4.1.1 Driving and retarding forces for DRX
The flow stress is determined by the competition between work hardening that increases the dislocation density on the one hand and strain softening from recovery through dislocation annihilation and dynamic re-crystallization (DRX), where in the latter there is the nucleation and growth of strain-free
γ grains.[
15] In case of
\( \dot{\varepsilon } = 0.1 {\text{s}}^{ - 1} \), the grain size is coarser than the starting as-HIP condition, Figures
4(a) and (b). The absence of stress decay following reaching the peak stress, as in Figure
1, clearly indicates the absence of DRX. Therefore, the increase in grain size occurs from grain growth at close to solvus temperature, where any Zener pinning from the precipitates is minimum. However, for
\( \dot{\varepsilon } = 1 \) and 10 s
−1, not only does the peak flow stress increase with increasing strain rate, but there is also concomitant greater stress decay observed at the higher strain rate. This is a direct manifestation of the occurrence of DRX. As a comparison, the extent of flow softening for
\( \dot{\varepsilon } = 1 \) and 10 s
−1 is markedly greater than that reported in References
5‐
7 but comparable to that reported in Reference
2 and encompassing comparable range of conditions, T = 1223 K to 1423 K, strain rates = 0.1 to 10 s
−1, and true strains up to 0.9. Furthermore, flow softening at typically,
\( \dot{\varepsilon } > 1\;{\text{s}}^{ - 1} \) is consistent with the flow stress behavior reported in low-stacking fault energy Ni-based alloys.[
8,
9]
γ grains that nucleate during DRX have different growth kinetics in case of
\( \dot{\varepsilon } = 1 \) and 10 s
−1 resulting in the variation in grain size, Figures
5(a), (b) and
6(a), (b). The principal driving force in case of DRX arises from the remnant dislocation density into which the nucleated strain-free grain grows, while the retarding force is provided by the pinning of the grain boundary by the precipitates. The relative magnitude of these two forces will dictate the kinetics of grain growth.
Under condition of high strain rate/low temperature,
i.e., akin to DRX, the variation of flow stress with dislocation density under high strain rate conditions is given by[
24]
$$ \sigma = \alpha Gb\sqrt \rho, $$
(1)
where
σ is the peak flow stress,
α ~ 0.45;
G is the shear modulus;
b is the Burgers vector; and
ρ the dislocation density. For
\( \dot{\varepsilon } = 1 \) and 10 s
−1,
σ = 325 and 225 MPa, respectively, (Figure
1) and for RR1000 at 1100 °C,
G = 46.2 GPa,
ν = 0.436,
Vm (molar volume) of
γ phase = 7.44 × 10
−6 m
3 and
b = 2.54 × 10
−10 m. From Eq. [
1], the dislocation densities are
ρ (
\( \dot{\varepsilon } = 1\;{\text{s}}^{ - 1} \)) ~ 1.82 × 10
15 m
−2 and
ρ (
\( \dot{\varepsilon } = 10\;{\text{s}}^{ - 1} \)) ~ 3.79 × 10
15 m
−2, which is ×10 greater than that reported in Reference
12 under comparable deformation conditions. The driving force is then given by[
24]
$$ P_{\text{d}} = \alpha Gb^{2} \rho. $$
(2)
Therefore, the driving force,
Pd (
\( \dot{\varepsilon } = 1\;{\text{s}}^{ - 1} \)) ~ 2.44 × 10
−12 J
μm
−3 and
Pd (
\( \dot{\varepsilon } = 10\;{\text{s}}^{ - 1} \)) ~ 5.08 × 10
−12 J
μm
−3. The driving force is expressed per unit volume, since it corresponds to the free energy per volume, which is released as the strain-free grains consume the strained matrix. However, it should be emphasized that the dislocation density can vary across grains, depending on whether the grain has most recently re-crystallized or not. Therefore, this estimate of the driving force is an average value.
The retarding force arises from pinning from the precipitates at the grain boundary. The inter/intragranular
γ′ precipitates in the as-HIP condition are coherent, as in Figure
4(f). Although adiabatic heating effects exist, as in Figures
2(a) and (b), the time for deformation, which is 1 and 0.1 seconds corresponding to strain rates of
\( \dot{\varepsilon } = 1 \) and 10 s
−1, respectively, is insufficient for dissolution of
γ′ precipitates[
25] and a similar observation has been reported in Reference
12. The approach for calculation of the driving force and comments on the retarding force for DRX have similarities with that reported in Reference
12. The driving force in both cases has shown to be proportional to the dislocation density, which is calculated from the flow stress. The main difference arises in Eq. [
1], where in the present case the peak stress is considered,
c.f., Figure
1. In Reference
12, however, a much lower flow stress of 50 MPa was considered, which accounts for the lower dislocation density. A similar approach is adopted for the retarding force, which arises from the presence of incoherent inter/intragranular primary precipitates, although unlike in Reference
12, no explicit calculations are made in this study.
