Since its inception in 1961, the cube root law developed on either side of the iron curtain by Lifshitz, Slyozov [
13] and Wagner [
29] (LSW) to quantitatively describe the phenomenon of spherical particle coarsening (Ostwald ripening), has been successfully applied to several systems [
9]. Furthermore, using an adaptation defined by Boyd et al. [
2] a decade later, the LSW approach has been shown to yield an apparently good agreement with experimental measurements of the change in size of non-spherical particles such as
\(\gamma ^{\prime \prime }\) (Ni
\(_{3}\)Nb–D
\(0_{22}\)) in Ni-base alloys [
4,
7,
8,
11,
23,
28,
30,
31]. Despite this apparent success, however, closer scrutiny of the results to which the latter formalism has been applied (such a coarsening of
\(\theta ^{\prime \prime }\) precipitates in Al–Cu alloys by Boyd et al. [
2] in the derivation paper) reveals a marked discrepancy between the real and calculated precipitate size distributions.
Considering the distribution of precipitates like
\(\gamma ^{\prime \prime }\) particles in Ni-base alloys and
\(\theta ^{\prime \prime }\) particles in Al–Cu, an immediate source for their distributions deviating from LSW can be identified as their high volume fraction,
viz. a small fraction (defined such that particle spacing >> particle dimensions) is necessitated by the LSW mechanism. To this end, it is clear that a more accurate description of the coarsening behaviour of these spheroidal precipitates can only be achieved through the use of one of the many different adaptations to the LSW formalism to describe the evolution of precipitates comprising a high volume fraction [
1].
Amongst the afore alluded to descriptions, almost all are based principally on approximate solutions/modifications to the solute diffusion description in LSW and, therefore, often require assumptions which are seemingly contradicted by experimental observations. In contrast, the Lifshitz–Slyozov-Encounter-Modified (LSEM) theory of Davies et al. [
6] is physically sound owing to its derivation being predicated on the mechanism of precipitate “encounter”; the broadening of the precipitate distribution from the classical LSW shape occurs by the coalescence of precipitates whose diffusion fields have overlapped, i.e. they are said to have “encountered” one another. Such behaviour has been observed for
\(\gamma ^{\prime \prime }\) precipitates in a number of studies [
11,
23,
26].
The governing precipitate growth/coarsening equation originally composed by Davies et al. is strictly applicable only to spherical precipitates but, following the work of Boyd et al. [
2] (in their original adaptation to the LSW model), an LSEM cube root law defining the growth of spheroidal particles can be composed according to Eq.
1 where,
\(\bar{L}_{\text {M}}\) is the average major axis of ellipsoidal particles initially (0) and at time
t,
\(X^e\) is the equilibrium solute concentration in the matrix,
\(\zeta \) is the precipitate–matrix interfacial energy,
\(V_\text {m}\) is the molar volume of the precipitate,
\(\alpha \) is the precipitate aspect ratio (minor axis length/major axis length),
R is the ideal gas constant,
D is the diffusion coefficient of the solute atoms in the matrix,
T is the absolute temperature, and
\(f_{\mathrm{LSEM}}\) and
\(C_{\text {LSEM}}\) are a function and system constant, respectively, defined by Davies et al. [
6]. As a result, it is evident that a more appropriate LSW description of the evolution of the size distribution for high volume fraction ellipsoidal precipitates, accounting for “encounter”, should be possible through LSEM.
$$\begin{aligned} {\bar{L}_{\text {M}}^3}(t) - {\bar{L}_{\text {M}}^3}(0) = \frac{32D\zeta V_{\text{m}} X_\gamma^{\text {e}}t }{\alpha \pi RT } \frac{f_{\text {LSEM}}(\bar{L}_{\text {M}}, L_{\text {M}}^*)^3}{C_{\text {LSEM}}} \end{aligned}$$
(1)
Owing to the importance of the precipitate population in determining the mechanical properties of the superalloys containing
\(\gamma ^{\prime \prime }\) [
15,
16], and in light of the aforementioned likely increase in the accuracy of the replicated size distribution that should result from adopting the LSEM theory
cf. LSW, the principle aim of this work was to assess the evolution of
\(\gamma ^{\prime \prime }\) populations with respect to LSEM. In the absence of any adaptation being made to the LSEM theory, a verbatim repeat of its specific intricacies as discussed by Davies et al. [
6] is avoided.