# Irradiation-induced defect evolution in concentrated solid-solution alloys: a molecular dynamics perspective

A Aligayev, Marie Landeiro dos Reis, A Chartier, U Jabbarli, F

Domínguez-Gutiérrez, Q Huang

► To cite this version:

A Aligayev, Marie Landeiro dos Reis, A Chartier, U Jabbarli, F Domínguez-Gutiérrez, et al. Irradiation-induced defect evolution in concentrated solid-solution alloys: a molecular dynamics perspective. Plasma Physics and Controlled Fusion, 2025, 67 (5), pp.055020. 10.1088/1361-6587/adcd2b. hal-05371678

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# Irradiation-induced defect evolution in concentrated solid-solution alloys: a molecular dynamics perspective

To cite this article: A Aligayev et al 2025 *Plasma Phys. Control. Fusion* **67** 055020

View the [article online](#) for updates and enhancements.

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# Irradiation-induced defect evolution in concentrated solid-solution alloys: a molecular dynamics perspective

A Aligayev<fnref n="1" />,<fnref n="2" />,<fnref n="3" />,* ID, M Landeiro Dos Reis<fnref n="4" />, A Chartier<fnref n="5" /> ID, U Jabbarli<fnref n="6" />, F J Domínguez-Gutiérrez<fnref n="1" />, * ID and Q Huang<fnref n="2" />,<fnref n="3" /> ID

E-mail: amil.aligayev@ncbj.gov.pl and javier.dominguez@ncbj.gov.pl

Received 10 January 2025, revised 27 March 2025

Accepted for publication 15 April 2025

Published 24 April 2025

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CrossMark

## Abstract

This computational study delves into the intricate interplay of alloying elements on the generation, recombination, and evolution of irradiation-induced defects. Molecular dynamics simulations were conducted for collision cascades at room temperature, spanning a range of primary knock-on atom energies from 1 to 10 keV. The investigation encompasses a series of model crystals, progressing from pure Ni to binary concentrated solid solution alloys (CSAs) such as NiFe₂₀, NiFe, NiCr₂₀, and NiFeCr₂₀ CSA. We observe that materials rich in Cr actively facilitate dislocation emissions and induce the nucleation of stacking fault tetrahedra in the proximity of nanovoids, due to Shockley partial interactions. This result is validated by molecular static simulations, which calculate the surface, vacancy, and defect formation energies. Among the various shapes considered, the spherical void proves to be the most stable, followed by the truncated octahedron and octahedron shapes. On the other hand, the tetrahedron cubic shape is identified as the most unstable, and stacking fault tetrahedra exhibit the highest

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<footnote n="1">¹ NOMATEN Centre of Excellence, National Center for Nuclear Research, 05-400 Swierk/Otwock, Poland</footnote>

<footnote n="2">² Key Laboratory of High Magnetic Field and Iron Beam Physical Biology, Institute of Intelligent Machines, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, People's Republic of China</footnote>

<footnote n="3">³ Science Island Branch of Graduate School, University of Science and Technology of China, Hefei 230026, People's Republic of China</footnote>

<footnote n="4">⁴ LaSIE UMR CNRS 7356, La Rochelle Université, Av. Michel Crépeau, 17042, La Rochelle Cedex 1, France</footnote>

<footnote n="5">⁵ Université Paris-Saclay, CEA, Service de recherche en Corrosion et Comportement des Matériaux 91191 Gif Sur Yvette, France</footnote>

<footnote n="6">⁶ Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093 Warsaw, Poland</footnote>

<footnote>* Authors to whom any correspondence should be addressed.</footnote>
formation energy. Notably, among the materials studied, NiCr$_{20}$ and NiFeCr$_{20}$ CSAs stood out as the sole alloys capable of manifesting this mechanism, mainly observed at high impact energies.

Supplementary material for this article is available online

Keywords: defects, stacking, fault, collision cascades, MD

# 1. Introduction

Nickel-based concentrated solid solution alloys (CSAs) have been extensively investigated for their potential use in extreme service conditions such as those encountered in nuclear fission reactors, where structural materials are subjected to high irradiation doses and elevated temperatures [1–6]. These alloys are considered promising candidates for next-generation nuclear fission systems, especially fast reactors, due to their potential for high radiation tolerance and robust mechanical performance [7–9]. Recent successes in the fabrication of CSAs have paved the way for a new research direction aimed at substantially enhance alloy performance [1, 10–12]. While historical alloy development focused on traditional alloys with unique microstructural heterogeneity to mitigate displacement damage [13–16], CSAs represent a shift, containing two to five or more elements at high concentrations, sometimes in equal or near-equal amounts. The random arrangement of multiple elemental species on a crystalline lattice results in atomic-level elemental alternation, creating disordered local chemical environments [16–18]. This intrinsic property leads to unique site-to-site lattice distortions, creating complex energy landscapes that affect defect migration [5, 19, 20].

Considering the challenging service conditions of next-generation nuclear fission power reactors, which involve intense radiation flux, higher operating temperatures, and high stress [21, 22], CSAs emerge as promising candidate materials for nuclear power applications [12, 20]. To be viable, these alloys must complement their superior mechanical properties with high radiation resistance [2, 6, 9, 10]. In fission reactor environments, irradiation-induced point defect formation, migration, and evolution are primary factors influencing microstructural changes that impact the performance of structural materials. Thus, controlling defect formation and migration in structural materials is crucial for designing materials with high radiation tolerance. The phenomenon of irradiation hardening is governed by the interactions between moving dislocations and irradiation-induced defects occurring at the atomic scale [14, 23]. For instance, at doses around 0.01 dpa, the observation of softening becomes apparent in constant strain rate tests for materials relevant to nuclear applications [24]. Upon the initiation of plastic deformation, it is observed that the applied stress initially decreases and subsequently stabilizes.

