could emanate from the change of intrinsicstrength of clusters as their size changes. This change can also have different origins: the change of chemicalenergy emerging from a change of topology of the clusters when they are sheared by dislocations [102], theelastic interaction between clusters and dislocations [99], and the change of stacking fault energy [101].26Chapter 3Scope and objectivesIn precipitation hardening, scaling laws are used to relate the dislocation-obstacle interaction force with thesize of precipitates. Clusters being a form of pre-precipitates, the question arises: can such laws be appliedto cluster strengthening? In this thesis, an empirical scaling law that relates the change of CRSS with thesize of clusters is first developed. This scaling law has been successfully applied in the case of the aluminumalloy AA6111 studied by Marceau et al. [57] to predict its yield stress as a function of different artificialand natural aging treatments. The subsequent question relates to the physical basis of such a scaling law, i.e.are there physical phenomena that could explain it? This problem will be investigated in the simpler case ofbinary Al-Mg alloys. The investigation uses numerical simulations as means of obtaining physically relevantdata.The objectives of the present work are as follows:• The first objective is to predict the yield stress of a cluster strengthened AA6111 alloy aged at roomtemperature, using a simple and empirical scaling law based on experimental data from atom probetomography.• The second objective is to identify which mechanisms of interaction between dislocations and soluteatoms are responsible for cluster strengthening in Al-Mg binary alloys. This will be done in two steps.1. Mesoscopic simulations will be used to investigate and discuss the influence of spacing betweensolutes on the strength of alloys.272. Analytical models taking into account elasticity and bond breaking will be used to predict thestrength of single solutes as well as small clusters (dimers and trimers). These predictions willthen be tested against results of atomistic simulations and their validity discussed.• Finally, the third objective is to use the previous results to compare the increase of strength due tothe clustering of solid solutions into dimers, and then trimers in binary Al-Mg with the scaling lawproposed in the case of the AA6111 alloy.It is important to identify the primary mechanisms responsible for cluster strengthening as it will allowfor the development of 1st order models of the strength of each class of clusters. These models are importantas they can be used to determine the strength of any cluster in any type of alloy with only a small fractionof the computations needed for atomistic simulations. As a result, this could then be used to help predictthe strength of cluster strengthened aluminum alloys, and in a longer term perspective aid the design of newmaterials that are needed to respond to tomorrow’s engineering challenges.28Chapter 4MethodologyThe present work uses two different tools operating at two different length scales for simulating the glide ofa dislocation through obstacles. An areal glide model is used to simulate the glide of a single dislocationthrough an array of obstacles. These simulations usually feature 104 to 105 obstacles. Therefore, if suchobstacles are single solute, the scale of these simulations would be similar to the scale of a single grain.This scale will be referred to in the following as the ‘mesoscale’. The areal glide model takes as inputs theposition of the obstacles as well as their strength. On the other hand, molecular statics simulations are usedto simulate the interaction force and energy between a single dislocation and a single obstacle (single solute,dimer or trimer) at the atomistic scale as function of their separation. These simulations typically featuredaround 105 atoms and were used to determine the stress required to bypass an obstacle.Figure 4.1 shows the linkage between the simulation tools used in this work. Atomistic simulations areused to examine what happens at the atomistic level. They provide a means to determine the maximumpinning force of an individual obstacle. It is then fed into the areal glide model, which is used to simulatethe interaction between a dislocation and an array of statistically significant obstacles, and predict the criticalresolved shear stress of an alloy.29Molecular StaticsSimulations(0 K)xzLzdxyττmaxdLlLrΦ=ΦcθlθrαAreal glide simulationsCritical Resolved Shear StressFigure 4.1: Schematic showing the different simulation tools used in this work as well as their link.4.1 Areal glide modelNogaret and Rodney [92] have shown that in order to predict the strength of a random distribution of obsta-cles, &10,000 obstacles on the glide plane must be resolved to avoid significant system size effects. Thisnumber increases with the aggregation level of the obstacles [105]. For instance, in the case of the AA6111alloy studied here [57], for all the studied aging conditions, most of the clusters are composed of 2 to 10atoms. The average spacing between 10-atom clusters is approximately 100 nm. If the distribution of clus-ters is random, the size of the smallest square to have the probability of always containing a 10-atom clusteris (100 nm)2. Therefore, the smallest simulation box having a probability to contain 10,000 of these clustersneeds to be as large as 10,000 of these small squares. It sides must then be at least 10 µm large. Suchsimulations are out of reach for atomistic simulations such as MD. However, DDD could be a candidate forsuch simulations. In fact, DDD box sizes can have sides up to 50 µm large [106]. Nevertheless, because therequired box size is close to the upper limit of DDD box sizes, such simulations would require days or weeksof computational time.30Due to their small size (r < 1 nm), clusters are weak obstacles [5], [57]. Thus, it is assumed that theclusters are easily shearable. The interest in this work is in low temperature deformation, therefore thecontribution of dislocation cross-slip or climb to the flow stress is neglected. Further, in the case of weakobstacles, it has been shown by Patinet [69] that dislocation segments are well approximated by arcs of acircle. Thus the line tension of the simulated dislocations is assumed to be constant. As the dislocationline simulated here is long compared to the size of obstacles, the obstacles can reasonably be modelled aspoints [57], [69], [92]. For these reasons, the simple but efficient areal glide model has been adopted topredict the strength of distributions of obstacles.The areal glide model used in the present work is written in C++ and is based on the model developedby Nogaret and Rodney [92]. It is a 2D model which assumes the dislocation to be a line with constanttension and the obstacles to be points. Unlike the classic ‘circle rolling method’ [97], [107], [108], Nogaretand Rodney’s method follows a set of rules inspired by dislocation dynamics simulations and thus leads toa more physically justifiable sampling of the obstacles by the dislocation, particularly in the case of strongobstacles [92]. The model differs from the original Nogaret and Rodney’s model as it was adapted to workfor all strengths of the obstacles including the limit of obstacles that are non-shearable.The areal glide model simulates the glide of a dislocation through an array of obstacles. The dislocationis modelled as a line which has a constant tension of Γ = 12µb2. Each obstacle is described by its (x,y)position on the plane, and a normalized strength parameter β (0 ≤ β ≤ 1) [79], which is a function of thecritical bowing angle Φc (Figure 4.2). This critical bowing angle corresponds to the angle Φ at which thedislocation has to bend around the obstacle in order to overcome it. As in previous work [79], stresses anddistances are all provided as normalized quantities. All distances are normalized by the equivalent squarespacing of obstacles, which in the case of N obstacles per unit area is,Ls = 1/√N. (4.1)while the dimensionless applied shear stress is given by,τ∗ =bLs2Γτ. (4.2)31Rmeetτ∗meet = (2R∗meet)−1(a)LlLrΦ=Φcθlθrατ∗break =|sin(α+Φc)|√2cos(α+Φc)L∗l L∗r+L∗l2+L∗r 2(b)τ∗Orowan = (L∗)−1LR= L2(c)Figure 4.2: Schematic of the three possible evolution mechanisms for a dislocation segment: (a) meetan obstacle, (b) break an obstacle, and (c) by-pass an obstacle. From Nogaret and Rodney [92].The simulation cell has periodic boundary conditions along the direction of the dislocation line. Due tothe constant line tension approximation, each segment of the dislocation is an arc of a circle of radius R (andnormalized radius R∗=R/Ls). This radius is inversely proportional to the shear stress, and if L is the distancebetween the two obstacles pinning the dislocation segment, the Peach-Koehler force exerted on the segmentis τbL [109]. When the dislocation segment is at equilibrium, this force is equal to the resultant force dueto the line tension, i.e. ΓL/R. Once the lengths and stresses are normalized, one obtains the following [92]:τ∗ =12R∗(4.3)All simulations commence with a single straight dislocation in contact with three widely spaced, weakobstacles (Φc = 179.1◦) at the very bottom of the domain. From this starting configuration the simulationproceeds by incrementally moving the position of the dislocation based upon the event that requires thelowest stress. At each step three actions are possible for each segment:• the bowing out of a pinned segment under a stress τ∗meet such that it touches another obstacle ahead ofit (Figure 4.2a),32• the breaking of an obstacle when the stress reaches τ∗break (Figure 4.2b),• the by-passing of an obstacle by an Orowan process governed by τ∗Orowan (Figure 4.2c).The decision on which of these three events will occur at a given step is determined by calculating foreach segment of the dislocation, the three stresses τ∗meet , τ∗break, and τ∗Orowan (Figure 4.2) and determining thesegment and event requiring the lowest stress. This event is then selected and the position of the dislocationupdated accordingly. As the dislocation advances, the stress required to accomplish the easiest event fromthese three options is recorded. The simulation is stopped when the most advanced portion of the dislocationhas traversed more than 95% of the glide plane and the highest shear stress encountered is taken as the criticalresolved shear stress (CRSS).The original model of Nogaret and Rodney was applied to problems having weak obstacles where β <0.3 (corresponding to Φ > 145◦) [92], [110]. To deal with the more complex topologies possible whenβ → 1, additional rules had to be implemented. Most importantly, to avoid the dislocation crossing itselfand touching the same obstacle twice, a rule was implemented that would remove any segment forming aloop between a twice contacted obstacle. This results in a dislocation loop being left surrounding a group ofobstacles. To confirm the validity of the simulation procedure, the model was used to obtain the CRSS forrandomly distributed obstacles. Several different obstacle strengths were tested, the results being shown inFigure 4.3.For weak obstacles the variation of τ∗c with β (see Figure 4.3) predicted with the new model closelymatches equation (2.9). For β > 0.8, however, the present simulations predict up to a 5% lower CRSSwhen compared with Equation (2.9). This is a consequence of the different way in which the two techniquessample the obstacle array. To make a direct comparison, the circle rolling method of Foreman and Makin wasimplemented. The two methods were applied to the same random array of obstacles with the same strengthβ = 1. Under the same shear stress level, the glide distance of the dislocation predicted by the new model(Figure 4.4a) is much larger than that predicted by the circle rolling method(Figure 4.4b). In this case, theresults of the new model are well described by an empirical expression with a slightly modified from (2.9):τ∗c,random = 0.9β3/2(1− β56). (4.4)3300.20.40.60.810 0.2 0.4 0.6 0.8 1τ∗ cStrength βSimulationsEquation (2.9)Equation (4.4)Figure 4.3: Simulated critical resolved shear stress of 10,000 randomly distributed obstacles as a func-tion of the obstacle strength. Each point represents the average of 10 simulations at that strengthlevel.(a) (b)Figure 4.4: Under the same shear stress, the distance a dislocation glides for the same array of 10,000randomly distributed obstacles whose breaking angle is Φc = 0◦ when simulated with (a) Nogaretand Rodney’s method and (b) the ‘circle rolling’ method. All obstacles bypassed by the dislo-cation and dislocation loops are not represented. Clusters of obstacles below the dislocation aregroups of looped obstacles (not represented in (b)).34The higher CRSS obtained by the ‘circle rolling’ method is a consequence of the artificial configurationsthat are considered. Rather than allowing weak configurations to be broken and for the dislocation topologyto adjust naturally to the obstacle distribution, the ‘circle rolling’ method over constrains the dislocation linethereby leading to a higher CRSS. Upon introducing their method, Nogaret and Rodney noted that it morenaturally mimics the dynamic nature of dislocation motion observed in fully discrete dislocation dynamicssimulations [92]. Therefore, the results obtained with our method are presumed to be closer to the trueCRSS.In this chapter, the fundamentals of the areal glide model have been presented. Each part of this workuses this model in a specific manner. For clarity, the specific methodologies used in each part of this workwill be explained in the corresponding chapters.4.2 Atomistic simulations: molecular staticsUnderstanding the mechanisms responsible for cluster strengthening cannot be achieved without studyingthe intrinsic interaction of a dislocation with individual clusters. The strength of a cluster has historicallybeen assumed to come primarily from the interaction with the core of the dislocation [84]–[86], [102], [103].For these reasons, the investigation of the intrinsic strength of clusters needs to be done using tools that canresolve the core of a dislocation.The core of a dislocation in aluminum can be resolved using ab initio or first principle calculations. Thesecalculations solve Schro¨dinger’s equation to compute the position of each atom [111]. They are the mostaccurate simulation tools available to study dislocations. However, the length scale in which they operate islimited to the extended size of the core of a dislocation. This scale is too small to capture the effects due tothe elastic interaction between dislocations and solute clusters [89].Simulations working at the next length scale above ab initio calculations are atomistic simulations. Suchsimulations are used in the present work. Molecular statics simulations are performed on systems that aresufficiently large to allow for the long range elastic field of the solute and dislocation to be captured. Theyalso provide full resolution of the atomic structure in the core of a dislocation [89]. These simulationsare preformed at 0 K. Contrary to MD (finite temperature atomistic simulations), they do not incorporatethe additional complexity of thermal effects. Thus, they cannot capture effects such as thermal activation.MS allows the measure of energies at a set of equilibrium positions, whereas MD allows the measure of35Table 4.1: Elastic constants and intrinsic stacking fault energy (ESF ) of aluminum: comparison betweenvalues obtained with Liu’s and Mendelev’s potentials and experimental ones..Property Experimental Value Liu’s potential Mendelev’s potentialC11 (GPa) 116 (at 4 K) [114] 119 110C12 (GPa) 64.8 (at 4 K) [114] 62.3 61.4C44 (GPa) 30.9 (at 4 K) [114] 34.9 37.4ESF (J/m2) 0.120 - 0.144 [115], [116] 0.135 0.227the dynamics of a system, thus the calculated energy also takes into account the effect of momentum, andthermal noise. Like Olmsted et al. [70], the interaction energy between a dislocation and individual clustersas a function of their distance is of interest here. Therefore, MS simulations are preferred to MD simulations.In contrast to ab initio calculations, the interaction between atoms in MS simulations is described by anempirical interatomic potential [89]. Among the existing potentials, EAM potentials are the ones usuallyused for the study of dislocations in aluminum alloys [69], [70], [80], [89], [103]. However, only a fewof these potentials have been developed for aluminum alloys [112]. For simplicity, binary alloys are theonly alloys of interest here in this thesis, i.e. the effects of co-clusters between different solutes are notconsidered. A few EAM potentials exist for binary alloys but only a few of these alloys show significantcluster strengthening and are of practical interest. The potential candidates were the Al-Cu and Al-Mgsystems. Experimentally, Al-Cu features faster cluster strengthening when aged at room temperature asnoted in Chapter 2. The clusters in Al-Cu alloys form initially as monoatomic layers on the {001} plane [24],whereas in many important commercial alloys such as 6000 and 7000 series, clusters are spherical [30], [31]as is the case of the binary Al-Mg alloys [16], [21]. Moreover, a large body of work on random solidsolution strengthening in Al-Mg alloys using atomistic simulations exists [69], [80], [84], [86], [88]–[90]which served as an excellent starting point for this study. Thus the Al-Mg binary alloys were chosen for thisstudy.There are two EAM interatomic potentials available for the Al-Mg system: namely Mendelev’s [113] andLiu’s potential [91]. To decide which potential to use, some material properties were calculated using bothpotentials, and then compared with known properties. These material properties are: the elastic constantsof pure aluminum, the intrinsic stacking fault energy of pure aluminum, and the enthalpy of mixing of Al-Mg. First, the elastic constants and intrinsic stacking fault energy were calculated. The predicted valuesby both potentials are listed alongside experimental data in Table 4.1. For these properties, Liu’s potential36gives results that are in better agreement with experimental values. Aluminum is a quasi-isotropic materialwith an anisotropic factor, 2C44/(C11−C22) [117], of 1.2 [118]. This is well captured by Liu’s potentialwhich gives an anisotropic factor of 1.23, whereas Mendelev’s potential gives 1.54. Next, the enthalpy ofmixing of Al-Mg for different concentrations were computed using both potentials. The results are plottedin Figure 4.5 alongside reference data obtained from CALPHAD [42]. Again, Liu’s predictions agree betterwith the reference data. Based on these results, Liu’s potential seems to be the best choice of potential forthis work. Further, Liu’s potential has been used by a number of other researchers to study the interactionbetween dislocations and obstacles [69], [84], [90], [119], [120]. In this work, therefore, the inter-atomicpotential used for all MS simulations is the EAM potential developed by Liu and coworker [91], and allthese simulations were performed using the software LAMMPS [121]. LAMMPS is an open-source large-scale parallel classical molecular dynamics/statics code developed at Sandia National Laboratory in NewMexico, USA. It is well suited to parallel computing on large supercomputers. All the MS simulations wereperformed using supercomputers Bugaboo and Jasper as part of the Canadian WestGrid network.Dislocations of both edge and screw character are present in aluminum alloys and both interact withsolutes. However, Patinet and Proville [84] have shown that the elementary interaction between edge andscrew dislocations, and the increase of flow stress from both dislocations are of the same order of magnitude.This can be attributed to the fact that both perfect edge and screw dislocations split into partial dislocationsof mixed characters. In order to decrease by a factor of two the number of simulations to run, it was decided00.020.040.060.080.10 0.2 0.4 0.6 0.8 1Enthalpyofmixing(eV/atom)Atomic proportion of MgLiuMendelevCALPHADFigure 4.5: Enthalpy of mixing: of aluminum: comparison between values obtained with Liu’s andMendelev’s potentials and CALPHAD data [42].37to focus here only on the interaction between edge dislocations and obstacles.4.2.1 Creation and glide of an edge dislocationThe simulation of dislocation glide with MS is not straightforward as it requires a precise description ofdislocations through careful positioning of atoms [89]. The creation of the dislocation is thus a critical task.It needs to be done such that the system does not experience undesired strains and stresses which lead toartifacts. The method used here is that described by Bacon and coworkers [89]. The simulation box (seeFigure 4.6) is oriented such that the glide planes (111) corresponds to the xy plane. The boundary conditionsare: periodic along the x direction ([1¯10]), shrink wrapped (free surface) along the y direction ([1¯1¯1¯]), andperiodic along the z direction ([1¯1¯2]). Once a perfect lattice is created, the crystal is composed of Nl p yzlattice planes with spacing b. The method to generate the dislocation involves the following four steps (seeFigure 4.6):1. the atoms in the bottom half of a (1¯10) plane are removed,2. the top and bottom half crystals are strained respectively by −1/2Nl p and −1/2(Nl p−1), in order forthe length of the top and bottom half crystals to be equal,3. the simulation box is re-sized to remove any gap in the lattice,4. the system is relaxed, i.e. the energy of the system is minimized using the conjugate gradient min-imization built in LAMMPS by first authorizing the boundaries to relax in order to minimize theexternal stresses, and then considering the boundaries fixed to minimize the internal stresses.Figure 4.7 shows a snapshot of the atomic structure in and around the dislocation obtained this way. Inthis figure, one can clearly see the two Shockley partial dislocations aligned in the z direction ([1¯1¯2]), as wellas the stacking fault.To cause the glide of a dislocation at 0 K, a shearing force needs to be applied to the box. In this work,this force is applied as shown schematically in Figure 4.8. The centre of mass of the top and bottom twolayers of atoms are attached to perfect springs of constant k. Their ends are displaced at a constant speed inthe opposite direction, thus creating shear. When the energy minimizer at 0 K was used, the obtained Peierlsstress was one order of magnitude higher than that of already published results such as those of Patinet [69],38[x= 1¯10][y= 1¯1¯1¯]b1. Half plane removal[1¯10][1¯1¯1¯]b2b22. Half slabs resizing[1¯10][1¯1¯1¯]3. Box adjustment[1¯10][1¯1¯1¯]4. RelaxationFigure 4.6: Schematics showing the different steps of the creation of an edge dislocation.