Voids

Voids are formed by an accumulation of vacancies above 0.3 _Tm, particularly in

fcc metals and alloys [67]. This leads to a volumetric expansion of the irradi- ated material, a phenomenon known as void swelling [25]. Voids are like empty bubbles; they do not depend on the availability of fission gases [24]. Neverthe- less, when He atoms are present as end-products of (n-_α) transmutation reactions with atoms in the metal, they are insoluble in the lattice and the cavities may contain He gas [27]. The presence of He-filled bubbles may weaken or strengthen a material, depending on He concentration. In _α-Fe, these bubbles are weaker obstacles than voids for low He content, whereas for high He content they are stronger [68].

The process of void swelling can be divided into three stages at a given dose rate and temperature: an incubation period, a transient regime and a steady state regime, where swelling increases monotonically with increasing dose [8]. Several theoretical treatments have been suggested to account for the generation of voids in irradiated metals and to explain the underlying physical processes [8]. Many variables can have an effect on void swelling. The most important are recoil en- ergy, irradiation dose and temperature, grain size of material and distance from grain boundaries, and crystal structure [25]. These variables affect void swelling by determining either the initial number of point defects formed, or the mobility of clusters formed by them, in accordance with the PBM model. For bcc metals (as well as fcc), there exists a characteristic temperature above which the defect

cluster density decreases rapidly. This is called Stage V recovery temperature [32]. Above this temperature vacancies leave clusters and they can either recom- bine with SIAs, contributing to interstitial cluster shrinkage, or promote void nucleation and growth, enhancing void swelling.

From an engineering point of view, determining the factors that control void swelling is of importance in order to identify and design swelling-resistant ma- terials [8]. An extensive number of experiments have been conducted for many different candidate alloys and have revealed that commercial ferritic-martensitic steels based on the 9-12% Cr composition exhibit the highest swelling resistance [43, 69, 70]. This low-swelling response seems to be a generic property of ferritic alloys as a class.

Another property of voids which is of significance, especially with respect to the subject of this dissertation, is the resistance they offer to dislocation motion, as a result of direct contact. It is energetically favourable for a dislocation to intersect a void, since its core and strain energy is zero within the cavity. However, the creation by shear of a surface step in the direction of the Burgers vector during the penetration of the void offsets this energy gain, at least partially. The counter-action of these two effects determine the obstacle strength of the void [27].

Two models have been proposed to describe dislocation-void interactions [28]. The Russell-Brown modulus-hardening model [71] assumes constant dislocation line tension, no dislocation self-stress and treats the boundary conditions where the line enters a void approximately without consideration of the surface step. The Scattergood-Bacon model [72] is more rigorous. It allows for computation of the self-stress of a flexible dislocation line, which it assumes to be constructed from piecewise segments using either isotropic or anisotropic elasticity theory. It also allows for a more realistic dislocation boundary condition at the void surface, enabling a surface step of lengthb to be created as the dislocation cuts through the void. Numerous atomistic simulation results have been compared with this model [26, 39], including the ones presented here.

The mechanism of edge dislocation-void interaction for _T = 0 K, as given by the Scattergood-Bacon model, can be investigated by means of MS and MD. The detailed methodology to do that is given in [26, 27] and will be discussed in the following chapter. For now, it suffices to say that a model crystal of sufficient

dimensions was created, using a suitable interatomic potential. Due to periodic boundary conditions, when reference is made to a dislocation interacting with a void of diameter_D, this actually means a periodic array of dislocations of periodic distance _Lx from each other along the x-axis interacting with a periodic row of

voids with centre-to-centre spacing _L along the y-axis from each other. In the following paragraphs, voids are implied to be centred on the slip plane of the dislocation.

Under increasing applied strainxz, the interaction mechanism can be divided

into four stages [27], as indicated by the regions in fig. 2.6.

Figure 2.6: Potential energy and applied stress versus applied strain in an Fe crystal at T=0 K containing an edge dislocation gliding through a 2 nm void [27].

• Stage I: The dislocation starts gliding towards the void, as soon as the applied strain results in a stress higher than some value specific for the material and the potential in use. This stress is called Peierls stress, and more discussion about it will follow in chapter 4.

the creation of a step of length _b on the entry surface. Its strain energy drops because of a decrease in line length, and stress becomes negative. This happens because the plastic shear strain due to dislocation motion is larger than the imposed strain. As the applied strain increases, shear stress increases again. Region II ends when the dislocation becomes straight and coincides with the centre-to-centre line of the periodically repeated voids. At this point, the minimun energy configuration is achieved and stress is exactly zero.

• Stage III: Under increasing strain, the dislocation bows between voids, even- tually creating a screw dipole. This dipole can reach lengths of several tens of nm for diameter _{D >}5 nm. As its length increases, the slope of the stress-strain plot_{dτ /d}decreases. The end of this region is when the dislo- cation breaks away at the critical resolved shear stress,_τc. As it does that,

it absorbs a few vacancies from the void, and acquires a pair of superjogs, whereas the void is sheared [39].

• Stage IV: Finally, region IV corresponds to motion of the jogged dislocation. This interaction mechanism is sensitive to certain simulation parameters. Sim- ulation of models with different distances_Lx between dislocations gave the same

critical stress but at different strains [26]. For different spacings_L between voids this was reversed: the critical strain was now the same, but the resultant stress was different [26]. Dependence on void size is also quite strong [27, 35]. Small voids are relatively weak obstacles. Large ones exert significantly stronger resis- tance and the dislocation line has to bow out a lot more before breakaway. This way, a screw dipole is created in the critical configuration when D≥2 nm. The mechanism resembles the Orowan mechanism for an edge dislocation to overcome an impenetrable obstacle, except that no Orowan loop is left behind, around the obstacle, after breakaway. More on the Orowan mechanism will be mentioned in the next subsection. It is the aforementioned self-stress that is allowed in the Scattergood-Bacon model that enables the segments of the dislocation to attract each other at the void surface [35]. This attraction assists their alignment in the screw dipole arrangement, the decrease of _{dτ /d} and their subsequent annihila- tion. When the dislocation breaks away, it climbs by vacancy absorption from the void. The amount of climb depends on void size.

For temperatures higher than 0 K, MD results have shown that the mechanism for the interaction is similar to the one observed at 0 K [27]. Initially, as the dislocation is attracted by the void, it bows forward towards it, implying the existence of an image force which increases as the dislocation approaches the interface (as if it was interacting with a virtual dislocation in the other side of the interface, hence the name) [68]. The image force is induced by the internal free surface, despite the nm size of the void. It is attractive due to zero elastic constants in the void.

There appears to be a weak dependence of critical stress on temperature: in general, with increasing _T, _τc decreases and the dislocation line in the critical

condition bows out less [35]. Climb occurs at all temperatures at breakaway, by vacancy absorption from the void, which shrinks, thus becoming a weaker obstacle for a following dislocation [27].