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Variations with oxygen concentration of titanium lattice parameters are obtained by means of ab initio calculations, considering the impact of oxygen ordering. The quasiharmonic approximation is used to take into account the thermal expansion at finite temperature. Results show that lattice parameters depend mainly on oxygen concentration and, to a lesser extent, on the ordering state. Knowing these theoretical variations, one can get insights into the composition of ordered compounds existing in Ti-O binary alloys from their lattice mismatch measured experimentally by x-ray diffraction. The approach is used in a binary alloy containing 6000 ppm in weight of oxygen. It is concluded that the ordered compounds, which are observed after a recrystallization heat treatment, do not have the expected Ti6O stoichiometry but have a composition close to the nominal concentration. Oxygen ordering proceeds, therefore, before oxygen partitioning in titanium.

Plasticity in hexagonal close-packed zirconium is controlled by screw dislocations which easily glide in the prismatic planes where they are dissociated. At high enough temperatures, these dislocations can deviate out of the prism planes to also glide in the first order pyramidal and basal planes. To get a better understanding of these secondary slip systems, we have performed molecular dynamics (MD) simulations of a screw dislocation gliding in a basal plane. The gliding dislocation remains dissociated in the prism plane where it performs a random motion and occasionally cross-slips out of its habit plane by the nucleation and propagation of a kink-pair. Deviation planes are always pyramidal, with an equal probability to cross-slip in the two pyramidal planes on both sides of the basal plane, thus leading to basal slip on average. Basal slip appears therefore as a combination of prismatic and pyramidal slip in the high stress regime explored in MD simulations. This is confirmed by nudged elastic band (NEB) calculations. But NEB calculations also reveal a change of glide mechanism for a decreasing applied stress. At low stress, kinks do not lie anymore in the pyramidal planes. They are now spread in the basal planes, thus fully compatible with a motion of the screw dislocation confined to the basal plane as seen in experiments. Basal slip, which is in competition with pyramidal slip, appears therefore favoured at low stress in pure zirconium.

The interplay of screw dislocations with carbon atoms is investigated in tungsten at high temperature using in situ straining experiments in a transmission electron microscope (TEM) and through ab initio calculations. When the temperature is high enough to activate carbon diffusion, above 1373 K, carbon segregates in the core of screw dislocations and modifies their mobility, even for a carbon concentration as low as 1 appm. TEM observations reveal the reappearance of a Peierls mechanism at these high temperatures, with screw dislocations gliding viscously through nucleation and propagation of kink-pairs. The mobility of screw dislocations saturated with carbon atoms is then investigated with ab initio calculations to determine kink-pair formation, nucleation and migration energies. These energies are used in kinetic Monte-Carlo simulations and in an analytical model to obtain the velocity of screw dislocations as a function of the temperature, the applied stress and the dislocation length. The obtained mobility law parametrised on ab initio calculations compares well with experiments.

The interaction between carbon and screw dislocations in tungsten is investigated using ab initio calculations. The presence of carbon atoms in the vicinity of the dislocation induces a reconstruction, with the dislocation relaxing to a configuration, the hard core structure, which is unstable in pure tungsten. The reconstruction corresponds to a strong binding of carbon in the prismatic sites created by the dislocation which is perfect for high concentrations of carbon segregated on the dislocation line. However, the reconstruction is only partial for lower atomic fractions, with the dislocation tending to fall back in its easy core ground state. This pinning by carbon atoms of the dislocation in an unstable position is well described by a simple line tension model. A strong carbon-dislocation attraction is also evidenced at larger separation distances, when the solute is in the fourth nearest neighbour octahedral sites of the reconstructed core. The equilibrium concentrations of carbon in these different segregation sites are modelled with an Ising model and using a mean-field approximation. This thermodynamic model evidences that screw dislocations remain fully saturated by carbon atoms and pinned in their hard core configuration up to about 2500 K.

The influence of magnetism on the properties of screw dislocations in body-centered cubic chromium is investigated by means of ab initio calculations. Screw dislocations having Burgers vectors 1/2 111 and 100 are considered, following experimental observations showing activity for both slip systems. At low temperature, chromium has a magnetic order close to antiferromagnetism along 100 directions, for which 1/2 111 is not a periodicity vector. Hence, dislocations with Burgers vectors 1/2 111 generate magnetic faults when shearing the crystal, which constrain them to coexist and move pairwise, leading to dissociated 111 super-dislocations. On the other side, 100 is a periodicity vector of the magnetic order of chromium, and no such magnetic fault are generated when 100 dislocations glide. Dislocation properties are computed in the magnetically ordered and non magnetic phases of chromium for comparison purposes. We report a marginal impact of magnetism on the structural properties and energies of dislocations for both slip systems. The Peierls energy barrier opposing dislocation glide in {110} planes is comparable for both 1/2 111 {110} and 100 {110} slip systems, with lower Peierls stresses in the magnetically ordered phase of chromium.

