Dynamic nanoindentation and short-range order in equiatomic NiCoCr medium entropy alloy lead to novel density wave ordering

Chemical short-range order (CSRO) is believed to be a key contributor to the exceptional properties of multicomponent alloys. However, direct validation and conﬁrmation of CSRO has been highly elusive in most compounds. Recent studies for equiatomic NiCoCr alloys have shown that thermal treatments (i.e., annealing/aging) may facilitate and manipulate CSRO. In this work, by using molecular simulations, we show that nanomechanical probes, such as nanoindentation, may be utilized towards further manipulation of CSRO, providing explicit validation pathways. By using well established interatomic potentials, we perform hybrid Molecular-Dynamics/Monte-Carlo (MD/MC) at room temperature to demonstrate that particular dwell nanoindentation protocols can lead, through thermal MC equilibration, to the reorganization of CSRO under the indenter tip, to a density-wave stripe pattern (DWO). We characterize the novel DWO structures, that are directly correlated to incipient SRO but are highly anisotropic and dependent on local, nanoindentation-induced stress concentrations, and we show how they deeply originate from the peculiarities of the interatomic potentials. Furthermore, we show that the DWO patterns consistently scale up with the incipient plastic zone under the indenter tip, justifying the observation of the DWO patterning at any experimentally feasible nanoindentation depth.