−25 25−2525x (A˚)y(A˚)[1¯10][1¯1¯1¯][1¯1¯2]-3.30-3.36eVFigure 4.7: Snapshot of the potential energy of atoms in and around the core of the dislocation.x=[1¯10]y=[1¯1¯1¯]−δkδkFigure 4.8: Schematics showing the method used to make the dislocation glide.39and Olmsted et al. [70]. This is believed to be due to the inability for the energy minimizer at 0 K to resolvethe core structure of the dislocation properly during shearing. To solve this, at each increment, the systemis equilibrated by heating it to 1 K (by giving an initial speed of random direction to each atom) and thencooling it down to 0 K. Once the temperature of the system is back to 0 K, the energy of the system isminimized using the conjugate gradient method built into LAMMPS. Since the focus of this thesis is on themaximum pinning force of single solutes and clusters, to save computational time, the stopping criterionof the energy minimization is a tolerance in force, not in energy. This does not have any influence on theresults as the heating temperature is extremely low. The compliance of the shearing setup is minimized bytaking k as high as possible without ‘losing’ atoms during the simulations. Indeed, if k is too high, a smalldisplacement of the atoms connected to the springs would generate high elastic forces and propel someatoms out of the simulation box. The selection of k was made by trial and error. The spring constant usedwas k = 500 eV/A˚2.The accuracy of results from atomistic simulations always depends on the box size used. Appropriatebox sizes have to be chosen to guarantee convergence of the simulation results, and minimization of thecomputation time required. The sizes in x and y need to be as large as possible to minimize the effects ofimage stresses, but as small as possible to minimize the computation time. Here, these dimensions weredetermined as the smallest sizes over which the maximum pinning force of a single solute laying on theglide plane does not change by more than 1%. Five different box configurations were tested: Lx = 5.6 nmand Ly = 14 nm, Lx = 11 nm and Ly = 14 nm, Lx = 11 nm and Ly = 28 nm, Lx = 22 nm and Ly = 14 nm,and Lx = 22 nm and Ly = 28 nm. The different maximum pinning forces obtained using these differentconfiguration are summarized in Figure 4.9. The convergence criterion was met by the simulations whichbox sizes were Lx = 11 nm and Ly = 28 nm and larger. Thus it was chosen to use a box of size Lx = 11 nm,and Ly = 28 nm with Lz to be determined as a function of the type and strength of obstacle present in the box.In order to ensure that the dislocation was created properly and that the shearing method was well im-plemented, two previously published results were reproduced:1. the Peierls stress in pure Al,2. the breaking angle of a single Mg solute laying on the atomic plane just above or below the glide plane.400.020.0220.0240.0260.0280.035.6x14 11x14 11x28 22x14 22x28Fmaxobs(nN)Box size (LxxLy in nm2)Figure 4.9: Maximum pinning force of a single solute laying on the plane just above the glide plane asa function of the simulation box size.4.2.2 Peierls stressThe Peierls stress is the minimum stress required for a dislocation to glide through a perfect lattice [109]. Inthis case, because no obstacles are present, the dislocation line does not bend when moving. The periodicityof the system is the same as that of the lattice. Thus Lz only needs to be as large as the smallest repetitivedistance of the lattice in the [1¯1¯2] direction. As a result, Lz = 0.99 nm.A simulation box containing a relaxed split edge dislocation and no solute was sheared at constant incre-ment of strain, i.e. 7×10−6. This increment was selected such that it is small compared to the period of thevariation of the shear stress τxy (see Figure 4.10). The variation of the shear stress τxy as a function of theengineering shear strain γxy during the simulation is presented in Figure 4.10. At the start of the simulation(i.e. γxy = 0), τxy = 0. Then, the shear strain increases progressively. At first, the response is elastic: the shearstress increases linearly with the strain and the dislocation does not move. Just before the maximum stress isreached, the response is observed to be no longer linear as the dislocation slowly starts to glide. At the peakstress, the dislocation already moved by 1/4 of a Burger’s vector. Then, when the peak stress is reached, i.e.≈ 2.8 MPa, the whole dislocation glides abruptly and stop after having travelled one Burger’s vector in total.This abrupt glide causes the stress to drop to 0 MPa. Then, the stress increases again until τxy = 2.8 MPa,the dislocation glides a full Burgers’ vector and stops; and the cycle starts again. From this simulation, itwas found that the Peierls stress, τp, of an edge dislocation dissociated into two partials obtained is 2.8 MPa.This value is close to that published in the literature by researchers using different MS codes as Patinet [69]found 2 MPa and Olmsted and coworkers [70] found the Peierls stress to be in a range of approximately 1.541to 2.5 MPa. This result attest that the dislocation was correctly created and that the method used to shear thebox was properly implemented.012340 0.0005ShearStressτ xy(MPa)Shear Strain γxyFigure 4.10: Simulated stress-strain response of a split edge dislocation gliding in a perfect aluminumlattice.4.2.3 Interaction with a single solute on the glide planeThe next result that was checked was the strength of a single Mg solute laying on the glide plane. After thecreation of split edge dislocations in the aluminum lattice, a single solute atom was placed in the plane justabove or just below the dislocation line. Along the x direction, the solute was placed as far away as possiblefrom the dislocation in order to minimize the elastic interaction between the dislocation and the solute atomat the start of the simulation. In this case, the distance between the centre of the two partial dislocations andthe solute is thus Ly/2. Due to the periodic boundary conditions in x and z, the simulated system is reallycomposed of a series of periodically spaced dislocations gliding through a rectangular array of solute atoms.The distance separating these solute atoms is Lx and Ly along the x and y axes, respectively (see Figure 4.11).The output of the simulations are the stresses on the box and the different energies of the system, ateach increment. Here the variable of interest is the pinning force exerted by the obstacle on the dislocation,namelyFobs. This pinning force can be calculated either (a) from the shear stress τxy, or (b) from the changein total energy Etot . To determineFobs from τxy let us isolate a single dislocation segment (see Figure 4.12).The forces applied on the segments are:• the Peach-Kohler force (FPK),• a drag force (Fp) due to the resistance of the lattice linked to the Peierls stress (τp),• and the pinning force due to the obstacle (Fobs).42xzdτLzLxdxyFigure 4.11: Effect of the periodic boundary conditions on the MS simulations: the dislocation glidesthrough a rectangular array of obstacles.fPK = τxybfP = τpbLzFobs/2Fobs/2xzFigure 4.12: Forces acting on a dislocation segment during glide.43The Peach-Kohler force is a force per unit length acting normal to the line direction of the dislocation, i.e. z.It results from the shear stress τxy applied on the box and is expressed as follows [109]:FPK = fPKLz = τxybLz. (4.5)The drag force fp due to the resistance of the lattice is similar to the Peach-Kohler force as it results from ashear stress τp, thus one finds:Fp = fpLz = τpbLz. (4.6)Each obstacle pins two segments of a dislocation. But one segment is also pinned by two obstacles separatedby a distance Lz. Thus from the equilibrium of all the forces applied on one dislocation segment, one gets:Fobs = τxybLz− τpbLz. (4.7)The other method to determine Fobs is from the total energy of the system, Etot . In this approach, thefirst step is to determine the position of the dislocation xd at each simulation step. This is done by lookingat the maxima of the first derivative of the disregistry with respect to x [109], using the Peierls-Nabarromodel [122]. If ∆x is the disregistry with respect to x, this model gives:∆x =b2pi(atan(x+dp/2ζ)+atan(x−dp/2ζ))+b2, (4.8)where dp = 1.2 nm is the distance between the two partial dislocations and ζ = 0.43 nm is the half-widthof the dislocation core. For a given simulation snapshot, the disregistry across the glide plane was first cal-culated for each atomic position along x. Then the Peierls-Nabarro model [122] was fitted to the disregistry(see Figure 4.13a), and finally its derivative was taken using the first order finite difference method. Anexample is given in Figure 4.13b. The position of the two highest maxima correspond to the position of eachpartials. For this example in Figure 4.13b, they are located at −6 nm and +6 nm respectively.The centre of the dislocation being at the middle of both partials, the position of the dislocation is thus theaverage of the position of the partials. In all the following, the position of the dislocation will be considered‘zero’ when the position of the solute corresponds to the centre of the dislocation. Since the system is at 0 K,the change in total energy corresponds to the work needed to move the dislocation. Thus, the pinning force440123-30 -20 -10 0 10 20 30Disregistryalongx(A˚)x (A˚)MS simulationsPeierls-Nabarro fit(a)00.020.040.060.080.10.120.14-30 -20 -10 0 10 20 30Derivativeofdisregistrywithrespecttoxx (A˚)(b)Figure 4.13: (a) Disregistry projected on the x direction of a split edge dislocation. The symbols corre-spond to the simulation values, and the solid line corresponds to the Peierls-Nabarro fit. (b) Thefirst derivative of the Peierls-Nabarro fit of the disregistry.is:Fobs =∂Etot∂xd− τpbLz. (4.9)The maximum value of Fobs is the maximum pinning force of the obstacle (Fmaxobs ). It corresponds tothe maximum force the dislocation has to overcome to shear through the obstacle. Then, using the constantline tension approximation, the strength β (or the breaking angle Φc) of the obstacle is thus linked toFmaxobsas follows [79]:β = cosΦc2=Fmaxobs2Γ, (4.10)where Γ is the line tension of the dislocation.The accuracy of the simulations, should not depend on the box size Lz. However, as the distance Lzincreases, the distance between the two solutes pinning the dislocation segment also increases (see Figures4.11 and 4.12), and the stress required to overcome the solute increases (see Equation 2.7). As the stressincreases, the curvature of the dislocation decreases (the dislocation bends more and more), and the disloca-tion moves further from its zero stress position, which makes it harder for the simulations to find the pointof convergence. The results of these simulations are assumed to converge when the two different ways ofcalculatingFobs give results that do not differ by more than half the force due to the resistance of the lattice.45In the case of a single solute, this criterion was met with Lz = 6.0 nm and Lz = 8.0 nm when the atom layson the atomic plane respectively just above and just below the glide plane. It was found that when the soluteis on the atomic plane just above and just below the glide plane, respectively, the breaking angle is 178.4◦and 178.6◦, respectively. These results are in good agreement with the previous calculations of Patinet [69]who obtained, 178.8◦ and 179.3. By being able to reproduce previously published results, it is with confi-dence that these simulations can be used in this work to study the interaction between dislocation and singlesolutes, dimers and large clusters.46Chapter 5Modelling the yield stress of a clusterstrengthened AA6111 alloy1Marceau et al. [57] performed APT experiments on AA6111 samples that were solution heat treated at 560◦Cfor 10 min and then aged under different conditions, i.e. 2h, 8h, 4 days 5h, 1 week, 8 days and 2 weeks atroom temperature, and 3 days at 60◦C, 24 hours at 90◦C and 20min at 150◦C. The alloy had a nominalcomposition of 0.81 wt.% Mg, 0.62 wt.% Si, 0.7 wt.% Cu, 0.21 wt.% Fe, 0.20 wt.% Mn, trace additionsof Cr (0.06 wt.%) and Ti (0.05 wt.%), and balance Al. From these experiments, the size, volume fraction,and chemical composition of clusters of a representative volume were extracted [57]. These clusters weredetermined from the atom probe data using a core-linkage cluster finding algorithm described in Ref. [123].As it can be seen on Tables 5.1 and 5.2, the majority of the clusters identified were Mg-Si and Mg-Mgclusters [57], as they respectively account for 40% and 25% of all the clusters detected [57]. Also, it is worthnoting that Mg-Si clusters are on average larger than Mg-Mg clusters. Other studies linked the formation ofthese clusters to the observed strengthening during early stages of aging [55], [124]–[127]. This raises thequestion of the possibility to predict the change of yield stress from the change of microstructure (i.e. theformation of clusters) in aluminum alloys.At later stages of aging, it was observed that precipitates form [13]. The strength of theses precipitateshave been supposed to scale proportionally with their radius until a critical radius is reached [128], [129].1The work presented in this chapter has been published: R.K.W. Marceau, A. de Vaucorbeil, G. Sha, S.P. Ringer,W.J. Poole,Analysis of strengthening in AA6111 during the early stages of aging: Atom probe tomography and yield stress modelling, ActaMaterialia, vol. 61, no. 19, pp. 7285 7303, 201347Table 5.1: Experimental number densities (1015 cm3) of clusters tabulated within certain size rangesfor the AA6111 samples after natural aging at room temperature [57].Cluster type Size (atoms) 2 h 8 h 4 days 5 h 1 week 8 days 2 weeksMg-Si-Cu 3 to 10 1158 ± 52 1941 ± 50 3509 ± 278 2890 ± 89 2914 ± 241 2016 ± 5411 to 25 43 ± 1 74 ± 7 116 ± 24 177 ± 16 62 ± 17 23 ± 426 to 50 0 0 0 16 ± 3 0 0Mg-Si 2 7021 ± 87 7882 ± 65 10230 ± 291 9741 ± 100 10360 ± 283 9773 ± 753 to 10 4117 ± 92 6420 ± 86 10826 ± 459 9112 ± 148 9158 ± 391 7596 ± 9811 to 25 0 56 ± 6 199 ± 37 120 ± 12 62 ± 12 31 ± 426 to 50 0 0 0 2 ± 0.4 0 0Cu-Mg 2 3417 ± 61 3294 ± 42 3791 ± 177 4198 ± 65 4008 ± 176 3967 ± 483 to 10 1385 ± 50 1707 ± 41 2351 ± 70 2395 ± 69 2436 ± 190 1908 ± 4511 to 25 0 1 ± 0.3 0 6 ± 2 0 026 to 50 0 0 0 0 0 0Si-Cu 2 2154 ± 54 2165 ± 34 3294 ± 165 2717 ± 53 3176 ± 156 2558 ± 383 to 10 674 ± 34 738 ± 26 1655 ± 139 888 ± 39 1187 ± 120 892 ± 2811 to 25 0 0 0 0 0 026 to 50 0 0 0 0 0 0Mg-Mg 2 6251 ± 82 7611 ± 64 11770 ± 312 9890 ± 100 9327 ± 268 8807 ± 713 to 10 1178 ± 43 2207 ± 44 3195 ± 206 3002 ± 72 2837 ± 189 2225 ± 4511 to 25 0 0 0 0 0 026 to 50 0 0 0 0 0 0Si-Si 2 2693 ± 54 3001 ± 40 4916 ± 202 3715 ± 62 4779 ± 192 3728 ± 463 to 10 376 ± 22 508 ± 19 1076 ± 104 688 ± 32 925 ± 189 609 ± 2211 to 25 0 0 0 0 0 026 to 50 0 0 0 0 0 0Cu-Cu 2 580 ± 25 467 ± 16 612 ± 71 621 ± 25 524 ± 64 552 ± 183 to 10 28 ± 5 30 ± 4 50 ± 17 33 ± 6 31 ± 17 28 ± 411 to 25 0 0 0 0 0 026 to 50 0 0 0 0 0 048Table 5.2: Experimental number densities (1015 cm3) of clusters tabulated within certain size rangesfor the AA6111 samples after artificial aging at for various times and temperatures [57].Cluster type Size (atoms) 60 ◦C 3 days 90 ◦C 1 day 150 ◦C 20 minsMg-Si-Cu 3 to 10 3398 ± 69 3872 ± 95 3009 ± 6411 to 25 128 ± 9 243 ± 17 328 ± 1626 to 50 1 ± 0.2 12 ± 2 58 ± 650+ 0 3 ± 0.5 13 ± 1Mg-Si 2 6776 ± 60 8747 ± 87 6197 ± 553 to 10 5697 ± 83 7997 ± 128 5456 ± 8011 to 25 47 ± 5 99 ± 9 140 ± 1026 to 50 0 2 ± 0.3 5 ± 0.9Cu-Mg 2 3054 ± 40 4128 ± 60 2873 ± 383 to 10 1376 ± 37 2076 ± 59 1417 ± 3611 to 25 0 0 026 to 50 0 0 0Si-Cu 2 2847 ± 39 2573 ± 47 2873 ± 383 to 10 1242 ± 34 969 ± 37 1417 ± 3611 to 25 1 ± 0.3 0 026 to 50 0 0 0Mg-Mg 2 4595 ± 49 8396 ± 85 5172 ± 503 to 10 1086 ±30 2316 ± 57 1498 ± 3511 to 25 0 2 ± 0.5 0.9 ± 0.326 to 50 0 0 0Si-Si 2 3765 ±45 3276 ± 53 2835 ± 373 to 10 777 ± 25 596 ± 28 562 ± 2011 to 25 0 0 026 to 50 0 0 0Cu-Cu 2 696 ± 19 688 ± 24 555 ± 163 to 10 43 ± 5 22 ± 3 29 ± 311 to 25 0 0 026 to 50 0 0 049Clusters can be considered to be a form of pre-precipitates, the question arises as to whether it is possibleto extend such a scaling law to the case of clusters. In this chapter, it is assumed that a scaling law existsfor clusters and that, as for precipitates, their strength scales proportionally with their radius, and is alsoindependent of their chemical composition. Using this scaling law and the cluster distribution obtained fromatom probe data by Marceau et al. [57], a prediction is given for the macroscopic yield stress of the samplesthey studied.In cluster strengthened alloys, the macroscopic yield stress arises from three different contributions: thegrain size of the alloy, σ0 (here taken as 20 MPa), the strength of the non-clustered solute atoms σ ss (hereprimarily Mg, Si and Cu), and the strength of clusters σclusters. The strength of the solid solutions, (i.e.non-clustered solute atoms) was determined first using the atom probe results for the proportion of Mg, Siand Cu not in clusters. Their individual contribution, σ ssMg, σ ssSi , and σssCu, respectively were obtained from thecalculations of Leyson et al. [86] who showed that the solid solution contribution for each element could beexpressed as:σ ssi =MkiC2/3i , (5.1)where i represents either Mg, Si or Cu, M is the Taylor factor (assumed to be 3.06), Ci, the concentration(in at.%), and ki a constant. The values of ki determined by DFT are 342, 137, and 348 MPa for Mg, Si andCu, respectively [86]. The question of how the σ ssi should be added has been investigated previously [102],[130] and depends on the breaking angle Φsolutei of each solute. These breaking angles were estimated usingthe empirical expression for the mesoscopic strength (Equation (4.4)) by solving the following expressionfor Φsolutei :σ ssi =0.9MµbLsoluteicos3/2(Φsolutei2)1− cos5(Φsolutei2)6 , (5.2)where Lsolutei is the average square spacing between solutes on the glide plane. Assuming randomly dis-tributed solutes outside of the clusters, a simple geometric calculation of the area occupied by a single atomon a {111} plane leads to:Lsolutei =31/42√Cib. (5.3)The values for Φsolutei obtained by solving Equation (5.2) are 177◦, 179◦ and 178◦ for Mg, Si and Cu,respectively. These breaking angles are all very large, meaning that the solutes are all very weak; thus the50results from a previous published work [130] suggest the use of a Pythagorean addition law:σss =√σ2Mg+σ2Si+σ2Cu=M√k2MgC4/3Mg + k2SiC4/3Si + k2CuC4/3Cu .(5.4)Next, the more complicated problem of predicting the strength of the clusters was considered. To sim-plify, a number of assumptions have been made:1. All clusters of the same size have been assumed to have the same strength, independent of their chem-ical composition.2. Each cluster is assumed to be of spherical shape.3. The radii of the clusters follow a truncated log-normal distribution with a minimum possible radiusassumed to be half the smallest bond length between first neighbours in the FCC Al crystal lattice(rmin = 0.143 nm). The probability density function of the distribution of clusters is then:f3D(r) =er f(ln(r)−µCDF√2σ2CDF)− er f(ln(rmin)−µCDF√2σ2CDF)1− er f(ln(rmin)−µCDF√2σ2CDF) , (5.5)where µCDF and σ2CDF are the mean and the variance, respectively, of the best fit to the cumulativedensity function of the average three-dimensional Guinier radius data obtained from the atom probeanalysis as reported in Table 5.3. Two examples of the fit of the cumulative density function to theexperimental data are given in Figure 5.1a and 5.1b.4. The distribution of cluster radii in the glide plane plane is the result of its interaction with a randomdistribution of spheres whose radii are obtained from the 3D distribution of clusters as illustrated inFigure 5.2. Following the work of Tallis [131], the distribution of radii on the glide plane is thus:f2D(r) = r∫ ∞r1t√t2− r2 f3D(t)dt. (5.6)5. The strength of each cluster is assumed to be directly proportional to its radius on the glide plane.51Table 5.3: Best-fit parameters of the average three-dimensional Guinier radius cumulative log-normalfunction at various aging conditions in the AA611 alloy with corresponding mean cluster radiusand variance. Also given for each condition is the cluster volume fraction f clusterstot as well as theaverage breaking angle of the distribution of clusters Φ¯clustersc . Data taken from Ref. [57]Aging condition µCDF σ2CDF Mean cluster radius (nm) Variance (nm) f clusterstot (%) Φ¯clustersc (◦)60◦C for 3 days -1.83 0.33 0.1638 ± 0.0004 0.0124 ± 0.0001 0.70 ± 0.05 16590◦C for 24 h -2 0.35 0.1745 ± 0.0004 0.0119 ± 0.0001 0.78 ± 0.06 166150◦C for 20 min -2.03 0.38 0.1781 ± 0.0004 0.0131 ± 0.0001 0.64 ± 0.05 166RT for 2 h -1.92 0.21 0.1566 ± 0.0004 0.0060 ± 0.0001 0.20 ± 0.02 168RT for 8 h -1.88 0.29 0.1613 ± 0.0003 0.0098 ± 0.0001 0.64 ± 0.04 166RT for 4 days 5 h -1.85 0.3 0.1603 ± 0.0011 0.0102 ± 0.0001 1.03 ± 0.2 166RT for 1 week -1.92 0.33 0.1672 ± 0.0004 0.0113 ± 0.0001 0.90 ± 0.07 166RT for 8 days -1.69 0.23 0.1481 ± 0.001 0.0081 ± 0.0001 0.93 ± 0.19 166RT for 2 weeks -1.91 0.28 0.1615 ± 0.0003 0.0089 ± 0.0001 0.69 ± 0.04 16700.20.40.60.811.20 0.2 0.4 0.6 0.8 1 1.2CumulativeprobabilityAverage Guinier Radius (nm)Atom probe dataTruncated log-normal CDF(a)00.20.40.60.811.20 0.2 0.4 0.6 0.8 1 1.2CumulativeprobabilityAverage Guinier Radius (nm)Atom probe dataTruncated log-normal CDF(b)Figure 5.1: Examples of best-fit of truncated cumulative density function for two aging conditions: (a)3 days at 60◦C and (b) 1 week at room temperature.