The occurrence of incoherent inter/intragranular primary precipitates following hot compression at 1100 °C (sub-solvus) at strain rates,
\( \dot{\varepsilon } = 1 \) and
\( \dot{\varepsilon } = 10\;{\text{s}}^{ - 1} \) requires an explanation. Coherent precipitates interact with the grain boundary in three ways[
26,
27]:
(a)
The precipitate retains its coherency if dissolution of the precipitate occurs at the grain boundary. The solute supersaturation (Al, Ti) following dissolution results in precipitation of coherent precipitates within γ matrix behind the boundary.
(b)
When the boundary is pinned by the precipitate partial coherency loss occurs accompanied by a change in shape of the precipitate at the grain boundary. The increased particle radius subsequently results in capillarity-driven coarsening of the intergranular precipitate.
(c)
Complete loss in coherency occurs when the grain boundary sweeps past the precipitate. If the initial coherent γ′ precipitate lies in γ-grain 1, then consumption of γ-grain 1 by γ-grain 2 through advancement of the grain boundary results in the γ′ precipitate being incoherent with γ-grain 2.
In case of (a), the rate of precipitate dissolution just ahead of the grain boundary should be sufficiently rapid and, able to keep pace with the mobility of the boundary. This has been reported in Reference
4, where the
γ grain re-crystallization front dissolves
γ′, which then precipitates discontinuously or continuously and coherently with
γ matrix phase. However, the predominant incoherent primary
γ′ precipitates for
\( \dot{\varepsilon } = 1 \) and 10 s
−1 indicate that complete dissolution of coherent
γ′ precipitates ahead of the grain boundary does not occur. The existence of incoherent primary
γ′ precipitates suggests two possible mechanisms.
Mechanism (i)—The remnant plastic strain within the
γ matrix also provides a driving force for post-dynamic or meta-dynamic re-crystallization (MDRX) of
γ′ precipitates homogeneously or heterogeneously (intergranular) within
γ matrix. The nucleation and growth rates are given by the following expressions[
28,
29]:
$$ \dot{J} = N_{0} Z\beta \exp \left( {\frac{ - \Delta G*}{RT}} \right)\exp \left( {\frac{ - \tau }{t}} \right), $$
(3a)
$$ v = \frac{{D{{\partial C_{\gamma } } \mathord{\left/ {\vphantom {{\partial C_{\gamma } } {\partial t}}} \right. \kern-0pt} {\partial t}}}}{{\left( {C_{\gamma '} - C_{\gamma } } \right)}}, $$
(3b)
$$ \Delta G* = \frac{{16\pi \sigma^{3} }}{{3\Delta G_{\text{v}}^{2} }}, $$
(3c)
where
N0 = density of active nucleation sites/volume (number m
−3);
Z = Zeldovich non-equilibrium factor that corrects for the equilibrium concentration of critical nuclei for the loss of nuclei to growth (unitless);
β = rate of atomic attachment to the critical nuclei (s
−1); Δ
G* is the activation energy for nucleation;
σ is the interfacial energy; Δ
Gv is the Gibbs volumetric free energy realized during nucleation of a precipitate phase;
Cγ′ and
Cγ is the elemental concentration in
γ′ and
γ at the
γ/
γ′ interface; and
D = elemental diffusivity in
γ phase (m
2 s
−1). Therefore,
\( {\dot{\text{J}}} \) has units of number m
−3 s
−1.
The number of active nucleation sites is dependent on the dislocation density and is given by[
30]
$$ N_{0} = \rho \left( {\frac{{6.023 \times 10^{23} }}{{V_{\text{m}} }}} \right)^{1/3}, $$
(4)
where
ρ is the dislocation density. Using
N0 and the remaining parameters in Eq. [
3a] are from Reference
31 from which the nucleation rates can be calculated and given by
-
\( {\dot{\text{J}}} \) (\( \dot{\varepsilon } = 1\;{\text{s}}^{ - 1} \)) ~ 7.9 × 1024 nuclei m−3 s−1
-
\( {\dot{\text{J}}} \) (\( \dot{\varepsilon } = 10\;{\text{s}}^{ - 1} \)) ~ 16.1 × 1024 nuclei m−3 s−1
The dislocation density can vary across grains, but importantly a large fraction has been consumed during DRX involving
γ grains. Therefore, this nucleation density is an upper bound, specifically when comparing with the simulations of slow continuous cooling in the case of the alloy UDIMET 720 Li, where
\( {\dot{\text{J}}} \) ~ (10
14 to 10
16) nuclei m
−3 s
−1 for a typical cooling rate of 0.5 K min
−1 from 1453 K (1180 °C).[
28] However, it must be pointed out that nucleation rate of
γ′ precipitates from Eq. [
3a] does not distinguish between coherent precipitates and incoherent precipitates.
