Hamiltion’s point of view and statistical mechanics

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Precipitation of intermetallic phases

In many Fe–Al–X systems the solid solubility for th e third element within the Fe–Al phases is limited and the possibility of strengthening Fe–Al- based alloys by precipitation of another intermetallic compound exists. In several systems this intermetallic phase is a Laves phase, e.g. in the Fe–Al–X systems with X=Ti, Zr, Nb and Ta. In order to study the effect of precipitates on strengthening, the Fe–Al–Zr system may be considere d as a prototype system as only limited solid solubility for Zr in the Fe–Al phases has bee n found, which is independent from temperature, at least between 800 and 1150 °C [21]. Fig. I-4 presents the partial isothermal section of the Fe-corner at 1000 °C, which is shown by means of example, for the phase equilibria in the temperature range 800–1150 °C. Th e isothermal section reveals that for-(Fe,Al)-based alloys strengthening by a Laves phase is possible in this temperature range while FeAl-based alloys may be strengthened by precipitates of the tetragonal phase (Fe,Al)12Zr (1). As the solubility for Zr in-(Fe,Al)/FeAl does not increase with temperature, no possibility exists for generating fine and evenly distributed precipitates from a solid solution.

Precipitation of carbides

Besides hardening by precipitation of intermetallic phases, carbides could also act as strengthening phases. Fig. I-7 shows two partial isothermal sections at 800 and 1000 °C of the Fe corner of the Fe–Al–C system [25]. At both temperat ures-(Fe,Al) and FeAl are in equilibrium with the cubic K-phase Fe3AlC. The solid solubility for carbon in-(Fe,Al)/FeAl changes only slightly between 800 and 1200 °C but is considerabl y lower at lower temperatures, e.g. drops from about 1 at.% C at 1000 °C to about 50 ppm at 3 20 °C [26]. This leads to the precipitation of fine needle shaped precipitates of the k phase at the grain boundaries during cooling. As the carbon diffusivity is even high at ambient temperatures, these precipitates at the grain boundaries are found at room temperature in all alloys of appropriate compositions even after quenching and they do strongly affect mechanical properties at low temperatures [26]. The effect of k phase precipitates on the mechanical behaviour of Fe–Al-b ased alloys with Al contents between 25 and 30 at.% has been studied in detail by Schneider et al. [27].
Figure.I- 7 Partial isothermal sections of the Fe–Al–C system at (a) 800 (b) and 1000 °C [15]. The exact course of the-(Fe,Al)/FeAl phase boundary has not been determined within the ternary system and therefore only its position in the binary Fe–Al system is indicated by a bar on the Fe–Al axis.
To control the precipitation end microstructures for the carburizing process of the Fe-Al alloys, it is necessary to rely on the thermodynamical properties of the iron riche phases Fe-Al-C system, and to know the fundamental properties of these phases. Several experimental and theoretical informations are present in the literature about the k carbide. The k phase is associated to the Fe3AlC structure, with the Strukturbericht Designation E21 (a perovskite-type structure). This carbide is based on the fcc ordered structure Fe3Al-L12 where the iron atoms are located in the center of each face, and the aluminium atoms sit on the corners of the cube (see Fig. I-8). The carbon atom occupies the central octahedral interstitial position formed by the six iron atoms as first nearest neighbours.
The stoichiometric Fe3AlC has, in fact, never been observed. Experimentally, the stoichiometry proposed for k is Fe4-yAlyCx where 0.8<y<1.2 and 0<x<1 [28]. Other results indicate that the composition of the different synthesized compounds is probably close to Fe3AlCx=1/2 [29, 30]. In addition, the experimental magnetic nature of the compound (ferro- or nonmagnetic) is not yet well established. Since the investigations of Morral (1934) [31], it has been stated several times that the Kappa phase is ferromagnetic. The given Curie temperature values would lie between 125 [32] and 290 °C [33]. However, the investigatio ns of Parker et al. [34] indicate that the k phase might not be magnetic. Later, the investigations of Andryushchenko et al. [28] seem to have confirmed these observations. These authors have observed that the distribution of aluminium on the corners of the cube and of iron on the faces of the cube is apparently not perfect. Antisites’ defects (aluminium atoms on iron sites and reciprocally iron on aluminium sites) seem to be at the origin of the reduced magnetic moment.
Ohtani et al. [35] have published a Fe-Al-C phase diagram based on ab initio calculations within an all electron approach, and Maugis et al. [36] have discussed the relative stability of various phases in aluminium- containing steels, through ab initio calculations using the VASP package. More recently, Connétable et al. [37] have investigated the influence of the carbon on different properties of the Fe3Al system using ab intio calculations. The authors have found that the insertion of the carbon atom decreases the magnetism of the iron atoms and modifies strongly the heat capacity and the elastic constant in k-phase compared to the Fe3Al-L12 structure. The interactions between the Fe and the C are is the main origin of these modifications. Kellou et al. have also investigated the structural and thermal properties of Fe3AlC k-carbide [78]. The authors show that The C addition has the highest effect in strengthening the cohesion of the Fe3Al base between several additions. These authors have found that the bulk modulus (166GPa) and cohesive energy (5.7eV/ atom) of the Fe3AlC (k-carbide) phase has been found to the highest from all the investigated Fe3AlX compounds (X=. H, B, C, N, O) [78].

