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Cr-induced fast vacancy cluster formation and high Ni diffusion in concentrated Ni-Fe-Cr alloys
Accepted Manuscript Cr-induced fast vacancy cluster formation and high Ni diffusion in concentrated Ni-FeCr alloys Debajit Chakraborty, Dilpuneet S. Aidhy PII:
S0925-8388(17)32505-7
DOI:
10.1016/j.jallcom.2017.07.140
Reference:
JALCOM 42556
To appear in:
Journal of Alloys and Compounds
Received Date: 9 May 2017 Revised Date:
1 July 2017
Accepted Date: 13 July 2017
Please cite this article as: D. Chakraborty, D.S. Aidhy, Cr-induced fast vacancy cluster formation and high Ni diffusion in concentrated Ni-Fe-Cr alloys, Journal of Alloys and Compounds (2017), doi: 10.1016/ j.jallcom.2017.07.140. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Cr-induced fast vacancy cluster formation and high Ni diffusion in concentrated Ni-Fe-Cr alloys Debajit Chakraborty and Dilpuneet S. Aidhy*
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Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071 USA *Corresponding author Email: [email protected]
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Abstract: We have performed molecular dynamics (MD) simulations on pure Ni, Ni-Cr, Ni-Fe and Ni-Fe-Cr single phase concentrated solid solution systems to elucidate the kinetics of point defects and vacancy cluster formation. We find that diffusion-based vacancy clustering leads to the formation of stacking fault tetrahedra (SFT) in all of these systems. The presence of Cr leads to faster SFT formation and high Ni diffusion in Ni-Cr binary systems than in pure Ni and Ni-Fe binary systems. The same trend is observed for the ternary systems that have high Cr composition. This Cr-induced effect is due to the low Cr migration barrier that first induces vacancy diffusion and later leads to faster clustering and SFT formation. We find that the fast Ni diffusion is also Cr-induced, where the binding of two vacancies provides much lower migration barrier pathways for Ni diffusion. Due to this ‘Cr-effect’, the vacancy-vacancy pair distribution shows higher peaks in Ni-Cr and Ni-Fe-Cr systems than in pure Ni and Ni-Fe binary systems. The low migration barriers of Cr compared to Ni and Fe are confirmed by density functional theory calculations.
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Highlights: 1. Molecular dynamics simulations show fast SFT formation in Ni-Cr compounds. 2. Lower migration barrier of Cr is the key factor for fast vacancy diffusion. 3. Ni-diffusion is also enhanced in the presence of Cr. 4. Ternary systems show higher vacancy and Ni-diffusion with higher Cr concentration.
Keywords: concentrated Ni alloys, molecular dynamics simulation, stacking fault tetrahedra, vacancy diffusion.
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1. Introduction
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Due to some of the much-improved properties over the conventional alloys, high entropy alloys (HEAs) have captured wide interest of the metal-alloys community in recent years. These alloys possess exceptional mechanical properties such as high strength-weight ratio, wear resistance, high-temperature strength and structural stability, fracture resistance, etc. [1-4] In the addition, these alloys have attracted interest in physical properties such as electrical, magnetic and thermal, thus opening a new research field in metallic materials [5]. HEAs are defined as the alloys containing multiple elements, usually five or more, present in equal or near-equal proportions where the elements are randomly present in HEAs without having any significant long-range order. In addition, these alloys form thermodynamically stable single–phase solid solutions. Recent work shows that some of these alloys show better performances even under extreme conditions [6-9].
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Due to the attractive mechanical properties, HEAs have also stimulated interest in the nuclear materials community, where the focus is on understanding the irradiation response of these materials. In nuclear reactors, cascade events create point defects that diffuse and form large defect clusters. The clusters cause changes in the dimensions of the fuel and cladding materials leading to degradation in mechanical and thermal properties [10, 11]. For example, defect clusters can lead to irradiation hardening, embrittlement, radiation-induced segregation, void swelling, and irradiation-induced creep [9,12-16] thereby causing microstructural degradation. Preventing such defect cluster formation is important in the design of materials for nextgeneration nuclear reactors that are required to withstand over 200 displacements per atom (dpa) and 800 °C [17]; these conditions are extremely severe compared to those in the current nuclear reactors. Early irradiation experiments strongly indicate that HEAs possess qualitatively superior irradiation response properties than the conventional alloys [18]. For damages up to 10 dpa, they display a high degree of phase stability, only minor radiation-induced segregation, and are essentially immune to void formation/swelling. For example, as reported by Kumar et al. [18], while at 10 dpa the average dislocation loop diameters grow to 110 nm at 650 °C in 316 stainless steel [19] they do not exceed 6 nm at 700 °C in Fe0.27-Ni0.28-Mn0.27-Cr0.18. Similarly, while a void density of 1022/m3 and swelling of 0.1-2.5% was reported in austenitic steels between 450–700 °C at 10 dpa, no void formation or swelling was observed in HEAs under the same conditions [18]. Furthermore, ~19% Ni enrichment and ~13% Cr depletion was reported in 304 SS at 500 °C under 10 dpa [20]. In contrast, only ~7% Ni enrichment and ~3% Cr depletion was observed in HEAs under 10 dpa at 600 °C [18]. In addition, Fe-Ni-Cr-Co HEAs exhibit complete phase stability after ion irradiation up to 10 dpa in the temperature range of 400–700 °C. Thus, these early observations, albeit limited to only 10 dpa, highlight a higher radiation tolerance and better performance of HEAs compared to the conventional steels. While still in the early stages of research, the superior radiation tolerance of HEAs possibly originates from the random distribution of elements, where due to the lack of atomic ordering, the irradiation-induced defects (e.g., an interstitial) can readily recombine with a vacancy at any site. Such recombination enhances defect annihilation probabilities, possibly leading to their higher radiation tolerance. The high radiation tolerance has been widely observed in various compositions from diverse sets of experiments in the past year [9, 21, 22]. It is interesting to note that not only single-phased multi-elemental HEAs but a subset of such
