The properties of radiation defects at high tempera­ture may change due to three possible contributions to the free energy: electronic, magnetic, and vibrational. These three effects can be well modeled in bulk bcc iron,59 but they are more challenging for defects. The electronic contribution, which exists only in metals, arises due to changes in the density of states close to the Fermi level. The electronic entropy difference between, for example, two configurations is, to first order, proportional to the temperature, T, and the change in density of states at the Fermi level. This electronic effect is straightforward to take into account in DFT calculations. It was shown in tungsten to decrease the activation free energy for self-diffusion by up to 0.4 eV close to the melting temperature. Thus, although this effect is relatively small in general, it cannot be neglected at high temperature.

The magnetic contribution is important in iron. Spin fluctuations were shown to be the origin of the strong softening of the C0 elastic constant observed as the temperature increases up to the a—g transition temperature,60 and it drives, for instance, the tempera­ture dependence of relative abundance of <100> and <111> interstitial loops formed under irradiation.61 It is also known to have a small effect on vacancy properties, but to the authors’ knowledge there is pres­ently no tractable method to predict this effect for point defects quantitatively from DFT calculations. This is probably one of the important challenges in the field.

Finally, vibrational entropy effects can in princi­ple be obtained either in the quasi-harmonic approximation from phonon frequency calculations or directly from first-principles molecular dynam­ics. There are very few examples of such calcula­tions in the literature. The vibrational modes of vacancies and self-interstitials in iron have been investigated by DFT calculations, and their forma­tion entropies have been estimated.62 As illustrated recently in Mo, it is also possible to calculate the temperature dependence of the vacancy formation enthalpy, from DFT molecular dynamics simula­tions, including anharmonic effects, as well as the defect jump frequency, going beyond the transition state approximation.63

In the density functional theory (DFT), the elec­tronic energy of a system can be written as a func­tional of its electron density:

U = F [p(r)]

The embedded atom model (EAM)7 postulates that in a metal, where electrostatic screening is good, one might approximate this nonlocal functional by a local function. And furthermore, that the change in energy due to adding a proton to the system could be treated by perturbation theory (i. e., no change in p). Hence, the energy associated with the hydrogen atom would depend only on the electron density that would exist at that point r in the absence of the hydrogen.

UH(r) = FH(p(r))

The idea can be extended further, where one con­siders the energy of any atom ‘embedded’ in the effective medium of all the others.8 Now, the energy of each (ith) atom in the system is written in the same form,

Ut = Fi (p(r))

To this is added the interionic potential energy, which in the presence of screening, they took as a short-ranged pairwise interaction. This gives an expression for the total energy of a metallic system:

Utot = Y< Fi(p(r,))+£ V(r, y)

iij

To make the model practicable, it is assumed that p can be evaluated as a sum of atomic densities f(r), that is, p(ri) = ^2j f(rij) and that F and V are unknown functions which could be fitted to empirical data. The ‘modified’ EAM incorporates screening of f and addi­tional contributions to p from many-body terms.

Copper concentrations as high as ^0.4atom% were found in early reactor pressure steels, largely due to both steel recycling and the use of copper as a corrosion-resistant coating on steel welding rods. Research that began in the 1970s demonstrated that this minor impurity was responsible for a signifi­cant fraction of the observed vessel embrittlement due to its segregation into a high density of very small (a few nanometer diameter) copper-rich solute clusters (Becquart and coworkers,126 Chapter 4.05, Radiation Damage of Reactor Pressure Vessel Steels). Becquart and coworkers employed MD cas­cade simulations to determine whether displacement cascades could play a role in the Cu-segregation process, for example, by coalescing with vacancies in the cascade core during the cooling phase. The set of interatomic potentials used is described in Becquart and coworkers.126 Cascade energies of 5, 10, and 20 keV were employed in simulations at 600 K, with copper concentrations of 0, 0.2, and 2.0 atom%. Similar to the case for Fe-C, no effect of copper was found on either stable defect formation or point defect clustering. The tendency for copper to be found bound with either a vacancy or an interstitial in solute-defect complex was observed. The copper-vacancy com­plexes may play a role in the formation of copper — rich clusters over longer times, but no evidence for copper clustering was observed in the cascade debris. Similar results were found in an earlier study by Calder and Bacon.127 Overall, the results of the Fe-Cu studies completed to date are consistent with the fact that Fe and Cu have similar masses and do not strongly interact.

