# Dislocation plasticity in equiatomic NiCoCr alloys: The effect of short-
range order

Amir H. Naghdi1 These authors made equal contributions. Kamran Karimi1∗
Corresponding author<EMAIL_ADDRESS>Axel E. Poisvert1 Amin
Esfandiarpour1 Rene Alvarez1 Pawel Sobkowicz1 Mikko Alava1,2 Stefanos
Papanikolaou1 1 NOMATEN Centre of Excellence, National Center for Nuclear
Research, ul. A. Sołtana 7, 05-400 Swierk/Otwock, Poland
2 Aalto University, Department of Applied Physics, PO Box 11000, 00076 Aalto,
Espoo, Finland

###### Abstract

Equiatomic NiCoCr solid solutions have been shown by recent experiments and
atomistic simulations to display exceptional mechanical properties, that have
been suggested to be linked to nanostructural short-range order (SRO) features
that may arise from thermal treatments, such as annealing or/and aging. Here,
we use hybrid Monte Carlo-Molecular Dynamics (MC-MD) simulations to gain
further insights of thermal effects on the SRO formation as well as the edge
dislocation plasticity mechanisms of equiatomic NiCoCr face-centered cubic
solid solution. For that purpose, we utilize two well-known NiCoCr interatomic
potentials, one of which displays well documented SRO, believed to be linked
to experimental evidence and labeled as the Li-Sheng-Ma potential, while the
other (Farkas-Caro) does not. We use these two potentials to discern short
range ordering (from inherent randomness in random solid solutions) and
understand how SROs influence dislocation depinning dynamics in various
thermal annealing scenarios. In this context, we used robust, scale-dependent
metrics to infer a characteristic SRO size in the Li-Sheng-Ma case by probing
local concentration fluctuations which otherwise indicate uncorrelated
patterns in the Farkas-Caro case in a close agreement with random alloys. Our
Voronoi-based analysis shows meaningful variations of local misfit properties
owing to the presence of SROs. Using relevant order parameters, we also report
on the drastic increase of chemical ordering within the stacking fault region.
More importantly, we find that the Li-Sheng-Ma potential leads to excellent
edge dislocation depinning strength with low stacking fault width. Our
findings indicate an enhanced roughening mechanism due to the SROs-misfit
synergy that leads to significant improvements in dislocation glide
resistance. We argue that the improvements in alloy strength have their
atomistic origins in the interplay between nanoscopic SROs and atomic-level
misfit properties.

Figure 1: A sketch of the shear setup including the partial dislocations.
Periodic boundary conditions are applied parallel to the dislocation lines and
the glide direction ($x$ and $z$ dimensions, respectively) and fixed
boundaries are implemented along the $y$ direction (rigid slabs in gray). To
shear, rigid atoms within the bottom surface are held fixed (i.e. zero
displacements) while the shear stress $\sigma$ is applied on the rigid top
plane.

##  I Introduction

Metallurgy of alloys is at the core of technological progress. Concentrated
solid solution alloys (CSAs) have recently emerged as major candidates for
novel alloys for extreme conditions applications [1, 2]. Out of all, the
equiatomic NiCoCr CSA represents a simple enough composition that has been
consistently reported to show exceptional mechanical properties [1]. These
include (among others) a relatively high (tensile) strength and ductility,
fracture toughness, as well as (micro-)hardness, that often exceed those of
the “Cantor” alloy [3], yet with a fewer number of principal components. This
is most likely rooted in the chemical composition and underlying sub-
structure. However, the microstructural origin of the exceptional mechanical
properties has been heavily debated, with a possible explanation being the
presence of nm-level (chemical/structural) short-range order (SRO) [4, 5, 6,
7] that arises from particular thermal processing and influences dislocation
pinning and stacking fault widths. In addition, lattice distortions and local
crystalline misfits, due to atomic size differences [8, 9], have been shown to
be correlated to the exceptional mechanical behavior of this alloy. Given the
apparent importance of local misfit volumes [8], it remains a challenge to
identify the role of SRO for exceptional mechanical properties. In this paper,
we focus on extensive molecular simulations to understand how SRO influences
dislocation plasticity and how it might depend on processing parameters (i.e.
annealing temperature), the properties of the ordered phase, as well as the
role of the MD atomic potentials in the ordering process. In this framework,
we investigated two case studies involving commonly-used NiCoCr interatomic
potentials i) Li-Sheng-Ma and ii) Farkas-Caro potential. While i) leads to the
formation of short range ordering upon aging, ii) displays chemical/structural
features, under the exact same thermal treatment, that are almost
indistinguishable from random solid solutions. Our multi-scale
characterization of local ordering were based upon the use of novel
descriptors that exhibit distinct structural/chemical signatures owing to the
presence of nanoscopic SROs. We study the effects of the SRO on mechanical
properties, showing that it significantly influences them, via the interplay
of dislocations and SRO structures. Such an interplay was quantified via a
detailed analysis of the dislocation substructure indicating enhanced
roughening properties due to combined SRO-misfit effects.

Short range order has been at the core of studies in CSAs across the board
[10, 11, 12]. Thermodynamically speaking, SROs’ ubiquity at low-temperature
alloys has been mainly attributed to dominant enthalpic effects that, in the
absence of entropy-driven mechanisms, do not favor idealistic perfect mixtures
of equimolar elements [13]. In this context, SROs typically refer to coherent
compositional deviations apart from (statistically) random distributions of
atoms within the solution matrix as in random solid solution alloys (RSAs).
More importantly, (thermal) processing parameters associated with annealing
and homogenization procedures (i.e. temperature and time) or irradiation may
have a drastic effect on the nucleation of SROs and associated substructural
features [8, 14, 15]. Owing to their nanoscopic scales, laboratory-based
observations of SROs are quite nontrivial involving intensive use of advanced
characterization techniques such as high-resolution electron microscopy and
atomic-resolution energy dispersive spectrometry mapping [16, 7]. The latter
are strongly tied to underlying physical mechanisms that govern fundamental
alloy properties. For instance, the formation of Ni-rich nano-precipitates and
associated inhomogeneities within the annealed NiCoCr matrix has been recently
suggested to tune the stacking fault width with evident consequences in terms
of the alloy strengthening [17, 13]. Similar conclusions were drawn
experimentally by Ritchie et al. [4] who reported on the emergence of SROs in
aged NiCoCr CSAs with significant impacts on the dislocation activation energy
and hardness. In studies of NiCoCr-based alloys, stacking fault energy,
hardness, and fracture toughness, as bulk properties, were recently shown to
strongly correlate with the degree of Ni-rich SROs and corresponding
structural features [18, 19, 20, 21].

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Figure 2: Chemical potential differences for a) Ni-Co and b) Ni-Cr pairs used
for variance constrained semi-grand canonical (VCSGC) ensemble.