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alloys containing two or three elements present in random distribution also show enhanced irradiation properties. Such alloys, often referred to as single phase concentrated solid solution alloys (SP-CSAs), have been synthesized and are found to be thermodynamically stable [7, 9]. Face-centered cubic (fcc) equiatomic alloys such as NiCo, NiFe, NiCoCr, NiCoFe, and Ni0.4Fe0.4Cr0.2 are thermodynamically stable and have experimentally shown high swelling resistance and fewer void formation after irradiation compared to pure Ni [7-9]. Addition of Fe in Ni to form a NiFe SP-CSA decreases the swelling by ~20 times (0.45%) than that of pure Ni (9.4%) [7]. Transmission electron microscopy (TEM) images show formation of large voids of ~200 nm in pure Ni after irradiation, whereas SP-CSAs like NiFe, NiCo, NiCoFe [7, 9] show very few and smaller void formation than pure Ni system. These results show that significantly improved irradiation tolerance properties can be achieved even in binary or ternary SP-CSA systems. In addition, by tuning the chemical composition of a single-phase system, defect formation can be controlled at the early stages of irradiation-induced microstructural evolution.
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Further scrutiny of the chemical complexity reveals that each element has a significant effect on the irradiation tolerance and phase stability of these alloys. Recent findings [7-9,21,22] from Rutherford backscattering spectroscopy channeling (RBS-C) show that the irradiation resistance can significantly vary between similar alloys. For example, while working on NiCoFe and NiCoCr, Zhang et al. [22], showed that the irradiation-induced defect accumulation is reduced to half from NiCoFe to NiCoCr, by simply replacing Fe with Cr. Similarly, recent experimental and theoretical investigations reveal that addition of a new or a substituting element can significantly affect the phase stability of SP-CSAs. For example, Otto et al. [22] show that by replacing Ni with Cu in CoCrFeMnNi, the alloy is unstable and phase separates. This is rather interesting because both Cu and Ni have similar properties, i.e., both are fcc, have the similar ionic radius, and have similar electronegativities. Thus, keeping in view the strong influence of each constituting element on the properties of SP-CSAs, it is important to understand and disentangle the individual atomistic effects of the constituting elements.
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A recent work by Aidhy et al. [23, 26, 28], elucidated the formation mechanism of stacking fault tetrahedra (SFT), which is one type of a large defect cluster, in pure fcc Ni. Here, we extend the understanding beyond pure Ni [23], to NiCr and NiFe and NiFeCr alloys using classical molecular dynamics (MD) simulation, and elucidate how Cr and Fe affect the SFT formation kinetics in the binary and ternary alloys. We perform simulations on Ni0.8Cr0.2, Ni0.9Cr0.1, Ni0.8Fe0.2, and Ni0.9Fe0.1 binary systems. The choice of 80-20 and 90-10 compositions is driven by the experimental phase stability of Ni-Cr alloys, where it is found that the Ni-Cr SP-CSA is stable up to Ni0.8Cr0.2 composition [24]. To make direct comparisons between Ni-Cr and Ni-Fe systems, the same compositions are chosen for the Ni-Fe system. Among the Ni-Fe-Cr ternary systems, only Ni0.4Fe0.4Cr0.2 has been experimentally synthesized and ion irradiation experiments has been performed [8]. It has been observed from experiments [8] and cascade simulation results [25] that Ni0.4Cr0.4Fe0.2 shows much better irradiation resistance than pure Ni and Ni0.8Fe0.2. To elucidate broader qualitative trends in the Ni-Fe-Cr system, we perform simulations on four compositions, i.e., Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.1Cr0.5. In the ternary Ni-Cr-Fe alloys, we have fixed the Ni composition to 0.4 and systematically varied the Fe and Cr composition to elucidate the individual effects of Cr and Fe. Our simulations on both binary and ternary systems reveal that the Cr migration barriers are significantly lower than Ni and Fe. Such low barriers induce faster vacancy diffusion that eventually lead to faster SFT formation in Ni-Cr alloys than in pure Ni. In contrast, due to 3
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similar migration barriers of Fe and Ni, there is no apparent difference in the SFT formation kinetics between Ni-Fe alloy and pure Ni. In addition, we show that the Ni diffusion is higher in alloys containing Cr, which is also attributed to the low Cr migration barriers. 2. Calculation details
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In this work, we use MD simulations to capture defect diffusion and SFT formation. The study is performed on pure Ni system, four binary systems, i.e., Ni0.9Cr0.1, Ni0.8Cr0.2, Ni0.9Fe0.1, and Ni0.8Fe0.2 and four ternary systems, i.e., Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.1Cr0.5. We use a 20 x 20 x 20 fcc supercell consisting of 32,000 atoms. Since these are random alloys, the Cr and Fe atoms are randomly distributed in all of the alloys. The interatomic potential is taken from Bonny et al. [26] which has been successfully used previously in various MD studies on radiation response of Ni-based alloys [27-30]. The validity of the potential can be found elsewhere [26-30]. The simulations are performed using the LAMMPS code [31]. The radiation-induced point defects are introduced by randomly distributing 0.25% vacancies which corresponds to 80 vacancies. The diffusion of vacancies is followed over the MD simulation time to elucidate the clustering mechanisms [27, 28, 30]. The alloying atoms, i.e., Fe and Cr, as well as the vacancies are introduced in these systems using a simple random seed generator program.