[111]Figure 5.2: Schematic showing the interaction between the glide plane and a distribution of clustersmodelled by spheres.52Thus, the breaking angle of a cluster is given by:Φclusterc = 2acos(rrc). (5.7)where rc is the critical radius at which clusters become non-shearable by dislocations, and also is the onlyfitting parameter of this model. A priori, the value of rc is unknown. Therefore, different ones were tried. Theappropriate value was only determined in the end, in order for the model to fit the macroscopic experimentalyield stress of the alloy for the naturally aged conditions as reported in Table 5.4.The critical resolved shear stress of the distribution of clusters was studied using the areal glide modeldescribed in Chapter 4. The simulations featured 30,000 point obstacles that are randomly distributed onthe simulation glide plane. The breaking angle of each obstacle was randomly assigned from a discretedistribution. This discrete distribution was generated using the inverse transform method [132]. From f2Dand a uniform (0,1) random variable, the discrete 2D radius distributions was first generated. Then, usingEquation (5.7) the distribution of breaking angles was obtained. From the output of the areal glide model,i.e. the normalized CRSS τ∗clusters, the strengthening contribution from the clusters are:σclusters =M2ΓbLclusterssτ∗clusters, (5.8)where Lclusterss is the average square spacing between clusters on the glide plane, expressed as [79]:Lclusterss =(2pi3 f clusterstot)1/2< r¯ >, (5.9)Table 5.4: Yield stress at 0 K extrapolated from experimental yield stresses at 293 K and 77 K [57].Aging condition Yield stress (MPa) at 0 K60◦C for 3 days 23290◦C for 24 h 236150◦C for 20 min 234RT for 4 h 116RT for 24 h 178RT for 48 h 197RT for 1 week 237RT for 2 weeks 24753where < r¯> is the average radius and f clusterstot is the total volume fraction of clusters obtained from the atomprobe analysis. In all the cases considered, Ls varies between 4.5 and 6.4 nm, and more than 90% of theclusters have radii < 0.5 nm, suggesting that the ratio of cluster size to cluster spacing is safely greater than10. Therefore, it was reasonable to approximate clusters as point obstacles.Finally, the overall yield stress of the alloy is obtained by summing the strength contributions as follows:σy = σ0+(σssq+σclustersq)1/q , (5.10)where q is the addition exponent determined from the relationship proposed in Ref. [130]. This exponentis function of the average breaking angle of the solid solutions, and the breaking angle of the distributionof obstacles of same strength that have the same CRSS as the distribution of clusters, Φ¯clustersc . As for solidsolutions, Φ¯clustersc is determined by inverting the following equation:σclusters =0.9MµbLclustersscos3/2(Φ¯clustersc2)1− cos5(Φ¯clustersc2)6 . (5.11)The resulting breaking angles are shown in Table 5.3. The range of the average breaking angle of clusters is165−168◦ whereas it is 177−179◦ for single solutes. Therefore, following previous work [130], the valueof the exponent q is 2.Figure 5.3a presents the results for artificial aging. In this case, the model gives a very good fit tothe experimental values. Even if the clusters are weak, with breaking angles varying between 165− 168◦,significant strengthening is observed due to their high number density. Figure 5.3b presents the results fornatural aging. In this case, a more complex behaviour is observed. For aging times between 2 and 100 h, themodel predicts well the trend observed in the experimental yield stress, and only overpredicts it by ≈ 50%.For longer aging times, contrary to the experimental results, the model predicts a decrease in yield stress,primarily due to the decreasing volume fraction of clusters ( f clusterstot ) as seen in Table 5.3. To have a betterunderstanding of the origin of the misfit between model and experimental results at longer aging times, letus discuss the assumptions made earlier.1. Clusters as point obstacles: the ratio between cluster spacing and cluster size is always higher than10:1, i.e this assumption is reasonable.5405010015020025030060◦C 3 days90◦C 1 day150◦C 20 minsσ y(MPa)ExperimentModel(a)0501001502002503001 10 100 1000σ y(MPa)Ageing time (h)Experimental stressModelled stress(b)Figure 5.3: Model results compared to the experimental yield stress (extrapolated to 0 K) for (a) artifi-cial aging and (b) natural aging in the AA6111 alloy.2. Clusters are randomly distributed: the clusters being weak obstacles with breaking angles between165− 168◦, this hypothesis is reasonable as it will be shown in the next chapter that for the case ofweak obstacles, the spatial distribution does not affect the CRSS.3. Strength of clusters independent of chemistry: it is possible that the effects of chemistry dominateat longer aging times. Studying the precise implication of the orientation of big clusters with respectto the glide plane and the moving dislocation with different chemistry would be valuable. However,this approximation seems to be reasonable for small clusters.From this study, it was shown that assuming a linear relationship between cluster size and cluster strengthgives a good 1st order approximation of the strengthening due to clusters, especially at low aging times whenonly small clusters are present. This begs the question of the physical meaning of such a scaling law. Arethere physical phenomenon that can explain it? The next chapters will investigate in more detail the physicalphenomenon behind cluster strengthening.55Chapter 6Clustering: effect of spacing between soluteatoms1As shown in Chapter 5, a linear relationship between cluster size and cluster strength seems to explainreasonably the results of the experiments. This raises the question as to whether this strengthening is purelydue to the geometrical re-arrangement of solute atoms from a random distribution to an aggregated (orclustered) distribution.When solutes atoms are aggregated, i.e. when their distribution is clustered, their density throughoutthe lattice is not constant. Indeed, some areas exhibit a solute density above average (called ‘clusters’) andothers a density below average. The result of such inhomogeneity is the existence of two competiting effectson the motion of dislocations. First, inside clusters, the decrease of obstacle spacing increases the resistanceto dislocation motion. If the obstacles within a cluster are closely spaced then the dislocation will not passthrough. Instead, the dislocation will loop around the group of obstacles. In this scenario it is as if clusteringincreases the ‘effective’ strength of the solutes. In contrast, outside clusters (elsewhere), the increase ofobstacle spacing due to the low obstacle density decreases the resistance to dislocation glide.To study how these two effects affect the CRSS, the areal glide model has been used. It is an appropriatetool as, it works at the scale where the collective behaviour of solutes matter. This collective behaviour isaffected by the size and local density of areas inside or outside clusters, i.e. the clustering or aggregation1The work presented in this chapter has been published: A. de Vaucorbeil, C. W. Sinclair, and W. J. Poole, Dislocation glidethrough nonrandomly distributed point obstacles, Philosophical Magazine, vol. 93, no. 27, pp. 36643679, 201356level of the obstacle distribution. To be able to link the aggregation level to the CRSS obtained from theareal glide simulations, a single parameter was first developed to properly quantify the level of aggregation(or clustering) of a distribution of obstacles. Then, to isolate the effect due to the increase of obstacle spacingbetween clusters, the CRSS of non-shearable obstacle distributions was studied. Finally, the coupled effectof both the increase of density in clusters and the increase of spacing between clusters was analyzed.6.1 Aggregation parameterAnalyzing the effect of the aggregation level of an obstacle distribution on the CRSS using simulationsrequires the ability to create well controlled obstacle populations. This is done by randomly positioningNc virtual circles of radius r in the glide plane, within which Nin obstacles were randomly distributed (seeFigure 6.1). Different levels of clustering were obtained by varying r when holding Nin constant. As r ismade large relative to the average distance between obstacles (Ls =√A/√NcNin) the spatial distribution ofobstacles approaches random. On the contrary, as r is decreased the degree of clustering increases. Figure6.2 illustrates four different examples of clustered obstacle distributions on which areal glide simulationswere run for non-shearable obstacles, i.e. β = 1 (see Equation (2.6)). The corresponding normalized averageCRSS values for each of the simulations is also given in Figure 6.2.Visually, one can recognize the increasing clustering level in Figure 6.2. However, to precisely relate thechange in CRSS to this increase, a quantitative measure of the degree of obstacle clustering in the systemis required. Many tools to measure the ‘randomness’ of a spatial distribution of points have been developedin the field of biology and botany. Indeed, scientists from these fields have long been interested in thequantification of the spatial distribution of plant or animal species [133]. Within this community a great dealof effort has been put into developing simple parameters capable of describing the ‘randomness’ of spatiallydistributed points using either quadrat or distance based methods [133]. Quadrat methods are based on thenumber of points inside each square cell dividing the domain. In contrast, distance based methods use thedistance between ‘nodes’ of reference and their neighbouring points.A dislocation is a 1-dimensional object, and, as such, it naturally senses and is affected by distances. Thisis why a distance based method of describing clustering or ‘aggregation’ has been used as a starting point fordeveloping a measure of clustering in the current work. This method was developed by Hopkins [134], andrelies on the comparison of a reference square grid of points, referred to here as ‘nodes’, to the distribution57rFigure 6.1: Schematic representing how distributions of obstacles with different levels of clusteringwere obtained. The 14 dashed circles are the virtual circles inside of which Nin = 5 obstacles arerandomly placed.(a) τ∗c = 0.75 (r = 0.1,P = 1) (b) τ∗c = 0.71 (r = 0.01,P = 1.3)(c) τ∗c = 0.57 (r = 0.005,P = 2.0) (d) τ∗c = 0.35 (r = 0.001,P = 4.2)Figure 6.2: Partial representation (400 obstacles) of four different distributions of 40,000 obstacles withdifferent level of clustering with their respective CRSS. The level of clustering increases from (a),a quasi random distribution, to (d). For all distributions Nin = 5.58of obstacles. The distinction between obstacles and nodes is illustrated in Figure 6.3 where obstacles areindicated by blue dots and nodes are indicated by red crosses. In this method, each node is selected inturn and the distance between it and its nearest neighbour obstacles is recorded. Then the average of thesedistances is computed. As the degree of clustering increases, so does the average spacing between the nodesand the obstacles.In the context of the current problem of a dislocation gliding through an array of obstacles, the Hopkinsdefinition of a ‘neighbour’ needs to be reconsidered. One must consider not just the distance to the nearestneighbour, but to any neighbouring obstacle that could be reached by a bowing dislocation. The questionof what constitutes neighbouring obstacles in the context of this problem has been previously considered byKocks [135]. Using Kocks’ definition an obstacle P is a neighbour of a node S if no other obstacle lies in thesmallest circle connecting these two (Figure 6.4). Combining this concept of neighbouring obstacles withHopkin’s method, a parameterP can then be developed that gives a scalar measure of clustering.Let Si denote a node taken at random and let di j be the distance between Si and its jth neighbouringobstacle, Pj (see figure 6.3). If the set of di j is calculated for each of the NS nodes, one can quantify thedegree of clustering or ‘aggregation’ [134] of the obstacle distribution as,P =11.2L2s1NSNS∑i=1(Mi∑j=1d2i jMi), (6.1)where Mi is the number of neighbouring obstacles of the ith node.This is nearly identical to the definition of Hopkins’ ‘aggregation parameter’ with only the definition ofdi j changed [134] and a constant value, i.e. 1.2, inserted so as to make P = 1 for a random distribution ofobstacles (see Appendix A for derivation). It can be easily shown that P → ∞ as the degree of clusteringincreases, the theoretical upper limit occurs when all obstacles lie on top of each other. A C++ code wascreated to measure P for any given distribution. Correct implementation of the algorithm was insuredby testing it on distributions for which P is known analytically: square arrays of obstacles, and randomdistribution of obstacles.59(a)Sidi1di2di3di4(b)Figure 6.3: Processes involved in the calculation of the clustering parameter P . Given a set of pointobstacles (dots), a regular grid of ‘nodes’ (crosses) are superimposed such that the average squarespacing is the same for both. A random node from the grid is selected (zoom in (b)) and thedistances di j from these nodes to their neighbours are computed. Then P is obtained as theaverage of the square of these distances (equation (6.1))Si PjQC1C2d jFigure 6.4: Example of both neighbouring and non-neighbouring obstacles as defined by Kocks [135].Pj is a neighbour of Si because no points lies inside C1, the smallest circle going through both Pjand Si. However, Q is not a neighbour since an obstacle lies inside C2.606.2 Effect of the spacing inter-clustersWe can isolate the effect due to the increase of less populated areas by performing areal glide simulationsfeaturing non-shearable obstacles (β = 1). In this case, since the individual obstacles are already impene-trable, no ‘effective’ strengthening of individual obstacles happens, i.e. only the increase of size of the lesspopulated areas affects the glide resistance. Thus, clustering is expected to lead to a monotonic decrease ofthe CRSS as increasingly weak regions are developed in the glide plane.Areal glide model simulations were run for distributions of obstacles featuring different values of theaggregation parameterP . The resulting CRSS for these distributions versusP is plotted in Figure 6.5. Onecan see that the results collapse onto a single line well described by,τ∗c =τ∗c,random√P(6.2)where τ∗c,random is the normalized CRSS for randomly distributed strong obstacles, i.e. the CRSS shown inFigure 4.3 for β = 1. This simple relationship is not fortuitous, as it can be shown that P has a physicalmeaning in the context of this problem, particularly as the degree of clustering increases.In the limit of highly clustered distributions, the probability for a randomly selected node to fall outsideof a cluster of obstacles is much higher than the probability for it to fall inside a cluster. Since the distancebetween a node outside a cluster and its neighbouring obstacles inside a cluster will be much larger than thedistance between a node falling inside a cluster and its neighbours, P will be dominated by the distancecalculated from the nodes outside of clusters. In this case, P will be approximately proportional to theaverage of the square of the distance between the nodes and obstacles inside of clusters, and therefore to thesquare distance between clusters. The CRSS in the current case of β = 1 is given by equation τ∗Orowan =(L∗)−1, L∗ being approximately the distance between two clusters and therefore to 1/√P . One can see inFigure 6.5 that the agreement between the simulated CRSS and equation 6.2 is excellent over the full rangeof P but particularly so for high values of P (P > 4) and low values of P (P → 1), as expected basedon the arguments given above.610.111 10 100τ∗ cPSimulationsEquation (6.2)Figure 6.5: Variation of the normalized CRSS with the degree of clustering as measured by P fromsimulations with 40,000 strong (β = 1) obstacles. These distributions were obtained by varying rfrom r = 0.1 to r = 0.001 and the number of obstacles per clusters from Nin = 5 to Nin = 20. Thenumber of clusters remained unchanged as Nc = 4000.6.3 Combined effect of the spacing inter-clusters and of the ‘effective’strength of solutesThe second effect of clustering on the resistance to dislocation glide is the increase of ‘effective’ strength inclusters compared to random distributions. To study this effect independently of the spacing inter-clusters,one would need to increase the number of obstacles per clusters without changing the spacing between them.This is not possible as changing the number of obstacles per cluster will change the total number of obstacles,thus affecting the spacing between clusters (due to normalization). Therefore, the combination of both effectsis now examined. Let us consider distributions of shearable obstacles, i.e. of strength β < 1. In the limitwhen the obstacle strength β → 0 it is expected that the effect of obstacle distribution should disappear andthat the CRSS should be independent of clustering. For very weak obstacles the dislocation bends very littleand the obstacle spacing computed using the Friedel [75] or Fleischer [136] models tend towards infinity.The question is how the degree of clustering, as measured by P effects the CRSS at intermediate obstaclestrengths.Simulations were performed to determine the CRSS of distributions with different degrees of clusteringP with normalized obstacles strengths ranging between β = 0.02 to β = 1. Figure 6.6 illustrates the result-ing CRSS variation with β for four levels of clusteringP (1, 2.5, 5.2 and 20). In this Figure, the CRSS can6200.20.40.60.810 0.2 0.4 0.6 0.8 1τ∗ cβP = 1Random distributionOrowan stress(a)00.20.40.60.810 0.2 0.4 0.6 0.8 1τ∗ cββwc βwcP = 2.1Random distributionOrowan stress(b)00.20.40.60.810 0.2 0.4 0.6 0.8 1τ∗ cββwc β scP = 4.3Random distributionOrowan stress(c)00.20.40.60.810 0.2 0.4 0.6 0.8 1τ∗ cββwcβ scP = 16Random distributionOrowan stress(d)Figure 6.6: Change in the relationship between normalized CRSS and obstacle strength (β ) with thedegree of obstacle clustering,P . The results forP = 1 corresponds to the data shown in Figure4.3. The distributions of obstacles in the glide plane were generated using {r = 0.005, Nin = 5}for P = 2.5, {r = 0.001, Nin = 5} for P = 5.2 and {r = 0.005, Nin = 20} for P = 20. Thecritical values βwc and β sc fall within the shaded regions.be broadly described as following one of two types of behaviour. Below a critical value of β , which dependson the degree of clustering, all CRSS collapse on one another. Thus, for β below a threshold value βwc , thetopological distribution of obstacles in the glide plane has little or no effect on the predicted CRSS. On theother hand, for β above the threshold β sc , the predicted CRSS becomes nearly constant and equal to the valuegiven by the results for β = 1. While these two regimes appear to dominate the results, a third transitionregime between the two (βwc < β < β sc ) can also be identified (Figure 6.7). An exact identification of βwc andβ sc is difficult from Figure 6.6, instead possible ranges for these critical strengths are shown with the shaded6300.20.40.60.81βwc β sc0 0.2 0.4 0.6 0.8 1τ∗ cβIII II Ir = 0.001, Nin = 5Random distributionOrowan stressFriedel’s modelFigure 6.7: An example of the data from Figure 6.6 forP = 5.2 plotted alongside the results expectedfor a random distribution (equation 4.4) illustrating the three regimes of CRSS dependence onclustering. In region I, the CRSS is found to be independent of the obstacles’ strength, β . RegionII represents a transition regime, while in region III it is found that the CRSS does not depend onthe level of clustering. The Orowan stress is calculated using equation (6.2) while the expressiondeveloped by Friedel [75] for randomly distributed weak obstacles is also shown.regions.While the above results match well with our expectations in the limits of very low and very high strength,the clear delineation of the data into a region where the CRSS is independent of clustering and a region wherethe CRSS is independent of obstacle strength is striking. Below, we will attempt to rationalize the results, andin particular the transitions in behaviour, based on simple statistical arguments regarding dislocation-obstacleinteractions.6.3.1 Behaviour of strong obstacles (Region I):The easiest behaviour to understand (Figures 6.6 and 6.7) is that occurring when the obstacles are strong.At fixed strength, the decrease of distance between the obstacles in one cluster increases their effective localstrength leading to a situation where, above a critical value β sc , the clusters become (as a whole) impenetrable.In this case the dislocations loop around the clusters, treating them as finite-sized impenetrable objects in theglide plane. At this point, further increase in the strength of the individual obstacles (β ) will have no effectand therefore the CRSS is constant. In this region, the strength is given by the Orowan stress calculated usingthe spacing between clusters (equation (6.2)).64The transition which occurs when β = β sc marks the critical obstacle strength above which no obstaclesare overcome within the cluster. This occurs when a dislocation is pinned by all of the obstacles formedat the periphery of the cluster, here called the envelope (see example Figure 6.8). This envelope is definedas the smallest convex polygon that includes all the obstacles forming the cluster. The configuration is onaverage stable (no obstacles are overcome) if the Orowan stress (equation (6.2)) is smaller than the averagestress required to break away from these obstacles, this being expressed as,τ∗overcome envelope =|sin(Φc− α¯)|l¯√2(1+ cos(Φc− α¯)). (6.3)where α¯ is the average of the envelope’s internal angles and l¯ the average length of the envelope’s edges.When these two stresses are equal, the transition β = β sc is reached,β sc = cos(12α¯− sin l¯√Nin). (6.4)Table 6.1 presents values of β sc measured from the areal-glide model for the data shown in Figure 6.6alongside the values for β sc calculated from the equation (6.4). In this case, the values of α¯ and l¯ have beenestimated numerically from the generated obstacle distributions. Most of the calculated values fall in thelower limit of their possible range estimated from the simulation results (Table 6.1).