We show here how density functional theory calculations can be used to predict the temperatureand orientation-dependence of the yield stress of body-centered cubic (BCC) metals in the thermallyactivated regime where plasticity is governed by the glide of screw dislocations with a 1/2 <111> Burgers vector. Our numerical model incorporates non-Schmid effects, both the twinning/antitwinning asymmetry and non-glide effects, characterized through ab initio calculations on straight dislocations. The model uses the stress-dependence of the kink-pair nucleation enthalpy predicted by a line tension model also fully parameterized on ab initio calculations. The methodology is illustrated here on BCC tungsten but is applicable to all BCC metals. Comparison with experimental data allows to highlight both the successes and remaining limitations of our modeling approach., Comment: Comptes Rendus Physique, Centre Mersenne, In press

Plasticity in hexagonal close-packed zirconium is mainly controlled by the glide of dislocations with 1/3<1-210> Burgers vectors. As these dislocations cannot accommodate deformation in the [0001] direction , twinning or glide of dislocations, i.e. dislocations with 1/3<1-213> Burgers vector, have to be activated. We have performed in situ straining experiments in a transmission electron microscope to study the glide of dislocations in two different zirconium samples, pure zirconium and Zircaloy-4, at room temperature. These experiments show that dislocations exclusively glide in first-order pyramidal planes with cross-slip being activated. A much stronger lattice friction is opposing the glide of dislocations when their orientation corresponds to the direction defined by the intersection of their glide plane with the basal plane. This results in long dislocations straightened along which glide either viscously or jerkily. This direction governs the motion of segments with other orientations, whose shape is merely driven by the minimization of the line tension. The friction due to solute atoms is also discussed.

Basal slip acts as a secondary deformation mode in hexagonal close-packed titanium and becomes one of the primary mechanisms in titanium alloyed with simple metals. As these solute elements also lead to a pronounced reduction of the energy of the basal stacking fault, one can hypothesize that they promote basal dissociation of dislocations which can then easily glide in the basal planes. Here, we verify the validity of this hypothesis using ab initio calculations to model the interaction of a screw dislocation with indium (In) and tin (Sn). These calculations confirm that these simple metals are attracted by the stacking fault existing in the dislocation core when it is dissociated in a basal plane, but this interaction is not strong enough to stabilize a planar configuration, even for a high solute concentration in the core. Energy barrier calculations reveal that basal slip, in the presence of In and Sn, proceeds without any planar dissociation, with the dislocation being spread in pyramidal and prismatic planes during basal slip like in pure Ti. The corresponding energy barrier is higher in presence of solute atoms, showing that In and Sn do not ease basal slip but increase the corresponding lattice friction. This strengthening of basal slip by solute atoms is discussed in view of available experimental data.

This chapter reviews the different methodological aspects of the ab ini-tio modeling of dislocations. Such simulations are now frequently used to study the dislocation core, i.e. the region in the immediate vicinity of the line defect where the crystal is so strongly distorted that an atomic description is needed. This core region controls some dislocation fundamental properties, like their ability to glide in different crystallographic planes. Ab initio calculations based on the density functional theory offer a predictive way to model this core region. Because dislocations break the periodicity of the crystal and induce long range elastic fields, several specific approaches relying on different boundary conditions have been developed to allow for the atomistic modeling of these defects in simulation cells having a size compatible with ab initio calculations. We describe these different approaches which can be used to study dislocations with ab initio calculations and introduce the different analyses which are currently performed to characterize the core structure, before discussing how meaningful energy properties can be extracted from such simulations.

Basal slip of a screw dislocations in hexagonal closed-packed titanium is investigated with ab initio calculations. We show that a basal dissociation is highly unstable and reconfigures to other structures dissociated in a first order pyramidal plane. The obtained mechanism for basal slip corresponds to the migration of the partial dislocations and of the associated stacking fault ribbon in a direction perpendicular to the dissocia-tion plane. Presented results indicate that both basal and pyramidal slip will operate through the Peierls mechanism of double-kink nucleation and will be equally active at high enough temperature.

