In nearly free electron (NFE) theory, the effects of the atoms are included via a weak ‘pseudopotential.’ The interatomic forces arise from the response of the electron gas to this perturbation. To examine the appropriate form for an interatomic potential, we consider a simple weak, local pseudopotential V0(r). The total potential actually seen at ri due to atoms at rj will be as follows:

V(ri) = X Vo(rj) + W(r)

j

where W(r) describes how the electrons interact with one another. Given the dielectric constant,

we can estimate W in reciprocal space using linear response theory:

W (q) = Vo(q)/e(q)

where e(q) is the dielectric function. In Thomas Fermi theory,

e(q) = 1 + (4kF=P00q2)

where a0 is the Bohr radius. A more accurate approach due to Lindhard:

4kF

%a0q2

where x = q/2kF accounts for the reduced screening at high q, and p0 is the mean electron density.

From this screened interaction, it is possible to obtain volume-dependent real space potentials.6

The contributions to the total energy are as follows:

• The free electron gas (including exchange and correlation)

• The perturbation to the free electron band structure

• Electrostatic energy (ion-ion, electron-electron, ion-electron)

• Core corrections (from treating the atoms as pseudopotentials)

In this model, interatomic pair potential terms arise only from the band structure and the electrostatic energy (the difference between the Ewald sum and a jellium model) and give a minor contribution to the total cohesive energy. However, these terms are totally responsible for the crystal structure.

A key concept emerging from representing the Lindhard screening in real space is the idea of a ‘Friedel oscillation’ in the long-range potential:

cos 2kFr (2kFr )3

This arises from the singularity in the Lindhard function at q = 2kF: physically, periodic lattice per­turbations at twice the Fermi vector have the largest perturbative effect on the energy. The effect of Friedel oscillations is to favor structures where the atoms are arranged with this preferred wavelength. It gives rise to numerous effects.

• Kohn anomalies in the phonon spectrum are par­ticular phonons with anomalously low frequency. The wavevector of these phonons is such as to match the Friedel oscillation.

• Soft phonon instabilities are an extreme case of the Kohn anomaly. They arise when the lowering of energy is so large that the phonon excitation has negative energy. In this case, the phonon ‘freezes in,’ and the material undergoes a phase transfor­mation to a lower symmetry phase.

• Quasicrystals are an example where the atoms arrange themselves to fit the Friedel oscillation. This gives well-defined Bragg Peaks for scattering in reciprocal space, and includes those at 2kF but no periodic repetition in real space.

• Charge density waves refer to the buildup of charge at the periodicity of the Friedel oscillation.

• ‘Brillouin Zone-Fermi surface interaction’ is yet another name for essentially the same phenome­non, a tendency for free materials from structures which respect the preferred 2kF periodicity for the ions — which puts 2kF at the surface.

• ‘Fermi surface nesting’ is yet another example of the phenomenon. It occurs for complicated crystal structures and/or many electron metals. Here, structures that have two planes of Fermi surface separated by 2kF are favored, and the wavevector q is said to be ‘nested’ between the two.

• Hume-Rothery phases are alloys that have ideal composition to allow atoms to exploit the Friedel oscillation.

NFE pseudopotentials enabled the successful predic­tion of the crystal structures of the sp3 elements. It is tempting to use this model for ‘empirical’ potential simulation, using the effective pseudopotential core radius and the electron density as fitting parameters; indeed such linear-response pair potentials do an excellent job of describing the crystal structures of sp elements.

For MD, however, there are difficulties: the elec­tron density cannot be assumed constant across a free surface and the elastic constants (which depend on the bulk term) do not correspond to long-wavelength phonons (which do not depend on the bulk term). Since most MD calculations of interest in radiation damage involve defects (voids, surfaces), phonons, and long-range elastic strains, NFE pseudopotentials have not seen much use in this area. They may be appropriate for future work on liquid metals (sodium, potassium, NaK alloys).

The key results from NFE theory are the following:

• The cohesive energy of a NFE system comes pri­marily from a volume-dependent free electron gas and depends only mildly on the interatomic pair potential.

• The pair potential is density dependent: structures at the same density must be compared to deter­mine the minimum energy structure.

• The pair potential has a long-ranged, oscillatory tail.

• These potentials work well for understanding crystal structure stability, but not for simulating defects where there is a big change in electron density.

