Abstract The cluster dynamics method is used to model at the atomic scale the kinetics of first order phase transformations. Clusters are embryos of the growing phase. Their formation kinetics from the solid solution are obtained by solving a set of master differential equations. This set is the continuity equation of the cluster size distribution function and is based on inter-cluster solute exchanges. These exchanges, absorption and emission of solute atoms, are controlled by the solute diffusivities and the cluster free energies. The model is applied here to the precipitation of Al3(Zr, Sc) dispersoids with the L12 structure in Al–rich Zr–Sc solid solutions. Several characteristic features are obtained and discussed: 1) In the ternary alloy, the fast (Sc) diffusing species always controls the nucleation, in contrast to classical thermodynamical descriptions. 2) A Zr–Sc thermodynamic coupling induces heterogeneous nucleation on Zr atoms. 3) A segregation on the dispersoids outer shell of the slow diffusing solute (Zr) occurs during the coarsening stage, slowing down their coarsening rate. Finally, some extensions and prospects of the method are considered.

The homogeneous deformation of a zirconium-based bulk metallic glass is investigated in the glass transition range. Compression and stress-relaxation tests have been conducted. The stress–strain curves are modeled in the framework of the free volume theory, including transient phenomena (overshoot and undershoot). This approach allows several physical parameters (activation volume, flow defect creation and relaxation coefficient) to be determined from a mechanical experiment. This model is able to rationalize the dependency of stress overshoot on relaxation time. It is shown that, due to the relationship between flow defect concentration and free volume model, it is impossible to determine the equilibrium flow defect concentration. However, the relative variation of flow defect is always the same, and all the model parameters depend on the equilibrium flow defect concentration. The methodology presented in this paper should, in the future, allow the consistency of the free volume model to be assessed.

A time-resolved small-angle X-ray scattering (SAXS) study was carried out to investigate the precipitation kinetics of L12 Al3(Zr,Sc) precipitates in aluminium at temperatures ranging between 400 and 475°C. It is shown that the chemical heterogeneity of the precipitates, which consist of a Sc-rich core and a Zr-rich shell, results in a characteristic SAXS signal, which can be fitted by a three-phase model to extract the chemical and morphological features of the precipitate size distribution. The experimental results show a strong effect of the heating rate on the precipitation kinetics, and a precipitate density strikingly constant with time in the investigated range. These results are discussed in view of the mechanisms proposed in the existing literature for the formation of the core–shell structure of these precipitates.

The homogeneous deformation of a zirconium-based bulk metallic glass is investigated in the glass transition region. Compression tests at different temperatures and strain rates have been conducted. The mechanical behavior is analyzed in the framework of the free volume model, taking into account the dependence of the flow defect concentration on deformation. The activation volume is evaluated and allows one to gather the viscosity data (for the different strain rates and temperatures) on a unique master curve. It is also shown that, due to the relation between flow defect concentration and free volume, it is not possible to deduce the equilibrium flow defect concentration directly from mechanical measurements. However, if this parameter is arbitrarily chosen, mechanical measurements give access to the other parameters of the model, these parameters for the alloy under investigation being of the same order of magnitude as those for other metallic glasses.

The precipitation kinetics path in multi-component alloys may involve a competition between atomic mobilities and precipitates thermodynamic stability. Cluster dynamics modelling (CDM) is a simulation method that allows to describe this competition without introducing any heuristic assumptions as, for example, in the classical theory of nucleation. CDM consists in solving numerically, for each time increment, the master equations expressing the balance of solute exchanges (absorption and emission) between clusters/precipitates. A key issue is the energetics of the nano-clusters in the nucleation range. The computation of the precipitate size distribution function allows the complete description of the precipitates kinetic evolution, in chemical composition and in size. The method is applied to the precipitation of the Al-3(Zr,Sc) L1(2) phase in Al solid solutions. The model predicts fairly well in the precipitation path some observed coupling effects between the two solutes, particularly during the nucleation stage.

Cluster dynamics simulations in multicomponents alloys emphasize the complexity of the respective roles of atomic mobility and thermodynamics in the precipitation path. Some heuristic assumptions, like the maximum driving force criterion, used in conventional treatments, may fail and need to be carefully revisited at the light of these results.

Different Zr-Ti-Al-Ni-Cu bulk metallic glasses have been processed. They have been characterized by neutron and X-ray diffraction in the glassy state and after DSC heating up to crystallization.The homogeneous visco-plasticity of the glasses has been studied in the under-cooled liquid state by compression tests at constant strain-rate and by differential mechanical tests with strain-rate jumps at different temperatures.The flow stress appears to be strongly thermally activated. Another characteristic feature of the deformation of the fully amorphous state is that the flow stress is independent of the strain-rate path followed to reach a given level of strain.The stress-strain curves and their dependence on temperature and strain-rate are then discussed in terms of Newtonian and non-Newtonian viscosity. It is shown that a reduced viscosity can be scaled on a master curve, independently of the strain-rate and temperature.A second analysis of the results is made, in terms of free volume. The temperature and stress variation of the plastic flow are compared with Spaepen model. A constitutive equation of the deformation is proposed.Finally, the effect of crystallization on mechanical behaviour of metallic glasses is treated trough a model involving backstress due to a composite behaviour. (C) 2004 Elsevier B.V. All rights reserved.