The kinetics and interaction of defects play a crucial role in controlling microstructural evolution, significantly impacting material properties over time [25–27]. We thus investigate the influence of pre-existing defects on collision cascades,

specifically examining common material defects such as vacancy clusters and stacking fault tetrahedra (SFT) in elemental face-centered cubic (fcc) Ni, the binary alloys NiFe$_{20}$ and NiCr$_{20}$, and the ternary alloy NiFeCr$_{20}$ CSAs. We scrutinize several vacancy cluster shapes, from spherical to octahedron (in accordance with Wulff's predictions) to determine if these shapes influence the emergence of plasticity during collision cascade [28–32]. SFT, largely observed in these alloys are also considered. It is worth noticing that dislocation emission always begins from such pre-existing defects and that a clear influence of Cr is observed. It is interesting to note that in vicinity of voids, Shockley dislocations interact to form SFT defects and stair-rod dislocations [28].

# 2. Computational methods

The atomistic behavior of the samples was modeled using the molecular dynamics (MD) method, as implemented in the Large-scale Atomic Molecular Massively Parallel Simulator (LAMMPS) [33]. Interatomic potentials based on the embedded atom model developed by Bonny et al [34] with the Ziegler-Biersack-Littmark potential are employed to model interatomic interactions, with the inner cut-off set at 1.0 Å cut-off corresponding to the equilibrium bond length or the shortest significant interatomic distance, while the outer cut-off is chosen as 6 Å, ensuring computational efficiency and accuracy. These values, selected based on simulation conditions and convergence tests, are consistent with prior studies on similar alloys [35, 36]. The selection of these potentials is based on their comprehensive characterization of point-defect properties in FeNiCr alloys, particularly at the target composition of 20% Cr. These potentials are specifically tailored to model the microstructural evolution of radiation defects, aligning closely with the focus of our research.

We first prepare a numerical cell for pure Ni with [001] crystal orientation using the Atomsk numerical tool [37], employing a lattice constant of 0.352 nm. This initial configuration was then duplicated to create a supercell with varying volumes for different calculations in this study. Subsequently, alloy configurations were generated by introducing random substitutions in the supercell. This involved introducing 50% and 20% Fe for NiFe and NiFe$_{20}$ CSAs, 20% Cr for NiCr$_{20}$, and 40% Fe with 20% Cr for NiFeCr$_{20}$ CSAs. Then, the Monte Carlo (MC) simulations were employed to explore the configurational space of the NiFeCr system, ensuring that the most representative atomic arrangements were considered for all the systems. 9000 random alloy configurations
are considered to achieve statistical convergence in capturing the thermodynamically relevant configurations at the given composition. Given the large configurational entropy and the nearly equiatomic composition of the some of the alloys, a sufficiently large number of sampled structures was required to avoid bias and ensure that possible local chemical clustering was adequately captured. The MC simulations were implemented using a Metropolis-based approach within a semi-grand canonical ensemble, allowing compositional fluctuations within a predefined chemical potential framework [14, 15, 38]. In this study, alloy compositions are denoted using a notation commonly employed in MD simulations. For example, NiFe20 indicates an alloy where Fe constitutes 20 atomic percent (at.%), with the remainder being Ni (80 at.%). Similarly, NiFeCr20 represents an alloy with 20 at.% Cr, while Ni and Fe are present in equal proportions (40 at.% each). This notation is used for clarity and consistency with other computational materials studies. The cells were relaxed using FIRE minimization algorithm until the force reach $1 \times 10^{-3}$ meV Å$^{-1}$ [39] allowing the samples to find their lowest energy structure. It is noteworthy that the optimization process for the CSAs' geometry aimed to reach the nearest local minimum of the energy structure, ensuring that the change in energy between successive iterations and the most recent energy magnitude remained below $10^{-5}$. Additionally, the global force vector length of all atoms was maintained at less than or equal to $10^{-8}$ eV Å$^{-1}$.

## 2.1. Molecular static simulations

Point defects, such as vacancies where an atom is missing from a lattice point, can impact a CSAs' chemical properties and mechanical behavior during collision cascades. These defects are analyzed by computing the formation energy ($E_v$) of a vacancy, which determines the energy needed to break bonds and remove an atom from the crystal. To achieve this, we employ a computational cell containing $N_v = 4000$ atoms, with a lateral dimension of 3.52 nm. The vacancy formation energy is then calculated using the following method:

$$E_v = E_v^f - E_0 - e_x, \quad (1)$$

where $E_v^f$ and $E_0$ are the total energy with the vacancy and the defect free cell, respectively. $e_x$ is the energy of the atom deleted to create the vacancy.

Then we checked 2D defects, surface and stacking fault. Surface energy of the main crystal orientations [001], [011] and [111] were computed following:

$$\Gamma = \frac{E_{\text{surf}} - E_0}{2S} \quad (2)$$

with $E_{\text{surf}}$ the energy of the cell with the defect, $E_0$ the energy of the pristine cell and $S$ the surface area.