In contrast, local misfit properties have been conventionally viewed as a key
solid solution strengthening mechanism [22]. In this framework, the inherent
yield strength of alloys (or Peierls stress) associates to the stress
threshold required for dislocation depinning and in the context of multi-
component high-entropy alloys, their intrinsic hetrogeneity and randomness
gives rise to somewhat uncorrelated perturbations to the local thresholds.
This concept was theoretically put forward in the seminal work by Labusch [23]
who hypothesized that the motion of dislocations within a random set of solute
obstacles leads to significant hardening effects in dilute solutions. In a
series of relevant papers, Varvenne and Curtin (VC) [24, 25, 26] further
investigated the RSA context in terms of elastic-type long-range interactions
between dislocation lines and residual strain fields resulting from atomic
size misfits. Along these lines, a mean-field theoretical framework was
proposed to make fairly accurate predictions of yield strengths solely based
on the effective medium elastic properties and, more importantly, local misfit
fluctuations. The proposed theory accounts for local compositional
fluctuations described by spatial distributions of misfit volumes which are
accessible through atomistic simulations and experimentation. The VC framework
was further generalized to additionally account for thermal and strain-rate
effects on the alloys’ strength and checked in the context of random alloys
[25, 8]. In the case of high-entropy alloys (HEAs), however, there may often
be a considerable degree of short range ordering and, therefore, HEAs cannot
be simply treated as fully random.

Short-range order and local misfits are not one-way streets but they combine
and interplay in ways that are somewhat unpredictable in advance and
inseparable [19]. Moreover, almost all crucial properties related to alloy
strengths are strongly dependent on the underlying microstructure and
processing methods used to synthesize CSAs. In an aged multicomponent alloy,
the effective Peierls stress will be influenced by both randomness in the
local composition distribution (giving rise to misfit fluctuations) and short
ranged (but still finite) spatial correlations introduced by SROs. Naturally,
this combined effect leads to an effective yield stress that typically exceeds
that of a random solid solution, lacking this finite range ordering component.
A naive picture in this context is that dislocations will move by locally
bending between pinning sites to overcome locally-fluctuating Peierls stresses
leading to extra strengthening [27]. Nevertheless, a systematic study
accounting for dislocations’ glide resistance and their substructure
discerning the separate roles of SROs and misfits seems necessary.

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3: Annealing temperature effects on NiCoCr based on the Li-Sheng-Ma potential.
a) Heat capacity $C_{p}$ b) thermal expansion coefficient $\alpha_{p}$ versus
anealing temperature $T_{a}$. The inset represents a $(111)$ cross-section of
Ni (grey), Co (blue), and Cr (red) atoms at $T_{a}=800$ K.

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Figure 4: Snapshots of NiCoCr samples a) RSA equilibrated at $T=400$ K, b)
annealed at $T_{a}=400$ K, c) annealed at $T_{a}=800$ K and d) annealed at
$T_{a}=1400$ K.

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Figure 5: Short range ordering in annealed NiCoCr CSAs based on the Li-Sheng-
Ma potential. Warren–Cowley SRO parameters including a) $p_{\text{NiNi}}$ b)
$p_{\text{NiCo}}$ c) $p_{\text{NiCr}}$ d) $p_{\text{CoCo}}$ e)
$p_{\text{CoCr}}$ f) $p_{\text{CrCr}}$ plotted against distance $r$ at
$T_{a}=400$ K. g) Pair correlation function $g(r)$ at $T_{a}=400$ K. The base
(red) dashdotted line indicates the random concentration.

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Figure 6: SRO variations with annealing temperature $T_{a}$. a)
$p_{\text{NiNi}}-p^{\text{rss}}_{\text{NiNi}}$ as a function of pair distance
$r$ b) root-mean-squared fluctuations $p^{\text{rms}}_{\text{NiNi}}$ plotted
against $T_{a}$. The (red) base line indicate zero correlations associated
with RSAs. The results are based on the Li-Sheng-Ma interatomic potential.

To this end, we use a versatile approach to investigate the micro-
structural/chemical origin of strengthening in NiCoCr including enthalpy-
driven ordering effects and local distortions. We perform hybrid Monte
Carlo/Molecular Dynamics (MC/MD) simulations at a range of annealing
temperatures based on two commonly-used NiCoCr interatomic potentials. We find
that the emergence of SROs is not a robust feature of annealed model NiCoCr
CSAs but, to a great extent, depends on the chosen potential energy. More
specifically, the two models generate microstructurally different alloys
(with/out SROs) with the exact same thermal processing. Following the
numerical framework in [13], we probed effects of SROs in terms of local
concentration fluctuations, stacking fault widths, dislocation glide
resistance, and misfit volumes of NiCoCr as well as thermodynamic properties
such as the specific heat and thermal expansion coefficient. Our analysis
indicates meaningful correlations of the above observables with varying
degrees of SROs in aged alloys, making them easily distinguishable from RSAs.
We further observe a marked growth in the population of SROs inside the
stacking fault region and remarkable strengthening behavior against
dislocation glide with the latter rooted in the interplay between short range
ordering and local misfit properties.

The organization of this paper is the following. In Sec. II, we describe the
numerical setup, sample preparation (including aging/annealing), loading
protocols, and relevant simulation details including the hybrid MD/MC model,
interatomic forces, and shear test description. Section III presents our
simulation results relevant to the chemical/microstructural characterization
of SROs and their potential effects on dynamics of dislocations. In this
context, Sec. III.1 introduces robust structural/compositional metrics
associated with local elemental fluctuations to characterize the temperature-
dependence of SROs and distribution of misfit volumes. Lattice distortions in
the presence of short range ordering will be discussed in Sec. III.2. In Sec.
III.3, we provide an in-depth analysis of partial dislocations and their
depinning mechanism in the presence of SROs. This includes auto-correlation
analyses associated with the dislocation line fluctuations as well as local
variations of the dislocation velocity . Section IV presents relevant
discussions and conclusions.

##  II Methods and Materials

Molecular dynamics simulations were carried out in LAMMPS [28] by implementing
atomistic samples of size $N=500,000$ and $1,700,000$ within a three-
dimensional periodic cell. In order to study SRO properties (in the absence of
dislocations), we prepared cubic samples with length $10$ nm along the
$x[100]$, $y[010]$, and $z[001]$ directions. The NPT ensembles were
implemented via a Nose-Hoover thermostat and barostat with relaxation time
scales $\tau_{d}^{\text{therm}}=10$ fs and $\tau_{d}^{\text{bar}}=100$ fs. We
also set the discretization time to $\Delta t\simeq 1.0$ fs. Samples were
initially prepared via an energy minimization at $T=0$ K (at a fixed volume)
and subsequently thermalized at different temperatures and constant pressure
$P=0$ bar for $100$ ps prior to annealing.

The interatomic forces are based on two commonly-used embedded-atom method
(EAM) potentials in the context of NiCoCr solid solution alloys: _1_) the Li-
Sheng-Ma potential proposed in [13] which has been utilized in recent SRO
studies, modeling dislocation nucleation and glide dynamics [29, 19], and
nanoindentation tests [18] _2_) the EAM Farkas–Caro potential [30] originally
developed to model equimolar high-entropy FeNiCoCrCu alloys but used here to
validate the SRO formation and its robustness against different potentials.