The interaction between screw dislocations and oxygen interstitial atoms is studied with ab initio calculations in hexagonal close-packed titanium. Our calculations evidence a strong repulsion when the solute atoms are located in the dislocation glide plane, leading to spontaneous cross-slip, which allows the dislocation to bypass the atomic obstacle. This avoidance process explains several experimental observations in titanium in presence of oxygen: (1) a larger lattice friction against screw dislocation motion, (2) a reduction of the dislocation glide distance in prismatic planes and (3) an enhancement of cross-slip in pyramidal planes.

Different descriptions used to model a point-defect in an elastic continuum are reviewed. The emphasis is put on the elastic dipole approximation, which is shown to be equivalent to the infinitesimal Eshelby inclusion and to the infinitesimal dislocation loop. Knowing this elastic dipole, a second rank tensor fully characterizing the point-defect, one can directly obtain the long-range elastic field induced by the point-defect and its interaction with other elastic fields. The polarizability of the point-defect, resulting from the elastic dipole dependence with the applied strain, is also introduced. Parameterization of such an elastic model, either from experiments or from atomic simulations, is discussed. Different examples, like elastodiffusion and bias calculations, are finally considered to illustrate the usefulness of such an elastic model to describe the evolution of a point-defect in a external elastic field.

The thermodynamic properties of the Zr-Nb alloy are investigated at temperatures below 890 K with ab initio calculations. The solution energies of the bcc Nb-rich and hcp Zr-rich solid solutions obtained within the framework of density functional theory are in good agreement with experimental data, although insufficient for a quantitative description of the miscibility gap, for which non configurational entropy has to be accounted for. Whereas electronic free energies can be neglected, we show, using the harmonic approximation and the density functional perturbation theory, that both solution free energies are strongly modified by the contribution related to atomic vibrations. Considering this vibrational free energy leads to a good description of the phase diagram.

Plasticity in zirconium alloys is mainly controlled by the interaction of 1/3 1210 screw dislocations with oxygen atoms in interstitial octahedral sites of the hexagonal close-packed lattice. This process is studied here using ab initio calculations based on the density functional theory. The atomic simulations show that a strong repulsion exists only when the O atoms lie in the dislocation core and belong to the prismatic dislocation habit plane. This is a consequence of the destruction of the octahedral sites by the stacking fault arising from the dislocation dissociation. Because of the repulsion, the dislocation partially cross-slips to an adjacent prismatic plane, in agreement with experiments where the lattice friction on screw dislocations in Zr-O alloys has been attributed to the presence of jogs on the dislocations due to local cross-slip.

Twin growth in hexagonal close-packed zirconium is investigated at the atomic scale by modeling the various disconnections that can exist on twin boundaries. Thanks to a coupling with elasticity theory, core energies are extracted from atomistic simulations and the formation energy of isolated disconnection dipoles is defined. For twin systems where several disconnections can exist, because of this core contribution, the most stable disconnection is not always the one with the smallest Burgers vector. Crystallographic parameters of the disconnection with the lowest formation energy correlate well with twin modes observed experimentally. On the other hand, disconnection migration, characterized here by computing their migration energy and Peierls stress, does not appear critical for twin mode selection.

The effect of solute hydrogen on the stability of vacancy clusters in hexagonal closed packed zirconium is investigated with an ab initio approach, including contributions of H vibrations. Atomistic simulations within the density functional theory evidence a strong binding of H to small vacancy clusters. The hydrogen effect on large vacancy loops is modeled through its interaction with the stacking faults. A thermodynamic modeling of H segregation on the various faults, relying on ab initio binding energies, shows that these faults are enriched in H, leading to a decrease of the stacking fault energies. This is consistent with the trapping of H by vacancy loops observed experimentally. The stronger trapping, and thus the stronger stabilization, is obtained for vacancy loops lying in the basal planes, i.e. the loops responsible for the breakaway growth observed under high irradiation dose., Comment: submitted

Atomistic simulations, based either on an empirical interatomic potential or on ab initio calculations, are used to study the pyramidal glide of a 1/3 <1-210> screw dislocation in hexagonal close-packed zirconium. Generalized stacking fault calculations reveal a metastable stacking fault in the first order pyramidal {10-11} plane, which corresponds to an elementary pyramidal twin. This fault is at the origin of a metastable configuration of the screw dislocation in zirconium, which spontaneously appears when the dislocation glides in the pyramidal plane., Comment: symposium "Multiscale perspectives on plasticity in hcp metals", TMS 2014 annual meeting, Metallurgical and Materials Transactions A (2014)

The stability properties of vacancy clusters in hexagonal close-packed Zr, cavities and dislocation loops, are investigated at the atomic scale, with a modeling approach based on density functional theory and empirical potentials. Considering the vacancy-vacancy interactions and the stability of small vacancy clusters, we establish how to build the larger clusters. The study of extended vacancy clusters is then performed using continuous laws for defect energetics. Once validated with an empirical potential, these laws are parameterized with ab initio data. Our work shows that the easy formation of loops can be explained by their thermodynamic properties.