The stacking fault was introduced by rigidly shifting the upper half of the cell in the X and Y directions with the orientation of $X = [1\bar{1}0]$, $Y = [1\bar{1}2]$, and $Z = [111]$ and dimensions ($d_x, d_y, d_z$) = (2.5, 2.5, 9) nm, as depicted in figure 1. Ten different samples, each with the same atomic percentages for

<figure><img src="image_5.png" /><figcaption>Figure 1. Simulation cells for GSF computations, with the orientation of $X = [1\bar{1}0]$, $Y = [1\bar{1}2]$, and $Z = [111]$ and dimensions ($d_x, d_y, d_z$) = (2.5, 2.5, 9) nm. The atoms are color-coded based on their chemical species, except for the blue atoms, which highlight surfaces, and the red atoms, which indicate the fault.</figcaption></figure>

the CSAs, were employed to ensure a reliable average of the Generalized Stacking Fault (GSF) energy, denoted as $\gamma_{\text{GSF}}$. The energy of the cell, denoted as $E_{\text{GSF}}$, was then computed based on the displacement of the cell in the Y direction (See figure 1)

$$\gamma_{\text{GSF}} = \frac{E_{\text{GSF}} - E_0}{A} \quad (3)$$

with $E_0$ the energy of the pristine cell without the stacking fault, i.e. the defect free cell, and $A$ is the GSF area.

In third part we studied SFTs and voids. It is well-established that there are various defects in fcc metals [40]. Particularly, vacancy clusters and SFT are prominently observed defects in these materials [41]. In this work, we aimed to evaluate the impact of such defects (voids and SFT) on the evolution of collision cascade. The procedure to introduce these defects is described in the following subsections.

Vacancy clusters were built by removing atoms within the geometric region defining the defect. Voids were created in spherical, octahedral, and truncated octahedral shapes (as predicted by the Wulff theory using surface energies presented in the results section). Additionally, voids were shaped as tetrahedrons and cubes for further analysis. The Wulff theory, which relies on surface energy calculations across primary surface planes allows us to anticipate the stable configuration of vacancy clusters. This theory is particularly accurate for atomic simulations involving sufficiently large clusters, where the cluster faces can be treated as free surfaces. However, the
theory may have limitations when dealing with small clusters where the ratio of edge to face is high. Therefore, we initially investigated the stability of defects prior to collision cascade by employing Wulff theory alongside the analysis of formation energy results by using a Python package for Wulff construction [42, 43] to determine the Wulff shape using $\Gamma_{100}$, $\Gamma_{100}$ and $\Gamma_{111}$. WulffPack has been developed at Chalmers University of Technology in Gothenburg in the Condensed Matter and Materials Theory division at the Department of Physics [42, 43].

The SFT was implemented employing the approach delineated in reference [44]. Following the same methodology used for voids, we built a triangular vacancy plate that relaxed into SFT defects with stair-rod dislocations, due to the low stacking fault energy (SFE) of the CSAs, as predicted by the Hirsch and Silcox mechanism [45].

To assess the stability of these defects prior to irradiation, we computed the mean formation energy of each defect. For these particular simulations, the cells involved $N_{at} \simeq 863000$ atoms, arranged within dimensions $(d_x, d_y, d_z) = (21, 21, 21)$ nm. The defects were introduced as described before, and the mean formation energy per vacancy, $E_f$, of these defects was computed as follows:

$$E_f = \frac{1}{N_{at} - N_d} \left( E_d - \frac{N_d}{N_{at}} E_0 \right) \quad (4)$$

where $E_d$ is the energy of the simulation cell with the defect, containing $N_d$ atoms and $N_{at} - N_d$ vacancy and $E_0$ is the energy of the simulation cell of the pristine CSA, containing $N_{at}$ atoms. We chose to rationalize the formation energy based on the number of vacancies, as defects of the same size can have different numbers of vacancies due to their unique shapes. In the case of multi-component materials, we assume that the defect is sufficiently large to justify averaging over the various types of vacancies.

## 2.2. Collision cascades

A pure fcc Ni sample with $(d_x, d_y, d_z) = (13.7, 14.1, 14.4)$ nm with 255 840 Ni atoms was initially created. After energy minimization, a 100 ps equilibration into the isobaric-isothermal ensemble was conducted at 300 K with a time constant of 100 fs [46–48]. As illustrated in figure 2, we considered two kind of defects commonly found in fcc metals: nanovoids and SFT [40]. The size of the defects was set to 2.0 nm in diameter. Regarding nanovoids, we analyzed the impact of their shape and performed tests on different geometries, encompassing spherical, octahedral, tetrahedral, and cubic shapes.

MD simulations for collision cascades start with randomly selecting a Fe, Ni, or Cr atom at the center of the numerical sample. The Primary Knock-On Atom (PKA) is imparted with kinetic energy ranging from 1 to 10 keV. We conduct 50 simulations, each considering a different velocity direction denoted as $v_i = (v \sin\theta \cos\phi, v \sin\theta \sin\phi, v \cos\theta)$, where $v$ represents the PKA value. To ensure an isotropic distribution of PKA directions, the angles $\theta$ and $\phi$ were sampled appropriately over the unit sphere. The polar angle $\theta$ was generated using the