The concentration of vacancies, (Cvq)vHl, in equilib­rium with the dislocation loop of radius R of vacancy (subscript ‘vl’) and SIA (subscript ‘il’) type can be obtained in the same way as in the previous subsec­tion (e. g., Bullough et at29)

(c:q)vl, il = cfexp ±- where gsf, Eel, and b are the stacking-fault energy, the interaction energy of PDs with dislocation and the dislocation Burgers vector, respectively. The ‘+’ and ‘—’ in the exponent correspond to the cases of vacancy and SIA loops, respectively. In the case

	where
Eb (x) = EV + (gsf + Eel(x))b2 [88] For vacancy loops, the rates Pvl(x) and Qvl(x) are given by Pvl(x) = WlX1=2ZL DvCv
where m and v are the shear modulus and Poisson ratio, respectively. Hence, the growth rates of vacancy and SIA loops are
dRvl dt
-Eb! (x) . квТ
Qvl(x) = щх1!1	ZL DiCi + ZL Dv exp
[89]
dRil dt
ZL DiCi — ZLDvCv
where
[82]
Eb(x)=Ev — [gsf + Eel(R)]b2 [90]
The equations given above have been obtained by neglecting mutual recombination between vacancies and SIAs. Accounting for recombination makes the diffusion equations for the concentrations of PDs nonlinear, an approximate solution for which has been obtained using a linearization procedure.1 The correction is, however, insignificant for condi­tions of practical importance.
Read More 22.08.2015   Comprehensive nuclear materials Irradiation-induced formation of void lattices The formation ofvoids in irradiated solids results from the clustering of vacancies, which can be assisted by vacancy clusters produced directly in displacement cascades and by the presence of gas atoms. Vacancy supersaturation under irradiation may locally reach a level large enough to trigger clustering owing to the biased elimination of interstitials on sinks, especially since interstitial atoms and small interstitial clusters usually migrate much faster than vacancies. Evans134 discovered in 1971 that under irradiation voids may self-organize into a mesoscopic lattice. The symmetry of the void lattice is identical to that of the underlying crystal, but with a void lattice parameter about two orders of magnitude larger than the crystalline lattice parameter (see also the reviews by Jager and Trin — kaus135 and Ghoniem et al.136 for irradiation-induced patterning reactions). It has been suggested that the formation of the void lattice results from the 1D migra­tion of self-interstitial atoms (SIAs) and of clusters of SIAs, although elastic interactions between voids could also contribute to self-organization.137 This 1D migra­tion of SIAs would stabilize the formation of voids along directions of the SIAs migration by a shadowing effect.138-140 The model proposed by Woo141 indicates, in particular, that the mean free path of SIAs needs to exceed a critical value for a void lattice to be stable. Atomistic KMC simulations have been performed142 to evaluate the dynamics of void formation, shrinkage, and organization during irradiation. Due to the large difference in mobility of vacancies and interstitials, the slow evolution of the microstructure, and the large range of length scales, assumptions had to be used, in particular, regarding the void position and size. Recently, Hu and Henager143 have approached the problem of void lattice formation in a pure metal using a PFM. Their model relies on the traditional approach presented in Section 1.15.2 for the evolution of the vacancy field, but it makes use of continuum­time random-walk kinetics for modeling the fast transport of interstitials. 2D simulations indicate that irradiation can stabilize a void lattice if the ratio of SIA to vacancy diffusion coefficients is large enough (see Figure 12) and if the defect production rate is not too large (see Figure 13). It would be interesting to extend this first model to include interstitial clusters. The model lacks an absolute length scale, for the rea­sons discussed in Section 1.15.2, and thus nucleation of new voids is treated in a deterministic and phenomeno­logical manner based on the local vacancy concentra­tion. It would clearly be beneficial to use a quantitative PFM of the type presented in Section 1.15.3 to treat void nucleation. This would also then make it possible to directly compare the void size stabilized by irradia­tion with experimental observations. We note also that Rokkam et al.144 recently intro­duced a simple PFM for void nucleation and coarsen­ing in a pure element subjected to irradiation-induced vacancy production. In addition to the local vacancy concentration, these authors introduced a noncon­served order parameter to model the matrix-void interface, similar to the nonconserved order parameter used for solid-liquid interfaces. It is shown144’145 that this model reproduces many known phenomena, such as nucleation, growth, coarsening of voids, as well as the formation of denuded zones near sinks such as free surfaces and grain boundaries. This phe­nomenological model is currently limited by the absence of interstitial atoms in the description. It may also suffer from the fact that the void-matrix interfaces are intrinsically treated as