• The reference state is a free electron gas: descrip­tion of free atoms is totally inadequate.

Calder and coworkers examined the effect of carbon on defect production in the Fe-C system with the carbon concentration between 0 and 1.0atom%.125 The Fe potential was developed by Ackland and coworkers.134 The form of this potential is similar to

the Finnis-Sinclair potential discussed throughout this chapter, but the absolute level of defect produc­tion is somewhat lower. Simulations were carried out at temperatures of 100 and 600 K for cascade energies of 5, 10, and 20keV. Thirty simulations were carried out at each condition to ensure a good statistical sampling. No systematic effect of carbon was observed on either stable defect formation or the clustering of
vacancies and interstitials. Analysis of the octahedral sites around vacancies and interstitials revealed a sta­tistically significant association of carbon atoms with both vacancies and SIAs. This indicates an effective trapping, which is consistent with the solute-defect binding energies. Although primary damage formation was not affected by carbon, the trapping mechanism could have an effect on damage accumulation.

x

20 keV, 100 K copper

a 20 keV cascade at 100 K in Fe (a) and Cu (b). Note larger SIA

The concentration of vacancies in equilibrium with a void of radius R, C[;q (R), which enters eqn [74], can be obtained by considering the free energy of a crystal with a void and a solution of vacancies. Let x be the number of vacancies taken from a solution of vacan­cies to make a spherical void of a radius R = (3 хО/4я)1/3. The associated free energy change is given by

4pR3 2 r n

DF = —— mv + 4P~R2 [75]

where mv = kBT ln(Cv/C*) is the chemical poten­tial of a vacancy (C* is the equilibrium concentration in a perfect crystal) and у is the void surface energy. By differentiating this equation with respect to radius and equating the result to zero, one can find the equilibrium vacancy concentration, which is given by

c?(r)=[76]

Absorption and emission of PDs change a void vol­ume on the basis of the flux of PDs dDV/dt = 4pR2(dR/dt) = (Jv — Ji — Jvem). With the aid of eqns [51], [52], [74] and keeping the leading term
proportional to R only and [76], the growth rate of a void due to absorption of vacancies and SIAs and vacancy emission can be written as ’ 2O~ ‘

RkBT

Neglecting the entropy factor for simplicity, one can find that CJh = exp(-Ef / kBT), where Ef is the vacancy formation energy. The last term in the square brackets on the right-hand side of eqn [66] can be then represented in the following form ’ 2O~ ‘

RkBT where [79] is a well-known equation for the binding energy of a vacancy with a void that is valid for large enough radius. For voids of small sizes, the value Eb has to be calculated by using ab initio or MD methods.

Equation [77] is used in calculations of void swelling. Note that the vacancy and SIA fluxes, the first and second terms, enter this equation sym­metrically and this is because of the neglect of the difference in the interactions of SIAs and vacancies with voids. Also, when the sum of the second and third terms in the right-hand side of this equation is larger than the first term, the voids shrink. Such a shrinking takes place during annealing of preirra­diated samples or, in some cases, during irradiation, if the irradiation conditions are changed. However, in the majority of cases, voids grow under irradiation because dislocations interact more strongly with SIAs than vacancies.

Many engineering alloys contain ordered phases or precipitates to optimize their properties, in particu­lar mechanical properties. It is thus important to