<figure><img src="image_6.png" /><figcaption>**Figure 2.** Schematics of the numerical cell utilized for conducting collision cascades, encompassing various defects such as cubic, spherical, octahedron, tetrahedron, truncated octahedron, and stacking fault tetrahedra. Initiated by assigning kinetic energy to a randomly selected atom-projectile, defects are characterized by identifying their atomic structure for enhanced visualization.</figcaption></figure>

transformation $\theta = \arccos(1 - 2\xi_1)$, where $\xi_1$ is a uniform random variable in [0,1], ensuring a probability density proportional to $\sin(\theta)$. The azimuthal angle $\phi$ was sampled uniformly in the range [0, $2\pi$]. This approach guarantees a uniform distribution over the solid angle. The Velocity Verlet integration algorithm models the cascade for 6 ps, followed by 4 ps relaxation; during the ballistic phase and heat spike, to improve computational efficiency, a variable time step (0.0001–1.0 fs) was used, limiting atom displacements to a maximum of 0.02 Å per step. The choice of a 10 ps simulation time for collision cascades is based on the timescale limitations inherent to classical interatomic potentials, which are parameterized primarily for short-term atomic interactions. In our simulations, most of the primary damage, including defect formation and initial recombination events, occurs within the first few picoseconds. Beyond this timeframe, the ability of empirical interatomic potentials to accurately describe defect diffusion and long-term recovery processes likely diminishes, as they are not designed to capture slow diffusion mechanisms or thermally activated events occurring over extended timescales. Additionally, at longer timescales, temperature control and artificial recombination effects due to periodic boundaries may
introduce artifacts that are not representative of the true evolution of radiation-induced defects. To mitigate cascade-induced shock waves, border cooling, and damping were implemented by applying a Nosé-Hoover thermostat on a 5 Å thick shell around the periodic borders of the cell. Pressure control is not applied during cascade simulations [26, 47, 49]. An electronic stopping correction is applied to address high PKA energies ranging from 8 to 10 keV, preventing the overestimation of ion range [50]. To achieve this, we use SRIM with default values for the alloy [51], which provides tabulated data for the energy loss of ions due to electron interactions in the material. This data is crucial for accurately modeling energy dissipation during high-energy cascade events. The electronic stopping power in SRIM is calculated based on the ion's velocity and the material's electronic properties, and it becomes particularly important at high PKA energies, where it surpasses nuclear stopping. This implementation ensures a more realistic representation of energy transfer to the electron system, influencing cascade dynamics and the overall material behavior under irradiation. The tabulated data file is then read by the LAMMPS script during the collision cascade simulations. electronic stopping powers encompass inelastic energy loss resulting from electronic collisions, effectively applying a friction force to decelerate fast-moving atoms. This methodology allows us to perform a full calculations for each collision cascade.

## 2.3. Analysis of collision cascades

The analysis of defects and dislocations formation during collision cascades is performed by computing the dislocation length as a function of simulation time for all the samples using OVITO software [52]. We utilized the polyhedral template matching (PTM) to identify different atomic structures and the Dislocation Extraction Algorithm (DXA) [53] which extracts dislocation structure and content from atomistic microstructures. Thus, we categorized the atomic structures as: BCC, FCC, hexagonal close-packed (HCP), icosahedral, simple cubic, and cubic diamond; the dislocations into several dislocation types according to their Burgers vectors as: $1/2\langle 110 \rangle$ (Perfect), $1/6\langle 112 \rangle$ (Shockley), $1/6\langle 110 \rangle$ (Stair-rod), $1/3\langle 100 \rangle$ (Hirth), $1/3\langle 111 \rangle$ (Frank). Then the dislocation density is obtained as

$$ \rho(t) = L(t)/V_c \quad (5) $$

where $L(t)$ is the dislocation length of different types and $V_c$ is the cell volume. In addition, the vacancies and voids are identified by the Delaunay tessellation to partition space into tetrahedral elements that are categorized as either solid or empty through the computation of an alpha parameter implemented into the surface mesh tool in OVITO [54].

# 3. Results

## 3.1. Surface and generalized stacking fault energy (GSF)

We computed the surface energy of the pristine CSAs using equation (1) which is displayed in table 1. Although the

<table><caption>**Table 1.** Surface energy $\Gamma$ of the Ni, NiFe and NiFeCr systems computed using equation (2) in (meV Å$^{-2}$).</caption><thead><tr><th></th><th>[001]</th><th>[011]</th><th>[111]</th></tr></thead><tbody><tr><td>Ni</td><td>48.4</td><td>64.5</td><td>43.0</td></tr><tr><td>NiFe<sub>20</sub></td><td>70 ± 2</td><td>85 ± 2</td><td>65 ± 2</td></tr><tr><td>NiFe</td><td>93 ± 2</td><td>106 ± 2</td><td>89 ± 2</td></tr><tr><td>NiCr<sub>20</sub></td><td>58 ± 2</td><td>74 ± 2</td><td>53 ± 2</td></tr><tr><td>NiFeCr<sub>20</sub></td><td>95 ± 2</td><td>74 ± 2</td><td>53 ± 2</td></tr></tbody></table>

<table><caption>**Table 2.** Monovacancy formation energy comparison between values obtained in this work (T.W.) and DFT or other empirical potentials (Emp. Pot.) results.</caption><thead><tr><th></th><th>T.W</th><th>DFT</th><th>Emp. Pot.</th></tr></thead><tbody><tr><td>Ni</td><td>1.48</td><td>1.42 [55]<br>1.62 [57]<br>1.92 [59]</td><td>1.46 [56]<br>1.48 [58]</td></tr><tr><td>NiFe<sub>20</sub></td><td>1.50 ± 0.2</td><td></td><td></td></tr><tr><td>NiFe</td><td>1.70 ± 0.2</td><td></td><td></td></tr><tr><td>NiCr<sub>20</sub></td><td>0.97 ± 0.2</td><td></td><td></td></tr><tr><td>NiFeCr<sub>20</sub></td><td>1.89 ± 0.25</td><td></td><td></td></tr><tr><td>FeNi<sub>10</sub>Cr<sub>20</sub></td><td></td><td>1.95 [60]</td><td></td></tr></tbody></table>

obtained surface energies for pure Ni do not match the density functional theory (DFT) results ($\Gamma_{111} = 122$ meV Å$^{-2}$, $\Gamma_{100} = 142$ meV Å$^{-2}$, $\Gamma_{110} = 145$ meV Å$^{-2}$ [63]), they nevertheless maintain the order of the most stable surfaces ($\Gamma_{111} < \Gamma_{100} < \Gamma_{110}$) [63]. Fe significantly increases the surface energy across all three surfaces, whereas the impact of Cr is comparatively less noticeable. However, neither Fe nor Cr altered the stability hierarchy among the three primary planes. To provide information about the predefined defects in the alloys, monovacancy formation energies were computed, showing good agreement with the data reported in table 2.

