As mentioned above in Section 1.11.4.2, Bacon and coworkers84,107 have shown that the number of stable
point defects produced in many materials follows a simple power-law dependence over a broad range of cascade energies (see eqn [3]). This behavior is shown in Figure 32 for several pure metals and Ni3Al.107 This figure also includes a line labeled NRT that is obtained from eqn [2] if the displacement threshold is taken as 40 eV, which is the recommended average value for iron.16 The difference between the NRT and Fe lines reflect the ratio plotted in Figure 9. As the displacement threshold is different for different metals (e. g., 30 eV is recommended for Cu16), the other lines should not be compared directly with the NRT values. When normalized using the appro­priate NRT displacements, the difference in the sur­vival ratio between Fe and Cu can be seen in Figure 33.61 Although the stable defect production in the other metals may be either somewhat lower or higher than in iron, the behavior is clearly similar across this group of bcc, fcc, and hcp materials. As the energies involved in displacement cascades are so much greater than the energy per atom in a perfect lattice or the vacancy and interstitial formation ener­gies, it is not surprising that ballistic defect produc­tion would be similar.

In-cascade clustering behavior shows a stronger variation between metals than does total defect sur­vival. The fraction of surviving interstitials contained in clusters is shown in Figure 34 for some of these

Ep (keV)

same metals as in Figure 32.107 Although defect formation does not seem to correlate with crystal structure in Figure 32, there is some indication that this may not be the case with interstitial clustering. The lowest clustering fraction is seen in bcc Fe, while the close-packed Cu (fcc) and hcp (Ti, Zr) materials yield higher values. Ti and Zr exhibit nearly the same value. Ni3Al, which is nominally close-packed is more similar to iron. This may be a result of the ordered structure and some impact of antisite defects on inter­stitial clustering. However, there is insufficient data available to make any definitive conclusions.

A further comparison of vacancy and interstitial clustering in Fe and Cu is provided by Figure 35,61 which provides histograms of the cluster size distri­butions for two different cascade energies at 100 K. The interstitial cluster size distributions are shown in (a) and (c) for Fe and Cu, respectively, and the corresponding vacancy cluster size distributions are shown in (b) and (d). Note the scale difference on the abscissa between Figure 35(a and b) and Figure 35(c and d). In addition to having a higher fraction of surviving defects in clusters, copper clearly produces much larger clusters of both types. It is clear that some of these differences are related to either the crystal structure and/or such basic parameters as the stacking fault energy. Many of the large vacancy clus­ters in fcc copper, which has a low stacking fault energy, are large stacking fault tetrahedra (generally imperfect). Similarly, large faulted SIA loops are observed in Cu. Figure 36 illustrates the difference observed between iron and copper in typical 20 keV cascades at 100 K. The final damage state is shown for Fe in (a) and Cu in (b). The simulation cells have an edge length of 50 lattice parameters in both cases. The copper cascade is clearly more compact and exhibits more point defect clustering.

While comparisons of iron and copper have been thoroughly explored in the literature,59,61,107,139 there have also been studies on materials such as zirconium, which is relevant to nuclear fuel cladding.70,107,140,141 Figure 37 provides an example of the differences in point defect clustering between Fe and hcp Zr. The average number of SIAs and vacancies in clusters per cascade as a function of cascade energy at 100 K is shown for (a) zirconium and (b) iron.107 Note the difference in scale on both the number in clusters and the cluster size, and that the highest cascade energy is 20 keV in (a) and 49 keV in (b). In both metals the probability of clustering increases with cascade energy, and the size of the largest cluster similarly increases. As indicated by the fact that there are more single vacancies than single interstitials, a greater fraction of SIAs are in clusters. Similar to the Fe-Cu comparison, there is significantly more clustering in close-packed Zr than in bcc Fe.

Dissociation of vacancies from voids and other defects is an important process, which significantly affects their evolution under irradiation and during aging. Similar to the absorption rate eqn [54], it has been shown that the dissociation rate is proportional to the void radius. Such a result can readily be obtained by using the so-called detailed balance con­dition. However, as the evaporation takes place from the void surface, the frequency of emission events is proportional to the radius squared. In the following lines, we clarify why the dissociation rate is propor­tional to the void radius and elucidate how diffusion operates in this case.

Consider a void of radius R, which emits ndiss = t-d. vacancies per second per surface site in a spherical coordinate system. Vacancies migrate 3D with the diffusion coefficient Dv = a2/6t, where a is the vacancy jump distance and t is the mean time delay before a jump. The diffusion equation for the vacancy concentration Cv is

r2Cv = 0 [69]

To calculate the number of vacancies emitted from the void and reach some distance R1 from the void surface, we use absorbing boundary conditions at this distance

CV(R1) = 0 [70]

An additional boundary condition must specify the vacancy-void interaction. Assuming that vacancies are absorbed by the void, which is a realistic scenario, the vacancy concentration at one jump distance a from the surface can be written as

Cv(R V a) Cv(R V 2a)

ndiss V

t 2t

The left-hand side of the equation describes the fre­quency with which vacancies leave the site. The first term on the right-hand side accounts for the produc­tion of vacancies due to evaporation from the void. The last term on the right-hand side accounts for vacancies coming to this site from sites further way from the void surface. After representing the latter term using a Taylor series, in the limit of R ^ a, the boundary condition, eqn [71], assumes the follow­ing form

CV(R)=2tndiss V arCY(R) [72]

Using this condition and eqns [69] and [70], one finds the vacancy concentration, Cv(r), is equal to

1 — (R1) 1 R-1 — (R1)-1

It can readily be estimated using the last two equations that the gradient of concentration in eqn [72] is smal­ler than the other terms by a factor of a/r0 and does not contribute to eqn [73]. This means that most vacancies emitted from the void return to it. As a result, the equilibrium condition for the concentration near the void surface is defined by the equality ofthe frequency of evaporation and the frequency of jumps back to the surface and is not affected by the flux of vacancies away from the surface. The vacancy equilib­rium concentration at the void surface is readily obtained from eqn [73] as Cvq(R) = Cv(R) = 2tVjiss.