investigate how these optimized microstructures evolve under irradiation. It is anticipated that, under appropriate conditions, ballistic mixing can lead to the dissolution of precipitates, and to the disordering of chemically ordered phases.129 Matsumura et at}30 used a 1D PF approach on a model binary alloy system to specifically investigate what evolution irra­diation may produce. In that model, the composition field is represented by the globally conserved order parameter X(r), while the degree of order is repre­sented by the nonconserved order parameter S(r). X is chosen to vary from — 1 to +1 for pure A and pure B composition, respectively, and S takes a value ranging from 0 to 1 for fully disordered and fully ordered phases, respectively. The free energy func­tional of the system is written as	f (X, S, T)
(X — Xm)2 — b(T ){X0(T )2 — X2}S2 + b(T )2 X1(T )2S4
[29;
where f0 is the mean field free energy of the disor­dered phase with composition xm, and a, b, xj2 are positive constants depending on temperature. The equilibrium phase diagram for this model system is given in Figure 10. Notice, in particular, that at low temperature and for compositions sufficiently far from the equiatomic composition, an ordered phase coexists with a disordered phase. The kinetic evolution of these two fields is gov­erned by
dr
[30]
[28]
and
where f(X, S, T) is the mean field free energy of a homogeneous alloy, and H and K are positive constants of the interfacial energy coefficients in the presence of varying field X and S, respectively. The homogeneous free-energy density is given by a Landau expansion
@S, 8F({X, S, T}) — =-£fS — M(T, f) ({8’s’ })
[31]
where f is the atomic displacement rate, m is the chemical potential, Dmix and e are positive coefficients characterizing the efficiency of mixing and disordering
Figure 10 Equilibrium phase diagram for model alloy system given by eqns [28] and [29]. The dotted lines correspond to the metastable extrapolation of the order-disorder transition into the miscibility gap. Reprinted with permission from Matsumura, S.; Muller, S.; Abromeit, C. Phys. Rev. B 1996, 54(9), 6184-6193. Copyright by the American Physical Society.

by irradiation, and L and M are the mobility coeffi­cients for the conserved and nonconserved fields. Note that here there is no kinetic coupling between these two fields. There is, however, thermodynamic coupling through the expression chosen for the homogeneous free-energy density, eqn [29].

Although point defects are not explicitly used as PF variables, the dependency of the mobility co­efficients M and L with temperature, irradiation flux, and sink density (c$) is obtained from a rate theory model for the vacancy concentration under

irradiation in a homogeneous alloy.1 The steady-state phase diagrams for two irradiation flux values are given in Figure 11. At the higher flux, the phase diagram is composed of homogeneous disordered and ordered phases only. At low enough temperature, the ballistic mixing and disordering dominate the evolution of the alloy, leading to the destabilization of the ordered phase at and near the stoichiometric composition X = 0, and to the disappearance of the two-phase coexistence domains for off-stoichiometric compositions.

The model has also been used to study the disso­lution of ordered precipitates under irradiation. In agreement with prior lattice-based mean field kinetic simulations,131 it is found that two different dissolution paths are possible, depending upon the composition and irradiation parameters. Ordered precipitates may either disorder first and then slowly dissolve or they may dissolve progressively while retaining a finite degree of chemical order until their complete dissolution. These two kinetic paths have indeed been observed experimentally in Nimonic PE16 alloys irradiated with 300-keV Ni ions.132,133

As mentioned above, one of the most spectacular consequences of RIS is that it can completely modify the stability ofprecipitates and the precipitate micro- structure.47 When the local solute concentration in the vicinity of a point defect sink reaches the solubil­ity limit, RIP can occur in an overall undersaturated alloy. RIP of the y’-Ni3Si phase is observed, for exam­ple, in Ni—Si alloys28 at concentrations well below the solubility limit (Ni3Si is an ordered L12 structure and can be easily observed in dark-field image in transmis­sion electron microscopy (TEM)). In this case, it is believed that RIS is due to the preferential occupation of interstitials by undersized Si atoms.28 The y’-phase

can be observed on the preexisting dislocation network, at dislocation loops formed by self-interstitial cluster­ing,28 at free surfaces45 or grain boundaries.48 The fact that the y’-phase dissolves when irradiation is stopped clearly reveals the nonequilibrium nature of the pre­cipitation. This is also shown by the toroidal contrast of dislocation loops (Figure 4(a)): the y’-phase is observed only at the border of the loop on the disloca­tion line where self-interstitials are annihilated; when the loop grows, the ordered phase dissolves at the center of the loop, which is a perfect crystalline region where no flux of Si sustains the segregation.

In supersaturated alloys, the irradiation can com­pletely modify the precipitation microstructure. It can dissolve precipitates located in the vicinity of sinks when RIS produces a solute depletion. For example, in Ni—Al alloys,49 dissolution of y’-precipitates is observed around the growing dislocation loops due to the Al depletion induced by irradiation (Figure 5), and in supersaturated Ni—Si alloys, Si segregation towards the interstitial sinks produces dissolution of the homo­geneous precipitate microstructure in the bulk, to the benefit of the precipitate layers on the surfaces28 (Figure 6) and grain boundaries.50

In the previous examples, RIS was observed to produce a heterogeneous precipitation at point defect sinks. But homogeneous RIP of coherent pre­cipitates has also been observed, for example, in Al-Zn alloys.51 Cauvin and Martin52 have proposed a mechanism that explains such a decomposition. A solid solution contains fluctuations of composition. In case of attractive vacancy-solute and interstitial — solute interactions, a solute-enriched fluctuation tends to trap both vacancies and interstitials, thereby favoring mutual recombination. The point defect concentrations then decrease, producing a flux of new defects toward the fluctuation. If the coupling with solute flux is positive, additional solute atoms

arrive on the enriched fluctuations, and so it con­tinues, till the solubility limit is reached.

