The previous examples have looked at the atomistic and mesoscale of radiation damage and defect formation. This information can be used by plasticity models and microstructural mechanics models of the effect of radiation on materials properties. Here an example is presented where the atomistic calculations are used to parameterize a viscoplasticity treatment of hardening in materials due to irradiation.

Hardening and embrittlement are controlled by interactions between dislocations and irradiation induced defect clusters. Radiation hardening and embrittlement that occurs in metals irradiated at low temperatures (below ~0.3 Tm, where Tm is the melting temperature) is a an important technical challenge for advanced nuclear energy systems(Zinkle and Matsukawa 2004). In this example, the Visco Plastic Self Consistent (VPSC) polycrystalline code (Lebensohn and Tome 1993) is employed in order to model the yield stress dependence
in ferritic steels on the irradiation dose. The dispersed barrier hardening model is implemented in the VPSC code by introducing a hardening law, function of the strain, to describe the threshold resolved shear stress required to activate dislocations. The size and number density of the defect clusters varies with the irradiation dose in the model. Such modeling efforts can both reproduce experimental data and also guide future experiments of irradiation hardening.

In order to describe the nature of the yield stress dependence on the irradiation dose, we implemented a new microstructural model at the grain level in the VPSC code. The model assumes that hardening is affected by the presence of the defects and defect clusters produced by irradiation. These defects interact with the pre-existing dislocations in the microstructure leading to an increase in the critical stress necessary to move the dislocations. This leads to an increase in the overall yield stress of the material.

Defects are treated as barriers to the motion of dislocations. Two approximate dislocation barrier models have historically been used to describe radiation hardening in metals (Zinkle and Matsukawa 2004) and are reviewed in (Koppenaal and Arsenault 1971; Kocks 1977). The dispersed barrier model (Seeger, Diehl et al. 1957) is based on straightforward geometrical considerations for obstacles intersecting the dislocation glide plane and it is most appropriate for strong obstacles. An alternative hardening relationship was developed by Friedel-Kroupa-Hirsch (FKH) for weak obstacles (Friedel 1955; Kroupa and Hirsch 1964), where the effective interparticle spacing is increased compared to the planar geometric spacing due to less extensive dislocation bowing prior to obstacle breakaway. Using the simple approximation for dislocation line tension, the functional dependence of polycrystalline yield strength increase on defect cluster size and density for these two limiting cases is given by the following equations:

Aa = Ma/jbJNd, (8)

2

Aa = 8 MjubN 3 d

Equation 8 corresponds to the dispersed barrier hardening model and Equation 9 to the FKH model. In the two equations, Aais the change in the yield stress, M is the Taylor factor (3.06 for non-textured BCC and FCC metals), a is the defect cluster barrier strength, u is the shear modulus, b is the Burgers vector of the primary glide dislocations, and N and d are the defect cluster density and diameter.

Most radiation hardening studies have used the dispersed barrier model (Equation 8) for data interpretation, and in this work we find that it provides a better representation of our experimental results. However, the FKH model (Equation 9) may be more appropriate for many radiation-induced small defect clusters which are weak obstacles to dislocation motion. According to some early analyses (Kocks_ 1977), the FKH model is adequate for barrier strengths up to 1/4 of the Orowan (impenetrable obstacle) limit, i. e., a < 0:25, and the dispersed barrier model is more appropriate for barrier strengths of 0.25<a <1. Typical experimental values of the defect cluster barrier strength for copper and austenitic stainless steel neutron-irradiated and tested near room temperature are a = 0.15-0.2 (Zinkle 1987). The reported barrier strengths for the visible defect clusters in BCC metals (Rice and Zinkle

1998) are a = 0.4 or higher. It is possible that hardening from atomic scale voids in the BCC metals might cause one to overestimate the reported barrier strength for the visible defect clusters.

It is possible to introduce a hardening law that is a function of the strain to describe the threshold resolved shear stress required to activate dislocations. In the present application, however, evolution is not simulated and only the initial threshold is required. We assume that the initial critical resolved shear stress (CRSS) in each grain is affected by irradiation according to the dispersed barrier hardening law and follows the Orowan expression,

t = t0 + aubyfNd, (10)

where t is the initial CRSS, t0 is the unirradiated initial CRSS and the other parameters have the same meaning as in Equations 8 and 9. Observe that the Taylor factor is not included in Eq. 10, since the geometric crystal orientation effects are accounted for by the polycrystal model. The critical stress t is assigned to the 12 (110)[111] and the 12 (110)[112] slip systems of the BCC structure. The initial texture of the rolled ferritic steel is represented using 1000 crystallographic orientations. Each orientation is treated as an ellipsoidal inclusion embedded in and interacting with the effective medium that represents the aggregate. An incremental strain is enforced along the rolling direction, while leaving the lateral strains unconstrained. The stress and the strain is different from grain to grain, and the macroscopic (yield) stress is given by the average over all orientations.

Through Eq. 10 the model includes a dependence of the yield stress on the damage created due to radiation. Radiation damage is usually expressed as a statistical quantity describing the average number of displacements for each atom (dpa). The dpa influences the yield stress by determining the number density and the size of the defect clusters (obstacles) that impede the path of the dislocations and increase the critical stress required to move the dislocation.