The total number of vacancies passing through a spherical surface of radius R and area S = 4nR2 per unit time, that is, the rate of vacancy emission from the void, is equal to

SD

Jvem = — VVCv(rV=R

D Cvq 4PR — A Cvq 1pR

O 1 — R/R1 ~ O

There are three points to be made. First, eqn [73] becomes independent of the distance r from the surface, when r ^ R. Thus, vacancies reaching this distance are effectively independent of their origin and can be counted as dissociated from the void. Second, despite the fact that the total vacancy

emission frequency is proportional to the void sur­face area, the total vacancy flux far away from the surface is proportional to the void radius. This is a well-known result of the reaction-diffusion theory40 considering the void capture efficiency. Third, as can be seen from eqn [74], significant deviation from the proportionality to the void radius occurs at distances of the order of the void radius.

As discussed above, most emitted vacancies return to the void. The fraction of vacancies which do not return is equal to the ratio of the frequency defined by eqn [63] and the total frequency of vacancy emis­sion ~ 4nR2Vnss/a2. It is thus equal to a/R. The same result can be demonstrated considering another, although unrealistic, scenario in which vacancies are reflected by the voids.102 We also note that the first nonvanishing correction to the proportionality of the vacancy flux to the void radius is positive and pro­portional to the void radius squared, see eqn [74], where R(1 — R/R1)-1 « R + R2/R1. The same result was obtained previously by Gosele40 when con­sidering void capture efficiency. Thus, with increasing volume fraction more and more vacancies become absorbed at other voids and the proportionality to the void radius squared would be restored. The first cor­rection term just shows the right tendency.

1.15.4.2.1 Effects of ballistic mixing on phase-separating alloy systems

Consider the simple case where the external forcing produces forced exchanges between atoms (such relocations are found in displacement cascades), and let us assume for now that these relocations are ballistic (i. e., random) and take place one at a time. For this case, one can use a 1D PFM to follow the evolution of the composition profile C(x) during irra­diation.118 This evolution is the sum of a thermally activated term, for which the classical Cahn diffusion model can be used, and a ballistic term:

^ = MirrV2 dF — Gj{c(x)-JWR(x — x’)C(x’)dx’} [25] where Mirr is the thermal atomic mobility, here accel­erated by the irradiation, F the free energy of the system, Gb the jump frequency of the atomic reloca­tions forced by the nuclear collisions, and wR is the normalized distribution of relocation distances, char­acterized by a decay length R. Since most of these atomic relocations take place between nearest neigh­bor atoms, in a first approximation one may assume that R is small compared to the cell size. In this case, the second ballistic term in eqn [25] reduces to a diffusive term:

dC(x) 2 dF 2 2 г і

= MirrV2 gC — Gya2V2 C [26]

In this case, the model thus reduces to the one initi­ally introduced by Martin,119 and the steady state reached under irradiation is the equilibrium state that the same alloy would have reached at an effective temperature Tf = T(1 + Gj/Gj."), where GJh is an average atomic jump frequency, enhanced by the point-defect supersaturation created by irradiation. In particular, in the case of an alloy with preexisting precipitates, depending upon the irradiation flux and the irradiation temperature, this criterion predicts that the precipitates should either dissolve or contin­uously coarsen with time.

Some relocation distances, however, extend beyond the first nearest neighbor distances,120,121 and it is interesting to consider the case where the characteris­tic distance R exceeds the cell size. An analytical model by Enrique and Bellon118 revealed that, when R exceeds a critical value Rc, irradiation can lead to the dynamic stabilization of patterns. To illustrate this point, one performs a linear stability analysis of this model in Fourier space, assuming here that the ballistic jump distances are distributed exponen­tially. The amplification factor w(q) of the Fourier coefficient for the wave vector q is given by

o(q)/M = — (92/ /dC2)q2 — 2kq4

— gR2q2/(1 + R2q2) [27]

where f(C) is the free-energy density of a homoge­neous alloy of composition C, к the gradient energy coefficient, and g = Gj/M is a reduced ballistic jump frequency. The analysis is here restricted to composi­tions and temperatures such that, in the absence of irradiation, spinodal decomposition takes place, that is, d2f/dC2 < 0.

The various possible dispersion curves are plotted in Figure 5. Unlike in the case of short R, it is now possible to find irradiation intensities g such that the

ballistic term in eqn [27] is greater than <@f /dC2 at small q, but smaller than that at large q. In such cases, the amplification factor is first negative for small q values, but it becomes positive when q exceeds some critical value qmin, while for larger q, the amplifica­tion factor is negative again. Therefore, decomposi­tion is still expected to take place, but only for wave vectors larger than qmin, that is, for wavelengths smal­ler than 2p/qmin. It can thus be anticipated that coars­ening will saturate, since at large length scales, the alloy remains stable with respect to decomposition.