Annealed configurations were obtained performing hybrid MC/MD simulations
based on the variance constrained semi-grand canonical (VCSGC) ensemble [31]
within the annealing temperature range $T_{a}=400-1300$ K. In order to
determine the values of $\Delta\mu_{X_{1}X_{2}}=\mu_{X_{1}}-\mu_{X_{2}}$ which
minimizes the composition errors we perform a set of semi-grand canonical
simulations varying the chemical composition at $T=1500$ K, and fitted the MC
data using the equation:
$\Delta\mu(X_{1},P,T)=T\ln(X_{1}/[1-X_{1}])+\sum_{i=0}^{n}A_{i}X_{1}^{i}$
(Fig. 2), where $X_{1}$ is the reference element (Ni in our work) and $A_{i}$
are the fitting parameters [32]. This allow us to perform hybrid MD/VCSGC-MC
with a fixed target composition. During the annealing process, we perform $1$
MC cycle consisting of $N/2$ attempts of transmutation every $20$ MD steps .
We carried out a total number of $800,000$ MC cycles at all annealing
temperatures $T_{a}$ to ensure that the configurations reach thermal
equilibrium and that the structure of SROs are statistically indifferent.

We also studied dynamics of a $\frac{1}{2}[\bar{1}10](111)$ edge dislocation
which, under an external perturbation, dissociates into two separate partials
with a stacking fault in face-centered cubic (fcc) crystals. To this end, we
constructed a simulation cell with dimensions $L_{x}\simeq 80$ nm,
$L_{y}\simeq 20$ nm, and $L_{z}\simeq 15$ nm (see Fig. 1) and performed
annealing at a desired temperature $T_{a}$. We subsequently equilibrated the
annealed alloy at a low temperature $T=5$ K and pressure $P=0$ for the
duration of $100$ ps. To create an edge dislocation within the aged sample, we
used the periodic array of dislocation (PAD) model proposed in [33] to ensure
a periodic setup along the Burgers vector ($x[\bar{1}10]$) and dislocation
line ($z[\bar{1}\bar{1}2]$). The dislocated sample was further relaxed using
the NPT framework with $P_{xx}=P_{zz}=0$ and $T=5$ K (for 100 ps) leading to
the dislocation dissociation into two Shockley partials. A stress-controlled
framework was employed within the NVT ensemble at $T=5$ K by applying
additional forces on the top plane (normal to $y$) with the bottom layer held
fixed (with a thickness $\simeq 4$ Å) during the course of shear simulations.
The applied stress was gradually increased from $\sigma=50$ MPa (in a quasi-
static fashion) above a critical (depinning) stress in order to move the
partial dislocations.

##  III Results

The above methodology to prepare and anneal model CSAs indicates that aged
NiCoCr alloys may exhibit varying degrees of chemical ordering under different
annealing temperatures. Using robust composition-based metrics in Sec. III.1,
we further confirm the formation of SROs and their strong dependence on the
thermal processing. Section III.2 presents the misfit volume analysis
indicating meaningful correlations with the degree of short range ordering. In
Sec. III.3, we analyze dynamics of partial dislocations in the presence of
SROs and discuss potential implications in terms of the SRO-induced CSA
strengthening.

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Figure 7: Local concentration fluctuations $p_{b}$ with $b$ being Ni, Co, and
Cr in an annealed CSA (based on the Li-Sheng-Ma potential) at $T_{a}=400$ K
and multiple lengthscales a) $r_{c}=5\leavevmode\nobreak\ $Å b)
$r_{c}=10\leavevmode\nobreak\ $Å c) $r_{c}=18\leavevmode\nobreak\ $Å. We note
that at low distance scales $r_{c}=5$ and $10$ Å, the segregation of Ni and
Co/Cr domains is very strong. The black scatter points represent the two
dimensional projections. The red dot denotes the equimolar concentration.

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\begin{overpic}[width=99.73074pt]{Figs/varRssRescaled.png}
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Figure 8: Root-mean-squared (rms) fluctuations $p^{\text{rms}}_{b}$ of local
Ni, Co, and Cr concentrations as a function of window size $r_{c}$ in a)
annealed NiCoCr CSAs at $T_{a}=400$ K b) NiCoCr RSAs at $T=400$ K based on the
Li-Sheng-Ma potential. The panels in c) and d) are the same as a) and b) but
with the $y$-axis rescaled by $r_{c}^{-\frac{3}{2}}$. The curves are shifted
for the sake of clarity.

###  III.1 Temperature-dependence of SROs

Figure 3 illustrates that the formation of SROs within the solid solutions may
strongly depend on the annealing temperature $T_{a}$. The color map of atoms,
the inset of Fig. 3(a), indicates a clear segregation of Ni-rich phases (gray
circles) in the NiCoCr CSA modeled via the Li-Sheng-Ma potential at
$T_{a}=800$ K. Overall, the observed clustering features tend to become less
pronounced at higher annealing temperatures as shown in Fig. 4.

The presence of SROs appears to have a direct relevance on the (constant-
pressure) heat capacity $C_{p}=\partial_{T}H$ in Fig. 3(a) featuring a maximum
around $T_{a}\simeq 800$ K. Here $H$ denotes the enthalpy 111We only report
the (excess) configurational heat capacity (and thermal expansion coefficient)
neglecting (ideal) kinetic contributions.. The data presented in Fig. 3
correspond to a sample equilibrated at $T=300$ K and subsequently aged at
multiple annealing temperatures. The emerging peak in $C_{p}$ may suggest a
dominant role of enthalpic interactions over entropic effects that rule out
the formation of an ideal random solid solution [13]. We note that the (ideal)
heat capacity associated with the latter rises monotonically within the
temperature range of interest. Such deviations seem to be less pronounced in
terms of the thermal expansion coefficient
$\alpha_{p}=\frac{1}{V}\partial_{T}V$, with $V$ being the system volume, as
shown in Fig. 3(b).

The notion of SROs typically refers to coherent compositional deviations from
(statistically) random distributions of atoms within the solution matrix.
Along these lines, we investigated the Warren–Cowley SRO parameters
$p_{ab}(r)$ [12] probing the concentration variations of type-$b$ atoms at a
distance $r$ from a center type-$a$ element. For an equi-molar random NiCoCr
solid solution, one obtains $p^{\text{rsa}}_{ab}=0.33$ (on average) at any $r$
—more precisely, between the successive valleys of the pair correlation
function $g(r)$ as in Fig. 5(g). The SRO parameters could be also determined
locally for individual atoms which will presumably show strong fluctuations in
the presence of SROs. Nevertheless, the “averaged” Warren–Cowley parameters
should be relevant as the system tends to be statistically homogeneous beyond
the mean SRO size.