diffuse, whereas real void-matrix interfaces are essentially atom­ically sharp. This problem is further discussed in the following section. Read More 22.08.2015   Comprehensive nuclear materials RIS in Austenitic and Ferritic Steels We have seen that RIS was first observed in austenitic steels on the voids that are formed at large irradiation doses and lead to radiation swelling. The depletion of Cr at grain boundaries is suspected to play a role in irradiation-assisted stress corrosion cracking (IASCC); this is one of the many technological con­cerns related to RIS. The enrichment of Ni and the depletion of Cr can also stabilize the austenite near the sinks, and favor the austenite ! ferrite transition in the matrix.29 The segregation of minor elements can lead to the formation of y’-precipitates (as in Ni—Si alloys), or various M23C6 carbides and other phases.1,29 The segregation of major elements always involves an enrichment of Ni and a depletion of Cr at sinks over a length scale that depends on the alloy compo­sition and irradiation conditions.5 The contribution of various RIS mechanisms is still debated. It is not clear whether it is the IK effect driven by vacancy fluxes, as suggested by the thermal diffusion coefficients DNi < DFe < Dcr,30 or the migration of interstitial — solute complexes, resulting in the segregation of undersized atoms,29 that is dominant. Some models of RIS take into account only the first mechanism,5 while others predict a significant contribution of interstitials.12 For the segregation of minor elements, the size effect seems dominant, with an enrichment of undersized atoms (e. g., Si27) and a depletion of oversized atoms (e. g., Mo53) (Figure 7). The effect of minor elements on the segregation behavior ofmajor ones has been pointed out since the first experimental studies29; the effect of Si and Mo additions has been interpreted as a means of increas­ing the recombination rate by vacancy trapping. As previously mentioned, oversized impurity atoms, such as Hf and Zr, could decrease the RIS.46 RIS in ferritic steels has recently drawn much attention, because ferritic and ferrite martensitic steels are frequently considered as candidates for the future Generation IV and fusion reactors.54 Distance from grain boundary (nm) Figure 7 Thermal and radiation-induced segregation profiles in 316 stainless steel. Reproduced from Bruemmer, S. M.; Simonen, E. P.; Scott, P. M.; Andresen, P. L.; Was, G. S.; Nelson, J. L. J. Nucl. Mater. 1999, 274, 299-314. Experimental studies are more difficult in these steels than in austenitic steels, especially because of the complex microstructure of these alloys. Identification of the general trends of RIS behavior in these alloys appears to be very difficult.55 Nevertheless, in some highly concentrated alloys, a depletion of Cr and an enrichment of Ni have been observed, reminding us of the general trends in austenitic steels54 (Figure 8). The RIS mechanisms are still poorly understood. The segregation of P at grain boundaries has been observed and, as in austenitic steels, the addition of Hf has been found to reduce the Cr segregation.5 Read More 22.08.2015   Comprehensive nuclear materials Advantages and Disadvantages of Irradiations using Various Particle Types Each particle type has its advantages and disadvan­tages for use in the study of radiation effects or for emulating neutron irradiation damage. Common dis­advantages of charged particle beams are the lack of transmutation reactions and the need to use a raster — scanned beam. With the exception of some minor transmutation reactions that can occur with light ion irradiation, charged particles do not reproduce the types of transmutation reactions that occur in reactor core materials due to the interaction with neutrons. The most important of these is the produc­tion of He by transmutation, particularly in alloys that contain elements such as Ni or B. But a second consideration is that of a raster-scanned beam in which any volume element of the target is exposed to the beam for only a fraction of the raster-scan cycle. For a typical beam scanner and beam para­meters, the fraction of time that any particular vol­ume element in the solid is being bombarded is ^0.025. Thus, the instantaneous dose rate during the beam-on portion of the cycle is 40 times that of the average, Figure 28. The result is that the defect production rate is very high and defects can anneal out in the remaining 0.975 portion of the cycle before the beam again passes through the volume element. As such, the effective defect production
60
fon=50ms toff = 2000 ps
50
40
30 * 20
10 —
K0,avg
5
Figure 28 The effect of a raster-scanned beam on the instantaneous production rate of point defects with the same time averaged rate as a continuous source. From Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65-98.
rate in raster-scanned systems will be less, and must be accounted for. One objective of ion irradiation is to emulate the effect of neutrons, and a second is to understand basic physical radiation damage processes, for which