αlFigure 6.8: Example of a 10-obstacle cluster and its envelope (dashed lines).65Table 6.1: Comparison between the transition strengths βwc and β sc predicted and their possible rangeestimated from the simulation results, these correspond to the shaded regions in Figure 6.6r Nin P β sc βwcEstimation Prediction Estimation Prediction0.005 5 2.1 0.87 to 0.94 0.84 0.57 to 0.76 0.570.001 5 4.3 0.57 to 0.72 0.71 0.42 to 0.57 0.470.001 20 16 0.39 to 0.42 0.41 0.30 to 0.39 0.29.6.3.2 Behaviour in the transition region (Region II):As the obstacle strength decreases below β sc , a regime is entered where the clusters are penetrable, but thepenetration occurs by the cluster becoming unstable triggering a collapse of the dislocation line around itsperiphery. This collapse is triggered by a ‘weak’ configuration (a location where the local spacing is largerthan average) at the periphery of the cluster. Once this weak configuration is overcome, the rest of the clusterbecomes unstable with respect to the dislocation allowing the dislocation to glide through the cluster.Under the application of a stress, a dislocation touching an obstacle at the periphery of a cluster will bowout until it breaks away from it. The area swept out for this condition (Figure 6.9) is defined as Aswept . Theobstacle is overcome when Φ corresponds to the critical obstacle breaking angle. This transition is reachedwhen the probability of finding no obstacles in the swept area (Aswept) (equal to the probability of breakingaway from an obstacle)P(Aswept = /0) = exp(−Aswept Ninpir2)(6.5)equals that of finding one within the swept areaP(Aswept 6= /0) = 1− exp(−Aswept Ninpir2). (6.6)Based on the geometry shown in Figure 6.9, this corresponds to a critical swept area,Aswept =pir2Ninln(2). (6.7)where Aswept is a unique function of β . An expression for Aswept can be readily derived from the geometry ofFigure 6.9 and used to solve equation (6.7) for β to give the transition obstacle strength βwc . The predictedvalue of βwc found in this way for the data from Figure 6.6 are given in Table 6.1. These values of βwc fall66LΦcClusterrAsweptFigure 6.9: Definition of Aswept , the maximum area of a cluster the dislocation can sweep before break-ing away from an obstacle lying at the cluster’s boundary.in the upper limit of their possible range estimated from the simulation results (Table 6.1). Therefore, for agiven distribution of obstacles, it is possible to reliably estimate the range of the three domains and predictthe CRSS of such a distribution.6.3.3 Behaviour in the cluster-insensitive limit (Region III):As noted above, it was expected that in the limit as β → 0 the effects of clustering should disappear due to thefact that the dislocation remains approximately straight as it moves across the glide plane. The results here,however, show that the insensitivity to obstacle clustering can extend to relatively strong obstacles (β > 0.5).In this region, the areal-glide simulations revealed that the propagation of the dislocation through aclustered set of obstacles is controlled by the dislocation overcoming one obstacle pinning two dislocationsegments; a long segment (length L1) connecting obstacles from two different clusters and a short segment(length L2) connecting obstacles in the same cluster.To simplify the following treatment, it is assumed that there is a high degree of clustering (P > 2). Theaverage distance between clusters is, in this case, much larger than the average distance between obstaclesin the same cluster (i.e. L1 >> L2). Since all the obstacles have a strength β < 1, the CRSS is governed byτ∗meet (see Figure 4.2). Under these conditions the CRSS can be estimated as,τ∗c =θL1/Ls(6.8)67where θ = pi−Φc.Intuitively, one would expect the distance L1 to be proportional to the average distance between clusters(L1 =C1Lcs whereC1 is a constant and Lcs = 1/√Nin the average normalized distance between clusters). Thiswould result in τ∗c = θC1√Nin, and therefore a dependence of the CRSS on the degree of clustering. What is notconsidered in this hypothesis is the possibility of a dislocation crossing a cluster without finding an obstaclewithin it. To properly account for the independence of the CRSS on clustering in this range of β , one mustinstead consider more carefully what determines the spacing L1 in equation (6.8).Figure 6.10 schematically illustrates a dislocation touching an obstacle (O) in the glide plane. We imag-ine that this dislocation segment is also touching a second obstacle belonging to an adjacent cluster (notshown). Let L be the distance (along the x-axis) between these two obstacles. The distance L1 is then themean value of L for all of the different possible configurations in the glide plane. If we know the probabilityP(L) of finding the closest obstacle to O between L and L+dL or−L and−(L+dL), then L1 can be obtainedas,L1 =∫ ∞0LP(L). (6.9)with P(L) being (see Appendix B for the derivation details):P(L) =22NcNinθpiA(1− NinNcθ2piAL2)LdL, if L<√2piAθNcNin0 otherwise(6.10)No obstaclesxL L+dL0θΦcClusterFigure 6.10: CRSS prediction at low obstacle strength. Two states of a dislocation are represented: oneat zero stress (along the x-axis) and the second at the stress required to overcome obstacle O .The dark grey shaded area is the swept area free of obstacles while the light grey shaded areacontains at least one obstacle. The closest obstacle to O is thus found between L and L+dL.68From this expression, the distance L1 is then obtained as,L1 =∫ ∞0lP(L) =∫ √ 2piAθNcNin04NcNinθpiA(1− NinNcθ2piAL2)L2dL (6.11)=1615√2piANcNinθ (6.12)≈ 2.6 Ls√θ(6.13)Substituting this expression for L1 into equation (6.13) for the CRSS gives,τ∗c = 0.35 θ3/2 = β 3/2. (6.14)This expression is identical to that developed by Friedel [75] considering randomly distributed weakobstacles. Equation (6.14) and (4.4) approach one another when β tends to zero. Comparing this predictionwith the simulation results in Figure 6.7 shows a good prediction of the simulated data in the weak limit.The results presented here, based entirely on the geometry of the obstacle distribution in the glide plane,suggest that in the case of solid solutions (weak obstacles), the obstacle distribution has no effect on theCRSS. This would suggest that the experimentally observed cluster strengthening is not a consequence ofthe simple geometric re-arrangement of solute in the glide plane. Instead, this strengthening should raisefrom the change of intrinsic strength of clusters due to chemical and/or elastic effects [110]. Thus, atomisticsimulations may be useful to understand the mechanisms responsible for cluster strengthening.69Chapter 7Intrinsic strength of individual obstacles:from single solutes to trimersAs shown in the last Chapter, the strengthening due to clustered distributions of solutes does not come fromthe change of spacing between solute atoms. Thus cluster strengthening must come from the change instrength of individual obstacles as they grow from single solutes to dimers, trimers, and even larger clusters.Here, this change has been studied by means of atomistic simulations performed at 0 K. First the interactionbetween a single solute atom with an edge dislocation was simulated and analyzed. Then, the same studywas conducted in the case of dimers and trimers.7.1 Strength of a single soluteThe question of the origin of the strength of a random solid solution needs to be examined before lookingat the origin of cluster strengthening. In random solid solutions, dislocations interact with single solutes. Asmall proportion of these solutes lie on the dislocation glide plane, but the majority does not. Although theinteraction of solutes with dislocations decreases rapidly as their distance to the glide plane increases [69],the added strength of the many atoms outside the glide plane might be non-negligible. In order to examinethis question, first the strength of a single solute as a function of its distance to the glide plane was analyzed.This data was used to develop an analytical model predicting the strength of single solutes whatever theirdistance to the glide plane. This model is based on elasticity, which describes well the interaction between70dislocation and solutes far away from the glide plane [89]. Second, the collected data on how the strengthof a single solute changes as a function of its distance to the glide plane was used to populate the areal glidemodel simulations to assess the relative importance of solutes on the glide plane versus solutes outside theglide plane, this will be presented in Chapter 8.7.1.1 Simulation resultsThe methodology for simulating the glide of a single dislocation past a single solute atom was described inChapter 4. The centre of the dislocation being located between two {111} atomic planes, when a solute islocated just above or just below the glide plane, the distance d between the solute and the glide plane is 0.5and -0.5 atomic plane spacings, respectively. When the solute is moved one atomic plane further away, ineach direction, d is respectively 1.5 and -1.5 atomic plane spacings, and so on,. . . This distance d was variedfrom -7.5 to 7.5 atomic plane spacings. The length of the box along the line direction, i.e. in the z direction,was determined for each d following the methodology presented in Chapter 4. When the solutes are on theglide plane (d =±0.5), Lz = 6 nm was selected. When the solutes are 1.5 atomic planes away from the glideplane, Lz = 4 nm was selected. Finally, when the solutes are further away, Lz = 2 nm was selected.As one can see in Figure 7.1, when d is positive, the solute is above the glide plane, thus in the com-pressive region of the dislocation. When d is negative, the solute is below the glide plane, and it interactswith the tensile side of the dislocation. Because the radius of a Mg atom is larger than that of Al [137], theinteraction between the solute and the dislocation varies depending on its position. On the one hand, whenthe solute is on the compressive side of the dislocation, the solute repels the dislocation. Figure 7.2a showsthe change of total energy of the system ∆Etot as a function of the position of the dislocation with respect tothe solute while Figure 7.2b shows the obstacle pinning force (Fobs). When the dislocation is far away fromthe obstacle, there is no interaction, hence ∆Etot = 0. As the dislocation starts to move and the leading partialgets closer to the solute, ∆Etot increases and thus Fobs becomes positive and increases too. The energy andforce reach a maximum when the solute is just above the leading partial. This is not surprising as this pointcorresponds to the point where the stress field of the dislocation is maximum [117]. Then, once the lead-ing partial has just passed the solute, the energy starts to decrease, and the pinning force changes sign, i.e.Fobs < 0, indeed the dislocation is now attracted by the position x = 0 as the solute, being repelled by bothpartials, wants to sit at equal distance to each partial dislocation. This is where the system reaches a local71dτ-+τFigure 7.1: Schematic of the pressure field due to a split edge dislocation. Red corresponds to positivepressure (compression) and blue corresponds to negative pressure (tension).stable equilibriumFobs = 0. In this position, the repulsive forces due to both partials balance each other out.As the motion of the dislocation continues, the energy and the force increase again up to their maximum.This point corresponds to the position where the solute is just above the trailing partial dislocation. Thenas the dislocation moves further, the energy drops as the pinning force changes sign, i.e. Fobs < 0. There,the dislocation is repelled by the solute and want to sit as far away from it as possible. In this case, it wasexpected that the solute would repel both partial dislocations. Indeed, the larger Mg atom prefers to sit inthe areas under tension versus areas under compression. Once this point passed, the pinning force becomesnegative as the dislocation is repelled by the solute and the energy decreases back to 0.The energy profile in Figure 7.2a presents discontinuities. This happens when one of the partials movespast the position of the solute atom. Before bypassing the solute, the dislocation is pinned, causing theapplied stress and the elastic strain energy to increase. Once the stress level is high enough for the dislocationto depin from the solute, the sign of the pinning force due to the solute changes, and all the stored elasticenergy is released, causing the dislocation to glide until equilibrium is reached again. The pinning forceprofile in Figure 7.2b also features serrations. These are due to the resistance of the lattice to the motion ofdislocation, i.e. the Peierls stress. The amplitude of these serrations are independent of the box size.On the other hand, when the solute is located on the tensile side of the dislocation, the change of totalenergy of the system is shown in Figure 7.3a, and the pinning force in Figure 7.3b. As before, when thedislocation is located far from the obstacle, there is no interaction, hence ∆Etot = 0. However, as the dis-72-0.0200.020.040.060.080.10.12-30 -20 -10 0 10 20 30∆Etot(eV)Dislocation position (A˚)(a)-0.03-0.02-0.0100.010.020.030.04-30 -20 -10 0 10 20 30Fobs(nN)Dislocation position (A˚)From τxy(b)Figure 7.2: Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides past asingle solute located 1/2 an atomic plane above the glide plane.-0.12-0.1-0.08-0.06-0.04-0.0200.02-30 -20 -10 0 10 20 30∆Etot(eV)Dislocation position (A˚)(a)-0.03-0.02-0.0100.010.020.030.04-30 -20 -10 0 10 20 30Fobs(nN)Dislocation position (A˚)From τxy(b)Figure 7.3: Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides past asingle solute located 1/2 an atomic plane below the glide plane.73location starts moving the energy decreases and this time the pinning force becomes negative. The energyreaches a local minimum when the solute is just below the leading partial dislocation. Before this minimum,the pinning force reaches its minimum and increases again to zero when the dislocation reaches the pointof local energy minimum. This point is also where the stress of the dislocation is minimum [117]. Thedislocation then overcomes a small energy barrier, correlated to the positive pinning force, until the pointof global energy minimum. This point corresponds to the position where the solute is exactly in the middleof both partials and Fobs = 0. As in the case where the atom is above the glide plane, in this position the(attractive) forces due to both partials balance each other out. Next, as the dislocation continues gliding, thetrailing partial gets closer to the solute resulting in an increase of energy. The pinning force then becomespositive as a result of the trailing dislocation getting closer to the solute atom. The next energy minimum isreached when the solute lays just below the trailing partial dislocation. From this point on, the energy risesall the way back to zero as the dislocation get out of the energy well. The pinning force is then positive andpasses through its maximum as the solute prevents the dislocation from gliding away from it.The values of the maxima of the pinning force when the solute is just above the glide plane are 0.022and 0.026 nN, for the first peak (corresponding to the interaction with the leading dislocation) and secondpeak (trailing dislocation) respectively. These values are close to the absolute values of the minimum andthe maximum pinning force in the case when the solute is just below the glide plane: 0.023 and 0.030 nNrespectively. However, the maximum pinning force is found to be a little bit higher in the latter case.It is interesting to compare ∆Etot in two different cases: (a) when the solute atom is just above or justbelow the glide plane (d = ±0.5), and (b) when the solute atom is further away for instance in the case ofd =±7.5. When the solute is just above or just below the glide plane, it is interacting directly with the core,i.e. it ‘sees’ both partial dislocations. This is attested by the shape of the ∆Etot vs. position curves presentedin Figures 7.2a and 7.3a. In this case, the interaction is strong, as corroborated by the absolute change ofenergy being around 0.15 eV. When the solute is far from the glide plane, such as for the case of d =±7.5,the solute is too far to feel the influence of the individual partial dislocations, as attested by the bell like shapeof the ∆Etot vs. position curve seen in Figures 7.4a and 7.5a. In this case, the interaction is also much weakeras attested by the absolute change of energy being one order of magnitude smaller with the absolute changeof energy being around 0.02-0.03 eV. This raises the question of how the pinning force of the obstacle varieswith its distance to the glide plane d.7400.010.02-50 -40 -30 -20 -10 0 10 20 30 40 50∆Etot(eV)Dislocation position (A˚)(a)-0.002-0.00100.0010.002-30 -20 -10 0 10 20 30Fobs(nN)Dislocation position (A˚)From τxy(b)Figure 7.4: Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides past asingle solute located 7.5 atomic planes above the glide plane.