Figure 5(a-f) illustrates deviations of $p_{ab}$ associated with the annealed
CSAs from $p_{ab}^{\text{rsa}}$ including the six (distinct) elemental pairs
at $T_{a}=400$ K. The order parameters reveal several interesting features
describing the SRO microstructure. The abundance of the Ni-Ni pairs beyond
random concentrations is remarkable and appears to persist up to $r\simeq 5$ Å
in Fig. 5(a). It should be noted that twice this lengthscale is in a rough
agreement with the visual impression one gets from the segregation map, the
inset of Fig. 3(a), regarding the mean SRO size. Below the base line, the dip
in $p_{ab}$ corresponding to Ni-Ni pairs should indicate their scarcity above
the mean size. The order parameter re-crosses the horizontal line beyond which
it features a fairly broad peak at $r\simeq 15$ Å before going asymptotically
to $p_{ab}^{\text{rsa}}$. We remark that the inferred lengthscale could
potentially relate to the average spacing between adjacent SROs. Figure 5(b-c)
associated with $p_{\text{NiCo}}$ and $p_{\text{NiCr}}$ feature fairly
identical properties but with opposite trends as seen in $p_{\text{NiNi}}$
since they must all add up to unity.

As opposed to Ni-Ni ordering, we observe less coherent patterns associated
with the identical (same-element) pairs Co-Co and Cr-Cr as in Figure 5(d) and
(f). In particular, $p_{\text{CoCo}}$ and $p_{\text{CrCr}}$ seem to indicate
ordering as well as anti-ordering (potentially due to repulsion) at the first
and next nearest neighbor distances. The strong bonding between Co-Cr in Fig.
5(e) at the first nearest neighbor is also remarkable (see also [13, 19, 18]).
We further note that, unlike $p_{ab}$, the pair correlation function $g(r)$
does not suggest any _structural_ differences between annealed and random
solid solutions as shown in Fig. 5(g).

\begin{overpic}[width=138.76157pt]{Figs/pdf_pNi_T.png}
\put(68.0,71.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>T_{a}$\scriptsize(K)}\end{turn}}}}
\put(50.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(-6.0,40.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\end{overpic} Figure 9: Probability distribution functions of the local
concentrations $p_{\text{Ni}}$ in an annealed CSA based on the Li-Sheng-Ma
potential at the window size $r_{c}=10$ Å and various $T_{a}$. .

Figure 6 quantifies the abundance of Ni-Ni elemental pairs upon annealing at
multiple temperatures between $T_{a}=400-1400$ K. As shown in Fig. 6(a), the
curves show quite complex nonmonotonic features with certain characteristic
lengthscales that seem to scale non-trivially with temperature. Nevertheless,
the SRO-related features in $p_{\text{NiNi}}-p^{\text{rsa}}_{\text{NiNi}}$
become less and less pronounced with an increase in $T_{a}$ continually
approaching their asymptote at the zero-valued base line. This is
mathematically quantified by the metric
$p^{\text{rms}}_{\text{NiNi}}=\langle(p_{\text{NiNi}}-p^{\text{rsa}}_{\text{NiNi}})^{2}\rangle^{\frac{1}{2}}$
as a root-mean-squared (rms) measure of deviations from RSAs. Figure. 6(b)
features a monotonic growth of $p^{\text{rms}}_{\text{NiNi}}$ upon decreasing
$T_{a}$. We also note the tendency for the saturation of
$p^{\text{rms}}_{\text{NiNi}}$ at $T_{a}\simeq 1400$ K or above potentially
due to residual SRO distributions at atomistic levels that preclude an ideal
RSA formation. The above analysis was repeated for NiCoCr alloys simulated
based on the Farkas-Caro potential. Interestingly, we found no clear signature
of clustering in these samples as opposed to those generated via the Li-Sheng-
Ma potential (see Fig. S1).

We probed fluctuations in _local_ concentrations of the constituent elements
in annealed NiCoCr alloys that, in the presence of SROs, should deviate from
those of random solid solution alloys. In this context, the entire space was
partitioned using sub-volumes of varying size $r_{c}$ and local molar
compositions $p_{\text{Ni}}$, $p_{\text{Co}}$, and $p_{\text{Cr}}$ were
determined within each cube. As illustrated in the scatter plot of Fig. 7
associated with NiCoCr CSAs annealed at $T_{a}=400$ K, the fluctuations tend
to self-average at larger $r_{c}$ which could be also understood in terms of
counting statistics.

We also investigated _local_ fluctuations in CSA elemental concentrations in
space that, in the presence of SROs, show marked deviations from those of
RSAs. Figure 8 shows rms fluctuations of local concentrations $p_{\text{Ni}}$,
$p_{\text{Co}}$, and $p_{\text{Cr}}$ and their decay with distance $r_{c}$. In
Fig. 8(a) and (c), rms fluctuations in annealed NiCoCr CSAs seem to self-
average but only above some certain lengthscale above which the decay is well-
predicted by the expected $r_{c}^{-3/2}$ power-law behavior. The latter is the
relevant scaling in purely random atomic configurations as illustrated in Fig.
8(b) and (d). We take the characteristic peak in Fig. 8(c) as a signature of
spatial correlations which may be interpretted as the average SRO size
$\xi^{\text{sro}}\simeq 10$ Å. Furthermore, the inferred lengthscale closely
agrees with the one extracted from the SRO order parameters in the preceding
paragraphs which is within the typical range of SRO size ($0.5-1.5$ nm) seen
experimentally [4, 16]. Figure 9 illustrates the probability distribution
functions (PDFs) of the local Ni concentrations $p_{\text{Ni}}$ at $r_{c}=10$
Å for different annealing temperatures. We note the marked deviation of the
low-$T_{a}$ PDFs from a standard Gaussian distribution which, otherwise, seems
to be the asymptotic limit for the $p_{\text{Ni}}$ distributions at higher
annealing temperatures. One could also observe a marked abundance of low and
high (local) Ni concentrations away from the average at $p_{\text{Ni}}=0.33$
for $T_{a}=400$ and $600$ K which is indicative of the strong segregation of
Ni phases.

\begin{overpic}[width=104.07117pt]{Figs/hist_vorVolumeTemp600.png}
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annealed}}}
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\put(-6.0,40.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\end{overpic}

\begin{overpic}[width=104.07117pt]{Figs/hist_vorVolumeTemp5Rss.png}
\put(19.0,89.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$b)$
RSA}}}
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Figure 10: Probability distributions of alloy atomic (Voronoi) volumes $V_{b}$
with $b$ denoting the Ni, Co, and Cr atoms in a) the NiCoCr alloy aged at
$T_{a}=600$ K and b) NiCoCr RSA. The volume measurements were carried out at
$5$ K.

\begin{overpic}[width=104.07117pt]{Figs/voronoiAverageTempSro.png}
\put(19.0,76.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$a)$}}}
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\put(83.0,4.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>RSA}\end{turn}}}}
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\begin{overpic}[width=104.07117pt]{Figs/voronoiStdTempSro.png}
\put(19.0,76.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$b)$}}}
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V_{b}\rangle$}\end{turn}}}} \end{overpic}

Figure 11: Dependence of a) average Voronoi volume $\langle V_{b}\rangle$ and
b) rms volume fluctuations scaled by the average value
$\text{var}^{\frac{1}{2}}(V_{b})/\langle V_{b}\rangle$ on the annealing
temperature $T_{a}$. The empty symbols in correspond to a NiCoCr RSA.