Enomoto and Sawa122 have investigated this model using a 2D PFM based on eqn [25]. The interest here is that the PFM, unlike the above linear stability analysis, includes both linear and nonlinear contribu­tions to the evolution of composition inhomogeneity and also permits following the morphology of the decomposition. Using this model, Enomoto and Sawa have confirmed the existence of the patterning regime, see Figure 6, and showed that this patterning can take place in the whole composition range. The PFM approach allows for a direct determination of the pat­terning length scale as a function of the irradiation conditions, as illustrated in Figure 7.

Similar results have also been obtained using a variational analysis of eqn [25], leading to the dyna­mical phase diagram displayed in Figure 8. As seen in this diagram, when the characteristic length for the forced relocation is smaller than the critical value Rc, the system never develops patterns at steady state. Above Rc, patterning takes place when the irradiation

conditions are chosen so as to result in an appropriate g value. Another result obtained from the KMC simulations is that the steady state reached by an alloy is independent of its initial state.

Experimental tests performed on a series of dilute Cu-M alloys, with M = Ag, Co, Fe, have confirmed some of the key predictions of the above simulations and analytical modeling. In particular, irradiation con­ditions that result in atomic relocation distances exceeding a few angstroms do lead to the dynamical stabilization of precipitates at intermediate irradiation temperature.123 These results provide also a compel­ling rationalization of the puzzling results reported by Nelson eta/.124 on the refinement of g precipitates in Ni-Al alloys under 100-keV Ni irradiation at 550 °C.

The origin of the above irradiation-induced com­positional patterning lies in the finite range of the atomic mixing forced by nuclear collisions.125 Enrique and Bellon126,127 have shown that the effect ofthis finite-range dynamics can be formally recast as effective finite-range repulsive interaction between like particles. It is interesting to note that PF

simulations of alloys with Coulomb interactions also predict a patterning of the microstructure.128 The parallel with the treatment of finite-range mixing is in fact quite strong, since a screened Coulomb repul­sion is described by a decaying exponential, as also assumed for the probability of finite-range ballistic exchanges in deriving eqn [27]. The contribution of the Coulomb repulsion to the linear stability analysis is thus proportional to 1/(q2 + qD ), where qD is the

Figure 9 Phase field modeling of the evolution of ordered precipitates in the presence of electrostatic (repulsive) interactions, with increasing time from A to F. The average precipitate size reaches a finite value at equilibrium. Reprinted with permission from Chen, L. Q.; Khachaturyan, A. G. Phys. Rev. Lett. 1993, 70(10), 1477-1480. Copyright by the American Physical Society.

screening wavelength. For reasons similar to the ones discussed in the case of finite-range mixing, it is then anticipated that these interactions will suppress coarsening. This is confirmed by PF simulations, as illustrated in Figure 9.

RIS can be observed for very small irradiation doses; an enrichment of ~10% of Si has been measured, for example, at the surface of an Ni—1%Si alloy, after a dose of 0.05 dpa at 525 °C.32 Such doses are much lower than those required for radiation swelling5 or ballistic disordering effects.42

Increasing the radiation flux, or dose rate, directly results in higher point defect concentrations and fluxes towards sinks. The transition between RIS regimes is then shifted toward a higher temperature. But because point defect concentrations slowly evolve with the radiation flux (typically, proportional to its square root43 in the temperature range where RIS occurs), a high increase is needed to get a signif­icant temperature shift.

Radiation dose and dose rate are usually estimated in dpa and dpas~ , respectively, using the Norgett, Robinson, and Torrens model,44 especially when a comparison between different irradiation conditions is desired. It is then worth noting that the amount of RIS observed for a given dpa is usually larger during irradiation by light particles (electrons or light ions) than by heavy ones (neutrons or heavy ions). In the latter case, point defects are created by displacement cascades in a highly localized area, and a large frac­tion of vacancies and interstitials recombine or form

0. 8 0.6

0.4

P:

0.2 0.0

10-6 10-5 10-4 10-3 10-2

K0 (dpas-1)

Figure 3 Temperature and dose rate effect on the radiation-induced segregation.

point defect clusters. The fraction of the initially produced point defects that migrate over long dis­tances and could contribute to RIS is decreased. On the contrary, during irradiation by light particles, Frenkel pairs are created more or less homo­geneously in the material, and a larger fraction sur­vive to migrate (Figure 3).45

1.18.2.3.3 Impurity effects

The addition of impurities has been considered as a possible way to control the RIS in alloys, for example, in austenitic steels. The most common method is the addition of an oversized impurity, such as Hf and Zr, in stainless steels,46 which should trap the vacancies (and, in some cases, the interstitials), thus increasing the recombination and decreasing the fluxes of defects towards the sinks.

1.01.7.4.1 Capillary approximation

Consider a spherical cavity of radius R with an inter­nal gas pressure of p.