Concentrated multi-component alloys, and in particular, the celebrated Cantor alloys [1][2][3], such as equiatomic single-phase CoCrFeMnNi, have been instrumental into guiding the exploration for the discovery of affordable, durable alloys, suitable for applications under extreme conditions [4,5].It has been conjectured that a major contributor to the exceptional properties of these alloys is the formation of chemical short-range (1nm) order (CSRO) that may pin or/and obstruct moving lattice defects, such as dislocations.While the observation of CSRO is quite common in such complex alloys, its causal connection to exceptional mechanical properties has been a subject of intense debate.Extensive investigations have culminated towards the focus shining on the curious case of single-phase equiatomic NiCoCr.This alloy has outstanding mechanical properties, namely hardness, strength and ductility [6], and there is plausible formation of CSROs in the alloys, especially after sample aging at high temperatures [7][8][9][10][11][12].In addition, it has been observed that dislocation stacking fault widths are sensitive to the formation of such short-range order [7], which is commonly (in metallurgy) associated with large(r) mechanical strength.Nevertheless, a deeper understanding of CSRO is required to identify how to control and possibly, augment CSRO features, and check its causal effects on mechanical properties.For this purpose, in this work, we theoretically consider dwell nanoindentation as a possible way to locally manipulate CSRO in equiatomic NiCoCr, and lead to causal connections between hardness and microscopy-resolvable nanostructural features.We utilize molecular simulation, as well as Monte Carlo methods, and we show that CSRO (if it exists) will be unstable to the formation of unconventional density-wave ordered stripe patterns (DWO), that are highly anisotropic and originate due to interatomic potential features.
Creep deformation studies in multi-component alloys are abundant [27][28][29][30][31][32][33].Here, we aim at utilizing creep deformation to comprehend in a deeper sense the character of CSRO by promoting non-trivial predictions for elemental density wave ordering at the nanoscale.While dynamical uniaxial testing [33][34][35][36][37][38] and nanoindentation, alongside microscopy, have been used for the elucidation of lattice defect deformation mechanisms [39], the constant-load dwell nanoindentation tests, between 1 arXiv:2211.05436v1[cond-mat.mtrl-sci]10 Nov 2022 minute to 10 hours of dwell time, have been solely popular for investigating the relation between hardness and indentation strain-rate over a, possibly wide, temperature range [40][41][42][43].We use this concept to generate predictions for CSRO nano-patterning at room temperature that may be directly observable using electron microscopy techniques.While this work is fully focused on modeling aspects and predictions for CSRO saturation regimes that may emerge at each loading depth, prior alloy studies [40][41][42][43] shall allow us to conclude that the proposed scenario is attainable at room conditions.
In this paper, we utilize hybrid MC-MD simulations, using LAMMPS [44], to demonstrate the plausible thermomechanical effects of a dwell nanoindentation scenario in single-phase equiatomic RSS NiCoCr.RSS samples were generated using random elemental sampling on appropriate face-centered cubic (FCC) lattices, with crystal orientations of  [21], who also showed that annealing of the samples (even at room temperature) leads to the formation of characteristic Ni-rich SRO patterning, which we will generally refer to as SRO-samples in this paper.The nanoindentation process was performed through an NVE ensemble.Furthermore, nanoindentation was performed along z, using a tip at radii of 3.5, 5 and 7nm.Periodic boundary conditions (BC) were implemented along x and y, while along z, BC was fixed at the bottom boundary, and free at the top.Moreover, a 3Å-thick layer of atoms was frozen at the bottom.The indenter tip is assumed to be a rigid sphere with force: 2 , where K = 1000 eV/Å 3 and R is the tip radius, moving along z direction with a speed of v = 20m/s [45,46].As shown in Fig. 1, when the indenter reached the target depth (1nm, 2nm or 3nm), v was set to zero and then MC thermal relaxation ('holding') process was performed using the variance-constrained semi-grand canonical (VCSGC) ensemble [21,47] at T = 300K.The chemical potential differences ∆µ Ni-Cr = −0.31and ∆µ Ni-Co = 0.021 and the variance constraint κ = 1000 are set as in Ref. [21].
The 'Holding' part (cf.Fig. 1) includes 1 MC cycle for every 20 MD steps within the VCSGC ensemble for a total number of 150, 000 MC cycles, ensuring that the thermalized configuration contains stable SRO patterning.
In the studied protocol (cf.Fig. 1), equiatomic NiC-oCr, which has been shown to display Ni-rich SRO patterns [21], is indented to a specified indentation depth in the 'Loading' step.The CSRO changes only during the The resulting pattern retains its shape even after the indenter is removed from the sample (d).This pattern is not observed during normal loading-unloading nanoindentation simulation like the process shown by the blue line (e).
'Holding' step, where the indenter is held fixed and thermomechanical MC-MD relaxation is performed at room temperature (cf.Fig. 1).Due to 'Holding', a load drop is commonly observed from maximum P m at time t l to P h at time t a .The 'Unloading' step consists in removing the load altogether at a velocity of v = −20nm/s, using MD, and then perform further MC-MD relaxation at the sample.We find that a characteristic DWO pattern emerges after 'Unloading' at t a-u (cf.Fig. 1), that would not appear without the 'Holding' stage.The whole protocol is also illustrated in the Load-Depth plot in the inset of Fig. 1.
The characterization of the resulting nanoindentationdriven DWO pattern observed in our simulations, is shown and compared to a RSS and a Ni-rich SRO sample in Fig 2 .The DWO and SRO rich samples happen to have a larger mechanical strength and hardness than a RSS, as their load-depth (P-d) curves illustrate in Fig 2(a).This could be due to the already observed Ni-dominated solute segregation, that is pinning and obstructing dislocation motion [48,49].This phenomenon may also be implied by a drastic dislocation density increase for DWO and SRO samples at smaller depths, compared to RSS ones, as shown in the inset of With this input, we compare the total energy of a DWO ansatz (see Supplementary Material for details) order that compares well with Fig. 3(a) and contains the interstripe distance as a free parameter.By comparing this DWO ansatz energy to the energy of RSS samples, we find that the optimal inter-stripe distance is very close to the one realized in MD simulations.In this way, we conclude that the DWO emergence is deeply linked to the energetic features of the interatomic potential, which is also the key cause of SRO emergence in equiatomic NiCoCr simulations.
The emergent DWO displays strong size effects [50], which are dependent on the indentation depth and indenter tip radius as a function of temperature (cf.Fig. 4).In our displacement-controlled tests, we find that loadtime (P-t) curves display a larger load drop [50] during 'Holding' as depth or tip radius increases, leading to spa-tially extended DWO (cf.Fig. 4), resembling the plastic zone size.While not studied here, we also expect  that size-dependent strain bursts should be observed in load-controlled tests.More specifically, the protocol discussed in Fig. 1 is implemented for two different indenter depth (1 and 3nm) while the temperature (300K) and the indenter tip radii (3.5nm) are kept fixed.Furthermore, increasing the indenter radii from 2 nm to 7 nm gives rise to a larger plastic zone and as a result, a larger DWO pattern.This effect can also be observed from the larger load drop observed in the P-t curves for the larger radii (cf.Fig. 4(c)).However, increasing the temperature (from 300K to 400K), while the indenter depth and radii are the same, also results in a more organized DWO pattern shown in Fig. 4(cf.F-g), but there is no pronounced size effect, as shown by the absence of a load-drop decrease (cf.Given the elusive character of SRO formation in advanced alloys [25], the described dynamic nanoindentation protocol appears to be a plausible candidate for nano-scale manipulation and control of CSRO patterns in equiatomic NiCoCr and possibly, other multicomponent alloys.By the investigation of thermomechanical features and size effects, we conclude that atomic scale Ni-rich segregation strongly influences the mechanical properties of equiatomic NiCoCr, in a way that can be quantified in dynamic nanoindentation.CSRO reorganization mechanisms in NiCoCr, are shown to be energetic in character (as opposed to entropic) and are highly anisotropic.The origin of DWO emergence is tracked back at the potential energy surface of a RSS crystal.Finally, the pronounced observed size effects of the emergent DWO, suggest that, for experimentally relevant nanoindentation depths and tip radii, the emergent DWO shall be visible under common electron microscopy tools at the nanoscale.