-0.03-0.02-0.010-50 -40 -30 -20 -10 0 10 20 30 40 50∆Etot(eV)Dislocation position (A˚)(a)-0.003-0.002-0.00100.0010.0020.003-30 -20 -10 0 10 20 30Fobs(nN)Dislocation position (A˚)From τxy(b)Figure 7.5: Variation of (a) total energy Etot and (b) pinning forceFobs as the dislocation glides past asingle solute located 7.5 atomic planes below the glide plane.75The variation of the maximum pinning force, Fmaxobs , for a single Mg solute atom as a function of itsdistance to the glide plane d obtained from MS simulations is plotted in Figure 7.6 (red squares). From thisplot, one can see that the strength of a solute is much higher when it lays just above or just below the glideplane (i.e. d = ±0.5). Figure 7.6 also shows that the simulation results decay as 1|d| as the distance to theglide plane increases, similar to the pressure field around an edge dislocation [117]. This would suggestthat the strength of a single Mg solute atom is dominated by the elastic interaction between solute and thedislocation even close to the core. This point will be examined in the next section.00.010.020.030.040.050.060.07-8 -6 -4 -2 0 2 4 6 8MaximumpinningforcePmaxobs(nN)Distance from the glide plane d (in number of atomic planes)MS simulationsA|d|−1Figure 7.6: Variation of the maximum pinning force Fmaxobs of a single Mg solute with its distance tothe glide plane.767.1.2 Dislocation-solute interaction: linear elasticityThe observations made in the previous section suggest that the strength of a single solute comes primarilyfrom its elastic interaction with the dislocation. This raises the question: can linear elasticity theory be usedto predict the strength of a single solute as a function of its distance from the glide plane? In the following,linear elasticity theory is used to calculate the theoretical strength of single solutes as a function of theirdistance to the glide plane. The accuracy of such predictions is then discussed.To a first order approximation, it is assumed that the solute behaves as a point defect. Within elasticitytheory, the elastic field of a point defect can be modelled by a tensor Pi j called the ‘elastic dipole’ [138].The elastic dipole of the Mg solute is determined using the method described by Clouet et al. [99]. In a MSsimulation, a box of volume V is created with a perfect aluminum crystal. Once the energy of the system isminimized, one Al atom is replaced by one Mg atom without changing the strains (εi j) on the box. Whilekeeping εi j = 0, the energy of the system is minimized and the stresses σi j recorded. The elastic dipole isdetermined by relating the change of elastic energy Edipole to the stresses on the box.Using Einstein’s notation, Edipole due to the presence of the Mg atom is [99]:Edipole =VCi jklεkl−Pi jεi j, (7.1)where Ci jkl is the stiffness matrix. The stresses therefore are [99]:σi j =1V∂Edipole∂εi j=Ci jlkεkl− 1V Pi j. (7.2)Because εi j = 0, thenσi j =− 1V Pi j. (7.3)This equation shows that stresses vary linearly with the inverse of the box volume and that the coefficient ofproportionality is the elastic dipole. The simulation is repeated for different box volumes V . Pi j is the slopeof a plot of stress versus 1/V . In the case of substitutional solute atoms, Pi j is described by a single value77which is related to the volume expansion of a solute used in classic theories:(Pi j) = P1 0 00 1 00 0 1 , (7.4)with P= 2.55 eV .The dislocation is assumed to be two straight Shockley split dislocations of mixed character and separatedby a stacking fault of length dSF . Because the solute acts as a pure dilation defect, it interacts only with theedge character of each partial dislocation. The screw character of the dislocation is thus not taken intoaccount here. As explained in Section 4.2, it is reasonable to consider aluminum to be elastically isotropic,thus the strain field due to a single partial edge dislocation at the location of the solute is [117]:(ε 1/2⊥i j ) =ε 1/2⊥xx ε1/2⊥xy 0ε 1/2⊥xy ε1/2⊥yy 00 0 0 , (7.5)with [117]:ε 1/2⊥xx (x±dSF/2,y) =−b2 y4pi(1−ν)((x±dSF/2)2+y2)2[(3−2ν)(x± dSF2)2+(1−2ν)y2]ε 1/2⊥yy (x±dSF/2,y) =−b2 y4pi(1−ν)((x±dSF/2)2+y2)2[−(1+2ν)(x± dSF2)2+(1−2ν)y2] (7.6)The variables x, y are the relative positions along the x and y axis between the centre of the split dislocationand the solute as defined in Figure 7.7. The position y is extracted from the MS simulations and correspondto the position of the atomic planes in the core of the dislocation.The solute-dislocation interaction energy is the sum of two contributions [101]:E int = Ebind+Eslip. (7.7)Ebind is the elastic binding energy and Eslip the energy due to the change of the generalized-stacking-faultenergy surface in the presence of solute(s).78Glide plane[1¯1¯1¯][1¯10][1¯1¯2]Mg solutexy= d dSFFigure 7.7: Variables used to model the interaction between dislocation and solute.The first contribution to the interaction energy, the binding energy is given by [99], [139]:Ebind(x,y) = Pi jε⊥i j (x,y)= P[ε 1/2⊥xx (x−dSF/2,y)+ ε 1/2⊥xx (x+dSF/2,y)+ ε 1/2⊥yy (x−dSF/2,y)+ ε 1/2⊥yy (x+dSF/2,y)].(7.8)The second contribution to the interaction energy, the slip interaction energy Eslip is defined as [101]:Eslip(x,y) = Aa(γAl−1Mg (C (x))− γAl (C (x)))i f y=±0.5d{111}0 otherwise, (7.9)where Aa is the area of one atom on the glide plane; γAl and γAl−1Mg are the generalized-stacking-fault energysurface of pure aluminum and aluminum with one Mg solute, respectively; C (x) is the disregistry vector andd{111} the distance between two {111} consecutive planes. Following the assumptions of Ma et al. [101], thegeneralized-stacking-fault energy surface is assumed to only change when the solute lies on the glide plane.Figure 7.8 shows both the generalized-stacking-fault energy surfaces , γAl and γAl−1Mg, obtained using MSsimulations. These surfaces were calculated from the difference of energy due to the displacement along the[1¯1¯2] and [11¯0] directions of the bottom half with respect to the top half of an aluminum lattice, when therewas no solute, and when there was a Mg solute located in the bottom plane of the top half of the lattice.Figure 7.8 also shows the disregistry vector as x goes from −∞ to +∞. The obtained variation of Eslip withthe position of the dislocation x is displayed in Figure 7.9.79Displacementalong[11¯0](A)Displacement along [1¯1¯2] (A)Energy (J/m2)CDisplacementalong[11¯0](A)0120 1 2 3 400.20.40.60.811.21.41.61.8(a) γAlDisplacementalong[11¯0](A)Displacement along [1¯1¯2] (A)Energy (J/m2)CDisplacementalong[11¯0](A)0120 1 2 3 400.20.40.60.811.21.41.61.8(b) γAl−1MgFigure 7.8: Generalized Stacking Fault energy surfaces for (a) pure aluminum and (b) aluminum withone Mg solute.-0.03-0.02-0.0100.010.020.030.04-40 -20 0 20 40Eslip(eV)Position of the dislocation (A)Figure 7.9: Energy due to the change of the generalized-stacking-fault energy surface in the presenceof solute on the glide plane (d =±0.5).Finally, as seen in chapter 4, the maximum pinning force of the obstacle predicted by linear elasticity is:Fmaxobs = max(∂E int∂x). (7.10)Figure 7.10 presents the predicted maximum pinning force of a single solute for all the simulated cases(green dots) alongside the results from the simulations (red squares). This shows that linear elasticity over-predicts the strength of a solute lying on the glide plane (i.e. d =±0.5) by a factor of about 4. However, theprediction gets better as the solute is further away from the glide plane and is better on the compressive side(right) than the tensile side (left). In fact, the simulation results present an asymmetry that is not expected8000.0050.010.0150.020.0250.030.035-8 -6 -4 -2 0 2 4 6 8MaximumpinningforcePmaxobs(nN)Distance from the glide plane (in number of atomic planes)0.140.1450.15MS simulationsLinear ElasticityFigure 7.10: Comparison between the maximum pinning force predicted by elasticity and the resultsof MS simulations.based on the linear elastic description. This asymmetry originates from the fact that away from the x and yboundaries (in the centre of the simulation cell), the dislocation bends the crystal [89]. The {111} planes arethus not planar, and the crystal not symmetrical along the y axis.Two examples are chosen to compare the predictions of the linear elastic model and MS simulations. Thefirst example is that of a solute atom 5.5 atomic planes above the glide plane. In this configuration, the atomdoes not interact with the core of the dislocation. The predictions of elasticity are thus expected to be in goodagreement with the simulations. Figure 7.11 presents the comparison between the linear elastic predictionsand the simulation results. The energy curve predicted by linear elasticity is in good agreement with thesimulations. However, the maximum pinning force is over-predicted by ≈ 15%. The second example isthat of a solute atom lying on the atomic plane just above the gliding plane (i.e. d = 0.5). In this case, theprediction is worse (see Figure 7.12). The maxima of predicted energy and pinning force are much higherthan those from the simulations. This can be explained by the divergence of the strain field in the core of thedislocation (see Equation (7.6)) [117]. This divergence, a consequence of the assumption of linear elasticity,is not physical as strains can only be finite in reality.81-0.00500.0050.010.0150.020.0250.030.0350.04-40 -20 0 20 40Energy(eV)Position of the dislocation (A)Linear ElasticityMS(a)-0.003-0.002-0.00100.0010.0020.0030.004-40 -20 0 20 40Fobs(nN)Position of the dislocation (A˚)Linear ElasticityMS(b)Figure 7.11: Comparison between the predictions obtained from linear elasticity and the MS results foran atom located 5 planes above the glide plane.-0.0500.050.10.150.20.250.30.35-40 -20 0 20 40Energy(eV)Position of the dislocation (A)Linear ElasticityMS(a)-0.15-0.1-0.0500.050.10.150.2-40 -20 0 20 40Fobs(nN)Position of the dislocation (A)Linear ElasticityMS(b)Figure 7.12: Comparison between the predictions obtained from linear elasticity and the MS results foran atom lying on the plane just above the glide plane.82Due to discontinuities present in the displacement field in the simulations, determining an accurate strainat each atom site is a challenging task. This is why the displacement fields from the MS simulations andfrom the linear elasticity theory will be compared, instead of the strain fields. The displacement fields ux anduy are showed in Figures 7.13 and 7.14, respectively.−56 56−6060x (A˚)y(A˚)[1¯10][1¯1¯1¯][1¯1¯2]0-2.86ux(a)x (A˚)y (A˚)ux-60 -40 -20 0 20 40 60-60-40-200204060-2.860(b)Figure 7.13: Comparison between the displacement field ux of a split edge dislocation obtained from(a) MS simulations and (b) linear elasticity.−56 56−6060x (A˚)y(A˚)[1¯10][1¯1¯1¯][1¯1¯2]0-0.9uy(a)x (A˚)y (A˚)uy-60 -40 -20 0 20 40 60-60-40-200204060-0.90(b)Figure 7.14: Comparison between the displacement field uy of a split edge dislocation obtained from(a) MS simulations and (b) linear elasticity.83One can see that discrepancies exist between the MS simulations and the linear elasticity theory. The uxfield obtained from the simulations is clearly asymmetrical unlike the prediction of elasticity theory. Thiscorresponds to the asymmetry observed in the maximum pinning force of the simulation results (see Figure7.11). This asymmetry can be explained by the imposed boundary conditions in x. Elasticity theory assumesthe media to be infinite. However, the MS simulations use periodic boundary conditions in x. This imposesthe x = ±5.6 nm planes to remain vertical, thus resulting in extra displacements on the atoms. Meanwhile,while the uy field shows the same symmetry as elasticity theory, its magnitude is lower in the core of thedislocation compared to elasticity theory. In addition, a discrepancy can be observed close to the boundaries.This is is due to the periodic boundary conditions in x and the free surfaces at y = ±14 nm. Returning tothe predictions from elasticity (Equation (7.8)), one can substitute the strain field directly observed in MSsimulations to see if this allows the discrepancy to be resolved. If so, it would prove that the strength of asolute comes primarily from elastic interaction with dislocations.7.1.3 Dislocation-Solute interaction: elasticity using the elastic strain field of thedislocation from MS simulationsFrom the MS simulations, the strain field can be obtained two ways: (a) from the gradient of the atomisticdisplacement field of the dislocation, or (b) from the virial stress field of the dislocation [140]. The atom-istic displacement field is the measure of the displacement of each atom caused by the introduction of thedislocation in the simulation box. The virial stress is a measure of the stress at each atomistic site which wasproven to be equivalent to the continuum Cauchy stress by Subramaniyan and Sun [141]. In order to obtainthe strain field from the displacement field, the gradient of the displacement field needs to be calculated. Thisis troublesome as the displacement field ux is discontinuous in the half space (x≥ 0), for y = 0 (see Figure7.13a). Therefore, this method does not allow the calculation of the strain field right on the half glide planein front of the dislocation, and thus also in its core. For this reason, it was decided to use the virial stressfield from the simulations to determine the strain field at each atom site. The obtained strain field of the splitedge dislocation determined in this way is:ε⊥i j = Si jklσ⊥i j , (7.11)84where σ⊥i j is the virial stress tensor due to the split edge dislocation at each atomic site, and Si jkl is thecompliance matrix of the aluminum lattice, which using Voigt’s notation is [117]:(Si j) =S11 S12 S12 0 0 0S12 S11 S12 0 0 0S12 S12 S11 0 0 00 0 0 S44 0 00 0 0 0 S44 00 0 0 0 0 S44, (7.12)where S11, S12 and S44 are three coefficients which vary with the local state of stress. Therefore, the interac-tion energy is given by:E int(x,y) = Ebind(x,y)+Eslip(x,y) = Pi jSi jklσkl(x,y)+Eslip(x,y)= P(S11+2S12)(σxx(x,y)+σyy(x,y)+σzz(x,y))+Eslip(x,y),(7.13)where Eslip is as defined is Equation (7.9). In the following, it will be assumed that S11 and S12 are indepen-dent of the state of stress and equal to their value in the zero stress aluminum matrix (bulk value) [117]:S11 =C11+C12(C11−C12)(C11+2C12) = 1.33×10−2 GPa, (7.14)S12 =− C12(C11−C12)(C11+2C12) = 4.57×10−3 GPa. (7.15)Considering S11 and S12 to be equal to their bulk value outside the core of the dislocation where linearelasticity works well is a good approximation [117]. In the core of the dislocation and in the atomic planejust above the glide plane, the virial pressure, i.e. −(σ11+σ22+σ33)/3, is negative and reaches a minimumvalue of −2000 MPa. There, it was found using Liu’s EAM interatomic potential that (S11 + 2S12) is 9%lower than its bulk value. Therefore, under this approximation, the absolute value of E int and Fobs will beunderpredicted when the solute is on the plane just above the glide plane. In this case, this approximationcould lead to an error between the model and the simulation results of less than 9%, which seems reasonable.On the other hand, in the core of the dislocation and in the atomic plane just below the glide plane, the virialpressure is positive and reaches a maximum value of 3100 MPa. There, it was found that (S11 + 2S12) is8520% higher than its bulk value. Therefore, under this approximation, the absolute value of E int andFobs willbe overpredicted when the solute is on the plane just above the glide plane. In this case, this approximationcould lead to an error between the model and the simulation results of less than 20%. This error seemsimportant, however, it seems reasonable compared to the overprediction of 380% achieved by the linearelastic model when the solute is located on the plane just below the glide plane (Figure 7.10).Figure 7.15 shows that there is good agreement between the simulation results and the maximum pinningforce predicted by the elastic model using MS stresses estimated directly from the virial stresses. This modelwill be referred to as the ‘hybrid elastic model’, or ‘hybrid elasticity’ in the following, and the simulationresults. Noting that the prediction does not agree with the simulations for the cases where the solute is (a)1.5 atomic planes above the glide plane, and (b) on the plane just below the glide plane.The first case of disagreement between theory and simulations is when the solute is 1.5 atomic planesabove the glide plane (d = +1.5). In this case, it was found that the core of the dislocation changes. Thisis illustrated by examining how the disregistry changes in this case compared to case where the solute justabove the glide plane (d = +0.5): see Figure 7.16. One can see that the curve for d = +0.5 has two stepscorresponding to each split dislocations, however, when d =+1.5, the disregistry curve presents four steps.This would indicate that the dislocation further splits. This change in the core structure of the dislocation maybe an artifact from the potential used, and therefore not a true representation of what happens in experiments.00.0050.010.0150.020.0250.030.035-8 -6 -4 -2 0 2 4 6 8MaximumpinningforcePmaxobs(nN)Distance from the glide plane (in number of atomic planes)MS simulationsHybrid elasticityFigure 7.15: Comparison between the maximum pinning force predicted by the hybrid elastic modeland the results of MS simulations.8600.511.522.533.54-40 -20 0 20 40Disregistry(A˚)Position of the dislocation (A˚)d =+0.5d =+1.5Figure 7.16: Disregistry: comparison between the case where the solute is just above the glide plane(d = +0.5) and the case where the solute is located 1.5 atomic planes above the glide plane(d =+1.5).DFT calculations could be performed to check this. This idea is supported by the fact that such behaviour isnot observed when the solute is in the tensile side of the dislocation (d < 0).The second case of disagreement between theory and simulations is when the solute is lying on theatomic plane just below the glide plane (d = −0.5). In this case, the hybrid elastic model considerablyunderpredicts the strength of the solute (see Figure 7.18), which is surprising since the approximation ofthe elastic coefficients S11 and S12 by their bulk value should have resulted in an over prediction of themaximum pinning force. This difference is obvious when the shape of the energy curves are compared (seeFigure 7.18a). Quantitatively, the shape obtained by the model is in good agreement with the simulations.However, the energy well is wider in the predictions of the hybrid elastic model. Also, a sharp peak in thesimulation energy profile is visible around x = 1 nm and is not captured by the model. Its origin may comefrom a change in the core of the dislocation, but it could also originate from the assumption that the elasticstiffness matrix is independent of the stress level. As explained above, this approximation leads to a possibleunderestimation of the absolute value of the strain, thus the absolute change of energy as observed here.Nevertheless, predictions for the case where the solute is just above the glide plane (d = +0.5) are in verygood agreement with the simulations (see Figure 7.17). In addition, if the simulation box size were largeenough to make the boundary effects disappear, the stress field should be symmetrical and antisymmetricalwith respect to the glide plane. Thus the maximum pinning force of a solute lying just above or just below theglide plane should be the same. Therefore, it would be sufficient to be able to predict correctly the maximum87-0.0200.020.040.060.080.10.120.14-40 -20 0 20 40Energy(eV)Position of the dislocation (A˚)Hybrid ElasticityEslipMS(a)-0.03-0.02-0.0100.010.020.030.04-40 -20 0 20 40Fobs(nN)Position of the dislocation (A˚)Hybrid ElasticityEslipMS(b)Figure 7.17: Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom lying on the plane just above the glide plane.-0.12-0.1-0.08-0.06-0.04-0.0200.020.04-40 -20 0 20 40Energy(eV)Position of the dislocation (A˚)Hybrid ElasticityEslipMS(a)-0.03-0.02-0.0100.010.020.030.04-40 -20 0 20 40Fobs(nN)Position of the dislocation (A˚)Hybrid ElasticityEslipMS(b)Figure 7.18: Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom located on the plane just below the glide plane.88pinning force of a solute lying just above the glide plane. Meanwhile, when the solute is outside of thedislocation core, the predictions are in good agreement with the simulations with errors of less than 10%.Two examples are given in Figure 7.19 and 7.20. This result was expected as explained in Section 7.1.1.It is important to note that the non-negligeable contribution of the energy due to the change of thegeneralized-stacking-fault energy, i.e. Eslip, when the solute is just above or just below the glide plane (seeFigure 7.17 and Figure 7.18). Indeed, one can see that the contribution of Eslip to the maximum absolutechange of energy and maximum pinning force is between 20 and 30%. When the solute is not on the glideplane, like in Figure 7.19 and 7.20, Eslip is not represented as it is assumed to have no effect.When comparing the simulation results and the predictions of the hybrid elastic model in Figure 7.17one can notice that, contrary to the predictions of the hybrid elastic model, the energy curves obtained fromthe MS simulations are not symmetrical. In Figure 7.18, it can be seen that the slope of the pinning force aspredicted by the model is higher than that obtained from the simulations, and that important disagreementcan exist between the minima. It is worth noting that such discrepancies between the simulation results andthe predictions of the hybrid model are not observed when the solute is located further away from the glideplane as shown in Figure 7.19 and 7.20. These observations could be explained by the fact that, contraryto the assumptions made in the hybrid elastic model, the dislocation does not remain straight when a shearstrain is applied (see Figure 7.21). When the dislocation is far away from the solute (Figure 7.21a), theinteraction between them is low, thus the dislocation is straight. As the dislocation approaches the solute, theincrease of Peach-Kohler force, coupled with the pinning force of the solute cause the dislocation to bend(Figure 7.21b). Then, bending increases as the dislocation is pinned by the solute (Figures 7.21c and 7.21d).The bending of the dislocation affects the measure of the position of the dislocation. Because the presence ofthe solute atom changes the disregistry, as seen in Figure 7.16, the measure of the position of the dislocation(see section 4.2.3) is done in the z= Lz/2 plane, which is the furthest xy plane from the solute atom (locatedin the z= 0 plane). Therefore, under high applied shear stress, the measured position of the dislocation doesnot correspond with its position in the z = 0 plane, causing the energy curve to by asymmetrical, and theslope of the pinning force curve to be lower than that predicted by the hybrid elastic model (which assumesthe dislocation to be always straight). The difference of slope is also responsible for the shift of the locationof the maxima of force. Eventually, when the maximum pinning force is reached and the dislocation movesfree from the solute (Figure 7.21e). All the stored elastic energy is released back into the system causing the8900.010.020.030.040.05-40 -20 0 20 40Energy(eV)Position of the dislocation (A˚)Hybrid ElasticityMS(a)-0.003-0.002-0.00100.0010.0020.0030.004-40 -20 0 20 40Fobs(nN)Position of the dislocation (A˚)Hybrid ElasticityMS(b)Figure 7.19: Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom located 5.5 planes above the glide plane.