To obtain the local equilibrium vacancy concen­tration, we take one atom from the cavity surface and place it in one of the vacancies in the crystal lattice next to the cavity surface. The cavity volume is thereby increased by one atomic volume, O, and its surface free energy is changed by

AFs(«) =Fs(« + 1) —Fs(«)

= (4я)1/3 (30п)2/3 [(1 + 1/п)2/3 — 1] g0 [105]

Here, g0 is the specific surface energy per unit area, assumed to be a constant, and n is the number of vacancies contained in the cavity. This number is related to the unrelaxed cavity volume by

4n 3

R3 = nfi [106]

3 1 J

When the cavity is large, that is, n ^ 1, we can use the approximation

1+n

and then obtain

AFs ffi 2у0(4я/3п)1/3(3П)2/3 ffi2R0O [107]

The growth of the cavity volume by O implies that the gas inside expands and thereby performs the work pO.

The overall change in Gibbs free energy is then

A G = -£Fv + TSfV + — O — pO — kT IndO R

Under local thermodynamic equilibrium, AG = 0, and we find that

= CV4exp{0R°—p)kT} [108]

The chemical potential ofvacancies at cavities is then

mV (r) = kT ‘n[cC/cVq] = —p) о [109]

This constitutes the well-known capillary approxi­mation, and it is often interpreted as the combined mechanical effect of surface tension and pressure. However, the above derivation shows that the surface tension 2g0/R is not a mechanical force, but a ther­modynamic or chemical force. The assumption that the surface tension acts like a mechanical pressure and generates a stress field whose radial component satisfies the boundary condition

Srr(R)=2R0 — p [110]

is incorrect. However, to clarify this point, we need to first introduce the correct definition of a surface stress.

exposure. The defect accumulation kinetics73 can be described by N = Nmax[1 — exp(—Aft)], where the parameter A is determined by the spontaneous recombination volume for point defects or the cas­cade overlap annihilation volume for defect clusters and ft is the product of the irradiation flux and time. Due to the lack of defect mobility, defect clus­ters resolvable by TEM are usually not visible in this irradiation temperature regime unless they are created directly in displacement cascades by ener­getic PKAs.74 Saturation in the defect concentration typically occurs after ^0.1 dpa as monitored by atomic disorder,75-77 electrical resistivity,78-82 and dimensional change.83-85 Due to the large increase in free energy associated with lattice disordering and defect accumulation, amorphization typically occurs

15,85,86

in this temperature regime in many ceramics and ordered metallic alloys87,88 for doses above ~0.1- 0.5 dpa. Figure 7 shows an example of the dose — dependent defect concentration in ion-irradiated ZnO at 15 K as determined by Rutherford backscat-

89

tering spectrometry.

1.04.8.1 Tensile Behavior

The refractory metals are the metals in groups V and VI of the periodic table: vanadium, niobium, and tantalum in group V and chromium, molybdenum, and tungsten in group VI. All have the characteristic of a high melting point, hence the term refractory. The group VI metals are typically brittle, even with­out irradiation. For example, chromium is almost never used pure or as a major alloy element, although it is invaluable as a minor alloying element. Molyb­denum and tungsten are both brittle in nature but can be made into useful structural alloys by controlling interstitial impurities and by the addition of minor elements. In contrast to the brittle behavior of the group VI metals, the group V metals are inherently ductile. Structural alloys based upon this group have been developed, primarily for very high temperature and space applications. The primary disadvantage of the refractory metals is their formation of volatile oxides as opposed to protective oxide layers. Vana­dium and molybdenum oxides have melting points below metal working temperatures so that the metals become wet and can have liquid oxide drip off them.

Unlike the tensile behavior of fcc metals, where there is a smooth increase in strength as plastic defor­mation proceeds and work hardening progresses, bcc metals typically exhibit a load drop, or yield point, almost immediately following the onset of plastic deformation, as discussed in Section 1.04.7.2.

In the case of refractory metals, mechanical prop­erties are largely determined by interstitial solutes. High purity refractory metals do not exhibit a yield point but behave more like fcc metals. Niobium alloys irradiated in Li at 1200°C for over three months in EBR-II had total elongations of about 60%. Despite any irradiation hardening, the near absence of oxygen resulted in a very soft material at these high temperatures.50 However, since intersti­tials are nearly always present, tensile behavior is more typically characteristic of bcc metals.

Irradiation-produced defects interact with inter­stitial elements, resulting, in some cases, in severe embrittlement. The tantalum alloy, Westinghouse T — 111 (Ta—8W—2Hf) is used in Figure 25 to illustrate a commonly observed phenomenon of plastic instabil — ity.51 Plastic deformation becomes local, with high levels of slip on closely spaced planes where disloca­tions sweep out the irradiation-generated defects giving rise to local channels of very high deformation. This phenomenon, called channel deformation, is very common in irradiated metals. The result is a sudden and severe load drop with the fracture surface showing what appears to be a completely ductile chisel point fracture.52 Addition of 405 wt ppm oxygen to T-111 results in a cleavage fracture with no measurable plastic deformation, as shown in Figure 26.51 In both Figures 25 and 26, corresponding unirradiated alloys are shown demonstrating ductile behavior. In the unirra­diated condition, the addition of 400 wt ppm oxygen has only minor effects on strength and ductility, as can be concluded by a comparison of Figures 25 and 26. However, irradiation hardening superimposed upon the oxygen interstitial hardening appears to raise the yield stress above the cleavage stress for the alloy. It is suggested that interstitial solutes such as oxygen dif­fuse to irradiation-produced defect clusters, enhanc­ing their hardening effect.53,54 All three behaviors are observed in irradiated refractory metals: ductile with hardening, plastic instability, and cleavage fracture in the elastic range.55