neutron irradiation is often less well suited. While ion irradiation can be conducted with great control over temperature, dose rate, and total dose, such control is a challenge to reactor irradiations. For example, instrumented tubes with active temperature control are expensive to design, build, and operate. Even so, frequent power changes can be difficult to handle as the flux-temperature relationship will change and this can result in artifacts in the irradiated microstructure.44 On the other hand, temperatures in cheaper irradiation vehicles that use passive gas gaps and gamma heating (such as ‘rabbit’ tubes) are known with even less certainty. While neutron dosimetry is used in some experiments, doses and dose rates are often determined by neutronic models of the core locations and are not verifiable. As such, ion irradiations enjoy the advantage of better control and verification ofirradiation conditions as compared to neutron irradiation. Table 3 provides a list for each of three particle types: electrons, heavy ions, and light ions (protons), and they are discussed in detail in the following sections.

1.07.5.1 Electrons

Electron irradiation is easily conducted in a high — voltage transmission electron microscope using either
a hot filament or a field emission gun as an electron source. An advantage is that the same instrument used for irradiation damage can be used to image the damage. Another advantage is that the high dose rate requires very short irradiation time, but will also require a large temperature shift as explained in the Section 1.07.3.

There are several disadvantages to electron irra­diation using a TEM. First, energies are generally limited to 1 MeV. This energy is sufficient to produce an isolated FP in transition metals, but not cascades. The high dose rate requires high temperatures that must be closely monitored and controlled, which is difficult to do precisely in a typical TEM sample stage. Another drawback is that as irradiations are often conducted on thin foils, defects are created in close proximity to the surface and their behavior may be affected by the presence of the surface. Perhaps the most serious drawback is the Gaussian shape to the electron beam that can give rise to strong dose rate gradients across the irradiated region. Figure 29 shows the composition profile of copper around a grain boundary in Ni-39%Cu following electron irradiation. Note that while there is local depletion at the grain boundary (as expected), the region adja­cent to the minimum is strongly enriched in copper because of the strong defect flux out of the irradiated

Table 3 Advantages and disadvantages of irradiations with various particle types

Disadvantages

Source: Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65-98.

zone defined by the horizontal line below the spec­trum. This outward-directed defect flux causes a reversal in the direction of segregation from that caused by a defect flux to the sink. Another often observed artifact in electron irradiation is very broad grain boundary enrichment and depletion profiles. Figure 30 shows that the enrichment profile for Ni and the depletion profiles for Fe and Cr in stainless steel have widths on the order of 75-100 nm, which is much greater than the 5-10 nm widths observed following neutron irradiation under similar conditions and model simulations of radiation — induced segregation. A similar effect was observed by Wakai45 using electron and D+ irradiation of the same alloy in which the segregation profile was much higher and narrower around the grain boundary in the deuteron-irradiated sample as compared to the electron irradiation (Figure 31).

Another major subject, which has attracted much interest for the hexagonal types of SiC, is related to the electronic properties of extended defects, sur — faces/interfaces, stacking faults, and dislocations. The reason why extended defects have been mainly studied in the hexagonal types of silicon carbide lies in the fact that electronic properties of dislocations and stacking faults are particularly important for understanding the degradation of hexagonal SiC devices144 and the remarkable enhancement of dislo­cation velocity under illumination in the hexagonal phase.145 Nevertheless, some studies have been done for cubic SiC on the electronic structure of stacking faults146-151 and various types of dislocations.152-155 Obviously, a lot of work remains to be done on the extended defects in p SiC.