Results for the GSF energies of the pristine CSAs are shown in figure 3 for all the systems a long the {112} plane. The intrinsic SFE represents the metastable point of the GSF energy (see magnification on figure 3). We show a consistent decrease in the intrinsic stacking energy (see magnification in figure 3) with increasing the concentration of Fe or Cr solutes. This reduction is particularly prominent with Cr solutes, showcasing a substantial drop from 116 mJ m$^{-2}$ for pure Ni to 38 mJ m$^{-2}$ when Ni incorporates only 20% Cr. Conversely, the impact of Fe solutes is more subdued at 104 mJ m$^{-2}$ and escalates with an increasing number of Fe solutes, reaching 60 mJ m$^{-2}$ for NiFe alloys.

In figure 4, the generalized stacking fault energy (GSFE) mapping for all alloys on the {111} $\gamma$-surface is presented, displaying the expected symmetry for the pure Ni case based on its crystallographic geometry. The minimum energy path for $\langle 112 \rangle$ Shockley partial dislocation nucleation is represented by the white dashed line. Notably, the addition of 20% Fe or Cr to the Ni matrix increases the GSFE relative to pure Ni, with Cr exhibiting a more pronounced effect compared to Fe. In the case of the NiFeCr$_{20}$ alloy, the GSFE reaches its highest value, highlighting the significant influence of Cr in altering the SFE landscape of the NiFe alloy.
<figure><img src="image_7.png" /><figcaption>**Figure 3.** Generalize the Stacking Fault energy across diverse alloy compositions compared to DFT [61, 62]. Uncertainties arise from the random distribution of solute atoms.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 4.** Generalize the Stacking Fault energy for γ-surface on the loosest packing {111} planes. Showing the effect of Cr on the Ni and NiFe alloy.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Mean formation energy per vacancy of defects according to their shape and their size (r) for the different alloys. The results are averages derived from 10 different atomic configurations of alloys, and the associated error bars represent the standard deviations.</figcaption></figure>

## 3.2. Formation energy of defects

The results depicted in figure 5 encompass all alloys and their respective defects. These findings represent averages derived from 10 distinct atomic configurations of the alloys, with associated error bars indicating standard deviations. Notably, the inclusion of Fe atoms correlates with heightened defect formation energy. Specifically, CSAs with higher Fe concentrations, such as NiFe alloys, exhibit the highest formation energies, whereas binary NiCr CSAs display the lowest. A lower formation energy indicates that the formation is more favorable. Consequently, our findings suggest that the presence of Fe may inhibit the formation of vacancy clusters compared to both pure Ni and NiCr₂₀, aligning with reported results in the literature [1, 64].

## 3.3. Collision cascade

Figure 6 presents the outcomes of collision cascades involving pristine Ni and its CSAs across different scenarios: a

defect-free sample in (a), a cubic defect in (b), and a spherical void in (c), focusing on atoms identified as non-FCC by the PTM method. The temporal evolution of total defects, defined here as non-FCC atoms exhibiting a structure different from FCC during collision cascades, progresses through phases of ballistic displacement, heat spike generation, and subsequent defect recombination, mirroring the behavior observed in pure Ni, consistent with previous findings [65]. Our results for pristine Ni also align well with existing literature [66], indicating that PKA events at 1 keV typically result in 4 ± 1 Frenkel Pairs (FPs), with 2 keV PKA events yielding 5 ± 1 FPs, and 5 keV PKA events averaging around 12 ± 2 FPs. Additionally, a notable effect observed in the CSAs, compared to Ni, is the duration of the ballistic and heat spike leading to the recovery phase. The influence of Fe and/or Cr atoms in the Ni sample is evident in the rapid production of defects by equiatomic NiFe and NiFeCr₂₀ alloys, where the inclusion of cubic and spherical defects modified the behavior of the material changing the recovery time after collision cascade. The introduction of Cr in the Ni sample accelerates defect production after 0.1 ps, attributed to Ni–Cr and Cr–Cr interactions and the significant decrease in the intrinsic SFE (figure 3), noticing a high defect production for the spherical void case. In the presence of Fe in NiFe₅₀.₂₀ and NiFeCr₂₀ CSAs, the material requires more time to recover. Conversely, the presence of Cr does not significantly impact these phases. These results are used as reference for our work.

To explore the impact of preexisting defects on point defect formation during collision cascades, MD simulations were conducted across various PKA values and materials. In figure 6, we present results at 8 keV PKA, showcasing defect count profiles over simulation time for the most unstable defect vacancy (cubic) in (b) and the most stable defect vacancy (spherical) in (c), which can be considered as a void in the CSAs [28]; however, these defects can respond variably to irradiation. A comparison with the pristine case reveals observable changes in the defect production mechanism across all MD simulations. Additional results are available in the supplementary material, including data on defect formation in Ni, NiFe₂₀, and NiFe₅₀ at PKA energies of 1, 2, 5, 8, and 10 keV. We also include DXA analysis tracking the formation of SFT in NiFeCr₂₀ at 10 keV PKA. Furthermore, we present observations of stair-rod dislocation formation following the heat spike during the defect recombination phase. This additional data further supports our findings on dislocation dynamics and defect evolution. Distinct effects of Cr on defect production in Ni and NiFe alloys were noted, leading to an increase in defect counts. Conversely, a small percentage of Fe in the Ni matrix resulted in reduced defect counts after the collision cascade in the pristine case. Intriguingly, when 50% of Fe is present in Ni, a drastic reduction in defects is observed for both cubic and SFT cases. This could potentially be an artifact of the Bonny potential [67].