The synergism between interstitial hardening and irradiation hardening does not necessarily lead to immediate catastrophic embrittlement. This behavior is shown in Figure 27 for vanadium containing a very high level of oxygen, 2100 wt ppm. Irradiation to a fluence level of 1.5 x 1019 (E > 1 MeV) leads to the familiar plastic instability but with several per cent plastic strain.53

Interstitial solutes, especially oxygen, may be controlled by the addition of gettering elements. In the vanadium system, titanium has been successful. Alloys in the V-Cr-Ti system have been studied for application to fusion reactors. In refractory metal
alloys, it is the oxygen in solution that is detrimen­tal, so that the oxygen must be combined with the titanium.56 This usually requires a heat treatment of sufficiently long times and high temperatures to pre­cipitate the oxygen. In the vanadium-titanium system,

a heat treatment of two hours at 950 °C has been found sufficient to achieve ductility, whether or not it is to be irradiated.57,58 Confusion over oxygen pres­ent in solution and oxygen combined in precipitates is believed to be one reason for the disparity in tensile data for this class of alloys and perhaps accounts for the relatively high level of ductility observed in Figure 27.

Upon irradiation of alloys in the range of V—3— 5Cr-3-5Ti in the HFIR, no cleavage fracture with­out plastic deformation was observed.59,60 However, plastic instability was commonly observed at irradia­tion and test temperatures below 400 °C. Irradiations in the range of 4-6 dpa in the HFIR produced uniform elongations from 0.2 to 0.6% and total elon­gations below 4%. Corresponding irradiations at 500 °C did not reveal plastic instability and produced uniform elongations in the range of 2-5%.59,60 Irra­diations to 3-5 dpa in the advanced test reactor (ATR) demonstrated plastic instability for irradiation and test temperatures of about 200 °C, with uniform elongations below 0.5%.61 Irradiations conducted in the high flux beam reactor (HFBR) at exposures of only 0.1 and 0.5 dpa corroborated these results and demonstrated a transition in the fracture mechanism between 300 and 400 °C, resulting in a significant increase in ductility at temperatures above 400 °C, Figure 28 62

For purposes of discussion and simplicity, the effects of cascade defect clustering and recombination are ignored, and we consider only single mobile vacan­cies and SIA defects in the simplest form of RT to illustrate the CBM. At steady state, isolated vacancies and SIA are created in equal numbers and annihilate at sinks at the same rate. Dislocation-SIA interactions due to the long-range strain field result in an excess flow of SIA to the ‘biased’ dislocation sinks and, thus, leave a corresponding excess flow of vacancies to other neutral (or less biased) sinks, (DvXv — DiXi). Here, D is the defect diffusion coefficient and X the corresponding atomic fraction. Assuming that the defect sinks are restricted to bubbles (b), voids (v), and dislocations (d), the DX terms are controlled by the corresponding sink strengths (Z): Zb («4ягЛ), Zv («4prvNv) for both vacancies and SIA; Zd («p) for vacancies and Zdi («p [1 + B]) for SIA. Here, r and N are the size and number densities of bubbles and voids, p is the dislocation density, and B is a bias factor. At steady state,

DvXv — DiXi = [BGdpaZdi]/{(Zb + Zv + Zd)

(Zb + Zv + Zd[1 + B])} + DvXve [2]

Here, DvXve represents thermal vacancies that exist in the absence of irradiation and p («1/3) is the ratio of net vacancy to dpa production. In the absence of vacancy emission, the excess flow of vacancies results in an increase in the cavity radius (r) at a rate given by

dr/dt+ = (DxXv — DiXi)/r [3]

However, cavities also emit vacancies, resulting in shrinkage at a rate given by the capillary approxi­mation as

dr/dt— = — DvXve exp[(2g/r — p)Q/kT]/r [4]

The Xveexp[(2g/r—p)O/kT] term is the concentra­tion of vacancies in local equilibrium at the cavity surface, and O is the atomic volume. Thus, the net cavity growth rate is

dr/dt ={DvXv — DiXi — DvXve

exp[(2g/rc — p)O/kT ]}/r [5]

Growth stability and instability conditions occur at the dr/dt = 0 roots of eqn [5], when

DvXv — DiXi — DvXve exp[(2g/r — p)O/kT] = 0 [6a]

Note that DvXve is approximately the self-diffusion coefficient, Dsd. The He pressure is given by

p = 3kmkT/4%r3 [6b]

Here, к is the real gas compressibility factor. Equa­tion [6a] can be expressed in terms of the effective vacancy supersaturation,

L =(DVXV — DiXi)/Dsd [6c]

The bubble and critical radius occur at

L — exp[(2g/r—p)O/kT ]=0 [6d]

In the absence of irradiation (or sink bias), L = 1 and all cavities are bubbles in thermal equilibrium, at p = 2g / rb. Assuming an ideal gas, к = 1, eqn [6d] can be written as

2g/r — (3 mkT )/(4nr3) — kT ln(L)/O = 0 [7 a]

Note that kT ln(L)/O is equivalent to a chemical hydrostatic tensile stress acting on the cavity. Rear­ranging eqn [7a] leads to a cubic equation with the form,

As shown in Figure 2(d) and 2(e), eqn [7b] has up to two positive real roots. The smaller root is the radius of a stable (nongrowing) bubble containing m He atoms, rb, and the larger root, r*, is the corresponding critical radius of a (m*,n*) cavity that transforms to a growing void. Voids can, and do, also form by classical heterogeneous nucleation

on bubbles between rb and r*.109,132,141 However, as shown in Figure 2(d) and 2(e), as m increases, rb increases and rv decreases, until rb = rv = r* at the critical m*. An example of the dr/dt curves assuming ideal gas behavior taken from Stoller133 is shown in Figure 11 for parameters typical of an irradiated AuSS at 500 °C with L = 4.57. The corresponding r* and m* are 1.50 nm and 931, respectively.