1.08.5.1 Uranium Oxide

1.08.5.2.1 Bulk electronic structure

Due to its technological importance and the com­plexity of its electronic structure, uranium oxide has become one of the test cases for beyond

LDA methods. Indeed, UO2 comes out as a metal when its electronic structure is calculated with LDA or GGA. This result has been found by many authors using many different codes or numer­ical schemes (the primary calculation being the work of Arko and coworkers156). The physical diffi­culty lies in the fact that UO2 is a Mott insulator. f electrons are indeed localized on uranium atoms and are not spread over the material as usual valence electrons are.

The first correction that has been applied is the LDA+U correction in which a Hubbard U term acting between f electrons is added ‘by hand’ to the Hamiltonian.157,158 This method allows the open­ing of an f-f gap.157 However, it suffers from the existence of multiple minima in the calculations, so the search for the real ground state is rather tricky as the calculation is easily trapped in metastable

159

states.

Hybrid functionals are another type of advanced methods that are very often used nowadays in the quantum chemistry community. Their principle is to mix a part of Hartree-Fock exact exchange with a DFT calculation; it has been applied to UO2 has been made by Kudin et a/.160 These methods are very promising for solid-state nuclear materials. However, the same problem of metastability as in LDA+U exists for such hybrid functionals,161 and the compu­tational load is much heavier than that in common or LDA+U calculations. Recently, an alternative to LDA+U has been proposed: the so-called local hybrid functional for correlated electrons162 in which the hybrid functional is applied only to the problematic f electrons. An application on UO2 is

available.163

Potentials based on bond stretching, bond bending, and long-ranged Coulomb interactions are widely used in molecular and organic systems. Chemists call these potentials ‘force fields.’ They cannot describe making and breaking chemical bonds, but by capturing molecular shapes, they describe the structural and dynamical properties of molecules well.

There are many commercial packages based on these force fields, for example, CHARMM35 and AMBER.36 They are primarily useful for simulating molecular liquids and solvation, but have seen little application in nuclear materials, on account of the long-range Coulomb forces, which are costly to eval­uate in large simulations.

1.10.10.2 Ionic Potentials

With no delocalized electrons, ionic materials should be suitable for modeling with pair potentials. The difficulty is that the Coulomb potential is long ranged. This can be tackled by Ewald or fast multipole meth­ods, but still scales badly with the number of atoms. The simplest model is the rigid ion potential, where charged (q) ions interact via long-range Coulomb forces and short-ranged pairwise repulsions V(r).

V (rij) +

For example, a common form of the pair potential in oxides consists of the combination of a (6-exp) Buckingham form and the Coulomb potential:

XA exp(-arj)-pi 4 + where a and b are parameters and r/ is the distance between atoms i and j.

As with other potential, various adjustments are needed in order to obtain reasonable forces at very short distance; see for example, recent reviews of UO2.37

For nuclear applications, the most commonly stud­ied material in the open literature is UO2, which is widely used as a reactor fuel. It adopts a simple fluorite structure with a large bandgap, which makes potential­fitting to get the correct crystal structure reasonably straightforward. Early work fitted the potentials to lattice parameter and compressibility, and later to elastic constants and the dispersion relation. The elas­tic constants are c11 = 395 GPa, c12 = 121 GPa, and c44 = 64 GPa.38 As previously described, the Cauchy relation generally applies to a pairwise potential,

C12 = C44, which is seldom true experimentally for oxides. However, the Cauchy relation is on the basis of the assumption that all atoms are strained equally, which is not the case for a crystal such as UO2 where some atoms do not lie at centers of symmetry. Thus, the violation of the Cauchy relation in UO2 can be fitted by attributing it to internal motions of the atoms away from their crystallographic positions. (The vio­lation of the Cauchy relation is similar in oxides with and without this effect, so it is debatable whether this is the correct physical effect.)

The earlier potentials were based on the Coulomb charge plus Buckingham described above; more recent parameterizations include a Morse potential. While this gives more degrees of freedom for fitting, having two exponential short-range repulsions with different exponents appears to be capturing the same physics twice. Comparison of the parameters39 shows that the prefactor for the U-UBuckingham repulsion varies by ten orders of magnitude when fitted. More­over, the original Catlow parameterization sets this term to zero. This difference tells us that the small U atoms seldom approach one another close enough for this force to be significant. Even the ionic charges vary between potentials by almost a factor of two, with more recent potentials taking lower values.