As differences in dose rates can confound direct comparison between neutron and ion irradiations, it is important to assess their impact. A simple method for examining the tradeoff between dose and temper­ature in comparing irradiation effects from different particle types is found in the invariance requirements. For a given change in dose rate, we would like to know what change in dose (at the same temperature) is required to cause the same number of defects to be absorbed at sinks. Alternatively, for a given change in dose rate, we would like to know what change in temperature (at the same dose) is required to cause the same number of defects to be absorbed at sinks. The number of defects per unit volume, NR, that have recombined up to time t, is given by Mansur14

t
Ci Cv dt	[12]
Nr
0
where kSj is the strength of sink j and Cj is the sink concentration. The ratio of vacancy loss to interstitial loss is
Nsv Nsi
[14]
Rs
where j = v or i. The quantity Ns is important in describing the microstructural development involving total point defect flux to sinks (e. g., RIS), while Rs is the relevant quantity for the growth of defect aggregates such as voids that require partitioning of point defects to allow growth. In the steady-state recombination dominant regime, for Ns to be invariant at a fixed dose, the follow­ing relationship between ‘dose rate (K) and temperature (Ti)’ must hold:
Figure 9 shows plots of the relationship between the ratio of dose rates and the temperature difference required to maintain the same point defect absorption at sinks (a), and the swelling invariance (b). The invariance requirements can be used to prescribe an ion irradiation temperature-dose rate combination that simulates neutron radiation. We take the example of irradiation of stainless steel under typical BWR core irradiation conditions of ^4.5 x 10-8 dpa s-1 at 288 °C. If we were to conduct a proton irradiation with a characteristic dose rate of 7.0 x 10-6dpas-1, then using eqn [15] with a vacancy formation energy of 1.9 eV and a vacancy migration
£M t
[15]
where Evm is the vacancy migration energy. In the steady-state recombination dominant regime, for Rs to be
Figure 9 Temperature shift from the reference 200 °C required at constant dose in order to maintain (a) the same point defect absorption at sinks, and (b) swelling invariance, as a function of dose rate, normalized to initial dose rate. Results are shown for three different vacancy migration energies and a vacancy formation energy of 1.5 eV. Adapted from Mansur, L. K. J. Nucl. Mater. 1993, 206, 306-323; Was, G. S. Radiation Materials Science: Metals and Alloys; Springer: Berlin, 2007.

energy of 1.3 eV, the experiment will be invariant in Ns with the BWR core irradiation (e. g., RIS) at a proton irradiation temperature of 400 °C. Similarly, using eqn [16], a proton irradiation temperature of 300 °C will result in an invariant Rs (e. g., swelling or loop growth). For a Ni2+ ion irradiation at a dose rate of 10~3 dpas, the respective temperatures are 675 °C (Ns invariant) and 340 °C (Rs invariant). In other words, the temperature ‘shift’ due to the higher dose rate is dependent on the microstructure feature of interest. Also, with increasing difference in dose rate, the AT between neutron and ion irradiation increases substantially. The nominal irradiation tem­peratures selected for proton irradiation, 360 °C and for Ni2+ irradiation, 500 °C represent compromises between the extremes for invariant Ns and Rs.

The interest in SiC as a large band gap semiconduc­tor for electronic applications has promoted works on typical dopants. Most of the calculations focus on hexagonal SiC, but one can reasonably assume that

the results would not be very different in cubic SiC. One can find calculations dealing with boron12 ’ as an acceptor and nitrogen131’132 or phosphorus133 as a donor. Other impurities were studied: transition metals,134-136 oxygen,137 important for the behavior of the SiO2/SiC interface, hydrogen,138-140 rare gases,141 and palladium.142 A systematic study of substitutional impurities has recently appeared,143 which focuses on the trends of carbon vs. silicon substitution according to the position of species in the periodic table.

