The CBM concept can also be applied to the effects of grain boundary He on creep rupture properties. Stress-induced dislocation climb also results in gen­eration excess vacancies that can accumulate at
growing voids. In particular, tensile stresses normal to GBs (s) generate a flux of vacancies to boundary cavity sinks, if present, and an equal, but opposite, flux of atoms that plate out along the boundary as illustrated in Figure 20(a). The simple capillary con­dition for the growth of empty cavities is the s > 2g/ r In this case of cavities containing He, the growth rate is given by

dr / dt = [(Dgbd)/(4TCr2)]

{1 — exp[(2g/r — P — s)Q/kT ]g [12]

Here Dgb and d are the grain boundary diffu­sion coefficient and thickness, respectively. The corresponding dr/dt = 0 conditions also lead to a stable bubble (rb) and unstably growing creep cavity (r*) roots. As noted previously, a vacancy supersaturation, L, produces a chemical stress that is equivalent to a mechanical stress s = kTln(L)/O. Thus, replacing ln(L) in eqn [8a] and [8b] with sO/kT directly leads to expressions for m* and r* for creep cavities

m* = [32FvTCg3 ]/[27kT s2] [13a]

r* = 0.75g/s [13b]

This simple treatment can also be easily modified to account for a real gas equation of state. Note that it is usually assumed that GBs are perfect sinks for both vacancies and SIA. Thus, it is generally assumed that displacement damage does not contribute to the for­mation of growing creep cavities.

Understanding HTHE requires a corresponding understanding of the basic mechanisms of creep rup­ture in the absence of He. At high stresses and short rupture times, the normal mode of fracture in AuSS is transgranular rupture, generally associated with power law creep growth of matrix cavities.181,182 However, at lower stresses IG rupture occurs in a

wide range of austentic and ferritic alloys. Although space does not permit proper citation and review, it is noted that a large body of literature on IG creep rupture emerged in the late 1970s and early 1980s. Briefly, this research showed that under creep condi­tions a low to moderate density of grain boundary cavities forms (10-1012m- ), usually in associa­tion with second-phase particles and triple-point junctions.183-184a Grain boundary sliding results in transient stress concentrations at these sites, and interface energy effects at precipitates also reduce the critical cavity volume (Fv ^ 4я/3) relative to matrix voids, as illustrated in Figure 20(b).

Once formed, however, creep cavities can rapidly grow and coalesce if unhindered vacancy diffusion and atom plating take place along clean GBs. Such rapid cavity growth rates lead to short rupture times in low creep strength, single-phase alloys. Thus, use­ful high-temperature multiphase structural alloys must be designed to constrain creep cavity nucle — ation and growth rates by a variety of mechanisms. For example, grain boundary phases can inhibit dis­location climb and atom plating.185

As illustrated in Figure 20(a), growth cavities, which are typically not uniformly distributed on all grain boundary facets, can be greatly inhibited by the con­straint imposed by creep in the surrounding cage of grains, which is necessary to accommodate the cavity swelling and grain boundary displacements.186 Creep
stresses in the grains impose back stresses on the GBs that result in compatible deformation rates. Thus, it is the accommodating matrix creep rate that actually controls the rate of cavity growth, rather than grain boundary diffusion itself. Creep-accommodated, constrained cav­ity growth rationalizes the Monkman-Grant relation187 between the creep rate (e0), the creep rupture time (tr), and a creep rupture strain (ductility) parameter (er) as

e0tr = er [14a]

Thus, in high-strength alloys, low dislocation creep rates (e0) lead to long tr. The typical form of e0

e0 = Acrexp(-Qcr/kT) [14b]

The effective stress power r for dislocation creep is typically much greater than 5 for creep-resistant alloys, and the activation energy for matrix creep of Qcr к 250-350 kJ mol-1 is on the order of the bulk self-diffusion energy.1 1 These values are much higher than those for unconstrained grain boundary cavity growth, with r к 1-3 and Qgb к 200 kJ mol — .