It has commonly been assumed that the defect cluster density in irradiated metals increases linearly with increasing dose, up to the onset of cascade overlap which causes a saturation in the cluster density (Makin, Whapman et al. 1962; Koppenaal and Arsenault 1971; Trinkaus, Singh et al. 1996). However, in several pure FCC metals the defect accumulation as measured by electrical resistivity (Makin et al. 1962; Zinkle 1987) or transmission electron microscopy (Zinkle 1987; Muroga, Heinisch et al. 1992) often appears to exhibit an intermediate dose regime where the defect cluster density is proportional to the square root of dose. The defect accumulation behavior was found to be linear at very low doses (<0.0001 dpa, where the probability of uncorrelated point defect recombination is negligible), and proportional to the square root of dose at higher doses. According to simple kinetic models such as the unsaturable trap model (Thompson, Youngblood et al. 1973; Theis and Wollenberger 1980), the critical dose for transition from linear to square root behavior depends on specimen purity. In this model, the transition to square-root accumulation behavior can be delayed up to high doses if impurity trapping of migrating interstitial — type defects is dominant compared to interstitial — interstitial or interstitial-vacancy reactions.

The dependence on irradiation dose (expressed as dpa) of the defect cluster density (N) and the defect diameter (d) are taken from atomic level kinetic Monte Carlo (kMC) simulations and experimental observations (Deo et al. 2006; Deo, Baskes et al. 2007) The kMC model takes atomic level information of the migration energies and jump attempt frequencies of irradiation induced defects (interstitials, vacancies) and transmutation products (e. g., helium under high energy proton irradiation), and evolves the microstructure according to the rates of migration of these defects. The defects are allowed to cluster, and new irradiation damage is introduced during the simulation according to the irradiation dose rate. Our kMC simulations predict that the number density varies as the square root of the displacements per atom for the case of bcc iron irradiated up to 1 dpa by high energy proton irradiation.

The size dependence on irradiation dose is more complicated as the kMC simulations provide an entire distribution of defect cluster sizes. A single value of d as a function of dose is still a simplification of the kMC results. . The defect size usually increases with increasing dose (dpa) and can be fit by a power law; however the exponent of the power law expression can vary from 0 to 0.5 depending on initial simulation conditions (dose rate, temperature) and the defect cluster size considered. At low dpa, the exponent of the power law dependence is small for all defect sizes and increases at higher dpa.

The density of defects N is assumed to vary as the square root of the dpa while two cases of size dependence are considered, one in which the size is invariant with the dose (dpa) while the other in which the defect size varies as the square root of the dose. Additional systematic work is needed to confirm the presence and to understand the physical mechanisms responsible for this square root fluence-dependent defect cluster accumulation regime.

The link between the atomic level simulations and the VPSC calculations was established using the dispersed barrier hardening model. In this model, the vacancy /interstitial clusters produced in radiation cascades are assumed to act as barriers to the gliding dislocation in the slip plane and are therefore taken to be the main source of radiation hardening. A different model of radiation hardening postulates the formation of defect clouds along the length of the grown-in dislocation( see [4,5] for review). These clouds prevent the dislocation from acting as Frank Read dislocation sources and emitting more dislocations. Singh, Golubov et al. (1997) proposed the cascade induced source hardening model which accounted for interstitial cluster formation during radiation cascade formation. Such cluster formation has been observed in molecular dynamics simulations. In the CISH model, glissile loops produced directly in cascades are assumed to decorate grown-in dislocations so they cannot act as dislocation sources. The yield stress is related to the breakaway stress which is necessary to pull the dislocation away from the clusters/loops decorating it. Various aspects of the model (main assumptions and predictions) have been investigated by these researchers using analytical calculations, 3-D dislocation dynamics and molecular dynamics simulations It is possible to investigate such recent radiation hardening mechanisms by including them to develop the links between the atomic level understanding of defect sizes and concentrations and the VPSC model of polycrystalline hardening. Such mechanisms may also be investigated by atomic level simulations of single dislocation motion in the presence of defect impurities.

In a manner similar to the approach of Arsenelis and co-workers (Arsenlis, Wirth et al. 2004), the VPSC model can be used to combine microstructural input from both experimental observations and model predictions to evaluate the contributions from multiple defect cluster types. Although not all of the relevant parameters are currently known, such parameter — studies that can inform future atomic-scale studies of dislocation — obstacle interactions. The VPSC model could also incorporate experimentally observed defect cluster distributions, number densities to assess the effect of multiple defect types and distributions. A detailed multiscale study, wherein the dislocation-obstacle strength and the number density and size of defects are correlated to the increasing strain and each other, would then further explain the effect of irradiation on mechanical properties of ferritic steels.

The VPSC calculations provide a means to link atomistic first principles calculations to macroscopic observables. The formulation of the irradiation hardening law allows for the introduction of parameters such as the defect size and number density that can be calculated from evolution models and simulations such as the kinetic Monte Carlo method. The interaction of the dislocation with defect clusters can be investigated by using atomistic molecular dynamics calculations. In this document we have provided a framework for performing physically based modeling and simulations of hardening behavior observed during irradiation. Such modeling efforts can both reproduce experimental data and also guide future experiments of irradiation hardening. Performing modeling and simulation studies before initiating an expensive neutron or proton beam experiment would prove invaluable and cost-effective.

5. Conclusions

In this chapter, an overview of multiscale materials modeling tools used to simulate structural materials in irradiation conditions is presented. Next generation nuclear reactors will require a new generation of materials that can survive and function in extreme environments. Advanced modeling and simulation tools can study these materials at various length and time scale. Such varied methods are needed as radiation damage affects materials in excess of 10 orders of magnitude in length scale from the sub-atomic nuclear to structural component level, and span 22 orders of magnitude in time from the sub­picosecond of nuclear collisions to the decade-long component service lifetimes. The inherently wide range of time scales and the "rare-event" nature of the controlling mechanisms make modeling radiation effects in materials extremely challenging and experimental characterization is often unattainable. Thus, modeling and simulation of such materials holds great promise if coupled with suitably designed experiments in order to develop and sustain materials for advanced nuclear energy.