The critical bubble parameters can be evaluated for a realistic He equation of state using master correction curves, 01(ln L) for m* and 02(ln L) for r*, based on high-order polynomial fits to numerical solutions for the roots of eqn [7b].143 A simpler ana­lytical method to account for real gas behavior based on a Van der Waals equation of state can also be used.1 1 The results of the two models are very simi­lar.143 Voids often form on critical bubbles located at precipitate interfaces at a smaller m* than in the matrix.142 This is a result of the surface-interface tension balances that determine the wetting angle be­tween the bubble and precipitate interface (see Figure 20(b)). Formation of voids on precipitates can be accounted for by a factor Fv < 4я/3, reflecting the smaller volume of a precipitate-associated critical

bubble at r*, compared with a spherical bubble in the matrix, with Fv = 4я/3. Note that the critical matrix and precipitate-associated bubble have the same r*. The m* and r* are given by

m* = [32Fv01g3O2]/[27(kT)3(ln(L)2)] [8a]

r* = [402gO]/[3kT ln(L)] [8b]

Figure 12 shows m*, r* as a function of temperature for typical parameters for SA AuSS steels taken from Stoller.13 More generally, L can simply be related to Dsd, V, Gdpa, B, and the sink’s various strengths. Assuming Zv « 0 during the incubation period,

L «{[vGdpaBZd]/[(Zb + Zd)

[9]

(Zd(1 + B) +Zb)]/Dsd} +1

Figure 13 shows the corresponding m* and r* as a function of the concentration of 1 nm bubbles, Nb, at 500 and 600 °C again using the AuSS parameters given in Stoller.133 Clearly, high Nb can lead to large critical bubble sizes requiring high He contents for void formation.

Thus, to a good approximation, the primary mechanism for void formation in neutron irradiations is the gradual and stable, gas-driven growth of bub­bles by the addition of He up to near the critical m*.

Although nucleation is rapid on bubbles with m close to m*, modeling void formation in terms of evaluating the conditions leading to the direct conversion of bubbles to voids is a good approximation.1 The corresponding incubation dpa (dpa*) needed for Nb bubbles to reach m* is given by

dpa* = [m* Nb]/[He/dpa] [10]

Figure 14 shows dpa* for He/dpa = 10appmdpa-1 and the same AuSS parameters used in Figure 13. Clearly high Nb increases the dpa*, both by increas­ing the neutral sink strength, thus decreasing A, and partitioning He to more numerous bubble sites. Indeed, in the bubble-dominated limit, Zb >> Zd and Zv the dpa* scales with N^l

The CBM also predicts bimodal cavity size dis­tributions, composed of growing voids and stable bubbles. Once voids have formed, they are sinks for both He and defects, and thus slow and eventually stop the growth of the bubbles to the critical size and further void formation. Figure 15 shows a bimodal cavity versus size distribution histogram plot for a Ni-He dual ion irradiation of a pure stainless steel,114 and many other examples can be found in
the literature111,112,114,133,153 Figure 16(a) shows low He favors the formation of large voids in a CW stainless steel irradiated in experimental breeder reactor-II (EBR-II) to 40 dpa at 500 °C and 43 appm He, resulting in «12% swelling, while Figure 16(b) shows that the same alloy irradiated in HFIR at 515— 540 °C to 61 dpa and 3660 appm He has a much higher density of smaller cavities, resulting in only

2% swelling.16

Thus, while He is generally necessary for void formation, very high bubble densities can actually suppress swelling for the same irradiation conditions as also shown previously in Figures 9 and 10. This can lead to a nonmonotonic dependence of swelling on the He/dpa ratio. One example of a model pre­diction of nonmonotonic swelling is shown in Fig­ure 17.1 Note that unambiguous interpretations of neutron-irradiation data are often confounded by uncertainties in irradiation temperatures and complex temperature histories.155,156 However, the suppression of swelling by high Nb is clear even in these cases.

Bubble sinks can also play a significant role in the post-incubation swelling rates. Neglecting vacancy emission from large voids, and using the same assump­tions described above, leads to a simple expression for the overall normalized swelling rate S, the rate of

increase in total void volume per unit volume divided by the displacement rate as

S = [v BZdZv/[(Zb + Zv + Zd)(Zd(1 + B) + Zv + Zb)]

[11]

Figure 18 shows S for B = 0.15 and v = 0.3 as a function of Zv/Zd, with a peak at Zd = Zv and

Zb ^ 1, representing the case when nearly all the bubbles have converted to voids and balanced void and dislocation sink strengths. The S decreases at higher and lower Zv/Zd. Figure 18 also shows S as a function of Zv/Zd for a range of Zb/Zv. Increasing Zb with the other sink strengths fixed reduces the S in the limit scaling with 1/Z2. These results again show that significant swelling rates require some

bubbles to form voids with a sink strength of Zv that is not too small (or large) compared with Zd. However, a large population of unconverted bubbles, with a high sink strength Zb, can greatly reduce swelling rates.