FIG. 1 .
FIG. 1. Nanoindentation protocol for Short-Range Order reorganization in equiatomic NiCoCr alloys.The process starts when the aged NiCoCr sample (a) is indented up to a certain depth (b).Afterwards, the indenter's velocity is set to zero, this is the moment when the Hybrid MD-MC process starts again and leads to a configuration in which Ni and Co-Cr segregations or no longer randomly distributed but are reorganized and form stripe patterns under the indenter tip (c).The resulting pattern retains its shape even after the indenter is removed from the sample (d).This pattern is not observed during normal loading-unloading nanoindentation simulation like the process shown by the blue line (e).

FIG. 2 .
FIG. 2. Characterization of nanoindentation-driven reorganization of Short-Range Ordering.The nanoindentation LD curves for a RSS, an annealed sample and a sample with stripe patterns are shown in (a).Obviously, the annealed and stripy samples are stronger.(b-c) are the pile up patterns for RSS, annealed and the sample with stripes.The strength of the segregated samples are also depicted in (e-f ) (RSS, annealed and striped sample, respectively.) in the sense that the plastic region is smaller for the two latter ones.

FIG. 3 .
FIG. 3. Origin of nanoindentation-driven reorganization of Short-Range Ordering.(a-b) are the SRO patterns found along the z[011] and z[001] orientations, respectively.(c-d) illustrate the correlation between the stripe patterns' orientation shown in (a-b) and the Von-Mises stress.The pairwise energy between Ni-Ni, Ni-Cr and Ni-Co atoms within each atom's neighbour list (e) is plotted in the same plane in (a) (but for separate Ni, Ni-Cr and Ni-Co crystals) with respect to the in-plane angle of the pair atoms (cf.F).As shown in (cf.F), the Ni-Cr pairs have an average energy lower by 3 orders of magnitude compared to Ni-Ni and Ni-Co pairs.

FIG. 4 .
FIG. 4. Size effects of nanoindentation-driven reorganization of Short-Range Ordering.Figure (a) shows the load drop is bigger for bigger holding-depths, which results in more pronounced stripe patterns (b).Also, the same correlation exists between the stripe patterns and the Von-Mises stress values (shown in inset snapshots of (b-c)).Following this, (c) shows by increasing the temperature by 100K the Von-Mises stress increases (more red spots in the inset of figure (c)) and leads to a more organized stripe pattern, although the indenter's depth and radius is the same for the two cases.As it is expected, a bigger indenter radius induces a bigger stress in the sample and leads to a bigger pattern as it is shown in (d).