Despite the huge disparity in parameters, the size of cascades is similar and the recombination rate is high.

Polarizability is not incorporated in rigid ion poten­tials; they will always predict a high-frequency dielec­tric constant of 1, which is much smaller than typical experimental values. The main consequence of this for MD appears in the longitudinal optic phonon modes.

The solution is that ions themselves change in response to environment. A standard model for this is the shell model in which the valence electrons are represented by a negatively charged shell, connected to a positively charged nucleus by a spring. (Typically this represents both atomic nucleus and tightly bound electrons.) In a noncentrosymmetric environ­ment (e. g., finite temperature), the shell center lies away from the nuclear center, and the ion has a net dipole moment — it is polarized.

u=EV to-) +

у

In this case, ry may refer to the separation between nuclei i and j or the centers of the shells associated with i and i. In MD, the shells have extremely low mass, and are assumed to always relax to their equilibrium posi­tion: this is a manifestation of the Born-Oppenheimer approximation used in DFT calculation.

Shell model potentials,40 which capture the dipole polarizability of the oxygen molecules, were devel­oped by Grimes and coworkers, and have been through many extensions and reparameterizations since then. Again, there have been many successful parameterizations with wildly differing values for the parameters; even the sign of the charge on the U core and shell changes.41

A particular issue with ionic potentials is that of charge conservation. A defect involving a missing ion will lead to a finite charge. If the simulation is carried out in a supercell with periodic boundary conditions, this will introduce a formally infinite contribution to the energy. The simple way to deal with this is to ignore the long wavelength (k = 0) term in the Ewald sum. This, under the guise of a ‘neutralizing homo­geneous background charge’ is routinely done in first principles calculation. Alternately, a variable charge approach can be used42 in which the extra charge is added to adjacent atoms. The original approach then involved minimizing the total energy with respect to these additional charges, which is computationally demanding. A promising new development is to limit the range of the charge redistribution.43 While this screening approximation is difficult to justify fully in an insulator, it is very computationally efficient or a system involving dilute charged impurities, and appears to reproduce most known features of AlO.

In contrast to the T= 0 K simulations above, mod­eling by MD provides the ability to investigate temperature effects in dislocation-obstacle interac­tion. (The limit on simulation time discussed in

Section 1.12.3.3 prevents study of the creep regime controlled by dislocation climb.) Results on the tem­perature dependence of tc from simulation of inter­action between an edge dislocation and 2 and 6nm voids in Fe,29’30’34 Cu-precipitates in Fe,27,29 and voids in Cu30’34 are presented in Figure 8. In general, the strength of all the obstacles becomes weaker with increasing temperature, although the mechanisms involved are not the same for the different obstacles. The temperature-dependence of void strengthening in Fe has been analyzed by Monnet et a/.44 using a mesoscale thermodynamic treatment of MD data in the point obstacle approximation to estimate activa­tion energy and its temperature dependence. In this way, the obstacle strength found by atomic-scale mod­eling can be converted into a mesoscale parameter to be used in higher level modeling in the multiscale framework. More investigations are required to define mesoscale parameters for more complicated cases such as voids in Cu and Cu-precipitates in Fe. Void strengthening in Cu exhibits specific behavior in which the temperature-dependence is strong at low T< 100 K but rather weak at higher T (for more details see Figure 8 in Osetsky and Bacon34). The reason for this is as yet unclear.

Interestingly, MD simulation has been able to shed light on thermal effects in strengthening due to Cu-precipitates in Fe, as in Figure 8 (for more details see Figure 5 in Bacon and Osetsky2 ). Small precipitates, D < 3 nm, are stabilized in the bcc coherent state by the Fe matrix, as noted above

for T= 0 K, and are weak, shearable obstacles. The resulting temperature-dependence of tc is small.