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The simulations are performed at 1200 K, and we ensured that no new vacancies are self-created during the course of a simulation at 1200 K. The temperature is kept high in order to capture defect diffusion within the MD simulation time [30]. Each simulation is run for 40 ns. The mean square displacement (MSD) is calculated for Ni atoms to understand Ni diffusivity. MSD is related to diffusivity D via the relationship, MSD = 6Dt, where t is the time. In this work, defect evolution is captured by using our recently-illustrated method of introducing a random distribution of vacancies [27, 30]. This method of vacancy distribution replicates the radiationinduced defect distribution during the ‘kinetic phase’ when a cascade event has already taken place. The justification of using this methodology to introduce defects in this specific manner has been discussed in detail in the previous work and can be found here [27, 30]. The defect migration barrier is calculated using nudged elastic band (NEB) method [32] as implemented in the LAMMPS code as well as using density functional theory (DFT) implemented in the Vienna Ab-initio Software Package (VASP) [33]. The VASP calculations are performed using PAW [34] method for Ni (10 valence electrons), Cr (12 valence electrons) and Fe (14 valence electrons) as implemented in the VASP code. The standard GGA-PBE exchange-correlation [35] function has been used. The Brillouin zone is sampled with a 4x4x4 Gamma-centered mesh with the 500 eV energy cut-off for the plane-wave basis set. The smearing of the Fermi surface is controlled by Methfessel-Paxton [36] with a smearing width of 0.2 eV. The magnitude of the forces acting on atoms is kept less than 0.001eV Å-1 for relaxation, and the energy difference between two consecutive steps is kept less than 1x10-6 eV. We have used five NEB images in between the two fixed end points. 3. Results 3.1 Vacancy clustering Previous MD simulations on pure Ni by Aidhy et al. [27, 30] revealed that diffusion of randomly distributed vacancies can lead to formation of the vacancy clusters, in particular, SFT. Here, we 4
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replicate the same simulation on pure Ni, and also on the Ni-Cr, Ni-Fe, and, Ni-Fe-Cr alloys. Figure 1 shows the snapshots of the structures at t = 5 ns for (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1 and, (e) Ni0.8Cr0.2 systems.
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We find that while most of the vacancies are present as isolated, single vacancies in pure Ni, Ni0.9Fe0.1, and Ni0.8Fe0.2, there is evidence of higher vacancy clustering in Ni0.9Cr0.1 and Ni0.8Cr0.2. Particularly, there are two SFT that have already formed in Ni0.8Cr0.2 that contain 13 & 21 vacancies, respectively. Similarly, there is much higher vacancy clustering in Ni0.9Cr0.1 than Ni0.9Fe0.1 and pure Ni. These clusters are 9 & 13 vacancies in the two SFT in Ni0.9Cr0.1. In contrast, the largest clusters in Ni, Ni0.9Fe0.1 and Ni0.8Fe0.2 have only 6, 8, and 6 vacancies respectively, indicating slower and similar defect dynamics between pure Ni and Ni-Fe alloys. These simulations show that Ni-Cr alloys have relatively faster vacancy diffusion that leads to faster SFT formation compared to the pure Ni and Ni-Fe alloys. In contrast, due to similar cluster sizes in Ni-Fe and pure Ni, we find that Fe has no significant effect on the defect dynamics. Furthermore, Figures 2 (a) and (b) show the number of mono-vacancies, and di- and higher vacancy clusters, respectively in pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1, and Ni0.8Cr0.2 systems at 1200 K for 1ns, 3ns and 5ns. Evidently, the number of mono-vacancies decreases and the number of larger vacancy clusters increases with simulation time for all these systems. However, the effect is much more prominent in Cr-based alloys. For example, in the case of Ni0.9Cr0.1, at t = 5 ns, most of the mono-vacancies have clustered to form small vacancy clusters, the largest being a 13-vacancy cluster. Similarly, in the case of Ni0.8Cr0.2, the largest cluster has 21 vacancies, forming an SFT cluster.