-0.05-0.04-0.03-0.02-0.0100.01-40 -20 0 20 40Energy(eV)Position of the dislocation (A˚)Hybrid ElasticityMS(a)-0.004-0.003-0.002-0.00100.0010.0020.0030.0040.005-40 -20 0 20 40Fobs(nN)Position of the dislocation (A˚)Hybrid ElasticityMS(b)Figure 7.20: Comparison between the predictions obtained from elasticity using strains from MS sim-ulations and the MS results for an atom located 5.5 planes below the glide plane.90xzτ(a)xzτ(b)xzτ(c)xzτ(d)xzτ(e)Figure 7.21: Schematic showing how the dislocation bends when interacting with an obstacle (here asolute atom).dislocation to move a great distance at once, this distance increasing with the magnitude of the maximumpinning force, it results with the impossibility for the dislocation to stabilize right after the maximum pinningforce peak, i.e. where the hybrid elastic model predicts the presence of the pinning force minima. This resultin the pinning force minima obtained from the simulations to be higher than those predicted by the hybridelastic model.. Finally, it can be observed that the model underpredicts the maximum energies. This couldbe due to the stopping criterion of the dislocation which uses a force tolerance as a stopping criterion insteadof an energy tolerance criterion (see Section 4.2.1).In this section, it was shown that the proposed hybrid elastic model, works well for predicting the maxi-mum pinning force in the simple case of a single solute atoms. The question arises as its validity in the caseof clusters; starting with the simplest case: dimers. This will be discussed in Section 7.2.917.2 Strength of dimersDimers are the smallest type of clusters as they are aggregates of only two solute atoms. The strength of adimer depends on three parameters, its crystallographic orientation, its orientation with respect to the glidingdislocation, and its distance from the glide plane. A set of unique crystallographic orientation and orientationwith respect to the dislocation line designate one particular dimer. Due to the symmetries of both the latticeand the dislocation, some dimers are equivalent. After looking at all the possible arrangements, fourteenunique dimers have been identified, each having been labelled with Roman numerals from I to XIV. Figure7.22 shows a stacking of three close packed atoms forming {111} planes: the plane just above the glideplane, i.e. close packed plane C is represented by atoms with black contours; the plane just below the glideplane, i.e. close packed plane B is represented by atoms with red contours; and the plane further below (1.5planes below the glide plane), i.e. close packed plan A is represented by atoms with blue contours. The sameschematic is used in Figure 7.23 to show all the possible arrangements of dimers with their correspondingstrength. Each dimer is composed of two solutes: the first is the grey filled atom, and the other is one of thelabelled atoms. For instance, Dimer I is composed of the grey filled atom and one of the red filled atomslabelled I. In Figure 7.23, one can see that there are four different dimers labelled I. Due to the symmetry ofthe system, these correspond to four equivalent arrangements. This is also the case of the two dimers labelledVII.The strength of a dimer lying on the glide plane is expected to be more than twice that of a singlesolute. The difference of strength between a dimer and two independent single solutes is expected to comefrom the presence of the chemical Mg-Mg interaction. The presence of this interaction could also explainthe difference of strength between dimers. The pinning force of a dimer whose configuration is changedwhen sheared by the dislocation raises not only from the interaction energy between each solute and thedislocation, but also from the change of its chemical energy. Therefore, the dimers whose configurations arechanged when sheared by the dislocation are expected to have a higher strength than the others. However,the problem is more complex as the interaction between dimers and dislocations involves multiple factorssuch as elastic interactions and changes in configurations.Molecular statics simulations have been performed for each of the arrangements in Figure 7.23. Thesesimulations were done following the methodology explained in Chapter 4, with a fixed box length alongthe dislocation line of Lz = 6 nm. The maximum pinning force Fmaxobs obtained from these simulations for92B BBA AACABC[111][112][110][110][112][111]Dislocation DislocationFigure 7.22: Schematic showing the position of the different close packs A (red), B (blue), and C(black) and the corresponding colour coding used in Figure 7.23.[110][112][111]Dislocation0.0240.043Fobs (nN)IIIIIIII IIV VIVII VIIVVIIIIXXXIXIIXIIIXIVmaxFigure 7.23: Arrangement and maximum pinning forceFmaxobs of dimers. Each dimer is formed by theatom filled in grey and one of the atoms labelled from I to XIV. Dimers are labelled as a functionof their strength: the lower the number the higher their maximum pinning force. Dimer I is thusthe strongest and Dimer XIV the weakest. The colour of the contour line of the atoms representsthe close pack they belong to as shown in Figure 7.22: blue for pack A, red for B, and blackfor C. Finally, the colour of the filling of the atoms depends on the magnitude of the maximumpinning force of the dimer they form with the grey filled atom. The glide plane is between the Band C close packs.9311.522.533.540 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5NumberofconfigurationsinintervalMaximum pinning forceFobs (10−2 nN)Figure 7.24: Distribution of maximum pinning force for all the different configurations of dimers de-termined from the results of MS simulations.[110][112][111]Dislocation(a) Before shearing[110][112][111]Dislocation(b) After shearingFigure 7.25: Arrangements of Dimer I (strongest dimer) lying just above the glide plane (a) beforeand (b) after passage of the dislocation. Both Mg atoms are on the same plane, therefore theconfiguration of Dimer I does not change.94each arrangement of dimers is presented in Figure 7.23. The distribution of strengths is shown in Figure7.24. The distribution of maximum pinning force shows that the strength of the clusters ranges from that ofa single solute when just above the glide plane, i.e. 0.024 nN, to a value of just less than twice this amount.The average strength of a dimer is found to be of 0.037 nN which is approximately 50% greater than thestrength of a single solute. The weakest dimer is only 8% stronger than a single solute, whereas the strongestdimer is approximately 1.7 times stronger than a single solute. Noting that the density of dimers for a fixedsolute concentration is half that of single solutes, when passing from single solutes to dimers, the effect ofthe increase of spacing between obstacles might be expected to compensate for the increase of their strength.This will be studied in detail in Chapter 8.Surprisingly, the strongest dimer was found to be Dimer I (see Figure 7.25), which does not experiencea change of configuration when the dislocation shears through it. This contradicts the assumptions that bondbreaking is critical to determine the strength [102]. This result is similar to that of Patinet and Proville [84]who found that the maximum pinning force for Dimer I was 0.047 nN [84] (0.043 nN in the current work).The strongest dimer whose configuration is changed by the passage of the dislocation is Dimer II (see Figures7.23 and 7.26).The weaker strength of Dimer II compared to Dimer I can be explained by the compensative interactionbetween the dislocation and each one of the atoms forming Dimer II as illustrated in Figure 7.27. Indeed, forthis arrangement one of the Mg atoms is on the tensile side of the dislocation, thus its interaction is attractive;whereas the other has a repulsive interaction due to its position on the compressive side. These forces do notexactly balance each other due to the angle the dimer forms with the direction of the dislocation line, leadingto a net elastic interaction between the dimer and the dislocation. It is difficult to assess what the contributionof the change of configuration to the strength of the Dimer II is, as it is not physically possible to simulatethe interaction between Dimer II and a dislocation without causing a change of configuration. However, thiscan be tested using the following analytical model.The hybrid elastic model used for single solute atoms in Section 7.1.3 is used here to treat the interactionbetween dimers and an edge dislocation. In contrast to the problem of the interaction between a single soluteand a dislocation, the problem of the interaction between a dimer and a dislocation is a three-body interaction.Indeed, each solute forming the dimer interacts with the dislocation and the other solute as illustrated inFigure 7.28. The interaction between each solute and the dislocation is the same as the one used in the case95[110][112][111]Dislocation(a) Before shearing[110][112][111]Dislocation(b) After shearingFigure 7.26: Arrangements of Dimer II (second strongest dimer) lying on the glide plane (a) beforeand (b) after passage of the dislocation. Mg atoms are on the planes respectively just above andjust below the glide plane. Therefore Dimer II experiences a change of configuration when thedislocation shears through it.-+MgMgFigure 7.27: Schematic of the compensative interaction that can exist between the two solutes of adimer when they are located just above, or just below the glide plane, respectively. In suchconfiguration, the solute above attracts the dislocation, while the solute below repels it.96ChemicalElastic ElasticSolute 1 Solute 2DislocationFigure 7.28: Three body interactionof single solutes. However, the question arises regarding the nature of the interaction between solutes. In theprevious section, it was shown that single solutes act as perfect centres of dilation. It can be shown that twocentres of dilation in an isotropic medium do not elastically interact with each other [100]. Therefore theonly interaction existing between the two solutes is a chemical interaction, which depends on the distancebetween them.The interaction energy between each solute forming the dimer and the dislocation is identical to thatbetween a single solute and the dislocation. However, the x= 0 and y= 0 reference points are now when thecentre of the dimer is at the middle of the dislocation and when the dimer is across the glide plane (i.e. whenboth solutes are on either side of the glide plane), respectively. Therefore, the interaction energy between adimer and the dislocation is:E intDimer(x,y) = Eintsolute1(x−12~dMg−Mg.~x,y− 12~dMg−Mg.~y)+E intsolute2(x+12~dMg−Mg.~x,y+12~dMg−Mg.~y), (7.16)with ~dMg−Mg being the distance vector between the two solutes forming the dimer, and E int the interactionenergy between a single solute and the dislocation.First, let’s look at the case of the strongest dimer: Dimer I. During the passage of the dislocation pastit, the distance between the Mg atoms of Dimer I experiences a small fluctuation of less than 2%, which isassumed to be negligible. Hence, only the interaction between each solutes and the dislocation will have anon-negligible influence on the total energy of the system as the dislocation glides past Dimer I. Therefore,from elasticity theory, the change of total energy is the sum of the interaction energy from solute 1 and solute2, respectively:∆EDimerI = E intDimer. (7.17)Figure 7.29 shows the difference between the simulation results and the predictions from the model in97-0.0500.050.10.150.20.25-40 -20 0 20 40Energy(eV)Position of the dislocation (A)Hybrid ElasticityMS(a)-0.06-0.04-0.0200.020.04-40 -20 0 20 40Fobs(nN)Position of the dislocation (A)Hybrid ElasticityMS(b)Figure 7.29: The case of Dimer I lying just above the glide plane. Comparison between the predictionsobtained from the hybrid elastic model and the MS results for (a) the change of total energy ofthe system as the dislocation glides past the dimer and (b) the pinning force due to the dimeronto the dislocation.the case where Dimer I is just above the glide plane. One can see that the change of total energy as well asthe pinning force due to the dimer are well predicted using the hybrid elastic model. More specifically, themaximum pinning force obtained from the model is in excellent agreement with that from the simulationswith values of 0.045 nN and 0.043 nN, respectively. However, one can see that the hybrid elastic modeloverpredicts the energy when the dimer is located in the middle of both partial dislocations, i.e. x= 0. Thiscould be explained by the fact that the slip energy is assumed to be the sum of the individual slip energiesfrom each atom, as if they were the only solute present in the lattice. In addition the distance between bothsolutes is supposed to be fixed, thus ignoring the complex interaction between both solutes together, as wellas with the stacking fault.In contrast to Dimer I, Dimer II experiences a change of configuration as presented in Figure 7.26.Therefore the change of total energy is the sum of the interaction energy form both solutes and the changeof chemical energy of Dimer II, ∆EchemicaldimerII , as follows:∆EdimerII = E intDimer+∆EchemicaldimerII . (7.18)98To determine the change of chemical energy of Dimer II, the change of distance between Mg atoms in thedirections [1¯1¯2] and [11¯0] as the dislocation moves was extracted from the MS simulations. The resultingvector was called CdimerII . Then, the generalized-stacking-fault energy surface of aluminum with Dimer IIγAl−dimerII was calculated. Finally, from γAl , γAl−1Mg, and γAl−dimerII , the change of chemical energy wascalculated as:∆EchemicaldimerII (x) = Aa[γAl−dimerII− γAl(CdimerII(x))]−2Aa [γAl−1Mg (CdimerII(x))− γAl (CdimerII(x))] . (7.19)The result is shown in Figure 7.30a. One can see that the energy of the dimer after passage of the dislocationis less than before its passage. This is not surprising as it is known that this potential reproduces a tendencyfor ordering at low temperatures. Experimentally, this leads to the formation of L12 Al3Mg [91], [119]. Itcan also be seen that the change of chemical energy is non negligible and significantly affects the changeof total energy predicted by the model. However, the change of chemical energy has little influence on themaximum pinning force as seen in Figure 7.30b. As in the case of a single solute (see Section 7.1.3), thedifference in the position of the peak force observed between the simulations and the hybrid elastic model-0.2-0.15-0.1-0.0500.050.10.15-40 -20 0 20 40Energy(eV)Position of the dislocation (A)Elasticity (w/o chemical interaction)Elasticity (w/ chemical interaction)Chemical interactionMS(a)-0.04-0.0200.020.04-40 -20 0 20 40Fobs(nN)Position of the dislocation (A)Elasticity (w/o chemical interaction)Elasticity (w/ chemical interaction)Chemical interactionMS(b)Figure 7.30: The case of Dimer II lying on the glide plane. Comparison between the predictions ob-tained from the hybrid elastic model when the chemical interaction between both Mg atoms isconsidered or not, and the MS results for (a) the change of total energy of the system as thedislocation glides past the dimer and (b) the pinning force due to the dimer onto the dislocation.99could be explained by the bending of the dislocation under the applied stress as described in Section 7.1.3for a single solute atom.This particular example shows that it is not so much the amplitude of change in the chemical energy thatmatters but the amplitude of its first derivative with respect to the dislocation position. But more importantlythe position of the maximum or minimum of this derivative is the most important. This example showsthat in spite of a significant change in chemical energy, the interaction between Mg atoms has a negligibleeffect on the strength of Dimer II. These two examples show that elasticity is the most important mechanismresponsible for the strength of dimers in the Al-Mg system when they are lying on the glide plane. It remainsto see how the strength of dimers varies with their distance to the glide plane?Simulations of the interaction between dislocations and Dimer I as a function of its distance to theglide plane have been performed. The results for the maximum pinning force obtained from this simulationare presented in Figure 7.31 alongside the predictions from the hybrid elastic model. One can see thatthe predictions from the hybrid elastic model using the strain field of dislocations from MS are in verygood agreement with the simulations for all cases but two. The cases for which predictions are not inagreement with the simulations are when the dimer is just below and 1.5 atomic planes above the glide plane,respectively. These correspond to the cases for which predictions were already in poor agreement with thesimulations for single solutes. Because the same model is used, it is not surprising that the predictions failfor these two cases as discussed in Section 7.1.3. One can also see from Figure 7.31 that away from the glideplane, the strength of a dimer decreases as 1/r, as predicted by classical linear elasticity theory. Also, theresults show that the strength of the dimer on the glide plane is at least twice that of dimers that are not onthe glide plane.10000.010.020.030.040.050.06-6 -4 -2 0 2 4 6MaximumpinningforcePmaxobs(nN)Distance from the glide plane (in number of atomic planes)MS simulationsHybrid ElasticityFigure 7.31: Variation of the maximum pinning force of Dimer I as a function of its distance to theglide plane.1017.3 Strength of trimersAfter dimers, the next bigger family of clusters in terms of size is that of trimers, which are aggregates of threesolute atoms. As for dimers, their strength depends on their crystallographic orientation, their orientationwith respect to the gliding dislocation, and their distance to the glide plane. In Chapter 8 it will be shown thatin the case of dimers 84% of the CRSS comes from clusters on or across the glide plane. Therefore, only thestrength of trimers on the glide plane will be discussed here. The question arises as to how many differentconfigurations of trimers exist? This is not an easy question. To help answer this question, trimers are hereconsidered to be made of two configurations of dimers joined together by the middle atom of the trimer(see Figure 7.32), which is shared by both ‘dimer’ segments. Let’s now explore the degrees of freedom of atrimer. The first degree of freedom is the position of the middle of the cluster, i.e. the middle atom, or atomB in Figure 7.32. This atom can either be located just above or just below the glide plane. However, due tothe crystal and dislocation symmetries, having the middle atom just above or just below the glide plane isequivalent. The other degrees of freedom are the type of dimer of the segment AB, and BC, respectively, andthe angle between them. From Figure 7.23, the number of possible configuration for each segment is 18. Itis not 14 as the configurations of dimers that are equivalent in the case of the interaction between dimers anddislocation will not necessarily be equivalent when combined with another one to form a trimer. Therefore,the total number of configurations found is (18× (18−1))/2 = 153.It was shown that the hybrid elastic model gives very good predictions for the maximum pinning force ofsingle solutes as well as dimers, but would it give as good predictions in the case of trimers as in the case ofABCSegment Dimer ISegment Dimer XIV[110][112][111]DislocationFigure 7.32: Trimers could be seen as being composed of two ‘dimer’ segments. For instance, heresolutes A and B form a dimer of type Dimer I; and solutes B and C form a dimer of type DimerXIV.102dimers? Given the large number of trimers, it would be impractical for all to be tested via MS simulations.Therefore two configurations were selected for a detailed analysis. The first configuration, called Trimer I isbuilt from two Dimer I configurations arranged ‘in line’ (see Figure 7.33). This configuration is expected tobe the strongest, or one of the strongest configurations of Trimer, as Dimer I is the strongest configurationamong the dimers. As this configuration does not involve any change in topology, the second configuration(Trimer II) was chosen such that a change in topology does occur. Trimer II is built as a combination ofDimer I and Dimer II as shown in Figure 7.34. This configuration was chosen as it is expected to also bestrong, though less than Trimer I, as it is built from the two strongest configurations of dimer. Molecularstatic simulations have been performed for these two configurations of trimers. These simulations are donefollowing the methodology explained in Chapter 4, with a box length along the dislocation line of Lz= 14 nm.The appliction of the hybrid elastic model for the case of trimers requires the consideration of four bodyinteractions. However, as the interactions between solute atoms are neglected, only the interactions betweenthe dislocation and the individual solutes are considered (see Figure 7.35). Therefore, the change of totalenergy of the trimers is given by:∆Etrimer =E intsolute1(x−12~dMg−Mg.