In figure 7, the defects that are atoms with a different structure than FCC following collision cascades in all materials are illustrated in a heatmap graph within the PKA energy range of 1 to 10 keV. Panel (a) represents the pristine case, (b) displays a cubic vacancy, and (c) showcases a spherical void.
<figure><img src="image_10.png" /><figcaption>**Figure 6.** Typical time profiles depicting the number of defects during 8 keV cascades in Ni and its solid solution alloys for a defect-free sample in (a), a cubic defect in (b), and an spherical void in (c). The three cascade stages-supersonic, sonic, and thermally enhanced recovery are illustrated by the background colors in (a).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 7.** Heat mapping displays the total number of defects after collision cascades for the pristine material in (a), the cubic defect in (b), and spherical one in (c); noticing the effect of Cr in the Ni matrix and NiFe alloys.</figcaption></figure>

It is evident that the presence of Fe atoms in the Ni sample reduces defect production compared to pure Ni, attributed to the soft bonding between Fe-Fe throughout the PKA range in

the pristine and cubic vacancy scenarios. Of particular interest is the behavior of the NiFeCr₂₀ alloy after collision cascades, where the total defect production is comparable to pure Ni.
<figure><img src="image_12.png" /><figcaption>**Figure 8.** Total dislocation density evolution in pure Ni depicted over simulation time in (a), and counts of single and clustering of vacancies in (b); for all defected materials at a PKA energy of 8 keV.</figcaption></figure>

This can be attributed to a compensation arising from the increase attributed to Cr solutes and the decrease due to Fe solutes.

In figure 8(a), results depicting dislocation density as a function of simulation time for a pure Ni sample with various preexisting defects at 8 keV of PKA are presented. In figure 8(b), the quantification of single vacancies and the clustering of vacancies to form voids is displayed for different vacancy volumes. We observed that total dislocation nucleation occurs around the heat spike and diminishes during the defect recombination phase, regardless of the defects present in the samples before cascades. However, the shape of the preexisting defect is crucial for the formation of vacancies and voids. In a pristine case, only single vacancies with a volume of ~35 Å³ and di/tri vacancies are created. An unstable cubic defect can produce more single defects, and the most stable defect spherical void can only create single and di-vacancies. Finally, both a tetrahedron and its truncated form demonstrate the ability to generate substantial voids following collision cascades. The former yields residual defects characterized by a significant volume, akin to those produced by the octahedron defect.

### 3.4. The influence of Cr on dislocation nucleation

To analyze the effects of the presence of Fe and Cr in the Ni matrices, figure 9 illustrates the results for the average of dislocation density over time for all the MD simulations in NiFe₂₀ in a), NiFe in b), NiCr₂₀ in c), and NiFeCr₂₀ in d), considering a cubic defect for total, Shockley partial, and Stair-rod dislocations. When 20% of Fe is in the Ni matrix, Shockley-type dislocations are nucleated around the heat spike in a similar way to the observed mechanisms for pure Ni. When the Fe concentration is increased to 50%, the Fe-Ni interaction and lattice mismatch make the Shockley partial dislocations more stable during the material's recovery. During the heat spike of the collision cascade, the nucleation of partial Shockley dislocations is observed for all samples. Indeed, as the solutes induce a significant decrease in intrinsic stacking fault energy,

they promote the nucleation of Shockley partials and a high density of stacking fault.

The addition of 20% Cr to pure Ni and NiFe alloy results in an interesting effect where the interaction between Ni, Fe, and Cr stabilizes the nucleation of SFT with a stair-rod dislocation during material recovery, as depicted in figures 10(a)–(d). The presence of Fe and Cr atoms in the Ni matrix stabilizes the Shockley partials for certain defect vacancies, and the mechanism of the interaction where Shockley partials nucleate a stair-rod junction persists. It is noteworthy that the nucleation of Shockley partials occurs during the material’s recovery, where Ni and Fe exhibit similar lattice constants, while Cr has a smaller lattice constant. This lattice mismatch is further accentuated in the NiFeCr₂₀ alloy, where the conversion of one defect into another geometry is more prominently observed. For instance, in the case of spherical vacancy defects, they are transformed into three SFTs simultaneously.

In figure 11, we present the average of the dislocation density over simulation time for NiFe₂₀ in (a), NiFe in (b), NiCr₂₀ in (c), and NiFeCr₂₀ in (d), considering a spherical void with 8 keV PKA. It is notable that the formation of SFT is more pronounced in spherical defects within Cr-rich alloys. The chemical effects and the decrease of the intrinsic SFE on the NiFe CSAs are again observed through the nucleation of stair-rod and Shockley dislocations during the material recovery phase. Specifically, the NiCr₂₀ alloy, which has the lowest SFE, nucleates a couple of SFTs after collision cascades, while the NiFeCr₂₀ alloy forms three SFTs where the spherical defect was initially located.