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The Cr-effect is observed over the longer duration of simulations. Figure 3 shows the snapshots of systems at t = 40 ns for the five systems, i.e., pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1 and Ni0.8Cr0.2. It is observed that pure Ni and Ni-Fe alloys show almost the same level of clustering as evidenced by the same level of single vacancies; in contrast, higher clustering and lower single vacancies are found in the Ni-Cr alloys. Evidently, one large SFT is seen in pure Ni and few small vacancy clusters are seen in Ni-Fe alloys. In contrast, there are two large SFT in Ni-Cr alloys. The SFT in pure Ni has 40 vacancies, and the Ni0.9Fe0.1 and Ni0.8Fe0.2 alloys contain 12 vacancies in each cluster. In contrast, the two SFT in Ni0.9Cr0.1 contains 14 and 22 vacancies, respectively. Due to the higher concentration of Cr in Ni0.8Cr0.2 than Ni0.9Cr0.1, the SFT sizes are even bigger; these SFT contain 22 and 52 vacancies, respectively. The progressive snapshots of vacancy clustering as a function of MD time from t = 0 ns to t = 40 ns in pure Ni and the alloys are shown in the supplementary section S-A1 to S-A5. To nullify the statistical nature of that defect clustering that can be common in these processes, we have performed five different simulations for each system. In each simulation, the initial random vacancy distribution is different. The outcome of each of these simulations is shown in the supplementary section (see S-A6 to S-A10). We find that same results are observed in these simulations as well, i.e., faster and larger SFT in Ni-Cr is observed than in pure Ni and Ni-Fe. Now we show the defect evolution in ternary systems. Figure 4 shows the snapshots of defect clustering at t = 5ns in (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.4Cr0.2, (d) Ni0.4Fe0.3Cr0.3, and (e) Ni0.4Fe0.1Cr0.5. Here, we have fixed the Ni concentration to 0.4, and have systematically varied the Fe and Cr concentration. We find that as the Cr concentration increases from 0.1 to 0.5, the size of the vacancy clusters (or SFT) increases. The increase in cluster sizes is clearly
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evident among Figures 4(b-e) that differ in Cr concentration by 0.4. There are more large clusters in Ni0.4Fe0.1Cr0.5 (Figure 4(e)) than in Ni0.4Fe0.5Cr0.1 (Figure 4(b)). We have counted the number of vacancies in SFT in all four compositions, and the number increases with increase in the Cr concentration; in Ni0.4Fe0.5Cr0.1 and Ni0.4Fe0.4Cr0.2, few relatively smaller SFT with the largest 10 and 19 vacancy SFT are observed respectively, whereas in Ni0.4Fe0.3Cr0.3 two SFT with 20-35 vacancies are observed. Finally, in Ni0.4Fe0.1Cr0.5 two big SFT containing 21 and 38 vacancies are present. Figure 5 (a) shows the mono-vacancy count, and Figure 5 (b) shows the count of diand higher vacancy clusters in pure Ni, Ni0.4Fe0.1Cr0.5, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.5Cr0.1 systems at 1ns, 3ns and 5ns. In these systems, as the simulation proceeds, the number of mono-vacancies decreases and the number of larger vacancy clusters increases with the increase in Cr concentration. It is noteworthy to mention that in Ni0.4Fe0.1Cr0.51 only one single vacancy and few large SFT clusters, the largest being 38 vacancies, observed at t = 5ns.
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These clusters grow with longer simulation time and the Cr effect becomes distinct in the ternary alloys as well. Figure 6 compares the snapshots of vacancy clustering at t = 40 ns for (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.3Cr0.3, (d) Ni0.4Fe0.4Cr0.2, and (e) Ni0.4Fe0.1Cr0.5. Ni0.4Fe0.5Cr0.1 and Ni0.4Fe0.4Cr0.2 show few SFT with the largest cluster size of 15 and 30 vacancies respectively. Ni0.4Fe0.3Cr0.3 shows relatively larger and fewer clusters with the largest cluster size of 45 vacancies. Ni0.4Fe0.1Cr0.5 shows a very large, single SFT containing 60 vacancies. Therefore, it is evident from these snapshots that the vacancy clustering is indeed influenced by the amount of Cr present in respective ternary systems. The consequence of increasing cluster sizes with Cr concentrations is the decrease in the number of isolated vacancies. Figure 6(a) shows a large number of isolated vacancies, and the number gradually decreases as the Cr concentration increases. Clearly, the number of isolated vacancies in Ni0.4Fe0.1Cr0.5 in Figure 6(e) is only four. Thus, the effect of Cr on defect clustering is prevalent even in the ternary systems, in accord with the binary systems. The progressive snapshots of defect evolution in all four ternary systems from t = 0 ns to t = 40 ns in shown in Supplementary section B (see S-B1 to S-B4). Similar to our simulations in the binary systems, we have performed five different simulations for each composition. The snapshots of each simulation at t = 40 ns are shown in Supplementary section (S-B5 to S-B8).