Although the favored glide planes in hexagonal close-packed Zr are prismatic, screw dislocations can escape their habit plane to glide in either pyramidal or basal planes. Using abinitio calculations within the nudged elastic band method, we show that, surprisingly, both events share the same thermally activated process with an unusual conservative motion of the prismatic stacking fault perpendicularly to itself. Halfway through the migration, the screw dislocation adopts a nonplanar metastable configuration with stacking faults in adjacent prismatic planes joined by a two-layer pyramidal twin.

Modeling point defects at an atomic scale requires careful treatment of the long-range atomic relaxations. This elastic field can strongly affect point defect properties calculated in atomistic simulations because of the finite size of the system under study. This is an important restriction for ab initio methods which are limited to a few hundred atoms. We propose an original approach coupling ab initio calculations and linear elasticity theory to obtain the properties of an isolated point defect for reduced supercell sizes. The reliability and benefit of our approach are demonstrated for three problematic cases: the self-interstitial in zirconium, clusters of self-interstitials in iron, and the neutral vacancy in silicon.

Plasticity in zirconium is controlled by 1/3<1-210> screw dislocations gliding in the prism planes of the hexagonal close-packed structure. This prismatic and not basal glide is observed for a given set of transition metals like zirconium and is known to be related to the number of valence electrons in the d band. We use ab initio calculations based on the density functional theory to study the core structure of screw dislocations in zirconium. Dislocations are found to dissociate in the prism plane in two partial dislocations, each with a pure screw character. Ab initio calculations also show that the dissociation in the basal plane is unstable. We calculate then the Peierls barrier for a screw dislocation gliding in the prism plane and obtain a small barrier. The Peierls stress deduced from this barrier is lower than 21 MPa, which is in agreement with experimental data. The ability of an empirical potential relying on the embedded atom method (EAM) to model dislocations in zirconium is also tested against these ab initio calculations.

The dislocation core field, which comes in addition to the Volterra elastic field, is studied for the <111> screw dislocation in alpha-iron. This core field, evidenced and characterized using ab initio calculations, corresponds to a biaxial dilatation, which we modeled within the anisotropic linear elasticity. We show that this core field needs to be considered when extracting quantitative information from atomistic simulations, such as dislocation core energies. Finally, we look at how dislocation properties are modified by this core field, by studying the interaction between two dislocations composing a dipole, as well as the interaction of a screw dislocation with a carbon atom.

Aside from the Volterra field, dislocations create a core field, which can be modeled in linear anisotropic elasticity theory with force and dislocation dipoles. We derive an expression of the elastic energy of a dislocation taking full account of its core field and show that no cross term exists between the Volterra and the core fields. We also obtain the contribution of the core field to the dislocation interaction energy with an external stress, thus showing that dislocation can interact with a pressure. The additional force that derives from this core field contribution is proportional to the gradient of the applied stress. Such a supplementary force on dislocations may be important in high stress gradient regions, such as close to a crack tip or in a dislocation pile-up.

The classical modeling of dislocation climb based on a continuous description of vacancy diffusion is compared to recent atomistic simulations of dislocation climb in body-centered cubic iron under vacancy supersaturation [Phys. Rev. Lett. 105 095501 (2010)]. A quantitative agreement is obtained, showing the ability of the classical approach to describe dislocation climb. The analytical model is then used to extrapolate dislocation climb velocities to lower dislocation densities, in the range corresponding to experiments. This allows testing of the validity of the pure climb creep model proposed by Kabir et al. [Phys. Rev. Lett. 105 095501 (2010)].

Dislocation climb mobilities, assuming vacancy bulk diffusion, are derived and implemented in dislocation dynamics simulations to study the coarsening of vacancy prismatic loops in fcc metals. When loops cannot glide, the comparison of the simulations with a coarsening model based on the line tension approximation shows a good agreement. Dislocation dynamics simulations with both glide and climb are then performed. Allowing for glide of the loops along their prismatic cylinders leads to faster coarsening kinetics, as direct coalescence of the loops is now possible.

