1. Introduction
Grain boundaries (GBs) are driven to migrate by various driving force sources, such as the GB energy, the surface energy anisotropy, the stored deformation energy, the chemical driving force, and the elastic energy [
1]. Especially for elastically anisotropic materials, inhomogeneities in the elastic strain energy are crucial for GB migration, which was theoretically investigated in Cu GBs by molecular-dynamics (MD) simulations [
2] and by phase field simulations [
3] and experimentally studied for a Ni GB by Lee et al. [
4]. Both Cu and Ni elastically anisotropic, whose Zener’s anisotropy factors
at 300 K amount to ~3.2 and ~2.2, respectively, where the elastic compliances,
,
, and
(in TPa
−1) for Cu [
5] and
,
, and
(in TPa
−1) for Ni [
6].
Using an MD method, Schönfelder et al. [
2] analyzed GB migration in a Cu bicrystal with two flat high-angle twist GBs, which thus enabled to exclude the possible source from GB curvature. They found that the elastic driving force was equal to the difference in the elastic energy density across the boundary [
2]. Using a phase field model, Tonks et al. [
3] also analyzed a Cu bicrystal under an applied strain similar to that studied by Schönfelder et al. [
2]. Two cases were considered: One case in which the strain was uniform throughout the bicrystal (isostrain), and the other in which the strains were heterogeneous (heterogeneous strain) with the average strain in the bicrystal equal to the applied strain.
Under the isostrain condition, the elastically softer grain had a lower strain energy density and it, thus, grew [
3]. In contrast, under the heterogeneous-strain condition, the softer grain had higher strain energy, but it grew [
3]. In this case, as the softer grain grew, the strain energies of both grains were calculated to be reduced. Therefore, the growth of the softer grain was explained in terms of the total strain energy reduction.
Lee et al. [
4] examined the migration of a Ni bicrystalline GB by in situ high-resolution transmission electron microscopy (TEM). A specimen for TEM was prepared using focused ion beam (FIB). A Ni lamella in the specimen was composed of two grains with surface normal directions of [1 0 0] and [1 1 0]. According to a finite element method (FEM), the stress state of the Ni lamella approximated to the isostress condition [
4]. The FE analysis showed that under the isostress condition, the total strain energy reduction favored the growth of the [1 0 0] grain. However, the stress development was not experimentally confirmed in the previous study. In the present study, we present an observation of stacking faults (SFs) with a length of 40–70 nm at the GB as direct evidence of the stress development. The long length of SFs observed is discussed in terms of the stress dependence of partial dislocation separation.
4. Discussion
The SFs shown in
Figure 4 originate from plastic deformation with heating to 600 °C. A SF forms between two partial dislocations, leading and trailing. An equilibrium separation between the two partials is determined by a balance between the repulsive forces of the two partials and the attractive forces due to the relevant SF energy. In Ni, the SF energy is estimated to decrease from 110 mJ m
−2 at room temperature to 64.3 mJ m
−2 at 600 °C (as read from a calculated curve from a result by Zhang et al. [
7]). At the SF energy of 64.3 mJ m
−2 [
7], the separation distance approximates to ~1 nm without any external stress. The larger separation distances (40–70 nm) shown in
Figure 4 are attributed to the applied stress developed during heating in the present work.
Byun [
8] related an applied stress to the increase in separation distance between the two partials. He formulated force balance equations for leading and trailing partial dislocations by considering not only the repulsive force between the two partials and the attractive force due to the SF energy, but also the Peach–Koehler force from an applied stress field and the friction force to the glide of the partial dislocations [
8]. The calculation results demonstrate that the separation distance increases with applied stress [
8]. We have extended Byun’s calculation [
8] to our case with the help of FEM and obtained a plot as shown in
Figure 5a.
Figure 5a indicates that the separation distance of 40–70 nm corresponds to an applied shear stress of ~900 MPa. Shear stresses plotted in
Figure 5b are obtained by processing FEA export data in consideration of a total of 12 slip systems in fcc Ni. Boxed slip systems in the legend of
Figure 5b, which correspond to the observed SFs, are shown to have enough shear stresses (~1 GPa) for the observed separation distance (
Figure 5b). Conversely, the long SFs (
Figure 4) are evidence that the stresses acting on the thin part of the Ni lamella have been high enough to produce the strain energy which drives the GB migration.
The nucleation of a full dislocation usually requires a much higher energy than dislocation propagation and thus its heterogeneous nucleation at the GB is the most plausible mechanism of dislocation formation. Normally in a homogenous matrix, the shear stress for the nucleation of a full dislocation is expected to amount to ~
G/10, where
G is the shear modulus of the system (76.7 GPa for Ni, see the caption of
Figure 5a). Our calculation of
Figure 5a reveals the shear stress of maximally 1 GPa acting on the slip systems, which is lower than the expected shear stress required for dislocation nucleation in the homogenous Ni matrix. However, our system is a GB. The GB is imperfect as compared with the homogeneous matrix and would require a much less shear stress for dislocation nucleation. We surmise that full dislocations were nucleated at the GB at much lower shear stresses or were already nucleated during the bicrystal fabrication. It is noted that
Figure 5 is regarding the shear stress required for the separation of a full dislocation.
The disappearance of the SFs during imaging at room temperature (
Figure 4a,b) is explained as follows: The SFs formed at 600 °C are expected to cause a decrease in volume to reduce the applied compressive stresses (
and
,
Figure 2). Therefore, after cooling to room temperature, conversely, both ends of the Ni lamella will apply tensile stresses to the Ni lamella, which can drive SFs to shrink or disappear to further reduce the total strain energy, as shown in
Figure 4a. Because no applied stress exists after cooling to room temperature, the long SFs shown in
Figure 4 could exist at room temperature probably due to lattice friction.
As shown in
Figure 4a,c, the long SFs are always located in the [1 1 0] grain side and their one end contacts the GB. As shown in
Figure 5b, some of the slip systems in the [1 0 0] grain also receive shear stresses enough for large separations. Therefore, the consideration of only the shear stress effect is not enough to explain why the long SFs are not observed in the [1 0 0] grain. The observation of one end of the long SFs touching the GB (
Figure 4) seems to indicate that the GB can stabilize the SFs. One of two partial dislocations bordering a SF, when contacting a GB, will be more stable than the other located in the matrix lattice because the GB has an open structure. Thus, the case in which one partial touches a GB with the other in the grain interior is likely to be more stable against the disappearance of SFs than the case in which both partial dislocations are located in the grain interior. This is the reason why such long SFs do not form inside the grains. Partial dislocations bordering SFs in the [1 1 0] grain are more probable to meet the GB because the GB was observed to migrate into the [1 1 0] grain [
4]. Therefore, SFs in the [1 1 0] grain side will remain long as it was at 600 °C, as shown in
Figure 4. While the GB migrated into the [1 1 0] grain, SFs in the side of the [1 0 0] grain, though one end initially touching the GB, would be detached from the GB, which is thus expected to make them unstable.