Zr–Ti–Al–Ni–Cu bulk metallic glasses have been prepared by copper mould casting and were characterized using neutron and X-ray diffraction and scanning calorimetry in the as-cast glassy state and during crystallization. Depending on chemical composition, the under-cooled liquid domain separating the glass transition temperature (Tg) from the onset of crystallization (Tx) ranges from 82 to 102 K, for a heating rate of 20 K/min.The visco-plastic behaviour of the material has been studied in the supercooled liquid state, between Tg and Tx, using constant strain-rate compression tests and strain-rate jump tests.The stress–strain curves display an overshoot at large strain-rate and low temperatures, followed by a constant flow stress up to very large plastic strains. The flow stress is strongly thermally activated. Another characteristic feature of the deformation of the fully amorphous state is the independence of the flow stress on the strain-rate path.The stress–strain curves and their dependence on temperature and strain-rate are first discussed in terms of glass viscosity. It is shown that the data obtained at various temperatures and strain-rates can be scaled on a master curve. The master curve is directly deduced from the stationary strain-rate proposed by Spaepen for homogeneous flow based on the free volume concept. Deformation constitutive laws are then established in the out of equilibrium general case, taking into account the strain induced free volume creation and relaxation. They allow an accurate description of the transients regimes commonly observed such as overshoot, stress oscillations and strain-rate changes.Finally, the hardening resulting from partial crystallization is described by the introduction of a back-stress originating in the undeformability of the crystallites.

The homogeneous precipitation kinetics of the ordered phases Al3Zr and Al3Sc in Al solid solutions has been simulated at the mesoscopic scale by the cluster dynamics method. The polyatomic clusters, embedded in the solid solution, have an L12 structure and exchange solute atoms via absorption and emission reactions controlled by interfacial and long range diffusion. The time evolution of the cluster distribution is determined numerically, solving a set of master differential equations coupled through monomer exchanges. Results are presented for different temperatures and solute supersaturations and are discussed in terms of the classical schemes of precipitation and of recent atomistic Monte-Carlo simulations.

The thermal and mechanical properties of a Zr–Al–Cu–Ti–Ni bulk metallic glass were investigated. The glass transition and the crystallization were studied by calorimetry and X-ray diffraction. It was found that the crystallization occurred in two steps. The precipitating phases, and the activation energies were determined. It was established that the crystallization was controlled by the diffusion of the alloying Cu and Ni atoms. The creep behavior was investigated by indentation tests. The viscosity and the activation energy of the deformation process determined from indentation were in reasonable agreement with those obtained by compression tests.

First-order transformations are first introduced. A distinction is then developed between heterophase and homophase statistical fluctuations. The basic elements of the classical theory of heterophase nucleation are the thermodynamic driving force and the-kinetics master equation. The nucleation, growth and coarsening stages emerge naturally from this theory. In the following section the kinetic ingredients of the classical theory, i.e. the monomer condensation and evaporation rates are examined in detail. A numerical solution of the master equation is given for dilute binary alloys. Cluster size distribution, nucleation current, growth and coarsening are analyzed and discussed. A generalized cluster dynamics framework, involving more complex inter-cluster exchange mechanisms is then considered.

Transient creep (1) tests have been performed at the yield point of icosahedral AlCuFe poly-quasicrystalline specimens in order to investigate the actual causes of this phenomenon. The experimental results strongly suggest that the yield point occurrence is essentially related to the lack of dislocation mobility and that a competition does exist between dislocation velocity and short range recovery. (C) 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

Constant strain-rate and creep tests have been performed on poly-quasicrystalline specimens of the icosahedral Al–Cu–Fe system. Three temperatures (870, 960 and 1000 K) have been investigated by creep for samples predeformed at constant strain-rate before and beyond the upper yield stress, over the stress range 30–320 MPa. The influence of the creep stress level and plastic strain amount on the stress–strain curve has also been studied when the samples are deformed again at constant strain rate. Depending on temperature, stress and initial predeformation states, single or two-stage creep curves have been observed. When performed before the upper yield stress, in the high work-hardening portion of the stress–strain curve, only a primary stage of continuous decreasing creep rate is observed. At higher stresses, close to the upper yield stress, the primary stage is followed by a second stage of work-softening leading to an increasing strain-rate. The shape of the constant strain-rate and creep curves, as well as the transients observed when creep is switched-back to constant strain-rate conditions, are in fair agreement with the predictions of the dislocation model recently proposed by Guyot and Canova on the basis of an irreversible strain dependent friction stress [P. Guyol, G. Canova, Phil. Mag. A 79 (1999) 2815].