While much of the work on structural materials has concentrated on metals, there are important issues involving nonmetallics for coatings, corrosion, and fuel. In this section, we review other types ofpotentials.

1.10.10.1 Covalent Potentials

Empirical potentials for covalent materials have been much less successful than for metals. As with the NFE pair potentials, the bulk of the energy is contained in the covalent bond, and potentials which well describe distortions from fourfold coordination tend to fail when applied to other bond situations, such as sur­faces, high pressure phases, or liquids.

A commonly used example, the Stillinger-Weber potential,29 is written as

Uj = e V (rj )+EI3

j j k V3 /

where V(rj) and F(rj) are short-ranged pair poten­tials. The form of the three-body term, with its mini­mum at 109°, ensures the stability of a tetrahedrally bonded network. The stacking fault energy (equiva­lently, the difference between cubic and hexagonal diamond) is zero, so the ground state crystal structure is not unique; however, this is not far from correct, and hexagonal diamond can be found in carbon and silicon. Moreover, it has little effect in many simulations since the large kinetic barrier against cubic-hexagonal phase transitions prevents them occurring in simulations. Solidification and recovery from cascade damage are counterexamples.

Stillinger-Weber works well for fourfold — coordinated amorphous networks, and vibrational properties of the diamond structures. It gives too low

density (and coordination) for the liquid and high — pressure phases, because it fails to reproduce the rebon­ding of atoms at the surface to remove dangling bonds.

An alternate approach to stabilizing diamond via the 109° angle is to do so through its tetravalent nature. The simplest type is the restricted bond pair potential30:

и = X A(r/) — X B(r/)

j j=M

where the attractive part of the potential is summed over at most four neighbors (one per electron). This formalism describes well the collapse of the network under pressure or melting, but lacks shear rigidity (only the repulsion of second neighbors provides shear rigidity). There is also some ambiguity over which four neighbors should be chosen, which makes implementation difficult.

An embellishment on this is the bond-charge model, in which the electrons in the bonds repel one another. This adds a three-center term of the form

X C(J)

i

where 1 and k are bonded neighbors ofi. This approach avoids explicitly introducing the tetrahedral angle into the potential. Note that although this term is asso­ciated with atom i (and is often interpreted as a bond­bending term at i), in the simplest form forces derived from this term are independent of i.

The problem of defining ‘bonded neighbors’ can be circumvented, in the spirit of the embedded atom method, by having an embedding function that effec­tively cuts off after the bonding reaches four neigh­bors worth31 as in the Tersoff approach:

U = X A(rJ )-X B(r/)

j j

where

B(riJ ) = f (ri/) X G(ri’b J; rij)

The bond ij is weakened by the presence of other bonds ik and jk involving atoms i and j. The pro­tetrahedral angular dependence is still necessary to stabilize the structure, and further embellishment by Brenner32 corrects for overbinding of radicals.

These potentials give a good description of the liquid and amorphous state, and have become widely used in many applications, in addition to elements such as Si and C, as well as covalently bonded compounds such as silicon carbide33 and tungsten carbide.34

1.12.4.1 Inclusion-Like Obstacles

1.12.4.1.1 Temperature T=0K

Voids in bcc and fcc metals at T = 0 K and >0 K are probably the most widely simulated obstacles of this type. Most simulations were made with edge disloca — tions.10,25-34 A recent and detailed comparison of strengthening by voids in Fe and Cu is to be found in Osetsky and Bacon.34 Examples of stress-strain curves (t vs. e) when an edge dislocation encounters and overcomes voids in Fe and Cu at 0 K are pre­sented in Figures 2 and 3, respectively. The four distinct stages in t versus e for the process are described in Osetsky and Bacon10 and Bacon and Osetsky. The difference in behavior between the two metals is due to the difference in their dislocation core structure, that is, dissociation into Shockley partials in Cu but no splitting in Fe (for details see Osetsky and Bacon34).