Strengthening by coherent precipitates

In the Fe–Al–Ni system a miscibility gap between di sordered-(Fe,Al) (A2) and ordered NiAl (B2) exists at temperatures below about 1200 °C [38 ]. The lattice mismatch of both phases is sufficiently small so that it is possible to produce very fine-scale coherent two-phase microstructures of disordered-(Fe,Al) (A2) + ordered (Ni,Fe)Al (B2). Except for the Fe–Al–Ni system, the mechanical properties of the coherent two-phase microstructures have not been studied in detail. The coherent precipitates have a strong strengthening effect and microstructures can be varied such that the hard (Ni,Fe)Al phase is either the matrix or the precipitate and in both cases a strengthening effect has been achieved. The deformation behaviour of ternary Fe–Al–Ni alloys at high temperatures has been studied [39]. These studies have been extended to quaternary Fe–Al–Ni–Cr alloys and first results, es pecially on the creep behaviour of these alloys, are reported by Stallybrass et al. [40].

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Strengthening by order

An additional possibility for strengthening of Fe–A l based alloys is to stabilise the D03 structure with respect to the B2 structure to higher temperatures. Nishino et al. [41, 42] have determined the D03- B2 transformation temperatures in (Fe1-xMx)3Al with M= Ti, V, Cr, Mn and Mo. In particular, the transformation temperatures T0 for M= Ti and V increase rapidly with increasing x, reaching T0 values as high as 1300 K for x=0.15 (approx. 11 at.% Ti) and x=0.25 (approx. 19 at.% V). Anthony and Fultz [43] have reviewed the solute effects on T0 in Fe3Al and also measured the changes in T0 for a large number of solutes only in the dilute limit (see Fig. I-9).
Among the transition elements, the addition of Ti gives rise to the sharpest increase in T0 at the rate of 55 K/at.% Ti [43, 41]. Likewise, the additions of V and Mo increase T0 but only at the rates of 35 K/at.% V [43, 41], and 25 [44, 42] or 30 [43] K/at.% Mo. An initial rise in T0 for M=Ti, V and Mo tends to moderate at a higher composition. An approximately linear dependence of TD03-B2 on ternary concentration is observed for all except for two elements of the transition metals. The two exceptions, Nb and Ta, have a limited solubility in Fe3Al, and the increase in TD03-B2 saturates at a 1% concentration [43] (as seen from Fig. I-9). In contrast, the Cr, Hf and the Zr additions have been reported to have no significant effect on T0 [43], except for a slight increase in T0 reported by Mendiratta and Lipsitt [45]. It is worthwhile mentioning here that an increase in the D03-B2 transformation temperature T0 can lead to an improvement in the high-temperature strength of Fe3Al-based alloys [2, 46]. Nishino et al. [42] have indeed demonstrated that a peak in hardness extends to higher temperatures in parallel with the increase in T0 for M= Ti, V and Mo. At present, there is no clear understanding of why these solutes have their characteristic effects on the transformation temperature. The effect of solute atom on T0 D03-B2 is expected to be related to its c istallographic site preference in the D03 structure.

Boron addition and grain boundary strength

In the case of many intermetallic alloys, small boron additions modify their ambient temperature properties. In fact, these alloys, which present an intrinsic intergranular brittleness in their ‘pure’ state, change their fracture mode, when boron-doped. In some cases – like in the B-doped Ni 3Al alloys – the fracture becomes ductile. In other cas es, like in FeAl-B2 alloys, even in the B-doped alloys a brittle fracture is observed, it takes place cleavage in a transgranular manner. If the first (intergranular) type of room temperature brittleness of intermetallic alloys is commonly considered as an intrinsic one, the second one (transgranular) seems in fact to be due to an extrinsic embrittling action of atomic hydrogen, created during the oxidation reaction on the sample surface. 2Al+3H2O→Al2O3+6H (Eq. I-1)
This phenomenon, first identified by Liu et al. [60], is known as the ‘environmental effect’. The boron effect in intermetallic alloys is typically attributed to its intergranular segregation. This hypothesis is a simple conclusion of experimental measurements of some intergranular boron enrichment, mainly in Ni3Al alloys, by the Auger Electrons Spectrometry (AES) method [61, 62]. This hypothesis was confirmed by the experimental results of Fraczkiewicz et al. [63].