A quantitative model of plastic deformation of icosahedral quasicrystals is established. It is based on shears occurring in planes of high atomic density via a dislocational activity. A single-slip dislocation friction stress is first derived on a microscopic basis and introduced into a viscoplastic Kocks-type constitutive law where the dislocation density is the single microstructural internal variable. Dislocations are assumed to be stored and annihilated dynamically during deformation, as in simple crystals. The specificity of quasicrystals with long-range but non-periodic atomic order is the introduction of a friction stress limiting the dislocation mobility which decreases with increasing strain, as evidenced for instance by molecular dynamic simulations. Such a constitutive plastic law leads to shear localization and strain softening. The mean-field single-slip behaviour is then introduced in a multiple-slip state, where quasilattice rotation and activation of the various slip systems are calculated. The role of the quasicrystal symmetry relating the available slip systems is examined as a function of the quasilattice orientation. The results and consequences of the model are compared with experimental data on single icosahedral Al-Mn-Pd quasicrystals deformed at high temperatures at a constant strain rate or constant stress in uniaxial compression and discussed.

The recovery of AlMg alloys cold-rolled to several strains between 0.1 and 3 has been studied by tensile tests. The study is based on the determination of the tensile strain-hardening properties at constant strain-rate, as a function of the initial prestrain and of the static recovery temperature and duration.A modeling of the tensile curves is performed through plastic strain constitutive equations where are taken into account at a mesoscopic scale dislocation storage and dislocation annihilation. It is shown that prestrain and static recovery effects may be analysed in terms of modifications of the storage and dynamic recovery parameters involved in the strain-hardening rate. These modifications are in agreement with electron microscopy observations of the microstructure.

The recovery of Al–2.5wt% Mg alloys cold-rolled to several strains between 0.1 and 3 has been studied essentially using tensile tests. The yield stress and strain-hardening properties are studied as a function of the initial prestrain, and of the temperature and the duration of annealing treatments. A theoretical model based on the dislocation structure is proposed. The kinetic evolution of the yield stress is related to the variation of the total dislocation density as a single structural parameter. The pseudo-logarithmic time decay is explained on the basis of a relaxation of the internal stresses by thermally activated dislocation motion. A strain-hardening model is proposed based on Kocks' constitutive law of plasticity, where the dislocation storage and dislocation annihilation parameters are adapted to a heterogeneous cell/subgrain dislocation structure. The adjustment of the model to the work-hardening behaviour is in agreement with TEM observations.

The energy dissipation and storage during strain hardening of metals have been investigated by means of complementary in situ techniques - infrared thermography (IRT), digital image correlation (DIC) and acoustic emission (AE). Inspired by experimental results obtained in the present work and data available in literature, we proposed the analytically tractable thermodynamic modelling approach, which is conceptually based on a single-variable dislocation evolution approach with a total dislocation density serving as a principal variable governing the strain hardening process. Unified by the kinetic approach involving generation, annihilation and motion of dislocations, the models accounting for the acoustic emission behaviour and heat dissipation under load have been proposed and verified experimentally. For the first time, we were able to demonstrate that the key parameters governing the dislocation storage and annihilation rates can be, in principle, recovered from independent AE and IRT measurements. These results agree favourably with the predictions of the dislocation-based constitutive strain hardening models of the Kocks-Mecking type. The self-consistency, versatility and predictive capacity of the proposed modelling approach are demonstrated in the examples of Cu–Zn alloys with different concentrations of Zn varied from 0 to 30% and correspondingly different stacking fault energies.

The influence of predeformation on the precipitation and on the resulting mechanical properties in an Al-Zn-Mg alloy was studied using a combination of complementary experimental techniques, namely transmission electron microscopy, differential scanning calorimetry, small-angle X-ray scattering and microhardness. The thermomechanical treatment was designed to show a significant effect of deformation on the precipitation process. It was observed that the presence of dislocations during precipitation promoted fast nucleation of the equilibrium eta phase in the eta(2) orientation instead of nucleation of the eta' phase with subsequent transformation to eta(1) and eta(4) outside dislocations. The large growth rate of precipitates lying on dislocations provoked a growing precipitate-free zone around dislocations. This resulted in a lower value of peak hardness and a faster overall coarsening kinetics.

We present here results concerning stored energy in cold rolled Al-2.5Mg alloys subsequently recovered. It is shown that stored energy is sensitive to dislocation structure and that the principle of similitude cannot be applied to relate stored energy to dislocation density.

Previous experimental studies by ''in-situ'' X-rays small angle scattering of the eta' precipitation in 7050 alloys during different aging heat treatments, are used to quantify the resulting flow stress hardening. This is done in terms of classical schemes of precipitate-dislocation interactions, like shearing or Orowan by-passing. It is shown that the observed peak-hardening can be closely related to these mechanisms combined with the precipitation kinetics.