Under static conditions, T = 0 K, voids are strong obstacles and at maximum stress, an edge dislocation in Fe bows out strongly between the obstacles, creat­ing parallel screw segments in the form of a dipole pinned at the void surface. A consequence of this is that the screw arms cross-slip in the final stage when the dislocation is released from the void surface and this results in dislocation climb (see Figure 4), thereby reducing the number of vacancies in the void and therefore its size. In contrast to this, a Shockley partial cannot cross-slip. Partials of the dissociated dislocation in Cu interact individually with small voids whose diameter, D, is less than the partial spacing (~2 nm), thereby reducing the obstacle strength. Stress drops are seen in the stress-strain curve in Figure 3. The first occurs when the leading partial breaks from the void; the step formed by this on the exit surface is a partial step 1/6(112) and the stress required is small. Breakaway of the trailing partial controls the critical stress tc. For voids with D larger than the partial spacing, the two partials leave the void together at the same stress. However, extended screw segments do not form and the dislo­cation does not climb in this process. Consequently, large voids in Cu are stronger obstacles than those of the same size in Fe, as can be seen in Figure 6 and the number of vacancies in the sheared void in Cu is unchanged.

Cu-precipitates in Fe have been studied exten­sively23,27-29 due to their importance in raising the yield stress of irradiated pressure vessels steels35 and

200

150

100

50

0

-50

0.0 0.5 1.0 1.5

Strain (%)

Figure 2 Stress-strain dependence for dislocation-void interaction in Fe at 0 K with L = 41.4 nm. Values of D are indicated below the individual plots. From Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2010, 90, 945. With permission from Taylor and Francis Ltd. (http://www. informaworld. com).

350 300 250

|f 200

§.

150

сл 0

^ 100

ra ф

50 0

-50 -100

[112], a

Figure 4 [111] projection of atoms in the dislocation core showing climb of a1/2(111){110} edge dislocation after breakaway from voids of different diameter in Fe at 0 K. Climb-up indicates absorption of vacancies. The dislocation slip plane intersects the voids along their equator. From Osetsky, Yu. N.; Bacon, D. J. J. Nucl. Mater. 2003, 323, 268. Copyright (2003) with permission from Elsevier.

the availability of suitable IAPs for the Fe-Cu sys­tem.36 These precipitates are coherent with the sur­rounding Fe when small, that is, they have the bcc structure rather than the equilibrium fcc structure of Cu. Thus, the mechanism of edge dislocation inter­action with small Cu precipitates is similar to that of voids in Fe. The elastic shear modulus, G, of bcc Cu is lower than that of the Fe matrix and the dislocation is attracted into the precipitate by a reduction in its strain energy. Stress is required to overcome the
attraction and to form a 1/2(111) step on the Fe-Cu interface. This is lower than tc for a void, how­ever, for which G is zero and the void surface energy relatively high. Thus, small precipitates (< 3 nm) are relatively weak obstacles and, though sheared, remain coherent with the bcc Fe matrix after dislocation breakaway. tc is insufficient to draw out screw seg­ments and the dislocation is released without climb.

The Cu in larger precipitates is unstable, however, and their structure is partially transformed toward

(D~1 + L“V, b

the more stable fcc structure when penetrated by a dislocation at T = 0 K. This is demonstrated in Figure 5 by the projection of atom positions in four {110} atomic planes parallel to the slip plane near the equator of a 4 nm precipitate after dislocation break­away. In the bcc structure, the {110} planes have a twofold stacking sequence, as can be seen by the upright and inverted triangle symbols near the out­side of the precipitate, but atoms represented by circles are in a different sequence. Atoms away from the Fe-Cu interface are seen to have adopted a threefold sequence characteristic of the {111} planes in the fcc structure. This transformation of Cu struc­ture, first found in MS simulation of a screw disloca­tion penetrating a precipitate,37,38 increases the obstacle strength and results in a critical line shape that is close to those for voids of the same size.34 Under these conditions, a screw dipole is created
and effects associated with this, such as climb of the edge dislocation on breakaway described above for voids in Fe, are observed.23,27