Larger precipitates were seen to be unstable at T= 0 K with respect to a dislocation-induced trans­formation toward the fcc structure. This transforma­tion is driven by the difference in potential energy of bcc and fcc Cu. The free energy difference between these two phases of Cu decreases with increasing T until a temperature is reached at which the transfor­mation does not occur. Thus, large precipitates are strong obstacles at low T and weak ones at high T. This is reflected in the strong dependence of tc on T shown in Figure 8. More explanation of this effect can be found in Bacon and Osetsky.23 These simula­tion results showing the different behavior of small and large Cu-precipitates suggest that the yield stress of underaged or neutron-irradiated Fe-Cu alloys, which contain small, coherent Cu-precipitates, should have a weak T-dependence, whereas that in an over­aged or electron-irradiated alloy, in which the popu­lation ofcoherent precipitates has a larger size, should be stronger. Some experimental observations support this.45 One is a weak change in the temperature dependence of radiation-induced precipitate harden­ing in ferritic alloys observed after neutron irradiation when only small (<2 nm) precipitates are formed. The other is the experimentally-observed temperature and size dependence of deformation-induced trans­formation of Cu-precipitates in Fe.46

Other obstacles with inclusion properties, such as gas-filled bubbles and other types of precipitates, have been studied less intensively, and we present just a few examples here.

The effect of chromium precipitates on edge dislocation motion in matrices of either pure Fe or Fe-10at.% Cr solid solution was studied by Terentyev et a/47 Cr and Cr-rich precipitates have the bcc structure and are coherent with the matrix. Unlike Cu-precipitates in Fe, G of Cr is higher than that of both matrices and so the dislocation is repelled by Cr precipitates. Under increasing strain, the dislocation moves until it reaches the precipitate- matrix interface where it stops until the stress reaches the maximum, tc, just before the dislocation enters the precipitate (see Figure 2 in Terentyev et a/.47). The t versus e behavior is similar to that for voids in Cu, but without stress drops associated with partial disloca­tions, and no softening effects similar to voids and Cu — precipitates in Fe were observed. At tc, the dislocation shears the obstacle without acquiring a double jog. Only 2.8 and 3.5 nm precipitates in the size range D = 0.6-3.5 nm had tc comparable with values given by eqn [2] (see Figure 4 in Terentyev et al47); the others were much weaker. Separate contributions from the chemical strengthening (CS) and shear mod­ulus difference (SMD) between Fe and Cr were esti­mated and their sum was found to be close to the tc found in simulation. It was also found that tc for the alloy with Cr precipitates in an Fe-Cr solid solution is the sum of tc for the same precipitate in a pure Fe matrix and the maximum stress for glide of the dislo­cation motion in the Fe-Cr solid solution alone.

Helium-filled bubbles created by vacancies and helium formed in transmutation reactions are com­mon features of the irradiated microstructure of structural materials (see Chapter 1.13, Radiation Damage Theory). However, there is a lack of infor­mation on the properties of He bubbles and their contribution to changes in mechanical properties. The main problem is the uncertainty regarding the equilibrium state of bubbles of different sizes and at different temperatures, that is, their He-to-vacancy ratio (He/Vac). A small (0.5 M-atom) model was used48-50 to simulate interaction between an edge dislocation and a row of 2 nm cavities with He/Vac ratios of up to 5 in Fe at T between 10 and 700 K. It was found that tc has a nonmonotonic dependence on the He/Vac ratio, dislocation climb increases with this ratio, and interstitial defects are formed in the vicinity of the bubble. Recent work to clarify the equation of state of bubbles using a new Fe-He three-body interaction potential51 has shown that the equilibrium concentration of He is much lower than expected; for example the He/Vac ratio is ^0.5 for a 2 nm bubble at 300 K in Fe.52 Simulation of an edge dislocation interacting with 2 nm bubbles using the new potential for He/Vac ratio in the range 0.2-2 and T between 100 and 600 K has now been per- formed53 and preliminary conclusions drawn. The dislocation interaction with underpressurized bub­bles (He/Vac < 0.5) is similar to that with voids described above, that is, the dislocation climbs up by absorbing some vacancies on breakaway and tc increases with increasing values of He/Vac ratio up to 0.5. The interaction with overpressurized bubbles (He/Vac > 0.5) is different. The dislocation climbs down and tc decreases with increasing value of He/Vac ratio. At the highest ratio, the dislocation stress field induces the bubble to emit interstitial Fe atoms from its surface into the matrix toward the dislocation before it makes contact. The bubble pres­sure is reduced in this way and interstitials are absorbed by the dislocation as a double superjog. Equi­librium bubbles are therefore the strongest. Some of these conclusions, such as formation of interstitial clusters around bubbles with high He/Vac ratios, are similar to those observed earlier,48-50 others are not. More modeling is necessary to clarify these issues.