Nevertheless, the role of MDRX is expected to be more prevalent in case of
\( \dot{\varepsilon } = 10\;{\text{s}}^{ - 1} \), where the twofold increase in P
d following from Eq. [
2] provides the driving force for nucleation and growth of strain-free
γ′ precipitates. In case of
\( \dot{\varepsilon } = 1\;{\text{s}}^{ - 1} \),
γ′ precipitates predominantly nucleate on grain boundaries and therefore will be incoherent. However, sporadic instances of intergranular
γ′ precipitates are observed, but which are coherent with
γ matrix phase, as highlighted by white arrows in Figure
5(d). This indicates that these precipitates have nucleated homogeneously within the bulk, but a situation akin to (b) is observed, where the boundary that intersects the precipitate is pinned and some loss of coherency at the boundary will occur. The intergranular incoherent precipitates also are predominantly associated with a local average misorientation < 0.5 deg, as shown in Figure
5(e), thereby indicating strain-free
γ′ precipitates and occurrence of MDRX. In case of
\( \dot{\varepsilon } = 10\;{\text{s}}^{ - 1} \), the precipitates are predominantly intragranular, which points to another mechanism related to (c).
Mechanism (ii)—If the
γ′ precipitates cannot pin the grain boundary owing to large driving force, then the boundary will sweep past the precipitates. The driving force is consumed during grain growth and advancement of the boundaries will be retarded and the grain boundary will eventually be pinned. The Zener pinning force is given by[
24]
$$ P_{\text{z}} = \frac{3f\sigma }{2r}, $$
(5)
where
σ is the grain boundary energy,
f = volume fraction of precipitates, and
r = precipitate radius. It is not straightforward to use Eq. [
5] for calculation of the pinning force, since there is a continuous growth of the precipitates during cooling with a progressive increase in both f as well as r, which affects Eq. [
5]. Further, the distribution of primary
γ′ precipitates on
γ grains are not random, with a distinct concentration at grain edges and corners. Nevertheless, as in the case of the driving force, the approximate pinning force in relation to the microstructure in Figure
5(d) can be made, where from image analysis,
f ~ 0.07 and
r ~ 0.8
μm, average
γ grain size ~ 1.46
μm and
\( \sigma \) ~ 1 J m
−2.[
12] From Eq. [
6], P
Z ~ 0.13 × 10
−12 J
μm
−3, which is 5 pct of the maximum driving force. In case of
\( \dot{\varepsilon } = 10\;{\text{s}}^{ - 1} \), owing to the twofold increase in driving force, the
γ grain boundaries initially breakaway from the primary
γ′ population, which explains the intragranular incoherent population. Only in the later stages, as in
\( \dot{\varepsilon } = 1\;{\text{s}}^{ - 1} \), the grain boundaries are pinned. Using Eq. [
6] and for
f ~ 0.05 and
r ~ 0.8
μm gives,
PZ ~ 0.09 × 10
−12 J
μm
−3, which is two orders of magnitude smaller than the driving force. The existence of a number of incoherent intragranular
γ′ precipitates, with local average misorientation ~ 3 deg (Figure
6(e)) is consistent with the homogenous nucleation of these
γ′ precipitates within the initially strained
γ grains. The subsequently strain-free nucleated
γ grains formed
via DRX (local average misorientation < 0.5 deg, as in Figure
6(e)) and their high mobility results in incoherency of
γ′ precipitates when the grain boundaries sweep past these precipitates.
In lieu of mechanisms (i) and (ii), some comments must be made in relation to heteroepitaxial re-crystallization (HERX) reported in Reference
3, where the focus is on
γ grains, which nucleate as a thin rim of
γ phase on primary
γ′ precipitates. This is distinct to this study, since the focus here is on the incoherent
γ′ precipitates that form. HERX presupposes the existence of incoherent
γ′ precipitates to account for the subsequent re-crystallization of
γ grains during DRX. Also, HERX occurs for low strains, typically,
ε < 0.6, which is smaller than the strains in this study,
c.f., Figures
3(a) and (b).