~x,y− 12~dMg−Mg.~y)+E intsolute2(x+12~dMg−Mg.~x,y+12~dMg−Mg.~y)+E intsolute3(x+12~dMg−Mg.~x,y+12~dMg−Mg.~y),(7.20)with E intsolutei, where i= 1,2 or3 being given by Equation (7.13).Figure 7.36 and Figure 7.37 present the results given by the hybrid elastic model compared to the resultsof the MS simulations, in the case of Trimer I and Trimer II, respectively. These figures show that the modeldoes not fit the simulation results as well as in the case of dimers. The predicted maximum pinning forceare, however, within 13%, and 1% or the simulated values, respectively. Surprisingly, the prediction is bet-ter in the case of Trimer II. This suggests that the effect of the change of topology is well captured by thehybrid elastic model. These two examples show that in the case of trimers, elasticity is the main mecha-nism responsible for the strength of these clusters and that the hybrid elastic model gives good predictions.Therefore, the model can be used to predict the strength of the clusters for which we have not performed MSsimulations. These predictions will then be used to populate the glide plane of the areal glide simulations to103[110][112][111]Dislocation(a) Before shearing[110][112][111]Dislocation(b) After shearingFigure 7.33: Arrangements of Trimer I lying just above the glide plane (a) before and (b) after passageof the dislocation. The three Mg atoms are on the same plane, therefore the configuration ofTrimer I does not change.[110][112][111]Dislocation(a) Before shearing[110][112][111]Dislocation(b) After shearingFigure 7.34: Arrangements of Trimer II lying on the glide plane (a) before and (b) after passage ofthe dislocation. Two Mg atoms are on the plane just above the glide plane, whereas the otheris on the plane just below. Therefore Trimer II experiences a change of configuration when thedislocation shears through it.DislocationElasticElastic ElasticFigure 7.35: Four body interaction104-0.0500.050.10.150.20.250.30.35-40 -20 0 20 40Energy(eV)Position of the dislocation (A)Hybrid ElasticityMS(a)-0.08-0.06-0.04-0.0200.020.040.060.08-40 -20 0 20 40Fobs(nN)Position of the dislocation (A)Hybrid ElasticityMS(b)Figure 7.36: The case of Trimer I lying just above the glide plane. Comparison between the predictionsobtained from the hybrid elastic model and the MS results for (a) the change of total energy ofthe system as the dislocation glides past the dimer and (b) the pinning force due to the dimeronto the dislocation.-0.15-0.1-0.0500.050.10.150.2-40 -20 0 20 40Energy(eV)Position of the dislocation (A)Hybrid ElasticityMS(a)-0.06-0.04-0.0200.020.040.060.08-40 -20 0 20 40Fobs(nN)Position of the dislocation (A)Hybrid ElasticityMS(b)Figure 7.37: The case of Trimer II lying just across the glide plane. Comparison between the predic-tions obtained from the hybrid elastic model and the MS results for (a) the change of total energyof the system as the dislocation glides past the dimer and (b) the pinning force due to the dimeronto the dislocation.105obtain the CRSS of a random distribution of trimers.Using the hybrid elastic model, calculations have been performed to determine the strength of all the153 different configurations of trimer identified. The resulting distribution of maximum pinning forces ispresented in Figure 7.38. The weakest trimer has a strength just a little below the average strength of dimerswith 0.033 nN. The maximum pinning force is 0.072 nN which is a little more than twice that of the averagestrength of dimers and a 50% increase over the maximum strength of dimers. From these results, one cansee that the strength of the strongest trimer is in the 0.07 to 0.075 nN range. However, the strength of TrimerI, which was expected to be the strongest trimer, is just 0.063 nN (see Figure 7.36b). Instead, the strongesttrimer is represented in Figure 7.39. The strongest configuration is however built from two segments of thestrongest dimer (Dimer I). It is also worth noting that about 15% of trimers are weaker than the strongestdimer.051015202530354045500 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8NumberofconfigurationsinintervalMaximum pinning forceFobs (10−2 nN)Figure 7.38: Distribution of maximum pinning force for all the different configurations of trimers de-termined using the hybrid elastic model.106[110][112][111]DislocationFigure 7.39: Configuration of the strongest trimer7.3.1 DiscussionIn this chapter, we focused on the strength of small Mg clusters. A model, the hybrid elastic model, basedon elasticity theory, but using strains obtained from the MS simulation of dislocations, was developed andsuccessfully used to predict the strength of individual solutes, dimers and trimers. Using this model, it wasshown that, for Mg clusters, the change of chemical energy resulting from the change of topology of a cluster(sometimes refered to as bond breaking) does not significantly affect its strength. However, it is legitimateto ask what would happen in ternary alloys such as Al-Mg-Si, since the change in the chemical energy ofa Mg-Si dimer whose configuration is identical to that of Dimer II is one order of magnitude higher thanthat of Dimer II (in Al-Mg) [102]. The maximum of the resulting force would therefore be about one thirdof the maximum pinning force of Dimer II, which is non-negligible. However, not only the maximum ofthe derivative of the change of chemical energy is important, but also where this maximum happens. If itis aligned with the peak of force due to the elastic interaction, then it could have a significant effect on themaximum pinning force, otherwise its effect could be negligible. If the chemical interaction profile is of thesame shape as the interaction profile in the case of Dimer II (see Figure 7.30b), for example, this maximumwould not coincide with the peak force due to the elasticity. There, the force due to the change of chemicalenergy would then be about 10% of the present maximum pinning force. In this case, neglecting the chemicalinteraction seems to be a good first order approximation, Nevertheless, rigorous calculations need to be doneto precisely determine the effect of multiple elements. To perform these rigorous calculations, one wouldneed to determine (a) the dipole moment of Si, and (b) the slip interaction energy Eslip. This could be doneusing DFT, for example, as no EAM potentials exist for Al-Mg-Si.107Chapter 8Relationship between cluster size andcluster strength in binary Al-Mg alloysThe problem of cluster strengthening is not only limited to understanding the strengthening at the scale of asingle cluster. Indeed, when a dislocation moves on its glide plane, it will interact with clusters of differentstrengths (even if the material is populated by clusters of the same size). As seen earlier in Chapter 7, thisis due to the different strength of clusters depending on their relative position and orientation with respectto the glide plane, as well as their topology, and internal arrangement (i.e. position of the solutes inside thecluster). As explained in the literature review, the problem of the strength of random solid solution has beenwidely studied. However, this is not true in the case of randomly distributed dimers, or trimers. Little isknow of the effect of the aggregation of a solid solution to first dimers, then trimers, and eventually biggerclusters.The objective of this section is to investigate the CRSS due to a random distribution of fixed solidsolutions, then dimers, and finally trimers, using the results presented earlier in this thesis and the areal glidesimulations. The areal glide model is the appropriate tool for this work as it captures the statistical effectof a large number of obstacles (20,000 and more), which is out of reach for MS or MD simulations. Theresulting critical resolved shear stresses are then used to compare the strengthening effect of solid solutions,dimers and trimers with the scaling law postulated in the study of the aged strengthened AA6111 alloys (seeChapter 5).1088.1 CRSS of random solid solutionsIf we imagine a volume of a matrix of aluminum randomly populated by Mg solutes, and we take a randomglide plane (i.e. a {111} plane), some solutes are going to be located on the glide plane, and others at adistance to the glide plane. The dislocation will then interact with the two types of obstacles: solutes onthe glide plane, and obstacles that are the orthogonal projection of the out-of-plane solutes on the glideplane. The strength of each obstacle will depend on their distance to the glide plane as shown in Section7.1.1. As solute atoms are weak obstacles, they would not likely cause a moving dislocation to cross slipor climb, noting also that the temperature is 0 K. Therefore, this problem can be modelled as a 2D problemof a dislocation gliding through a random distribution of point obstacles whose strength vary according tothe distance to the glide plane of the 3D distribution of solute. Using our areal glide model, this problem issimulated as the interaction of a moving dislocation with a random distribution of N obstacles on the glideplane, and the strength of each obstacle being randomly picked from the distribution of strength of a singlesolute as a function of its distance to the glide plane.A convergence analysis similar to that of Nogaret and Rodney [92], was performed in order to determinethe number N of obstacles required in the simulations. Figure 8.1 shows the variation of the normalizedCRSS of randomly distributed solute atoms when Nplanes = 24 as a function of N. Convergence was assumedto be reached when the error between two consecutive points was less than 2%. This criterion was found tobe satisfied when N = 40,000.The following assumptions have been made for the simulations. First, the solutes are assumed to be pointobstacles. For a solute concentration of 2%, the average square distance between solutes is one order of mag-nitude higher than the radius of an atom, i.e Ls = 1.3 nm, and ra = 0.2 nm, thus this approximation seemsreasonable. Under this approximation, the force-distance profile interaction with dislocations is reduced to aDirac function whose maximum corresponds to its maximum pinning force. The second assumption is thatthe dislocation is a single line. The areal glide model does not account for dissociated dislocations. Theaverage square distance between solutes is close to the distance between partial dislocations. Therefore, as-suming the dislocation as a single line seems to be un-reasonable, as both partial dislocations could interactwith different solutes at the same time. This problem could be seen as each partial dislocation interactingwith the same distribution of obstacles, one after the other. The obstacles being randomly distributed, andassuming that the distance between partials remains constant, this is equivalent to having a single disloca-10900.00010.00020.00030.00040.00050.00060.00070.00080 10000 20000 30000 40000 50000 60000NormalizedCRSSτ∗ SSNumber of obstacles NFigure 8.1: Normalized CRSS of randomly distributed solute atoms when Nplanes = 24 as a function ofthe number of obstacles in the areal glide model.tion interacting with twice as many obstacles. As a result, following Patinet and Proville [84], the thirdassumption is that the concentration of obstacle is twice that of the nominal solute concentration. The fourthassumption is that the dislocation has a constant line tension. Again, Patinet and Proville showed that theshape of the line of a dislocation under stress and pinned by solute atoms is well approximated by arcs of acircle even when the pinning solute atoms are as close as only a few Burgers’ vectors [84]. Thus the constantline tension approximation is reasonable. It is also assumed that the probability of finding a solute atom is in-dependent of the plane considered, thus considering the solid solution perfectly random. Finally, the obstaclestrengths used in the simulations are the ones obtained from the MS simulations for distances up to 8 planesabove the glide planes, and from the elasticity model when located further away. This is appropriate, as itwas shown in section 7.1.3 that when the solute is located 4 planes away from the glide plane, the elasticitymodel used gives results similar to that of MS simulations.Due to the long range interaction between single solutes and dislocations (see Figure 7.15), the questionarises as to how many planes above and below the glide plane one should consider here. This was done bypopulating the glide plane of the areal glide model simulations with a population of N randomly distributedobstacles. Each obstacle was then randomly associated with one of Nplanes possible classes, such that thenumber of obstacles in each class was the same. This number corresponds to the number of atomic planes(on either side of the glide plane) to be considered, each class being the equivalent of one {111} plane. Toeach obstacle in the same class, the same breaking angle was assigned. For instance, the breaking angle of110024681012140 4 8 12 16 20 24CRSSτ SS(MPa)NplanesFigure 8.2: CRSS of randomly distributed single solutes as a function of the number of planes aroundthe glide plane considered Nplanes, and for a Mg concentration of 1 at%.obstacles of the pth class was the breaking angle of a solute atom located at a distance d = p−Nplanes/2from the glide plane, which is, using (4.10):Φc(d = p−Nplanes/2) =Fmaxobs (d = p−Nplanes/2). (8.1)The value ofFmaxobs is taken from the results of the MS simulations when (|d|< 7.5) and from the predictionsof the hybrid elastic model elsewhere. The accuracy of the underlying assumptions made here could bequestioned as the point obstacle approximation corresponds to reducing the entire pinning force profile of asolute atom to a simple Dirac function of same maximum amplitude. The pertinence of the transition fromMS results to the areal glide model will be discussed in Chapter 8.Figure 8.2 shows the variation of CRSS (τSS) obtained for Nplanes varying between 2 (only the plane justabove and below the glide plane are considered) and 24, and for a concentration c = 1 at.%. The CRSSis calculated using Equation (4.2) from the normalized CRSS τ∗SS given by the areal glide simulations asfollows:τSS = τ∗SSLsµb, (8.2)where τ∗SS is the normalized CRSS given by the areal glide simulations. The square spacing Ls between the111atoms in the areal glide simulations depends on the number of planes considered. In the {111} plane of aFCC crystal the density of atoms is ρ = 2/(√3b2), therefore square spacing between obstacles on the glideplane (i.e. the projection of every solute atoms whose distance to the glide plane is less or equal to Nplanes/2planes) is [130]:Ls =31/42√2cNplanesb, (8.3)where c is the concentration of solutes. Therefore, Equation (8.2) yields:τSS = 2µ√2cNplanes31/4τ∗SS. (8.4)These results show that, as expected, most of the CRSS arises from the solutes in the glide plane as theycontribute for 87% of the total CRSS. This value is higher than that obtained by Patinet [69] with 80%. Thedifference of result can be explained by the fact that Patinet used MS simulations that cannot simulate 40,000randomly distributed solutes at concentrations lower or equal to 10 at.%. The results presented here showthat considering only single solutes in the glide plane is a reasonable approximation. If all the solutes atomslocated on all the Nplanes planes are considered, the CRSS for a random distribution of Mg solute atoms isτSS = 184 MPa×√c. If only single Mg atoms on the glide plane are considered, τSS = 158 MPa×√c.The question of the strength of a solid solution of Mg atoms in aluminum was studied in depth by Patinetusing MS simulations as described in his thesis [69]. He used MS simulations to calculate the CRSS ofdifferent solute concentrations: 2%, 6%, and 10%. On the other hand, Leyson et al. used DFT coupledwith Labusch’s model. They found that τSS = kMgc2/3, where kMg = 342 MPa [86]. The results of theareal glide model are shown in Table 8.1 alongside those of Patinet and Leyson et al, and also results fromexperiments published by Lloyd and Court [142], for the same concentrations 2, 6, and 10at.%. One cansee that, the results from the areal glide simulations are closer to those of Leyson et al. [86] than those ofPatinet [69]. Nogaret and Rodney [92] showed that before convergence is reached, the CRSS decreasesmonotonically with the number of obstacles in the simulation box. The number of obstacles to dislocationmotion in Patinet’s simulations [69] (see Section 2.2.2.4), the real CRSS for random solute atoms of Mg inaluminum are expected to be lower or equal to the results of Patinet’s MS simulations as it is the case for theresults from the areal glide simulations described above. For the concentrations 6% and 10% the results fromthe areal glide simulations are closer to the experimental results published by Lloyd and Court [142] than112Table 8.1: Critical resolved shear stress obtained from the areal glide simulations compared with se-lected simulation and experimental results. The values in parenthesis correspond to the case whereonly solute atoms on the glide plane are considered.Mg concentration (at. %)Critical Resolved Shear Stress (MPa)Areal Glide Simulations Patinet [69] Leyson et al. [86] Experiments [142]2 25 (22) 56 25 116 45 (39) 148 52 3210 59 (50) 254 74 53.the results of either Patinet or Leyson et al. Though, EAM potentials being an approximation of the complexinteratomic interactions, the simulation results are not expected to be in perfect quantitative agreement withthe experimental values.8.2 CRSS of randomly distributed dimersA study on the relative importance of the dimers outside the glide plane compared to that of dimers in theglide plane on the strength of the material was thus performed following the same procedure as used for thecase of single solutes (Section 8.1). Figure 8.3 shows the variation of CRSS τc obtained in the case wherethe obstacles are all Dimer I for Nplanes. These results show that as expected, most of the CRSS is due to thedimers in the glide plane as they contribute for 84% of the total CRSS. Hence, as in the case of single solutes,considering only dimers in the glide plane seems to be a reasonable approximation. Therefore, for the sakeof simplicity, only the contribution from dimers on the glide plane will be considered in the following.0510152025300 4 8 12 16 20 24CRSSτ c(MPa)NplanesFigure 8.3: CRSS of randomly distributed Dimer I as a function of the number of planes around theglide plane considered Nplanes, and for a Mg concentration of 1 at%.113The problem of the interaction between a dislocation and a random distribution of dimers is thus reducedto a 2D interaction. It is considered that each dimer has the same probability to exist on the glide plane.Our areal glide model is used to simulate the CRSS of a random distribution of dimers under the sameassumptions as in the case of solute atoms. Compared to the solid solution case, the only differences beingthe increase of size of the physical obstacles and their spacing. The radius of a dimer is about 0.25 nm,but the average square distance between obstacles has increased as well to about 1.8 nm, therefore, the pointobstacle approximation is still reasonable. The simulated glide plane is populated by a random distribution ofpoint obstacles. The obstacles are divided into 14 different sets, each set corresponding to one configurationof dimers. The 14 contain the same number of obstacle, except the set corresponding to Dimer I whichcontains 4 times as many, and the set corresponding to Dimer VII which contains twice as many. This is dueto the fact that there exist 4 different arrangements of Dimer I, 2 of Dimer VII, and only one for the rest ofdimers. The breaking angle of the obstacles of a certain set is defined as:Φc = 2acos(Fmaxdimeri2T), (8.5)where Fmaxdimeri is the maximum pinning force of the corresponding dimer lying on the glide plane. Thesimulation featured 40,000 in order to achieve convergence of the results.From the areal glide simulations, it was found that the CRSS for a random distribution of dimers is:τdimers = 173 MPa×√c, (8.6)where c is the concentration of Mg solute atoms. This represents a small (9%) increase over the CRSS ofsingle solutes (when compared to the CRSS of random single solutes on the glide plane). Here, the increaseof strength of the obstacles is almost compensated by the effect of the increase of spacing between them.Indeed, on the one hand, the square spacing between dimers is√2 times the spacing between solutes, thuscausing a decrease of strength of√2 of the distribution. On the other hand, the average (as seen by thedislocation) strength of dimers is 1.55 times that of single solutes.1148.3 CRSS of randomly distributed trimersAs for dimers, the contribution of trimers out of the glide plane is neglected. All the different configurationsof trimers are supposed to have the same probability to exist. Similarly, our areal glide model is usedto simulate the CRSS of a random distribution of point obstacles representing trimers. The obstacles aredivided into 153 different sets, each set corresponding to one configuration of trimers. The simulationsfeatured 40,000 in