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From these simulations on both binary and ternary systems, we conclude that Cr enhances vacancy clustering at very early stages of the defect evolution, which has a significant effect on the cluster size over the long time scale of MD simulations. While our MD simulations are limited to 40 ns, these results highlight stark differences at such early stages; such difference can significantly impact the overall microstructure evolution. 3.2 Ni diffusion
In addition to faster SFT formation in the Ni-Cr alloys, Cr also affects Ni diffusion. We find that the Ni diffusion in Ni0.9Cr0.1 and Ni0.8Cr0.2 is considerably higher compared to that in pure Ni, and the Ni-Fe systems. Figure 7(a) compares the MSD profile of Ni among pure Ni, binary Ni-Cr and Ni-Fe alloys. Evidently, Ni diffusion is much faster in the Ni-Cr alloys than the rest. In addition, the Ni diffusion increases by increasing the Cr concentration, as observed by comparing Ni0.9Cr0.1 and Ni0.8Cr0.2 curves. In contrast, the Ni diffusion is almost same between the two Ni-Fe alloys, and that in the pure Ni. In addition, increasing Fe concentration has no
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obvious effect on the Ni diffusion. Figure 7(b) shows the Ni diffusion in the ternary systems. Again, the Ni diffusion increases with increasing the Cr concentration. Interestingly, in ternary 0.1 and 0.2 Cr systems, the Ni diffusion is almost twice of the Ni diffusion in pure Ni. Similarly, it is almost three times in 0.3 and 0.5 Cr systems that in the pure Ni. From these results, we find that the addition of Cr to Ni accelerates Ni diffusion. In contrast, addition of Fe has essentially no effect on Ni diffusion. 3.3 Cr-effect
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Now we elucidate the underlying reason behind the Cr effect. The low Cr migration barrier is the key underlying reason that is common to both the faster SFT formation and higher Ni diffusion. We calculate the migration barriers of Ni, Fe, and Cr via vacancy diffusion mechanism as shown in Figure 8. The black filled circles show the fcc lattice, and the square represents the vacancy. The diffusing atom (Ni, Fe, or Cr) is shown in red filled circle. Figure 8(a) shows the vacancy jump when only one vacancy is present in the unit cell, whereas Figure 8(b) shows the vacancy jump in the presence of an additional vacancy as a common first nearest neighbor to both the original vacancy and the diffusing lattice atom. The migration barriers are calculated using the interatomic potential and DFT at 0K. Table 1 shows the migration barriers for Ni, Fe, and Cr for the vacancy diffusion mechanism as shown in Figure 8(a). The barriers for Ni, Fe, and Cr are 1.08, 1.02, and 0.71 eV respectively. We find that these barriers calculated from the interatomic potential [26] agree very well with the DFT results. In addition, the Ni barriers agree with the experimental findings [37] (1.04±0.04 eV) as well as with the DFT-based literature calculations [26, 38].
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We find that the Ni and Fe barriers are very similar, i.e., 1.08 and 1.02 eV respectively. This indicates that Ni and Fe would have very similar diffusivities. In contrast, the Cr migration barrier is significantly lower (i.e., 0.71 eV). We find that the lower Cr migration barrier is the reason behind the fast vacancy cluster formation and Ni diffusion. The process can be explained as follows: the lower Cr migration barrier leads to faster Cr diffusion in the Ni-Cr systems via the vacancy mechanism. Conversely, the faster Cr diffusion causes faster vacancy diffusion, which prompts faster accumulation of vacancies, eventually facilitating faster SFT formation. In contrast, Fe has almost similar diffusion barrier to that of Ni, which does not induce faster vacancy diffusion. Therefore, due to the sluggish diffusion of Fe, the vacancy diffusion is a fairly retarded process in the Ni-Fe system. As a result, slow SFT formation is observed in the Ni-Fe system. Once the vacancy diffusion is enhanced by Cr, there are more chances that a vacancy would come in close vicinity to another vacancy, i.e., lead to the formation of vacancy-vacancy first nearest neighbors, similar to Figure 8(b). This step is the early stage of vacancy cluster (or SFT) formation. We find that when such vacancy-vacancy configuration is formed, the migration barrier of Ni is significantly lowered. The Ni migration configuration when the nearest neighbor vacancy is present corresponds to Figure 8(b). The corresponding Ni migration barrier is given in Table 1, where it drops from 1.08 eV to 0.44 eV [39]. The DFT barrier of 0.61 eV is somewhat higher than from the interatomic potential, but the overall trend is the same, i.e., the migration barrier drops significantly in the presence of a nearest-neighbor vacancy. This difference could
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be possibly because such migration barrier in the presence of an additional vacancy has not been used to fit the classical potential. Such low Ni migration barrier leads to faster Ni diffusion that is revealed in the MSD plots in Figures 7(a) and (b). In comparison, the higher Fe migration barrier does not induce significant vacancy migration; consequently, the nearest neighbor vacancyvacancy configurations are less likely to form, thereby causing no change in Ni diffusion. Thus, the faster Cr diffusion is the common underlying reason that indirectly causes faster SFT formation and higher Ni diffusion.
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It is to be noted that, the temperature dependence on the migration barriers has not been mentioned throughout this analysis. Although the migration barriers are sensitive to temperature, we know that the observed trend of the vacancy migration barriers remain almost same within a wide range of temperature. A more detail discussion on the temperature dependence of the migration barrier has been discussed later in the ‘Discussion’ section. 3.4 Vacancy-vacancy pair distribution
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We now provide the evidence of vacancy-vacancy configurations from our MD simulations by plotting the pair distribution of vacancies. The vacancy distribution is calculated by averaging the number of nearest vacancies around every vacancy at different times, i.e., t = 0 ns and 5 ns. Figures 9(a)-(e) show the distribution for pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1, and Ni0.8Cr0.2 respectively, plotted in lattice parameter units.