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y=90$ \scriptsize\r{A}}}}
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y=10$ \scriptsize\r{A}}}}
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\end{overpic}

\begin{overpic}[width=117.07924pt]{Figs/wc_aging_dislocated_index0.png}
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Stacking fault}}} \end{overpic}

Figure 12: SRO microstructure in the presence of (partial) dislocations in
aging NiCoCr at $T_{a}=600$ K. a) cross section containing the stacking fault
b) cross section at $y=10$ Å c) the SRO parameter $p_{\text{NiNi}}$ as a
function of pairwise distance $r$. The different curves in c) correspond to
the entire configuration as well as the two dimensional stacks depicted in a)
and b). The black segments denote dislocation lines. The dashdotted (red) line
indicates RSAs. The stacking fault region in a) lies between $x\simeq 25$ and
$220$ Å.

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\end{overpic}

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\end{overpic}

Figure 13: Realizations of partial dislocations in a) NiCoCr annealed at
$T_{a}=600$ K b) RSA.

###  III.2 Lattice distortions

To characterize local distortion properties, we analyzed fluctuations in
atomic Voronoi cell volumes and associated temperature-dependence in annealed
alloys. The aged solid solutions were equilibrated at $5$ K upon annealing in
order to suppress thermal fluctuations. Figure 10(a) and (b) shows alloy
atomic volumes and associated PDFs for the Ni, Co, and Cr atoms in annealed
and random solid solutions. Both alloys feature quite narrow distributions
with well-defined mean values $\langle V_{\text{Ni}}\rangle$, $\langle
V_{\text{Co}}\rangle$, and $\langle V_{\text{Cr}}\rangle$ that show slight
variations with $T_{a}$ as in Fig. 11(a). The measured mean atomic volume in
annealed NiCoCr samples is $\langle V\rangle\simeq 11.3$ Å3 —which is
equivalent to the average lattice constant of $a=3.56$ Å— in very close
agreement with experimental observations reported in [8]. Figure 11(a)
indicates features near a characteristic annealing temperature $T_{a}\simeq
800$ K below which the mean atomic volumes seem to accelerate, potentially a
signature of remarkable enthalpy-driven ordering [13]. This observation is in
accordance with the heat capacity $C_{p}$ developing a characteristic peak in
Fig. 3(a).

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\end{overpic}

Figure 14: Realizations of (immobile) partial edge dislocations in an annealed
NiCoCr subject to the shear stress a) $\sigma=500$, b) $600$, c) $650$, and d)
$750$ MPa. Here the two-dimensional stack denotes the $(111)$ glide plane.

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\end{overpic}

Figure 15: Realizations of (immobile) partial dislocations in a NiCoCr RSA
subject to the shear stress a) $\sigma=400$, b) $500$, c) $550$, and d) $600$
MPa.

The misfit volumes $\Delta V_{b}=\langle V_{b}\rangle-\langle V\rangle$ of Ni,
Co, and Cr are determined as $+0.04$, $-0.03$, and $-0.01$ Å3, respectively.
The estimated atomic misfits appear to be at least one order of magnitude off
from precise experiments [8] but are, otherwise, reasonably close to ab
initio-based estimates in [35]. We further explored the scaled fluctuation
$\text{var}^{\frac{1}{2}}(V_{b})/\langle V_{b}\rangle$ as a relevant measure
of atomic distortions in Fig. 11(b) with $b$ denoting Ni, Co, and Cr.
$\text{var}^{\frac{1}{2}}(V_{b})/\langle V_{b}\rangle$ shows a (fairly)
monotonic increase for Ni as a function of $T_{a}$ till it saturates at a
limiting value that appears to be lower than the one set by the RSA. This is
in line with our SRO analysis that the abundance of Ni-Ni pairs within the
first nearest neighbor distance and, therefore, ordering tends to curtail
local atomic misfits or randomness in aged systems. As for Co and Cr, we
observe a non-monotonic evolution with a pronounced peak at $T_{a}\simeq 600$
K that, in the case of the former, even exceeds the associated RSA limit.
Apart from the observed peaks, the relative variance for aged Co/Cr seem to
always fall short of those of RSAs with a more dramatic decrease associated
with Cr (the green diamonds). This, we conjecture, might be attributed to the
favored formation of Cr-Co regions and less tendency for Cr-Ni as well as Cr-
Cr bonding as evidenced by the behavior of the first-nearest-neighbor order
parameters in Fig. 5. We conclude this subsection by stating that short range
order will have strong bearings on misfit volumes of NiCoCr as our data
suggest direct correlations between the latter and the order parameters
presented in Sec. III.1. One may infer a characteristic scale based on the rms
fluctuations analysis presented in Fig. 11(b) which we interpret as the (mean)
misfit size $\xi^{\text{misfit}}\simeq 1$ Å. Together with nanoscopic SROs
($\xi^{\text{sro}}\simeq 10$ Å), atomic-level misfit fluctuations will
determine the dislocation glide resistance as discussed in Sec. III.3.

###  III.3 Interplay between SROs and dislocations

We follow two different approaches to address the dislocation-SRO interplay in
NiCoCr CSAs: _1_) study of dislocation effects on the nucleation of SROs in
_aging_ alloys _2_) investigations of strengthening mechanisms at play in _as-
aged_ SRO-rich alloys driven out of equilibrium. In _1_), we aged samples with
a dislocation allowing for both the dislocation dissociation process and
spatio-temporal evolution of SROs while annealing. In _2_), on the other hand,
we embedded partial dislocations in as-annealed alloys and performed shear
depining tests, with no appreciable change in the SRO microstructure.

In line with _1_), Fig. 12 compares the structure of SROs within the
dislocation dissociation bounds and outside in aging NiCoCr at $T_{a}=600$ K.
In Fig. 12(a), the denser population of SROs within the stacking fault is
visually apparent in comparison with a dislocation-free two-dimensional stack
at $y=10$ Å illustrated in Fig. 12(b). We quantified the observed trend in
Fig. 12(c) where the SRO parameter $p_{\text{NiNi}}$ associated with the
former reveals a shallower decay relative to that of atoms outside the fault
plane. To our knowledge, the drastic increase in chemical ordering within the
stacking fault region has not been previously reported in the literature. As
one possible mechanism at play, we speculate that the long-range stress field
and mutual interactions between the two partials [36] might favor the SRO
nucleation and its growth within the dissociation zone. Figure 13(a) and (b)
illustrates that such interactions at $T_{a}=600$ K lead to a notable
reduction in the stacking fault width which implies the enhanced fault energy
due to SROs [13, 17]. It is expected that the fault dimension, and therefore
the associated formation energy, will be strongly controlled by the annealing
temperature as well. We note that a detailed description of the SRO kinetics
and dynamics of the dislocation dissociation as well as their (dynamical)
interplay during the aging processs is outside the scope of our current study.