order to achieve convergence of the results.From the areal glide simulations, it was found that the CRSS for a random distribution of trimers is:τtrimers = 219 MPa×√c, (8.7)where c is the concentration of Mg solute atoms. The CRSS of a random distribution of trimers is therefore27% higher than that of a random distribution of dimers, and 38% higher than that of single solutes (on theglide plane).8.4 CRSS as a function of cluster sizeFigure 8.4 shows the evolution of the CRSS as Mg solute atoms go from solid solution (Natoms = 1) todimers (Natoms = 2), then trimers (Natoms = 3), when the total concentration of solute is of 1 at%. The bluedots correspond to the case where all the different configurations of a same class clusters have the sameprobability of being found; whereas the red dots correspond to the case where only strongest configurationcan be found, thus representing an upper bound for the strength. For now, let us assume that the probabilityto find each configuration of dimer is the same (blue dots). The upper bound case will be discussed later.One can see, as explained earlier, that the strengthening effect due to the transition from a solid solution toa set of dimers is low. However, in the case where all configurations are present, when clusters grow fromdimers to trimers, the increase in CRSS is much higher. Indeed, it is 4 times higher than the increase whenpassing from solute atoms to dimers. This can be compared to the results for the scaling law used in the caseof the study on cluster strengthened in AA6111 presented in Chapter 5.The scaling used in the case of the AA6111 in Chapter 5 assumes that the strength of clusters β varieslinearly with their radius. It also assumes that when the radius of clusters reaches a critical radius rc, clusters115051015202530350 1 2 3CRSSτ c(MPa)Number of atoms per obstacleAll configurationsStrongest configurationFigure 8.4: Critical resolved shear stress of randomly distributed solute atoms, dimers and trimers, fora total Mg concentration of 1 at%. The blue dots correspond to the stress of distributions featuringall configurations of clusters. Whereas red dots correspond to the stress of distributions featuringonly the strongest configuration.become unshearable and thus their strength reaches 1. Therefore, the scaling law was expressed as follows:β =rrc. (8.8)In order to compare this scaling law with the results from the areal glide simulations, one needs to expressthe radius of a cluster as a function of its number of solute atoms Nin. In FCC metals, there are 4 atoms perunit cell. Simple geometry yields to the volume of a unit cell as being Vcell = 2√2b3. Thus the volume peratom in an FCC metal is:Va =b3√2. (8.9)Clusters in Al-Mg being spherical, it is assumed that even very small cluster have the same topology. Thus,from the previous equation, the radius of a cluster of Nin atoms is given by:r = b(4Nin3√2pi)1/3. (8.10)The strength of the random distributions of clusters simulated using the areal glide model is calculated fromthe normalized CRSS τ∗clusters obtained from the simulations by numerically solving Equation (4.4) for thestrength.116The change of strength of random distributions of clusters as a function of their radius obtained from theareal glide model coupled with the elasticity model developed here is shown in Figure 8.5 along the empiricalscaling law used in the AA6111 study. These results show that, as mentioned before, the increase of strengthof Mg clusters smaller than quadrimers (not included) is not exactly linear. Confirmation of this observationfor a wider range of clusters would require data for bigger clusters, eg. quadrimers and pentamers. Theincrease of strength with cluster size is however lower than what was seen in the case of the AA6111 (seealso Chapter 5). Indeed, the slope of the AA6111 scaling law is about 5 times higher than that of the bestlinear fit to the simulation results when all configurations are assumed to be equally present in the medium.This could be explained by the higher strengthening observed in the AA6111 compared to binary Al-Mgalloys: it takes about a day of natural aging for AA6111 alloys to the strength of the AA6111 alloy doublesafter 24h [5], when it takes binary Al-Mg alloys a long 8.5 years to see their strength double [16] as explainedin Chapter 2. Another explanation could come from the way clusters are identified during the analysis of theatom probe data. The efficiency of the detection of atoms being limited, this could cause Mg atoms presentin clusters to not be detected, thus artificially underestimating the size of clusters. This would lead to anunder-prediction of the critical radius rc at which clusters become non-shearable by dislocation. The resultwould be an over-prediction of the slope of the scaling law used in the case of the AA6111 (Chapter 5).00.010.020.030.040.050 0.05 0.1 0.15 0.2 0.25 0.3 0.35Strengthβ clusterObstacle radius (nm)Single solutesDimersTrimersSimulations - All config.Simulations - Strongest config.Best linear fitAA6111 Scaling lawFigure 8.5: Comparison between the scaling law used in the study of a clustered strengthened AA6111and the strength of random distributions of solute atoms, dimers, or trimers as obtained fromthe areal glide simulations.The blue dots correspond to the strength of distributions featuring allconfigurations of clusters. Whereas red dots correspond to the strength of distributions featuringonly the strongest configuration.117The analysis of the AA6111 atom probe data showed that ≈ 60% of the detected clusters were Mg-Mg or Mg-Si clusters [57], and that the number of Mg-Si clusters was ≈ 50% higher than that of Mg-Mgclusters [57]. As explained in Section 7.2, in the case of a Mg dimer, when one Mg atom is located on thetensile side of the dislocation, and the other one is located on the compressive side, as in the case of Dimer IIor Dimer VIII, both Mg atom would have an opposite interaction with the dislocation (one repulsive, and theother attractive) which would partially compensate each other (see Figure 8.6a). If one of the Mg atoms wereto be replaced by an atom with a negative dipole moment P (see Figure 8.6b), both atoms would have eithera repulsive or an attractive interaction with the dislocation which would make the dimer much stronger. Thiswould be the case if this atom is a Si atom, since a Si atom is 12% smaller than an Al atom [137]. However,when both the Mg and the Si atom would be on the same side of the dislocation, as in the case of Dimer I,their interaction with the dislocation would be opposite and partially compensate each other. The numberof configurations where both solute atoms are on either side of the dislocation being twice as many as thosewhere the solute are on the same side of the dislocation (see Figure 7.24), replacing a Mg atom by a Siatom is expected to increase the strength of a random distribution of dimers. The important difference in theslope of the scaling law between Al-Mg and AA6111 alloys suggests that the strengthening effect of Mg-Siclusters is much more important than that of Mg-Mg clusters and is key to explain the difference of strength-+ττMgMg(a)-+ττMgSi(b)Figure 8.6: When atoms on either side of the glide plane are both bigger than an Al atome.g. both Mg(a), their individual interactions with the dislocation nearly balance each other out. When one isbigger, and the other smaller, e.g. Mg and Si, respectively (b), their individual interactions withthe dislocation reinforce each other. The shades of red represent compressive stresses, and shadesof blue tensile stresses.118between Al-Mg and AA6111 alloys.The dipole moment of vacancies being also negative (−3.18 eV), the strength of a distribution of Mg-vacancy clusters is expected it be even higher than that of a distribution of Mg-Si clusters. However, the con-centration of vacancies in aluminum alloys being a few orders of magnitudes lower than that of solutes [13],[143], the role of vacancies on the strength of clusters is expected to be negligible.The hypothesis made earlier that all cluster configurations have the same probability to exist can alsobe discussed. The solutes atoms being able to diffuse, the configurations of clusters could evolve with timeand the presence of dislocations as argued by Curtin and coworkers [144]. They showed using Monte Carlosimulations that in the presence of dislocations, the Mg solutes diffuse across the core of the dislocationsto move from the energy unfavourable compression side of the dislocation to its energy favourable tensionside. As seen in the case of dimers and trimers, the strength of clusters is on average higher when the solutesare on all on the tensile side of the dislocation. The same simulations have been performed when only thestrongest clusters where present. The results are shown in both Figure 8.4, and 8.5. In this case, the slopeof the increase between dimers and trimers is now in fairly good agreement with the scaling law used inAA6111. However, it is not sure that all cluster will see their configuration change as they interact with agliding dislocation as vacancies are needed for diffusion to occur. Nevertheless, it is possible that not allconfigurations of cluster would be found in real alloys. One should see this as being an upper bound for thestrength of a distribution of clusters. Further simulations could be done to study this problem more in depth.119Chapter 9Conclusion and future work9.1 Thesis summaryThe first objective of this thesis was to predict the yield stress of an aged AA6111 alloy from the 3D distri-bution of clusters extracted from atom probe tomography data. The strengthening contribution of the distri-bution of clusters was determined using an areal glide model assuming the clusters with perfectly sphericalshape and that their strength depends linearly on the radius of their intersection with the glide plane. The 0 Kyield stress was then estimated by summing the strengthening contribution of the clusters, the solute atomsleft in solid solution and the grain size of the alloy. It was found that, in the case of naturally aged samplesaged for times shorter than 100 hours, the trend of the predicted 0 K yield stress was in good agreement withthat of the 0 K yield stress extrapolated from experiments. The relatively good predictions obtained usingthese simple assumptions motivated a study of the origin of such a scaling law.The second objective was to identify which mechanisms of interaction between dislocations and soluteatoms are responsible for cluster strengthening. From the literature, it was identified that cluster strength-ening would arise from the change in the way solute atoms are distributed as they cluster, or the intrinsicchange in the local interaction between dislocations and obstacles when they grow from single atoms toclusters. First, the effect of the distribution of solute atoms was studied. The areal glide model was usedto simulate the interaction between a dislocation and point obstacles whose distribution progressively variedfrom random to highly aggregated (or clustered). The results of these simulations showed that for weakobstacles, such as solute atoms, the level of aggregation has no effect on the strength of the distribution.120Second, the effect of the local interaction between dislocations and obstacles (single solute atoms, dimers,and trimers) was studied in the simple case of Al-Mg binary alloys. In the literature, it was argued thatthe strength of single solute atoms and single clusters would come from their elastic interaction with dislo-cations, the change in stacking fault energy when they are on the glide plane, and the change of chemicalenergy (or bond breaking) in the case of clusters. In addition, the strengthening effect of solute atoms orclusters located outside of the glide plane was usually supposed to be negligible.Molecular static simulations were used to study and analyze the interaction between dislocations anda single solute Mg atom as a function of its distance to the glide plane. It was found that classical linearelasticity would give good predictions of the maximum pinning force of a solute but only when it is locateda few {111} planes away from the glide plane. A new ‘hybrid’ model, based on classical elasticity but usingthe elastic strain field calculated from the results of MS simulations, and taking into account the change ofstacking fault energy when the solute is located on the glide plane was developed to obtain better predictionswhen solute atoms interact with the core of dislocations. Good predictions for the strength of a single soluteatom were obtained with this new model called hybrid elasticity, in particular when the solute is locatedclose to the glide plane.Next, molecular static simulations were used to study the interaction between dislocations and dimers,then trimers. The hybrid elastic model was successfully used to predict the strength of all the differentcombinations of dimers, as well as the few configurations of trimers simulated. Moreover, it was shownthat, in the case of Al-Mg alloys, the change of chemical energy (or bond breaking) due to the passage ofdislocation through a cluster had only a negligible influence on the strength of the cluster.The results of these molecular static simulations, coupled with the predictions of the hybrid elastic modelwere used to study with the areal glide model the strengthening effect of single solutes, and clusters locatedoutside of the glide plane. It was found that, as argued in the literature, neglecting their strengtheningcontribution was reasonable.Finally, all these results were employed in the areal glide model to study the strengthening effect of smallclusters. The results of this study gave three data points (for single solutes, dimers and trimers, respectively)on a strength versus size plot which were compared to the scaling law used to determine the yield stress ofan AA6111 alloy. The strengthening effect predicted for Al-Mg was found to be around 5 times less thanin the case of an AA6111 alloy. It was speculated that this could be explained by the fact that Si atoms are121smaller than Al atoms, therefore, when Mg and Si atoms are on either side of the glide plane, they havesimilar interaction with the dislocation, causing the strength of Mg-Si clusters to rise compared to that ofMg clusters. However, the resulting three points on the strength versus size plot were not sufficient to beable to make firm conclusions about the slope, and shape of a scaling law (linear, quadratic, or other) inAl-Mg alloys. However, tools and models have been developed that would allow future investigations of thestrengthening effect of distributions of bigger clusters, such as quadrimers, pentamers, and so on. . .The challenge associated to the study of bigger clusters is the rapidly growing number of possible con-figurations with the number of atoms present per cluster. Moreover, not every configurations would havethe same thermodynamic stability; some would be more likely to exist than others. Therefore, an exhaustivestudy of every configurations might be irrelevant. Instead, analyses should be performed in order to deter-mine which configuration exist as well as probability of existence. This could be done using tools such asMonte Carlo simulations, for example.In this thesis, a hybrid elastic model was developed and shown to be adequate to predict the strength ofsingle solutes and clusters in Al-Mg alloys. This model, coupled with the areal glide model, could be used inother works to derive physically based scaling laws for other alloys such as ternary alloys which are of greatinterest today.9.2 Suggestions for future workStudying the strengthening effect of clusters larger than trimers.In this thesis, atomistic simulations were used to study the interaction between dislocations and clustersconstituted with a maximum of three atoms. The limitation to trimers was due to the rapid increase of thenumber of possible configurations a cluster can take as its size increases. However, not all the differentconfigurations would be found in real systems. In a future work, Monte Carlo simulations could be used todetermine which cluster configurations that are thermodynamically stable as a function of the cluster size.Subsequently, hybrid elasticity technique and the areal glide model described in Chapter 8 could be used todetermine the strengthening effect of clusters as a function of their size. For large clusters, the tendency ofMg solute atoms to order following the L12 structure as seen in Section 2.1.1 might prevail. Again, MonteCarlo simulations could be used to determine the cluster size at which ordering becomes prevalent. Then,122molecular static simulations could be used to determine how ordering affects the maximum pinning force ofa cluster.Strengthening effect of clusters in ternary alloys.As suggested in Chapter 7, the hybrid elastic model developed this same chapter could be used to studyternary alloys such as Al-Mg-Si, or Al-Mg-Zn. In order to do so, one will have to determine the dipolemoment of Si, and Zn and how these solute atoms modify the slip interaction energy. As no EAM potentialsexist for Al-Mg-Si or Al-Mg-Zn, this could be done using DFT, for example. Monte Carlo simulations couldbe used to determine the thermodynamically stable configurations of cluster existing in these alloys. Theresults of these simulations could then be used as in Chapter 8 to populate the glide plane of the areal glidemodel and obtain a physically based prediction for the scaling law in the case of ternary alloys.ExperimentsIt is difficult to make a direct comparison between the results obtained in Chapter 8 and experimental datapublished for the distribution of clusters in naturally aged Al-Mg alloys. It would be valuable to performatom probe tomography and tensile experiments on naturally aged binary Al-Mg alloys in order to comparethe slope of an experimentally fitted scaling law with that obtained in Chapter 8. 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The circle of centre O and radius E(rn) containing exactly n−1 points, the probability forQn to be a neighbour is thereforePn =(34)3/4. (A.2)By definition,P =11.2L2S∑+∞n=1E(rn)2Pn∑+∞n=1Pn(A.3)which numerical solution is:P = 1. (A.4)133Appendix BDetail of the derivation of P(L)The detail of the derivation of the probability P(L) of finding the closest obstacle to a pinned obstacle, i.e.here the obstacle O (see Figure B.1), between L and L+dL or −L and −(L+dL) is as follows.No obstaclesxL L+dL0θΦcClusterFigure B.1: CRSS prediction at low obstacle strength. Two states of a dislocation are represented: oneat zero stress (along the x-axis) and the second at the stress required to overcome obstacleO . Thedark grey shaded area is the swept area free of obstacles while the light grey shaded area containsat least one obstacle. The closest obstacle to O is thus found between L and L+dL.If the closest obstacle to O is located at L then there should be no obstacle for x< L. There are two waysthat this can be true. First, it could mean that there is no cluster in this range. Given that the clusters arerandomly distributed, the probability for the axis x to cross no cluster in the range 0 < x< L is:P([0,L] 6= /0cluster) = 1− e−2NcRcLA . (B.1)The second possibility is that a cluster is crossed in x < L but that no obstacles are found within thecluster in that range. Computing the probability for this to be true requires that the bowing of the dislocationbe accounted for. In Figure B.1 the angle θ represents the angle swept by the dislocation when the stress is134increased. In the weak obstacle limit the resulting swept area is small and, therefore, the probability for thedislocation to meet no obstacles in a crossed cluster before overcoming O is approximately,P( /0 at x) = 1− Ninθx2piRc. (B.2)Considering a small dx along x, and based on equation (B.1), the probability for the x axis to intersect acluster between x and x+dx is,P([x,x+dx]> 0) =2NcRcAdx (B.3)and the probability to intersect no cluster between x and x+dx based on equation (B.2), isP([x,x+dx] = /0cluster) = 1− 2NcRcA dx. (B.4)The probability to find no obstacles in the area swept by the dislocation between x and x+ dx is thengiven as the probability that a cluster is crossed in x and x+dx but that no obstacle in that cluster is crossed,P([x,x+dx] = /0) = P([x,x+dx] 6= /0cluster)P( /0 at x)+P([x,x+dx] = /0cluster) (B.5)= 1− NinNcθpiAxdx. (B.6)Integrating this expression between 0 and L gives the probability of finding no obstacles in 0 < x< l,P([0,L] = /0) = 1− NinNcθ2piAL2 (B.7)The probability that one cluster is intersected between l and l+dl comes directly from equation (B.3),P([L,L+dL] = /0cluster) =2NcRcAdL (B.8)while the probability for that cluster to not be empty within the swept area is,135P(Cluster at L 6= /0) = NinθLpiRc(B.9)Based on these expressions, the probability of the nearest neighbour to O being between l and L+dL isgiven as the product of the probability of i) finding a cluster at L (equation (B.8)), ii) the probability that thecluster at L is not empty (equation (B.8)) and iii) the probability that there are no other obstacles between 0and L (equation (B.7)),P(L) =22NcNinθpiA(1− NinNcθ2piAL2)LdL, if L<√2piAθNcNin0 otherwise(B.10)where the condition that L<√2piAθNcNin is a consequence of the fact that P(L)≥ 0.136

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