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While focusing on the first nearest neighbor (1st NN), i.e., between 0.7-0.8 ao (where ao is the lattice parameter), we find that the peaks increase with time in all five cases. For example, in pure Ni system (Figure 9(a)), at t = 0 ns, there are essentially zero nearest neighbors, whereas at t = 5 ns there are at least 0.5 average nearest neighbors. At t = 0 ns, almost no vacancy-vacancy nearest neighbors are expected because, at the beginning of the simulation, all vacancies are randomly distributed. In addition, since the concentration of vacancies is very low, i.e., 80 in a system of 32,000 atoms, every vacancy is far away from its nearest neighbor. However, as the simulation proceeds, the vacancies start to diffuse and form small clusters. The appearance of the peak at t = 5 ns indicates the formation of small vacancy-vacancy clusters. We find that the peaks in Ni-Cr system in Figures 9(d) and 9(e) are significantly higher than Ni and Ni-Fe (Figures 9(a), (b) and (c)). In contrast, the peak size of Ni-Fe is almost same as that of the pure Ni. These results show that the vacancy-vacancy clustering is higher and much faster in Ni-Cr alloys than the Ni-Fe alloys and pure Ni. Moreover, the peak at 2nd and 3rd NN, i.e. between 0.9-1.1 ao and between 1.2-1.4 ao respectively indicate SFT formation. By comparing the 2nd NN and 3rd NN peaks between Ni-Cr and Ni-Fe, we find that the Ni-Cr peaks are much higher indicating that the SFT have much larger size than in pure Ni and Ni-Fe alloys. In contrast, the smaller peak sizes in Ni-Fe alloys indicate that most of the vacancies are still isolated. This vacancy-vacancy pair distribution is consistent with the observations in Figure 1. Now we compare the rate of vacancy-vacancy nearest neighbor formation. Figure 10(a) is the comparison of the average vacancy accumulation, i.e., the summation of the absolute numbers at each data-point under the average vacancy-vacancy distribution, at the first nearest neighbor from the same data in Figures 9 (a), (b) and (d), i.e. Ni, Ni0.9Fe0.1, and Ni0.9Cr0.1. It is evident that even at 0.2 ns the vacancy clustering in Ni-Cr is statistically much higher than the pure Ni and
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Ni-Fe system. At 5 ns, it is higher by a factor of four. Figure 10 (b) shows the similar comparison among Ni, Ni0.8Fe0.2, and Ni0.8Cr0.2. Here, by increasing the Cr concentration, the difference between Ni-Cr and pure Ni further increases, and by 5 ns the average vacancy accumulation is higher by a factor of five than the pure Ni. In addition, by comparing Ni0.9Cr0.1 and Ni0.8Cr0.2 in Figures 10(a) and 8(b), we find that the average accumulation in Ni0.8Cr0.2 is higher. This indicates that increasing the Cr concentration promotes vacancy-vacancy clustering. Such increase is anchored on the enhanced vacancy migration induced by Cr, as discussed above.
4. Discussion
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Figure 11 shows the average vacancy-vacancy distribution in ternary alloys. The vacancy pair distribution for the ternary systems is given in the Supplementary figure S-C1 under section C. Figures 11(a), (b) and (c) show the average number of vacancy neighbors individually at 1st NN, 2nd NN and 3rd NN with time, respectively. In Figure 11(a), we find that the number of vacancies at 1st NN increases as the Cr concentration increases. The neighbors are highest for Ni0.4Fe0.1Cr0.5 system (shown in red color), and lowest for Ni0.4Fe0.5Cr0.1 (shown in green color) system among all of the four ternary alloys. The rate of increase in the average number of vacancy neighbors is the highest for the Ni0.4Fe0.1Cr0.5 system which contains the highest Cr concentration reinforcing the observation that the increase in the Cr concentration increases the vacancy-vacancy clustering, and promotes faster SFT formation. Similar observations are made for the 2nd NN and 3rd NN peaks in Figures 11(b) and (c). With longer time, the SFT in these ternary systems grow in size depending on the amount of the Cr present in the respective systems. Figures 11(d) and (e) show average vacancy accumulation at the 1st, 2nd, and, 3rd NN at longer times, i.e., t = 20 ns and t = 40 ns, respectively. Even at these longer times, we find that the peak sizes are higher for alloys containing higher Cr concentrations. Thus, it is evident from the vacancy-vacancy distribution analysis for both binary and ternary systems that the amount of Cr influences the vacancy clustering in the respective systems, and larger SFT are observed in systems containing high Cr concentrations.
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The migration barrier is the key to estimate diffusion process in a system. Our observations on the ‘Cr-effect’ in these binary and ternary systems agree with previous experimental [40-43] and theoretical [22, 38, 44, 45] diffusion studies on HEAs and Ni-Fe-Cr austenitic steels. Experimental measurements [40 - 42] on diffusion coefficients of Fe, Cr, and Ni in fcc Ni-Fe, Ni-Cr systems, and austenitic Ni-Fe-Cr steels show that Cr diffuses faster than Fe and Ni. A recent comparative study on the diffusion kinetics of Fe, Cr and Ni in HEAs and austenitic steels by Tsai et al. [43] shows the similar diffusion trend in HEAs as well, although the diffusion of these elements in HEAs are much slower than austenitic steels. Similarly, the interatomic potential used in this work [26] that was originally developed for fcc Ni-Fe-Cr austenitic steels, predicts the diffusion coefficients of the constituent elements in Ni-Fe-Cr alloys in the same order, as found from experimental studies, i.e. DCr > DFe > DNi. From these studies, the diffusivity trend among Cr, Fe, and Ni in fcc Ni supports our observation on the effect of Cr addition on the defect dynamics, and the overall cluster formation in concentrated Ni-Fe-Cr alloys. The low migration barrier of Cr is the controlling factor for faster kinetics in these alloys. Due to the lower migration barrier, faster migration of Cr leads to faster vacancy diffusion which is leading to formation of large SFT in a very short time.