Nucleation is the onset of a first-order phase transition by which a metastable phase transforms into a more stable one. Such a phase transition occurs when an initial system initially in equilibrium is destabilized by the change of an external parameter like the temperature or the pressure. If the perturbation is small enough, the system does not become unstable but rather stays metastable. In diffusive transformations, the system then evolves through the nucleation, the growth and the coarsening of a second phase. Such a phase transformation is found in a lot of situations in materials science like condensation of liquid droplets from a supersaturated vapor, solidification, precipitation from a supersaturated solid solution, ... The initial stage of all these different processes can be well described within the same framework. Since its initial formulation in 1927 by Volmer, Weber and Farkas and its modification in 1935 by Becker and D\"oring the classical nucleation theory has been a suitable tool to model the nucleation stage in phase transformations. In this article, we first describe this theory. A kinetic approach, the cluster dynamics, can also be used to describe nucleation. This constitutes the second part of this article. The links as well as the difference between both descriptions are emphasized. Since its initial formulation, the classical nucleation theory has been enriched, so as to take into account the fact that clusters other than monomers can migrate and react. It has been also extended to multi-component systems. These generalizations of the initial formalism are also presented.

We derive an expression of the core traction contribution to the dislocation elastic energy within linear anisotropic elasticity theory using the sextic formalism. With this contribution, the elastic energy is a state variable consistent with the work of the Peach-Koehler forces. This contribution needs also to be considered when extracting from atomic simulations core energies. The core energies thus obtained are real intrinsic dislocation properties: they do not depend on the presence and position of other defects. This is illustrated by calculating core energies of edge dislocation in bcc iron, where we show that dislocations gliding in {110} planes are more stable than those gliding in {112} planes.

Ab initio calculations in bcc iron show that a <111> screw dislocation induces a short-range dilatation field in addition to the Volterra elastic field. This core field is modeled in anisotropic elastic theory using force dipoles. The elastic modeling thus better reproduces the atom displacements observed in ab initio calculations. Including this core field in the computation of the elastic energy allows deriving a core energy which converges faster with the cell size, thus leading to a result which does not depend on the geometry of the dislocation array used for the simulation., Comment: DOI: 10.1103/PhysRevLett.102.055502

The interaction of C atoms with a screw and an edge dislocation is modelled at an atomic scale using an empirical Fe-C interatomic potential based on the Embedded Atom Method (EAM) and molecular statics simulations. Results of atomic simulations are compared with predictions of elasticity theory. It is shown that a quantitative agreement can be obtained between both modelling techniques as long as anisotropic elastic calculations are performed and both the dilatation and the tetragonal distortion induced by the C interstitial are considered. Using isotropic elasticity allows to predict the main trends of the interaction and considering only the interstitial dilatation will lead to a wrong interaction.

The aim of this paper is to give a short review on cluster dynamics modeling in the field of atoms and point defects clustering in materials. It is shown that this method, due to its low computer cost, can handle long term evolution that cannot, in many cases, be obtained by Lattice Kinetic Monte Carlo methods. Indeed, such a possibility is achieved thanks to an important drawback that is the loss of space correlations of the elements of the microstructures. Some examples, in the field of precipitation and irradiation of metallic materials are given. The limitations and difficulties of this method are also discussed. Unsurprisingly, it is shown that it goes in a very satisfactory way when the objects are distributed homogeneously. Conversely, the source term describing the primary damage under irradiation, by nature heterogeneous in space and time, is tricky to introduce especially when displacement cascades are produced.

By means of low-temperature expansions (LTEs), the nucleation free energy and the precipitate interface free energy are expressed as functions of the solubility limit for alloys which lead to the precipitation of a stoichiometric L12 compound such as Al-Sc or Al-Zr alloys. Classical nucleation theory is then used to obtain a simple expression of the nucleation rate whose validity is demonstrated by a comparison with atomic simulations. LTEs also explain why simple mean-field approximation like the Bragg-Williams approximation fails to predict correct nucleation rates in such an ordering alloy.

Electrical resistivity evolution during precipitation in Al-Sc alloys is modeled using cluster dynamics. This mesoscopic modeling has already been shown to correctly predict the time evolution of the precipitate size distribution. In this work, we show that it leads too to resistivity predictions in quantitative agreement with experimental data. We only assume that all clusters contribute to the resistivity and that each cluster contribution is proportional to its area. One interesting result is that the resistivity excess observed during coarsening mainly arises from large clusters and not really from the solid solution. As a consequence, one cannot assume that resistivity asymptotic behavior obeys a simple power law as predicted by LSW theory for the solid solution supersaturation. This forbids any derivation of the precipitate interface free energy or of the solute diffusion coefficient from resistivity experimental data in a phase-separating system like Al-Sc supersaturated alloys.