Following approach _2_), the notion of “plastic flow” in solute strengthening
theories directly links to the existence of the intrinsic friction stress
$\sigma_{c}$ beyond which dislocations tend to glide rather smoothly at a non-
negligible (mean) velocity $\langle v\rangle$. Below this critical stress, by
contrast, the migration of dislocations within CSAs (with a severely distorted
energy landscape) typically occurs in a very intermittent manner with long
periods of quiescent states (i.e. $\langle v\rangle\simeq 0$) interrupted by
bursts of displacements [37]. In the absence of thermal activation, this
depinning transition is phenomenologically described as [38]

$\langle v\rangle\propto(\sigma-\sigma_{c})^{1/\beta},$ (1)

at $\sigma\geq\sigma_{c}$ and $\langle v\rangle=0$ otherwise, with $\beta\geq
1$ reflecting a marked discontinuity at $\sigma_{c}$ [39]. Here $\sigma$ is
the applied shear stress resolved in the glide plane.

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\begin{overpic}[width=117.07924pt]{Figs/h0_crltn_0.png}
\put(44.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(70.0,80.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{turn}{0.0}{$\sigma$\tiny(MPA)}\end{turn}}}}
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\end{overpic}

Figure 16: a) Configurations of (immobile) partial edge dislocations b)
associated correlations $c_{h}(|z-z^{\prime}|)=\langle\leavevmode\nobreak\
\hat{h}_{x}(z).\hat{h}_{x}(z^{\prime})\leavevmode\nobreak\ \rangle$ as a
function of distance $|z-z^{\prime}|$ in an annealed NiCoCr under different
applied stresses (below the depinning transition) at $T_{a}=600$ K. This
includes $\sigma=500$, $600$, $650$, $700$, and $750$ MPa. The shear tests
were carried out at $5$ K. Here, the two-dimensional $x-z$ glide plane denotes
a $(111)$ cross section. The dislocation configurations are shifted vertically
for the clarity.

\begin{overpic}[width=117.07924pt]{Figs/h0_1.png}
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\put(-6.0,35.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{turn}{90.0}{$z$(\r{A})}\end{turn}}}}
\put(17.0,90.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$a)$
\scriptsize RSA}}}
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\begin{overpic}[width=117.07924pt]{Figs/h0_crltn_1.png}
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\end{overpic}

Figure 17: a) Configurations of (immobile) partial edge dislocations b)
associated correlations $c_{h}(|z-z^{\prime}|)=\langle\leavevmode\nobreak\
\hat{h}_{x}(z).\hat{h}_{x}(z^{\prime})\leavevmode\nobreak\ \rangle$ as a
function of distance $|z-z^{\prime}|$ in a NiCoCr RSA under different applied
stresses below the depinning transition. This includes $\sigma=400$, $500$,
$550$, $600$, and $650$ MPa

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\end{overpic}

\begin{overpic}[width=104.07117pt]{Figs/DislVelocItime30000_id1_load1200.png}
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\end{overpic}

\begin{overpic}[width=104.07117pt]{Figs/DislVelocItime35000_id1_load1200.png}
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\end{overpic}

Figure 18:  (z-scored) velocity of partial edge dislocations $\hat{v}_{x}(z)$,
illustrated by the black arrows, in annealed NiCoCr at $T_{a}=600$ K and under
the applied stress $\sigma=1200$ MPa (above $\sigma_{c}$). The shear tests
were carried out at $5$ K. Here, the panels indicate different realizations
associated with the gliding dislocations and the two-dimensional plane denotes
a $(111)$ cross section. The arrows denote the velocity field.

\begin{overpic}[width=117.07924pt]{Figs/mobility.png}
\put(44.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\scriptsize(MPa)}\end{turn}}}}
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\end{overpic}

\begin{overpic}[width=117.07924pt]{Figs/v0_crltn_0_multipleLoads.png}
\put(44.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(-6.0,35.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>c_{v}(|z-z^{\prime}|)\rangle_{\text{ens}}$}\end{turn}}}}
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\scriptsize annealed}}} \end{overpic}

\begin{overpic}[width=117.07924pt]{Figs/v0_crltn_1_multipleLoads.png}
\put(44.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(-6.0,35.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>c_{v}(|z-z^{\prime}|)\rangle_{\text{ens}}$}\end{turn}}}}
\put(68.0,80.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{turn}{0.0}{$\sigma$\tiny(MPA)}\end{turn}}}}
\put(17.0,15.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$c)$
\scriptsize RSA}}} \end{overpic}

Figure 19: Stress-dependence of the mean dislocation velocity and associated
fluctuations. a) Mobility rule describing the mean dislocation velocity
$\langle v_{x}\rangle$ as a function of applied stress $\sigma$. Mean velocity
auto correlations $\langle c_{v}(|z-z^{\prime}|)\rangle_{\text{ens}}$ as a
function of distance $|z-z^{\prime}|$ in b) annealed NiCoCr c) NiCoCr RSA
subject to multiple applied stresses $\sigma$ above $\sigma_{y}$.

To estimate $\sigma_{c}$, we performed atomistic simulations of dislocation
properties in fcc-based NiCoCr CSAs by studying dynamics of
$\frac{1}{2}\langle{1}10\rangle\\{111\\}$ edge dislocations which, under an
external drive, dissociate into two mixed partials and a stacking fault in
between [36]. To measure the dislocation velocity and its spatial-temporal
evolution, we first identified all dislocation line defects in the atomistic
crystal, along with their Burgers vectors, and output a line representation of
the dislocations by using OVITO [40]. Due to inherent lattice distortions,
dislocation lines are not straight but show local fluctuations with respect to
the average line direction along $z$. We describe line fluctuations projected
along the glide direction $x$ by the function $h_{x}(z)$ discretized via a
fine grid of size $2$ Å across the glide plane parallel to the $z$ direction.
We obtain the dislocation velocity $v_{x}(z)=\delta h_{x}(z)/\delta t$ by
considering successive dislocation snapshots that are apart by the time window
$\delta t\simeq 4$ ps. The latter is chosen to be at least three orders of
magnitude longer than the discretization time $\Delta t$ yet short enough to
resolve displacements down to atomistic scales. The subsequent correlation
analysis was performed on CSAs initially annealed at $T=600$ K and sheared,
along with the RSAs, at $5$ K.

Figure 14 and 15 illustrate configurations of (frozen) dislocations in an
annealed NiCoCr as well as a NiCoCr RSA under different loads well below the
depinning transition ($\sigma<\sigma_{c}$). The local curvatures associated
with the dislocation segments in Fig. 14 indicate fairly coherent pinning
effects that somewhat correlate with the spatial locations of SROs. Such
features might be also present in RSAs, as in Fig. 15, but to a very limited
extent in space. The local line curvature, and its positive sign with respect
to the glide direction, should potentially indicate how effectively
dislocations are pinned near SROs and/or due to local atomic misfits. In this
context, line fluctuations associated with the aged alloy in Fig. 16(a) appear
to be correlated over larger lengthscales than those of the random alloy in
Fig. 17(a). Similar trends could be also inferred from the associated
correlation functions

$c_{h}(|z-z^{\prime}|)=\langle\leavevmode\nobreak\
\hat{h}_{x}(z).\hat{h}_{x}(z^{\prime})\leavevmode\nobreak\ \rangle,$ (2)

with the $z-$scored fluctuations $\hat{h}_{x}=(h_{x}-\langle
h_{x}\rangle)/\text{var}^{\frac{1}{2}}(h_{x})$. Here the angular brackets
$\langle.\rangle$ denote averaging in space. Overall, the slower decay of
correlations $c_{h}(|z-z^{\prime}|)$ in Fig. 16(b), in comparison with Fig.
17(b), may indicate additional SRO-induced pinning effects in annealed alloys.
We note that the absence of SROs does not necessarily rule out long-range
fluctuation patterns in RSAs, as in Fig. 17(b), and, therefore, coherent
pinning patterns due to atomic-scale distortions [41, 42].