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Apart from the migration barrier, there are other factors, such as the concentration and the random distribution of alloying elements that can significantly influence atomic diffusivities. An ab-initio study by Tucker et al. [38] showed that the Ni diffusion significantly increases when the Cr concentration increases from 1% to 10% in fcc Ni matrix. However, the Ni diffusion is almost unaffected with similar Fe addition. This supports our observation that the presence of Cr enhances the diffusion of Ni. Previous experimental studies [40, 41] on the diffusion coefficients of Cr, Ni and Fe in fcc Ni-based compounds supported the same trend. Thus, there is a common consensus among various studies that Cr enhances Ni diffusivity. This is due to the formation of favorable vacancy configurations as shown above. In contrast, due to relatively higher Fe migration barrier, vacancy and Ni diffusion remain unaffected by Fe addition.
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Although the use of DFT and MD methods to calculate diffusivities of dilute solutes is a common practice, understanding similar mechanisms in concentrated materials such as random alloys and HEAs is complex due to varying local atomic environments. Recent attempts have been made via ab-initio MD modelling [38, 44, 45] and kinetic Monte Carlo (kMC) [46, 47] approaches to elucidate such complex effects. In particular, Tucker et. al [38] showed that the binding between a vacancy and the Cr atom becomes attractive with increasing Cr concentration. In contrast, the vacancy and Fe atom binding becomes repulsive with increasing Fe concentration. Such solute-defect interactions will have significant impacts on the overall defect evolution. Therefore, developing an in-depth understanding of the local effects of alloying elements on defect evolution in random concentrated alloys is needed in future. 5. Conclusions
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In conclusion, this work shows that in the development of radiation resistant concentrated solid solution alloys, each element could have a significant effect on the defect dynamics. Our work highlights that increasing the concentration of Cr in the Ni-Fe-Cr alloys promotes vacancy diffusion leading to faster and larger SFT formation. In addition, Cr also enhances the Ni diffusion. These effects are due to the low migration barrier of Cr, which is the controlling factor for vacancy accumulation and high Ni diffusion. On the contrary, due to similar migration barrier as Ni, the effect of Fe towards vacancy accumulation and Ni diffusion is minimal in Ni-Fe alloys and in Ni-Fe-Cr alloys. Therefore, we propose that Cr addition to concentrated alloys may not be helpful in preventing larger cluster formation, Finally, this work emphasizes that the irradiation resistance of the binary and ternary SP-CSAs is deeply enrooted in the complexity of the alloying elements, and understanding the contribution of each element is critical towards the development of high radiation resistant alloys for nuclear applications. Acknowledgement: This work was supported as part of the Energy Dissipation to Defect Evolution (EDDE), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences. We acknowledge the support of computational resources from Advanced Research Computing Center (ARCC) at University of Wyoming. References: [1] B. Cantor, I.T.H. Chang, P. Knight, A.J.B. Vincent, Microstructural development in equiatomic multicomponent
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alloys, Mater. Sci. Eng. A 375-377, (2004) 213-218. [2] J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes, Adv. Eng. Mater. 6 (2004) 299-303. [3] Y.F. Ye, Q. Wang, J. Lu, C.T. Liu, Y. Yang, High entropy alloy: Challenges and prospects, Mater. Today, 19(6) (2016) 349-362. [4] D.B. Miracle, O.N. Senkov, A critical review of high entropy alloys and related concepts, Acta Mater. 122 (2017) 448-511. [5] K. M. Youssef, A. J. Zaddach, C. Niu, D. L. Irving, C. C. Koch, A Novel Low-Density, High-Hardness, Highentropy Alloy with Close-packed Single-phase Nanocrystalline Structures, Mater. Res. Lett., 3(2) (2015) 95-99. [6] B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P. George, R. O. Ritchie, A fracture-resistant highentropy alloy for cryogenic applications, Science, 345 (6201) (2014) 1153-1158. [7] K. Jin, C. Lu, L.M. Wang, J. Qu, W.J. Weber, Y. Zhang, H. Bei, Effects of compositional complexity on the ionirradiation induced swelling and hardening in Ni-containing equiatomic alloys, Scrip. Mater. 