A number of creep rupture and grain boundary cavity growth models were proposed based on these concepts.186,188,189 Note that there are also conditions, when grain boundary vacancy diffusion and atom plating are highly restricted and cavities are well separated, where matrix creep enhances, rather than constrains, cavity growth. As noted above, power law creep controls matrix cavity growth at high stress,
leading to transgranular fracture.181’182 Models of the individual, competing, and coupled creep and cavity growth processes have been used to construct creep and creep rupture maps that delineate the boundaries between various dominant mechanism regimes. How­ever, further discussion of this topic is beyond the scope of this chapter.

Accumulation of significant quantities of grain boundary He has a radical effect on creep rupture, at least in extreme cases. First, at high He levels, the number density of grain boundary bubbles (Ngb) and creep cavities (Nc) is usually much larger than the corresponding number of creep cavities in the absence of He; the latter is of the order 1010-1012m~2.181,190 Figure 21 shows the evolution of He bubbles and grain boundary cavities under stress.191 Indeed, Ngb of more than 1015m~2 have been observed in high-dose He implantation studies.10 , Although Ngb is not well known for neutron-irradiated AuSS, it has been esti­mated to be of the order 1013 m~2 or more.193,194

At high He levels, a significant fraction of the grain boundary bubbles convert to growing creep cavities, resulting in high Nc. Of course, both Ngb and Nc depend on stress as well as many material parameters and irradiation variables, especially those that control the amount of He that reaches and clus­ters on GBs. As less growth is required for a higher density of cavities to coalesce, creep rupture strains,

er roughly scale with N~1/2. Bubble-nucleated creep cavities are also generally more uniformly distributed on various grain boundary facets. More uniform distributions and lower er decrease accommodation constraint, thus, further reducing rupture times asso­ciated with cavity growth.

Equation [13a] suggests that m* scales with 1/s2. If the GB bubbles nucleate quickly and once formed the creep cavities rapidly grow and coalesce, then creep rupture is primarily controlled by gas-driven bubble growth to r* and m*.93-95,97 In the simplest case, assuming a fixed number of grain boundary bubbles Ngb and flux of He to the grain boundary, JHe, the creep rupture time, tr is approximately given by

tr = {[Fv32py3]/[27krff2]}[Ngb/jHe]~Ngb/[GHeff2] [15]

Note that this simple model, predicting tr / 1/s2 scal­ing, is a limiting case primarily applicable at (a) low stress; (b) when creep rupture is dominated by He bubble conversion to creep cavities by gas-driven bub­ble growth to r*; and (c) when diffusion (or irradiation) creep-enhanced stress relaxations are sufficient to pro­duce compatible deformations without the need for thermal dislocation creep in the grains. More gener­ally, scaling of tr / 1/sr, r > 2 is expected for bubbles containing a distribution of m He atoms. For example, if Ngb scales as m~q, then Nc would scale as s2q.194,195 Further, at higher s, hence lower tr, there is less time for He to collect on GBs. Thus in this regime, intragranular dislocation creep, with a larger stress power, r, may return as the rate-limiting mechanism controlling the tr — s relations.

Equation [15] also provides important insight into the effect of both the grain boundary and matrix microstructures. Helium reaches the GBs (JHe) only if it is not trapped in the matrix. Matrix bubbles are, by far, the most effective trap for He.95,97 If it is assumed that the number of matrix bubbles, Nb, is proportional to V GHe while the grain boundary bub­ble number density (Ngb) is fixed, a scaling relation for tr can be approximated as

tr / Ngb/[v/GHes2] [16]

if the number of grain boundary bubbles also scales with VjHe, tr scales with G^7"4. At the other extreme, if Nb and Ngb are both independent of the He con­centration, tr scales with Gj^ (eqn [15]). Thus, micro­structures with high Nb (and Zb) that are resistant to void swelling are also likely to be resistant to HTHE.

HTHE models for AuSS were developed based on these concepts and various elabora — tions93-96,182,190,194,196,197 as well as methane
gas-driven constrained growth of grain boundary cavities.198 The HTHE models developed by Trinkaus and coworkers were closely integrated with the extensive He implantation and creep rupture studies discussed further below. It should be empha­sized that the HTHE models cited above are only qualitative and primarily represent simple scaling concepts that must be validated and calibrated using microstructural, creep rate, and creep rupture data. For example, more quantitative models require detailed treatment of He accumulation and redistri­bution at the GBs into a stably growing population of bubbles, with a time-dependent fraction that ultimately converts to growing creep cavities.