In figure 12 shows the visualizations of spherical voids, vacancies, and SFTs identified by HCP atoms post-collision cascades. For NiFe CSAs, the formation of voids, vacancies, and some self-interstitial atoms is distributed throughout the sample deforming the defect without forming SFTs. In NiCr₂₀, the presence of Cr atoms has a significant impact on material recovery. Lattice mismatch and interactions between Cr-Cr and Cr-Ni promote the generation of stair-rod dislocations, ultimately leading to the creation of coupled SFTs defect post-collision cascades.
<figure><img src="image_13.png" /><figcaption>**Figure 9.** Dislocation density as a function of simulation time for (a) NiFe$_{20}$, (b) NiFe, (c) NiCr$_{20}$, and (d) NiFeCr$_{20}$ alloys, considering a cubic defect with an 8 keV PKA. The curves represent the evolution of total dislocations, Shockley partial, and Stair-Rod dislocations over time. The presence of Cr in both Ni and NiFe alloys promotes the nucleation of Stair-Rod dislocations, while Fe additions influence the stability of Shockley partial dislocations during material recovery [68].</figcaption></figure>

In figure 13 shows the single vacancy and voids volume at the end of the collision cascade for cube defect in (a) and a spherical void in (b) at 8 keV. It is observed that NiFe CSAs for cubic voids are able to create sets of 3 vacancies, while Cr presence in Ni and NiFe CSAs tends to deform the cube defect into several single vacancies. For the spherical void, the NiFe CSAs deforms the void into sets of single and di vacancies, while Cr generates sets of single vacancies and a large void (roughly 6 vacancies) located at the center of the initial void. This effect is due to the GSF energy associated to the Cr-rich alloys as presented in figure 1.

It is noteworthy that Cr and Fe have the capability to modify the SFE, a critical parameter in understanding dislocation behavior and the recovery process [62, 69–71]. Materials with low SFE promote the formation of deformation twins, partial dislocations with a wide stacking fault ribbon, and a high stacking fault density [72, 73]. This stacking fault acts as a barrier against cross-slip or climb mechanisms, leading to slower recovery and the development of materials with high strength

<figure><img src="image_14.png" /><figcaption>**Figure 10.** Visualizations depict vacancies, void formation, and SFTs identified by HCP atoms after collision cascades at 10 keV of PKA. In panels (a) and (b), concerning NiFe alloys, while the influence of Cr is shown in panel (c) for NiCr$_{20}$ and NiFeCr$_{20}$ (d) alloys by promoting stair-rod dislocation formation during material recovery. Additionally, a blue line dotted cube is included in the figures to illustrate the geometry of the cubic defect at the onset of the collision cascade, serving as a reference point.</figcaption></figure>

and good ductility. In contrast, materials with high SFE exhibit more rapid cross-slip and climb, resulting in a higher recovery rate [17, 74].

Our results indicate that both Fe and Cr solutes independently contribute to a decrease in intrinsic SFE. However, in the ternary alloy NiFeCr$_{20}$, the introduction of Fe solutes induces a notable increase in intrinsic SFE, up to 47 mJ m$^{-2}$, compared to NiCr$_{20}$ alloys. To assess validity of the utilized interatomic potential, our results were compared to those provided in reference [61, 62], for which more precise atomistic simulations were performed, employing DFT method (figure 3) reaching a good agreement. Thus, according to our molecular statics simulations spherical shape is the most stable, following by truncated octahedron and octahedron shape. The most unstable shape is tetrahedron and cubic. SFT exhibit the highest formation energy; however, a direct comparison with voids is challenging because SFTs are composed of dislocation stair rods, not solely vacancies.

From our collision cascades findings, we notice that defects formation energy increased with Fe concentration (figure 5). However, for spherical voids, defect production increases due to the higher surface energy associated with their curved geometry. The reduction in defect production for certain conditions has been a topic of ongoing research. Potential mechanisms include those discussed in [67]. Studies by Ullah *et al* further demonstrate that electron–phonon coupling can induce defect recovery and strain relaxation in NiFe alloys [71]. Conversely, the presence of Cr in the Ni sample increases defect production due to Ni–Cr interactions, especially at high
<figure><img src="image_15.png" /><figcaption>**Figure 11.** Dislocation density profiles over time for the spherical void case at 8 keV PKA, illustrating the influence of presence of Fe in Nickel matrices (a)–(b), and Cr in Ni and NiFe alloy (c)–(d). The curves represent the evolution of total dislocations, Shockley partial dislocations, and Stair-Rod dislocations during the simulation. The presence of Cr-rich alloys promotes the formation of SFT, particularly in NiCr₂₀ and NiFeCr₂₀, where the lowest stacking fault energy facilitates their nucleation.</figcaption></figure>

impact energies above 5 keV. This effect is independent of the pre-existing defect or void shape. In general, the presence of Cr in the Ni matrix is observed to impact the material’s recovery, this is primarily due to lattice mismatch and interactions between Ni–Cr, Cr–Fe, and Cr–Cr in the sample as reported during the analysis of dislocation nucleation. Furthermore, the interactions between different Shockley dislocations, as observed for the NiCr₂₀ and NiFeCr₂₀ alloys, can lead to the nucleation of Stair-rod dislocations in some cases, as outlined below:

$$ \frac{1}{6}[110] = \frac{1}{6}[\overline{121}] + \frac{1}{6}[\overline{211}] \quad \text{Stair-rod, (6)} $$

and other symmetrical cases. This was illustrated in figures 9(c) and (d), where the Stair-rod dislocation stabilizes, and Shockley partials are annihilated during the thermally enhanced recovery phase. Crucially, single collision cascades can transform unstable defects into stable ones, thereby significantly influencing plastic deformation mechanisms observed