A significant advantage of the CBM is that it requires a relatively modest number of parameters, and parameter combinations, that are generally rea­sonably well known, including for defect production, recombination, dislocation bias, sink strengths, inter­face energy, and Ds& Potential future improvements in modeling bubble and void evolution include better overall parameterization using electronic-atomistic models, a refined equation of state at small bubble sizes, and precipitate specific estimates of Fv based on improved models and direct measurements. Further, it is important to note that the CBM parameters can be estimated experimentally as the pinch-off size between the small bubbles and larger voids.114,124,157

Application of CBM to void swelling requires treatment of the bubble evolution at various sites, including in the matrix, on dislocations, at precipitate interfaces, and in GBs. Increasing the He generation rate (GHe) generally leads to higher bubble concen­trations, scaling as Nb / GHe.m,112,131-133,140,144,172 The exponent p varies between limits of ~0, for totally heterogeneous bubble nucleation on a fixed number of deep trapping sites, to >1 when the dominant He fate is governed by trap binding ener­gies, large He bubble nucleus cluster sizes (most often assumed to be only two atoms), and loss of He to other sinks. Assuming the dominant fate of He is to form matrix bubbles, p has a natural value of ~ 1/2 for the condition that the probability of diffusing He to nucleate a new matrix bubble as a di-He cluster is equal to the probability of the He being absorbed in a previously formed bubble.158

Bubble formation is also sensitive to temperature and depends on the diffusion coefficient and mech­anism, as well as He binding energies at various trapping sites. Substitutional He (Hes) diffuses by vacancy exchange with an activation energy of Ehs ~ 2.4 eV.159 For bimolecular nucleation of matrix bubbles, Nb scales as exp(—Ehs/2kT). Helium can also diffuse as small n > 2 and m > 1 vacancy-He complexes, but bubbles are essentially immobile at much larger sizes. Helium is most likely initially created as interstitial He (Hei), which diffuses so rapidly that it can be considered to simply partition to various trapping sites, including vacancy traps, where Hei + V! Hes. Note that, for interstitial dif­fusion, the matrix concentrations of Hei are so low that migrating Hei-Hei reactions would not be expected to form He bubbles. Thermal detrapping of Hes from vacancies to form Hei is unlikely because of the high thermal binding energy160 and see Section 1.06.5 for other references) but can occur by a Hes + SIA! Hei reaction, as well as by direct displacement events.152,161 If Hei and Hes maintain their identities at trapping sites, they can detrap in the same configura­tion. Clustering reactions between Hes, Hei, and vacancies form bubbles at the trapping sites.

Thus, He binding energies at traps are also critical to the fate of He and the effects of temperature and GHe. Traps include both the microstructural sites noted above as well as deeper local traps within these general sites, such as dislocation jogs and grain boundary junctions136 (and see Section 1.06.5 for other references). If the trapping energies are low, or temperatures are high, He can recycle between various traps and the matrix a number of times before it forms or joins a bubble. However, once formed bubbles are very deep traps, and at a significant sink density, they play a dominant role in the transport and fate of He.

In principle, the binding energies of He clusters are also important to bubble nucleation. Recent ab initio simulations have shown that even small clus­ters of Hei in Fe are bound, although not as strongly as Hes-V complexes. Indeed, the binding energies of small HemVn complexes with n > m are large (2.8­3.8 eV),134’135 suggesting that the bi — or trimolecular bubble nucleation mechanism is a good approximation over a wide range of irradiation conditions. Further, for neutron-irradiation conditions with low GHe and Gdpa that create a vacancy-rich environment, it is also reasonable to assume that He clusters initially evolve along a bubble-dominated path.

As discussed previously, the effects of higher bubble densities on overall microstructural evolutions are complex. The observation that N scales as GHe relation has been used in many parametric studies of the effects of varying bubble and void microstructures. Bubble nucleation and growth and void swelling are sup­pressed at very low GHe. However, as noted above, swelling can sometimes decrease beyond a critical GHe due to higher Nb. Indeed, void formation and swelling can be completely suppressed by a very high concentration of bubbles. High bubble concentrations can also suppress the formation ofdislocation loops and irradiation-enhanced, induced, and modified precipi­tation associated with solute segregation, by keeping

excess concentrations of vacancies and SIA very

low 16,26,111,112,162

Rudy Konings is currently head of the Materials Research Unit in the Institute for Transuranium Elements (ITU) of the Joint Research Centre of the European Commission. His research interests are nuclear reactor fuels and actinide materials, with particular emphasis on high temperature chemistry and thermodynamics. Before joining ITU, he worked on nuclear fuel-related issues at ECN (the Energy Research Centre of the Netherlands) and NRG (Nuclear Research and Consultancy Group) in the Netherlands. Rudy is editor of Journal of Nuclear Materials and is professor at the Delft University of Technology (Netherlands), where he holds the chair of ‘Chemistry of the nuclear fuel cycle.’

Roger Stoller is currently a Distinguished Research Staff Member in the Materials Science and Technology Division of the Oak Ridge National Laboratory and serves as the ORNL Program Manager for Fusion Reactor Materials for ORNL. He joined ORNL in 1984 and is actively involved in research on the effects of radiation on structural materials and fuels for nuclear energy systems. His primary expertise is in the area of computa­tional modeling and simulation. He has authored or coauthored more than 100 publications and reports on the effects of radiation on materials, as well as edited the proceedings of several international conferences.