Table of contents :

Chapter I: Backround for the iron aluminides based intermetallics analysis
I. Intermetallic compounds
II. Iron aluminides
III. Bulk strengthening
III.1 Strengthening by solid-solution hardening
III.2. Strengthening by incoherent precipitates
III.2.1. Precipitation of intermetallic phases
III.2.2. Precipitation of carbides
III.3. Strengthening by coherent precipitates
III.4. Strengthening by order
III.4.1. Site preference
III.4.2. Solute effects on D03 ordering
III. 5 Objective of our work for bulk analysis
IV. Effect of alloying elements on ductility
IV. 1 Boron addition and grain boundary strength
IV. 2. Transition metal additions
IV. 3 Modelling approach and objectives of our G.B. simulations
References
Chapter II: Theoretical tools
Part A: Ab Initio Molecular Dynamics
I.1. Introduction
I.2. Quantum Molecular Dynamic
I.2.1. Deriving Classical Molecular Dynamics
I.2.2. « Ehrenfest » Molecular Dynamics
I.2.3. « Born-Oppenheimer » Molecular Dynamics
I.2.4. « Car-Parrinello » Molecular Dynamics
I.3. Integration of the equations of motion
I.3.1. Hamiltion’s point of view and statistical mechanics
I.3.2. Microcanonical Ensemble
I.3.3. The molecular dynamics propagators
I.3.4. Extended System Approach
I.3.4.1. Barostats
I.3.4.2. Thermostats
References:
Part B: The Electronic Structure Methods
II. 1. Introduction
II. 2. Density Functional Theory
II. 3. Energy functionals
II. 4. The plane wave pseudopotential method
II.4.1. Plane waves
II.4.1.1. Supercell
II.4.1.2. Fourier representations
II.4.1.3. Bloch’s Theorem
II.4.1.4. k–Point Sampling
II.4.1.5. Fourier representation of the Kohn-Sham equations
II.4.1.6. Fast Fourier Transformation (FFT)
II.4.2. Pseudopotentials
II.4.2.1. Norm conserving Pseudopotentials
II.4.2.1.1. Hamann–Schluter–Chiang conditions
II.4.2.1.2 Bachelet-Hamann-Schluter (BHS) form
II.4.2.1.3. Kerker Pseudopotetials
II.4.2.1.4. Trouiller–Martins Pseudopotentials
II.4.2.1.5. Kinetic Energy Optimized Pseudopotentials
II.4.2.2. Pseudopotentials in the Plane Wave Basis
II.4.2.2.1. Gauss–Hermit Integration
II.4.2.2.2. Kleinman–Bylander Scheme
II.4.2.3. Non-linear Core Correction
II.4.2.4. Ultrasoft Pseudopotentials Method
References
Chapter III: Static ab initio calculations (0K)
I. Computational details
I.1. Computational method
I.2. Structural properties
I.3. Energetics
II. Point defects in bulk Fe3Al
II.1. Importance of relaxation
II.2. The site preference of point defects in the bulk D03-Fe3Al
III. Impurity segregation at grain boundaries
III.1. Crystal structures and location of structural defects
III.2. Site preference and effect of Ti and Zr on the grain boundary cohesion
III.3. Charge density distribution
III.4. Impurities induced bonding charge density
III. 5. The relaxation of the clean grain boundary
III.5. The relaxation of the doped grain boundary
IV. Summary and Conclusion
References
Chapter IV: Ab initio molecular dynamics calculations
I. Calculation details
I. 1. Computational methods
I. 2. Preliminary calculations
I.3. Energetic
II. Transition metal impurities in the bulk D03-Fe3Al
II.1. Site preference of the Ti and Zr substitutions
II.2. Structural and stability results
II.2.1. Equilibrium lattice parameters
II.2.2. Pair distribution function
II.2.2.1. Pair distribution functions for D03-Fe3Al
II.2.2.2. Pair distribution functions for doped Ti and Zr-Fe3Al
III. Transition metals segregation in 5 (310) [001] grain boundary
III.1. Site preference of Ti and Zr in the 5(310)[001]
III.2. The effect of temperature on the structural relaxation of 5 grain boundary
III.2.1. Relaxation of the clean grain boundary
III.2.2. Relaxation of the doped grain boundary at 300K
IV. Summary and conclusion
Reference

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