As noted in Section 1.12.2.2, impenetrable obsta­cles such as oxide particles and incoherent pre­cipitates represent another class of inclusion-like features. Although these obstacles are usually preex­isting and not produced by irradiation, they are con­sidered to be of potential importance for the design of nuclear energy structural materials and should be considered here. Atomic-level information on their effect on dislocations is still poor, however, and we can only refer to some recent work on this. The interaction between an edge dislocation and a rigid, impenetrable particle in Cu was simulated by Hatano54 using the Cu-Cu IAP as for the Fe-Cu system36 and a constant strain rate of 7 x 106s-1 at T = 300 K. The particle was created by defining a spherical region in which the atoms were held immo­bile relative to the surrounding crystal. The Hirsch mechanism2 was found to operate. In the sequence shown in Figures 1 and 2 of Hatano,54 several stages can be observed such as (1) the dislocation under stress approaching the obstacle from the left first bows round the obstacle to form a screw dipole; (2) the screw segments cross-slip on inclined {111} planes at tc; (3) they annihilate by double cross-slip, allowing the dislocation, now with a double superjog, to bypass the obstacle; (4) a prismatic loop with the same b is left behind and (5) the dragged superjogs pinch-off as the dislocation glides away, creating a loop of opposite sign to the first on the right of the obstacle. tc varies with D and L as predicted by the continuum modeling that led to eqn [1], but is over 3 times larger in magnitude. Hatano argues that this could arise from either higher stiffness of a dissociated dislocation or a dependence of tc on the initial posi­tion of the dislocation. It is also possible that the requirement for the dislocation to constrict and the absence of a component of applied stress on the cross­slip plane results in a high value of tc.

Simulation of 2 nm impenetrable precipitates in Fe has been carried out by Osetsky (2009, unpub­lished). The method is different from that used by Hatano54 in that the precipitate, constructed from Fe atoms held immobile relative to each other, was trea­ted as a superparticle moving according to the total force on precipitate atoms from matrix atoms. The interaction mechanism observed is quite different from those reported earlier for Cu,54 for the Hirsch mechanism and formation of interstitial clusters does

not occur. Instead, the mechanism observed was close to the Orowan process, with formation of an Orowan loop that either shrinks quickly if the obstacle is small and spherical, or remains around it if D is large (>4nm) or becomes elongated in the direction per­pendicular to the slip plane. It is interesting to note that if the model of a completely immobile precipi­tate in Hatano54 is applied to Fe, the same Hirsh mechanism is observed as that in the earlier study.

Comparison of strengthening due to pinning of a 1/2(111}{110} edge dislocation by 2 nm spherical obstacles of different nature simulated at 300 K is presented in Figure 9. One can see that the coherent Cu precipitate is the weakest whereas a rigid impen­etrable precipitate when Orowan loop is stabilized by its shape is the strongest. A surprising result is that an equilibrium He-bubble is a stronger obstacle than the equivalent void. The reason for this is not clear yet.

Little is known on the interactions between screw dislocations and inclusion-like obstacles. In fact, we are aware of only one published study of screw dislocation-void interaction in fcc Cu.55 It was found that voids are quite strong obstacles and their strength and interaction mechanisms are strongly temperature dependent. Thus at low temperature, the 1/2(110) {111} screw dislocation keeps its original slip plane

when it breaks away from the void. However, at high T> 300 K, cross-slip is activated and plays an im­portant role in dislocation-void interaction. Several depinning mechanisms involving dislocation cross­slip on the void surface were simulated and formation of DLs was observed in some cases. Interestingly, the void strength increases with increasing temperature and the authors explain this by changing interaction mechanisms. Intensive cross-slip was observed54 that propagated through the periodic boundaries along the dislocation line direction, with the result that the void was interacting with its images. Similar effects have been observed in the interaction of a screw dislocation and SIA loops and SFTs, and the possible significance of this for understanding the simulation results has been discussed elsewhere.4