\begin{overpic}[width=216.81pt]{Figs/Elastic.png}
\put(91.0,7.5){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>RSA}\end{turn}}}} \end{overpic} Figure 20: Elastic constants and their
dependence on the annealing temperature $T_{a}$. Measurements were carried out
at $T=5$ K. The rightmost data point represents elastic properties of a random
solid solution. Samples were generated based on the Li-Sheng-Ma potential.

\begin{overpic}[width=117.07924pt]{Figs/v0_0.png}
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\scriptsize annealed}}}
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\end{overpic}

\begin{overpic}[width=117.07924pt]{Figs/v0_crltn_0.png}
\put(44.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(-6.0,35.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{turn}{90.0}{$c_{v}(|z-z^{\prime}|)$}\end{turn}}}}
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\end{overpic}

Figure 21:  a) (Scaled) velocity of partial edge dislocations $\hat{v}_{x}(z)$
b) associated correlations $c_{v}(|z-z^{\prime}|)=\langle\leavevmode\nobreak\
\hat{v}_{x}(z).\hat{v}_{x}(z^{\prime})\leavevmode\nobreak\ \rangle$ as a
function of distance $|z-z^{\prime}|$ in an annealed NiCoCr at $T_{a}=600$ K
under the applied stress $\sigma=1200$ MPa (above $\sigma_{c}$). The shear
tests were carried out at $5$ K. Here, the numbers indicate different
realizations associated with gliding dislocations and the two-dimensional
plane denotes a $(111)$ cross section. The velocity profiles are shifted
vertically for the clarity.

\begin{overpic}[width=117.07924pt]{Figs/v0_1.png}
\put(44.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{turn}{0.0}{$\hat{v}_{x}(z)$}\end{turn}}}}
\put(-6.0,35.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{turn}{90.0}{$z$(\r{A})}\end{turn}}}}
\put(17.0,90.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$a)$
\scriptsize RSA}}}
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\end{overpic}

\begin{overpic}[width=117.07924pt]{Figs/v0_crltn_1.png}
\put(44.0,-3.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(-6.0,35.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{turn}{90.0}{$c_{v}(|z-z^{\prime}|)$}\end{turn}}}}
\put(17.0,15.0){\hbox{\pagecolor{white}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$b)$}}}
\end{overpic}

Figure 22:  a) (Scaled) velocity of partial edge dislocations $\hat{v}_{x}(z)$
b) associated correlations $c_{v}(|z-z^{\prime}|)=\langle\leavevmode\nobreak\
\hat{v}_{x}(z).\hat{v}_{x}(z^{\prime})\leavevmode\nobreak\ \rangle$ as a
function of distance $|z-z^{\prime}|$ in a NiCoCr RSA under the applied stress
$\sigma=1200$ MPa (above $\sigma_{c}$).

We repeated the above analysis by probing velocity fluctuations $v_{x}(z)$
associated with the gliding dislocations (at $\sigma>\sigma_{c}$) in annealed
as well as random alloys (see Fig. 18). Figure 19 illustrates the shear stress
dependence of the mean dislocation velocity as well as (mean) velocity
correlations (averaged over different configurations) for the aged and random
alloys. As shown in Fig. 19(a), the observed behavior of $\langle
v_{x}\rangle$ versus $\sigma$ at $\sigma_{c}$ marks the dislocation pinning-
to-depinning transition which is in agreement with the expected generic
dependence around the transition. This seems to be fairly insensitive to
annealing except for a meaningful shift of $\sigma_{c}$ to larger strengths.
The estimated critical shear stresses are $\sigma_{c}\simeq 950$ and $650$ MPa
associated with aged and random samples, respectively. The (mean) velocity
auto-correlations, averaged over different realizations, $\langle
c_{v}(|z-z^{\prime}|)\rangle_{\text{ens}}$ (see the definition of
$c_{v}(|z-z^{\prime}|)$ below) are shown at different stress levels beyond
$\sigma_{c}$ in Fig. 19(b) and (c), both indicating a finite correlation
length.

The marked increase of $\sigma_{c}$ is despite (relatively) insignificant
variations of elastic properties (Fig. 20) and, therefore, improving yielding
properties against RSAs cannot be naively attributed to the enhancement in
elasticity of aged alloys (see Sec. IV). The elastic constants we probed in
this study include $C_{11}$, $C_{12}$, and $C_{44}$ (based on the Voigt
notation) as well as the bulk modulus $B$ and Poisson’s ratio that were
determined by using the Li-Sheng-Ma interatomic potential. Here the $x$, $y$,
and $z$ dimensions are parallel to $[100]$, $[010]$, and $[001]$ crystal
directions, respectively. The overall trend we observe in Fig. 20 is
consistent with the study of Li et al. [13] which reported the change of
elastic properties with increasing SROs (upon decreasing $T_{a}$). The elastic
constants seem to develop features near $T_{a}\simeq 800$ where the dominant
presence of chemical ordering is expected.

We further investigate individual dislocation configurations and associated
fluctuations of local velocities in Fig. 21 and 22 where the dislocations move
at an average speed $\langle v_{x}\rangle\simeq 1000\leavevmode\nobreak\
\text{ms}^{-1}$ in both systems subject to the applied shear stress of
$\sigma=1200$ MPa, well above the corresponding depinning thresholds. The
($z$-scored) velocity profiles $\hat{v}_{x}(z)$ in Fig. 21(a) and 22(a)
correspond to five different snapshots of gliding dislocations that are
shifted for a better view of variations across the (average) dislocation line
direction $z$. We remark that the regions to the left of the dashdotted lines
indicate local velocities below the average speed $\langle v_{x}\rangle$, as
depicted by the left-headed arrows in Fig. 18 . Statistically speaking, the
segments with $v_{x}(z)<\langle v_{x}\rangle$ somewhat correlate with the
positively-bent segments of dislocation lines which are mostly influenced by
the existence of SROs and/or atomic misfits. Nevertheless, velocity
fluctuations quantified by the velocity auto correlations

$c_{v}(|z-z^{\prime}|)=\langle\leavevmode\nobreak\
\hat{v}_{x}(z).\hat{v}_{x}(z^{\prime})\leavevmode\nobreak\ \rangle,$ (3)

not appear to be statistically different in annealed and random alloys in Fig.
21(b) and 22(b).