<figure><img src="image_16.png" /><figcaption>**Figure 12.** The recovery phase reveals nucleation of stair-rod and Shockley dislocations, with NiCr₂₀ alloy forming a couple of stacking fault tetrahedra (SFTs), and NiFeCr₂₀ alloy forming three SFTs at the location of the initial spherical defect. Notably, SFT formation is exclusively observed for cubic and spherical defects among all considered defect types. A blue line dotted sphere is added into the figures to show the geometry of the void at the beginning of the collision cascade, serving as a reference point.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 13.** Variation in vacancy counts as a function of their volume for different materials. Panel (a) illustrates the scenario with a cube defect, while panel (b) represents a spherical void, both simulated at 8 keV. It is evident that the presence of Cr in the Ni and NiFe CSAs significantly influences both defects, leading to the production of a greater number of single vacancies.</figcaption></figure>

during irradiation experiments. Notably, the formation of SFTs is exclusively observed for these cubic and spherical defects compared to other defect types considered. The effect
of elemental composition on the elastic properties of NiFe, NiFe₂₀, NiCr₂₀, and NiFeCr₂₀ can help to understand their mechanical behavior and provide insights to our results [75]. The addition of Cr plays a significant role in modifying the elastic response, as it increases the shear modulus and elastic anisotropy due to stronger bonding and local lattice distortions in the FCC structure [76]. Comparing NiFe and NiCr₂₀, the latter exhibits higher stiffness, which reflects resistance to shear deformation. In NiFeCr₂₀, the presence of Cr enhances the bulk and shear moduli, contributing to increased strength, though it may also introduce local distortions that affect mechanical stability. Meanwhile, NiFe₂₀, with an increased Fe content, shows an enhancement in C₄₄, suggesting improved resistance to shear strains while maintaining relatively high ductility. To further contextualize these findings, the phase stability of these alloys was evaluated in relation to the Fe–Ni–Cr phase diagram [75]. NiFe and NiFe₂₀ exhibit stability within the FCC solid solution phase across a broad temperature range due to the strong mutual solubility of Ni and Fe. In contrast, NiCr₂₀ and NiFeCr₂₀ introduce Cr, which increases the likelihood of phase decomposition at lower temperatures, potentially leading to Cr–rich precipitation or spinodal decomposition. However, at high temperatures, the high configurational entropy in NiFeCr₂₀ stabilizes the single-phase FCC structure [76, 77]. These results provide insights into the mechanical properties and phase stability of these alloys, highlighting the role of Cr and Fe in influencing their structural integrity, as observed in our previous results.

Finally, recent advancements in atomistic modeling provide a valuable computational approach for exploring magnetic effects in defect formation [78, 79]. In our future studies, we will leverage this capability to assess how spin polarization influences defect stability, dislocation behavior, and phase transformations in complex alloys [75, 77, 80]. By incorporating spin-dependent interactions, we aim to deepen our understanding of the interplay between magnetism and defect evolution, which could lead to more accurate predictions of material behavior under extreme conditions.

## 4. Concluding remarks

Summarizing, our computational exploration of single–phase CSAs (SP-CSAs) has provided valuable insights into the complex dynamics of irradiation-induced defects. By conducting MD simulations across a spectrum of CSAs, ranging from pure Ni, passing through binary alloys NiFe₂₀,₅₀ and NiCr₂₀, to Ni₄₀Fe₄₀Cr₂₀, and varying PKA energies, we unraveled the interplay between alloying elements and defect evolution. The modeling framework, considering a diverse array of vacancy defects, illuminated a remarkable transition from spherical vacancy (voids) defects to SFTs. This transformation was intricately linked to the material’s chemical composition, on the SFE of the alloys, by an analysis of the dislocation nucleation and evolution as a function of the dynamics time, with only Cr-rich alloys exhibiting the nucleation of stair-rod dislocations, ultimately leading to the formation of stable SFTs. Notably, NiCr₂₀ and NiFeCr₂₀ emerged as

the exclusive materials capable of this intriguing mechanism. Indeed, both alloys exhibit very low SFEs, facilitating the nucleation of SFTs validated by molecular static simulations to compute the surface, vacancy, and defect formation energies. These findings underscore the pivotal role of alloy chemistry in dictating irradiation-induced defect dynamics. The observed mechanisms, including the conversion to SFTs, hold implications for the irradiation resistance of SP-CSAs.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgments

Research was funded through the European Union Horizon 2020 research and innovation program under Grant Agreement No. 857470 and from the European Regional Development Fund under the program of the Foundation for Polish Science International Research Agenda PLUS, Grant No. MAB PLUS/2018/8, and the initiative of the Ministry of Science and Higher Education ‘Support for the activities of Centers of Excellence established in Poland under the Horizon 2020 program’ under agreement No. MEiN/2023/DIR /3795. Computational resources were provided by the High Performance Cluster at the National Centre for Nuclear Research and Polish high-performance computing infrastructure PLGrid (HPC Center: ACK Cyfronet AGH) Grant No. PLG/2024/017084. We would like to express our gratitude to NOETHER computing facilities at La Rochelle University and MCIA (Mésocentre de Calcul Intensif Atlantique). Additionally, we extend our gratitude to GENCI - (CINES/CCRT), under Grant number A0110913037.

## ORCID iDs

A Aligayev https://orcid.org/0000-0002-6900-4419

A Chartier https://orcid.org/0000-0001-9760-8415

F J Domínguez-Gutiérrez https://orcid.org/0000-0002-1429-0083

Q Huang https://orcid.org/0000-0002-8884-2063

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