The results above were obtained at T = 0 K by MS, in which the potential energy of the system is minimized to find the equilibrium arrangement of the atoms. The advantage of this modeling is that the results can be compared directly with continuum modeling ofdislocations in which the minimum elastic energy gives the equilibrium dislocation arrangement. An early and relevant example of this is provided by the linear elastic continuum modeling of edge and screw dislocations interacting with impenetrable Oro — wan particles39 and voids.40 By computing the equilib­rium shape of a dislocation moving under increasing stress through the periodic row of obstacles, as in the equivalent MS atomistic modeling, it was shown that the maximum stress fits the relationship

f~*~L

+ L—^ W

where G is the elastic shear modulus and D is an empirical constant; A equals 1 if the initial dislocation is pure edge and (1 — n) if pure screw, where n is Poisson’s ratio. Equation [1] holds for anisotropic elas­ticity if G and n are chosen appropriately for the slip system in question, that is, if Gb2/4я and Gb2/4яА are set equal to the prelogarithmic energy factor of screw and edge dislocations, respectively.39,40 The value of G obtained in this way is 64 GPa for (111){110} slip in Fe and 43 GPa for (110) {111} slip in Cu.41

The explanation for the D — and L-dependence of tc is that voids and impenetrable particles are ‘strong’ obstacles in that the dislocation segments at the obsta­cle surface are pulled into parallel, dipole alignment at tc by self-interaction.39,40 (Note that this shape would not be achieved at this stress in the line-tension approximation where self-stress effects are ignored.) For every obstacle, the forward force, tcbL, on the dislocation has to match the dipole tension, that is, energy per unit length, which is proportional to ln(D) when D ^L and ln(L) when L ^ D.39 Thus, tcbL correlates with Gb2ln(D—1 + L—1)—1. The correlation between tc obtained by the atomic-scale simulations above and the harmonic mean of D and L, as in eqn [1], is presented in Figure 6. A fairly good agreement can be seen across the size range down to about D < 2 nm for voids in Fe and 3-4 nm for the other obstacles. The explanation for this lies in the fact that in the atomic simulation, as in the earlier contin­uum modeling, obstacles with D > 2-3 nm are strong at T = 0 K and result in a dipole alignment at tc.

Smaller obstacles in Fe, for example, voids with D < 2 nm and Cu precipitates with D < 3 nm, are too weak to be treated by eqn [1]. Thus, the descriptions above and the data in Figure 6 demonstrate that the atomic — scale mechanisms that operate for small and large obstacles depend on their nature and are not pre­dicted by simple continuum treatments, such as the line-tension and modulus-difference approximations that form the basis of the Russell-Brown model of Cu — precipitate strengthening of Fe,42 often used in pre­dictions and treatment of experimental observations.

The importance of atomic-scale effects in inter­actions between an edge dislocation and voids and Cu-precipitates in Fe was recently stressed in a series of simulations with a variable geometry.43 In this study, obstacles were placed with their center at different distances from the dislocation slip plane. An example of the results for the case of 2 nm void at T = 0 K is presented in Figure 7. The surprising result is that a void with its center below the disloca­tion slip plane is still a strong obstacle and may increase its size after the dislocation breaks away. This can be seen in Figure 7, where a dislocation line climbs down absorbing atoms from the void surface. More details on larger voids, precipitates, and finite temperature effects can be found in Grammatikopoulos et al4

As shown in Section 1.13.5, in the framework of FP3DM, the swelling rate depends on the dislocation density and becomes small for a low dislocation den­sity, dS/df « Bdpd/k2 ! 0 at pd! 0 (see eqn [96]). Thus, it was a common belief that the swelling rate in well-annealed metals has to be low at small doses, that is, when the dislocation density increase can be neglected. Under neutron irradiation, the effect of dislocation bias on swelling is even smaller because of intracascade recombination: (dS/df)^^ =

(dS/dfED(1 — er) < (dS/df)Zf. It has been

found experimentally, however, that the void swelling rate in fully annealed pure copper irradiated with fission neutrons up to about 10—2dpa (see Singh and Foreman18) is of ~1% per dpa, which is similar to the maximum swelling rate found in materials at high enough irradiation doses. This observation was one of those that prompted the development of the PBM. The production bias term in eqn [138] allows the understanding of these observations. Indeed, at low doses of irradiation, the void size is small, and therefore, the void cross-section for the inter­action with the SIA glissile clusters is small (Krc2Nc/Lg ^ 1). As a result, the last term in the pro­duction bias term is negligible and thus the swelling rate is driven by the production bias:

 

dS df

 

[138] where f = GNRTt is the NRT irradiation dose. The first term in the brackets on the right-hand side of eqn [138] corresponds to the influence of the dislocation bias and the second one to the production bias. The factor (1 — er) describes intracascade recombination of defects, which is a function of the recoil energy and may reduce the rate of defect production by up to an order of magnitude that can be compared to the NRT value: (1 — er)! 0.1 at high PKA energy (see Section 1.13.3). As indicated by this equation, the swelling rate is a complicated function of dislocation density, dislocation bias factor, and the densities and sizes of voids and PD clusters. It also demonstrates the dependence of the swelling rate on the recoil energy, determined by eig, which increases with increasing PKA energy up to about 10-20 keV. The main predictions of the PBM are dis­cussed below.