Todd Allen is an Associate Professor in the Department of Engineering Physics at the University of Wisconsin — Madison since 2003. Todd’s research expertise is in the area of materials-related issues in nuclear reactors, specifi­cally radiation damage and corrosion. He is also the Scientific Director for the Advanced Test Reactor National Scientific User Facility as well as the Director for the Center for Material Science of Nuclear Fuel at the Idaho National Laboratory, positions he holds in conjunction with his faculty position at the University of Wisconsin.

vi Editors Biographies

B1 Kanzaki Forces

In a perfect crystal, the equilibrium positions of all atoms are such that they exert no net forces on each other. However, when the crystal is subject to either external forces or internal forces originating from crystal defects, mutual interaction forces arise. For example, the atoms surrounding a vacancy move to new positions in response to the missing interaction forces from the atom that would normally occupy the vacant site. One may imagine that these missing forces are applied to atoms in a perfect crystal and given such magnitudes and directions that they pro­duce the same strain and stress fields as exist in a crystal with a real vacancy. These fictitious forces that are applied to a perfect crystal are known by the name of their inventor,26 as the Kanzaki forces.

For a localized defect in an elastic medium, the elastic strain or stress field can be generated with a finite number of point forces fa) acting at the lattice sites r0 + R(a), where e center of

the defect region and R(a) is the lattice vector from this center to the adjacent atom on which the force is to be applied. In harmonic crystal lattices, or equiva­lently, in solids that deform according to linear elas­ticity theory, the displacement field created by all these point forces is then given by

Z

rn (r) = £ Gj (r, r’ + R(a>)ff [B1]

a

where Gj is either the lattice Green’s function when the solid is described by a harmonic crystal lattice or the elastic Green’s function when it is described as an elastic continuum.

For distances |r — r0| >> |R(a)|, we can expand the elastic Green’s function into a Taylor series around the point r0. Using the notation

д r n

Gj ’k = ~dXk Gj [B2]

one obtains

i (r) = Gj (r: r)Y f/ + Gjk (r’ r) 53 Rka)f +2G„,k-,(r, r’)J2rS“)r!“)/;m + — m

The set of z forces must of course be self- equilibrating so as to not impose any net force or net force moment on the solid medium. Hence,

z

Y fya) = 0 [B4]

a=l

and

z

Y J R^ — fk“)Rj“)) = 0 [B5]

a=1 ‘ ‘

As a consequence of eqn [B3], the first term in the multipole expansion (B3) vanishes. The next term contains the first moment of the forces, which is called the dipole tensor of the defect, and is denoted by

z

Pjk = f (“)Rka) = Pkj [B6]

a=1

In an infinite medium, the elastic Green’s function has the form

G» = gj(o) ij |r — r’

where o is the solid angle of the unit vector parallel to (r-r’). It is then permissible to change the differ­entiations with respect to r to differentiations with respect to r, taking into account changes in sign. For example,

j = — G»k [B12]

The multipole expansion for the displacement field of a point defect can then finally be written as

U(r) = — G»k(r — r’)Pjk + (r — r’)Pjki

— 3TG;^kim(r — r’) Pjkim +••• [B13]

Using eqn [B11], we find that the first term falls off as 1/r2, the second as 1/r3, etc.

B2 Volume Change from Kanzaki Forces

and as indicated, it is a symmetric tensor by virtue of eqn [B5]. The second moment of the forces z Pjki = f;a)Rka)Rf) [B7] a=1
and the total volume change is then
1
o„ (r)d3r
is called the quadrupole tensor, and the third moment	[B15]
3K
We now transform this volume integral by using the following identity:
[B8]
Pa

the octupole tensor.

In terms of these multipole tensors, the displace­ment field of the defect region can be written in the series expansion

U (r) = Gj k (r, r’) Pjk + Gj ki’ (r, r’) Pjki + [B9]

If the crystal lattice defect has certain symmetries, some terms in this multipole expansion may not be present. For example, if the defect possesses an inver­sion symmetry, then for each force there exists an equal but opposite force at a position equal to but opposite to the position vector belonging to the mir­ror force. For such a defect,

Pa> = (-1)nPjki……… [B10]

n indices

and all multipole tensors of odd rank vanish.

s ii simdmi (simxi );m sim, mxi (simxi );m +fixi [B16]

Here, f(r) represents the distribution of internal forces that enter into the equilibrium equations that the stresses must satisfy, namely

s im, m + fi 0

With the formula [B15], the total volume change can be written as the sum of two terms. The first one contains as its integrant the divergence of the vector field simxj, and it can therefore be converted with the Gauss theorem into a surface integral. As a result

DV ък{ $xisimnmdS +

The surface integration is over the surface tractions simnm, where n is the surface normal vector. If no

external loads are applied on the surface of the solid, then the surface tractions vanish, and the first term in [B16] is zero. When the internal body forces are Kanzaki forces, then

f(r) = £f(a)d(r — R(a)) x,- = R

and using (B6) one obtains

DV = ^ (P11 + P22 + P33) [B19]

It is immediately obvious from this derivation that when more than one point defect is present in the solid, the volume change is simply the sum of indi­vidual traces of their dipole tensors divided by three times the bulk modulus of the solid.