Table 1: Cubic elastic constants and depinning stress $\sigma_{c}$ at $5$ K for NiCoCr with and without annealing using the Farkas-Caro potential.  |  $T_{a}=1100$(K) |  $T_{a}=600$(K) | RSA | Experiment
---|---|---|---|---
$C_{11}$(GPa) | 269.3 | 251.4 | 270.1 | 255.3 [43]
$C_{12}$(GPa) | 146.8 | 149.1 | 147.9 | 159.4 [43]
$C_{44}$(GPa) | 126.7 | 126.8 | 126.7 | 146.7 [43]
$\sigma_{c}$(GPa) | 0.075 | 0.070 | 0.080 | 1.3 [3]

##  IV Discussions & Conclusions

Our atomistic simulations of NiCoCr CSAs under special thermal treatments have
revealed the formation of nanostructural local chemical ordering and enhanced
dislocation glide resistance in close agreement with recent SRO-based studies
in simulated and _real_ NiCoCr experiments [13, 4]. On the ordering effects,
we made use of the Li-Sheng-Ma potential function that has been validated in
terms of detailed and accurate modelling of Ni, Co, and Cr interatomic
interactions [13]. Our direct measurements of local lattice strains agree very
closely with a recent ab initio study [35] but failed to fully reproduce
experimental findings [8]. By using the Farkas-Caro potential, we find very
limited relevance to real annealed NiCoCr alloys. The simulated alloys, in
this context, exhibit no ordering (beyond statistical fluctuations), but also
no notable improvement in yield strengths or elastic properties as reported in
Table 1. We have interpreted the physical origin of such differences by using
robust (experimentally-relevant) SRO descriptors in various thermal annealing
scenarios. We find that the Li-Sheng-Ma potential, under the proper aging
process, leads to an exceptional dislocation depinning strength with low
stacking fault width that falls short of that of RSAs. The latter is
associated with the enhanced stacking fault energy which might, in part,
relate to improved elasticity as a result of SROs but, compared to the yield
strength, the ordering effects appear to be less pronounced. The intrinsic
strengthening mechanism is mostly dominated by coherent SROs-induced pinning
effects, but random spatial distributions of misfit volumes and the resulting
roughening seem to be also at play.

Our correlation analyses of the dislocation structure and its spatial-temporal
evolution allow for inferring a characteristic pinning length $\xi_{p}$ and
optimal displacement $w_{p}$ [24]. We interpret the latter as being the rms
fluctuations in the dislocation height (with respect to the mean), i.e.
$w_{p}=\langle h^{2}_{x}-\langle h_{x}\rangle^{2}\rangle^{\frac{1}{2}}$,
whereas the former is determined as the shortest distance where the height
correlations cross zero, e.g. $c_{h}(|z-z^{\prime}|)=0$ at
$|z-z^{\prime}|=\xi_{p}$. At $T_{a}=600$ K and $\sigma<\sigma_{y}$, it follows
that $w_{p}=5.6-11.2$ Å and $\xi_{p}=25-31$ Å associated with the annealed
alloy. We conjecture that these two quantities should both correlate with the
observed increase in the depinning stress. Based on our present data, however,
we are not able to quantify such (anti-)correlations numerically. Both
observables $w_{p}$ and $\xi_{p}$ Å are also expected to show meaningful
associations with the average SRO size $\xi^{\text{sro}}$ as well as the
amplitude of misfit fluctuations (characterized by $\xi^{\text{misfit}}$) and
are relevant ingredients in _mean-field_ solute models that make yield
strength predictions based on dislocation line properties (e.g. line tension
$\Gamma$ , length $L$, and Burgers vector $b$). In this mean-field picture,
SROs introduce the characteristic scale $\xi^{\text{sro}}$ that _effectively_
decreases the pinning length $\xi_{p}$ leading to a reduction of the optimal
displacement $w_{p}$ and, therefore, an extra strengthening. Within these
mean-field model frameworks, the depinning stress should scale with the line
tension $\Gamma$, which itself is proportional to the shear modulus and, based
on our findings, annealing is not expected to boost $\sigma_{y}$ simply
because of such elasticity-based contributions but instead variations in the
disorder strength and the pinning field are the key factors. To validate such
theories in simulations, one must be vigilant to use appropriate mesoscopic
lengths (beyond atomistic scales) where continuum-like concepts such as line
tension and local curvature are well-defined [24]. To explore the full
dislocation waviness in MD, it is also necessary that the dislocation length
$L\gg\xi_{p}$ and associated deformation $w_{p}\ll\xi_{p}$ [44]. However, the
above separation of scales is typically a limit beyond atomistic modeling
assumptions including the present study.

Complications might also arise in the application of strengthening theories
(e.g. the VC model) due to SRO-induced correlations. The latter are at odds
with the “randomness” hypothesis taken as granted based on solutes’
arrangements in RSAs. Given that the VC theory is constructed exclusively on
misfit information, an _effective_ treatment in the presence of spatial
correlations is the assumption that SROs should alter dramatically local
misfit distributions and, in that case, an alternative length
$\xi^{\text{misfit}}_{\text{eff}}$ may be used to describe the strength of
distortions. Alternatively, Zaiser and Wu (ZW) [42, 45] formulated a more
relevant approach based on the fact that pinning forces caused by obstacles
are not fully random but rather correlated over some certain length $a$ [46]
that we tend to interpret as the effective SRO size
($a\simeq\xi^{\text{sro}}$). In their formulation of dislocation dynamics, ZW
introduced a order/disorder length that, along with the strength of misfit
fluctuations, may describe the SRO- and misfit-induced noise field in a more
accurate way than the VC methodology. Similar efforts were made by Zhang et
al. [47] along these lines who developed a stochastic Peierls-Nabarro model to
incorporate the role of both CSA randomness and short range ordering effects
on glide dynamics of roughened dislocations.

The existing literature reports the emergence of (varying degrees of) SROs as
a rather generic feature across a broad range of high-entropy alloys (see [5]
and references therein). Nevertheless, the focus has been placed on different
variants of NiCoCr-based alloys (including the well-studied Cantor alloy) and,
owing to similar atomic size and electron negativity, such compositions might
tend to favor SRO nucleation [48]. In terms of mechanical properties, the SRO-
induced enhancement in the dislocation glide resistance may also constitute
fairly universal mechanisms associated with it, i.e. coherent pinning and
enhanced roughening, not specific to particular chemical compositions but
their robustness over a broader range of compositionally complex solid
solutions has yet to be fully explored.

There is a large multitude of results in this work, some of which including
the dislocation roughening and SRO emergence could be potentially validated
experimentally through the in/ex-situ electron microscopy analysis or other
image-based characterization techniques. Furthermore, our finding will have
important implications for Discrete Dislocation Dynamics (DDD) models and
associated mobility rules that additionally consider spatial correlations
within the rough potential energy landscape [49]. This is conceptually similar
to intrinsic Peierls stresses that are locally distributed in space but also
correlated over certain microstructural scales. Incorporating and tuning SROs’
structural features as model ingredients will potentially lead to further
improvements in DDD predictive capabilities and design-level hardening
features in the context of NiCoCr-based CSAs with dense and complex networks
of interacting dislocations.

###### Acknowledgements.

This research was funded by the European Union Horizon 2020 research and
innovation program under grant agreement no. 857470 and from the European
Regional Development Fund via Foundation for Polish Science International
Research Agenda PLUS program grant no. MAB PLUS/2018/8.

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## Supplementary Materials

In this Supplementary Materials we will present further discussions on local