 

1.13.6.2 Main Predictions of Production Bias Model As can be seen from eqn [138], the action and con­sequences of the two biases, the dislocation and production ones, is quite different. As shown in Section 1.13.5, the dislocation bias depends only slightly on the microstructure and predicts indefinite void growth. In contrast, the production bias can be positive or negative, depending on the microstruc­ture. The reason for this is in negative terms in eqn [138]. The first term decreases the action of the

sessile vacancy and SIA clusters, the swelling rate is given by dS/df ~ 1/2(1 — er)eg where the sink strength ratio, k//(k2 + Z/pd), is taken to be equal to 1/2, as frequently observed in experiments. Taking into account the magnitude of the cascade para­meters er and eg estimated in Golubov et a/.24 and neglecting the dislocation bias term in eqn [138], one may conclude that the maximum swelling rate under fast neutron irradiation may reach about 1% per dpa. As pointed out in Section 1.13.5, in the case of FP production, that is, in the FP3DM, the maximum swelling rate is also ~1% per dpa. This coincidence is one of the reasons why an illusion that the FP3DM model is capable of describing damage accumula­tion in structural and fuel materials in fission and future fusion reactors has survived despite the fact that nearly 20 years have passed since the PBM was introduced.

Note that the production bias provides a way to understand another experimental observation, namely, that the swelling rate in some materials decreases with increasing irradiation dose (see, e. g., Figure 5 in Golubov et a/.24). Such a decrease is predicted by eqn [138], as the negative term of the production bias, %r Nc/Lg, increases with an increase in the void size. As the first term in the 10-4

Figure 5 Experimentally measured133 and calculated24 levels of void swelling in pure copper after irradiation with 2.5 MeV electrons, 3MeV protons, and fission neutrons. The calculations were performed in the framework of the FP3DM for the electron irradiation and using the production bias model as formulated in Singh etal.22 for irradiations with protons and fission neutrons. From Golubov etal.24

production bias is proportional to the void radius and the second to the radius squared, the swelling rate may finally achieve saturation at a mean void radius equal to Rmax ~ 2яг/.19,30,35

Finally, the cascade production of the SIA clusters may strongly affect damage accumulation. As can be seen from eqn [132], the steady-state sink strength of the sessile SIA clusters is inversely proportional to the fraction of SIAs produced in cascades in the form of mobile SIA clusters, thus k2d ! 1 when eg! 0. This limiting case was considered by Singh and Foreman18 to test the validity of the original frame­work of the PBM,16,17 where all the SIA clusters produced by cascades were assumed to be immobile (hereafter this case of eg = 0 is called the Singh— Foreman catastrophe). If for some reasons this case is realized, void swelling and the damage accumulation in general would be suppressed for the density of SIA clusters, hence, their sink strength would reach a very high value by a relatively low irradiation dose, f ^ 1dpa, (see Singh and Foreman18). Thus, irradiation with high-energy particles, such as fast neutrons, provides a mechanism for suppressing damage accumulation, which may operate if the SIA clusters are immobilized. In alloys, the interaction with impurity atoms may provide such an immobili­zation. The so-called ‘incubation period’ of swelling observed in stainless steels under neutron irradia­tion for up to several tens of dpa (Garner32,33) might be due to the Singh—Foreman catastrophe. A possible scenario of this may be as follows: during the incubation period, the material is purified by RIS mainly on SIA clusters because of their high density. At high enough doses, that is, after the incu­bation period, the material becomes clean enough to provide the recovery of the mobility of small SIA clusters created in cascades that triggered on the production bias mechanism. As a result, the high number density of SIA clusters decreases via the absorption of the excess of vacancies, restoring con­ditions for damage accumulation.