119 (2016) 65–70, 2016. [8] Y. Zhang, S. Zhao, W. J. Weber, K. Nordlund, F. Granberg, F. Djurabekova, Atomic-level heterogeneity and defect dynamics in concentrated solid-solution alloys, Curr. Opin. Solid State Mater. Sci., (2017) (in press). [9] C. Lu, K. Jin, L.K. Béland, F. Zhang, T. Yang, L. Qiao, Y. Zhang, H. Bei, H.M. Christen, R.E. Stoller, L. Wang: Direct observation of defect range and evolution in ion-irradiated single crystalline Ni and Ni binary alloys. Sci. Rep. 6 (2016) 19994. [10] R. A. Holt, "Mechanisms of Irradiation Growth of Alpha-Zirconium Alloys," J. Nucl. Mater., 159 (1988) 310337. [11] T. Allen, J. Busby, M. Meyer, D. Petti, Materials challenges for nuclear systems, Mat. Today, 13 (12) (2010) 14-23. [12] D. H. Gurinsky, G. J. Dienes (Eds), Nuclear Fuels, D. Van Nostrand Co., Princeton, New York, 1956. [13] T. H. Courtney, Mechanical Behavior of Materials, McGraw-Hill, New York, 1990. [14] A. Boltax, J. P. Fostar, R. A. Weiner, A. Biancheria, Void swelling and irradiation creep relationships, J. Nucl. Mater., 65 (1977) 174-183. [15] W. Han, E.G. Fu, M. J. Demkowicz, Y. Wang, A. Misra, Irradiation damage of single crystal, coarse-grained, and nanograined copper under helium bombardment at 450 °C, J. Mater. Res., 28 (20) (2013) 2763-2770. [16] J. E. Epperson, R. W. Hendricks, K. Farrell, Studies of voids in neutron-irradiated aluminium single crystals: I. Small-angle X-ray scattering and transmission electron microscopy, Phil. Mag., 30(4) (1974) 803-817. [17] S. J. Zinkle, L. L. Snead, Designing radiation resistance in materials for fusion energy, Annu. Rev. Mater. Res., 44 (2014) 241-267. [18] N.A.P. Kiran Kumar, C. Li, K.J. Leonard, H. Bei, S.J. Zinkle, Microstructural stability and mechanical behavior of FeNiMnCr high entropy alloy under ion irradiation, Acta Mater., 113 (2016) 230-244. [19] J.A. Hudson, Void formation in solution-treated AISI 316 and 321 stainless steels under 46.5 MeV Ni6+ irradiation, J. Nucl. Mater., 60 (1976) 89-106. [20] A. Etienne, B. Radiguet, N.J. Cunningham, G.R. Odette, R. Valiev, P. Pareige, Comparison of radiationinduced segregation in ultrafine-grained and conventional 316 austenitic stainless steels, Ultramicroscopy, 111 (2011) 659-663. [21] Y. Zhang, G.M. Stocks, K. Jin, C. Lu, H. Bei, B.C. Sales, L. Wang, L.K. Beland, R.E. Stoller, G.D. Samolyuk, M. Caro, A. Caro, W.J. Weber: Influence of chemical disorder on energy dissipation and defect evolution in nickel and Ni-based concentrated solid-solution alloys. Nat. Commun. 6 (2015) 8736 (pp 9). [22] Y Zhang, K. Jin, H. Xue, C. Lu, R. J. Olsen, L. K. Beland, M.W. Ullah, S. Zhao, H. Bei, D. S. Aidhy, G. D. Samolyuk, L. Wang, M. Caro, A. Caro, G.M. Stocks, B. C. Larson, I. Robertson, A. Correa, W. J. Weber, Influence of chemical disorder on energy dissipation and defect evolution in advanced alloys, J. Mater. Res. 31 (2016) 23632375. [23] F. Otto, Y. Yang, H. Bei, E.P. George, Relative effects of enthalpy and entropy on the phase stability of equiatomic high-entropy alloys, Acta Mater., 61(7) (2013) 2628–2638. [24] P. Nash, The Cr-Ni (Chromium-Nickel) System, Bull. Alloy Phase Diag., 7, 5, 466, 1986. [25] M. W. Ullah, H. Xue, G. Velisa, K. Jin, H. Bei, W. J. Weber, Y. Zhang, Effects of chemical alteration on damage accumulation in concentrated solid-solution alloys, Sci. Rep., 7(1) (2017) 4146. [26] G. Bonny, N. Castin, D. Terentyev, Interatomic potential for studying ageing under irradiation in stainless steels: the FeNiCr model alloy, Model. Simul. Mater. Sci. Eng. 21 (2013) 085004 (15pp). [27] D. S. Aidhy, C. Lu, K. Jin, H. Bei, Y. Zhang, L. Wang, W. J. Weber, Formation and growth of stacking fault tetrahedra in Ni via vacancy aggregation mechanism, Scr. Mater. 114 (2016) 137-141.
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[28] M. W. Ullah, D. S. Aidhy, Y. Zhang, W. J. Weber, Damage accumulation in ion-irradiated Ni-based concentrated solid- solution alloys, Acta. Mater. 109 (2016) 17-22. [29] L. K. Béland, G. D. Samolyuk, R. E. Stoller, Differences in the accumulation of ion-beam damage in Ni and NiFe explained by atomistic simulations, J. Alloy Compd. 622 (2016) 415-420. [30] D. S. Aidhy, C. Lu, K. Jin, H. Bei, Y. Zhang, L. Wang, W. J. Weber, Point defect evolution in Ni, NiFe and NiCr alloys from atomistic simulations and irradiation experiments, Acta. Mater. 99 (2015) 69-76. [31] S. Plimpton, Fast Parallel Algorithms for Short-Range Molecular Dynamics, J Comp Phys 117 (1995) 1-19. [32] G. Henkelman, G. Jónannesson, H. Jónsson, Methods for finding saddle points and minimum energy paths, in: S. D. Schwartz (Eds.), Theoretical Methods in Condensed Phase Chemistry, Kluwer Academic Publishers, Springer Netherlands, 2000, pp. 269-302. [33] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci., 6 (1996) 15-50. [34] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B, 50 (1994) 17953-17979. [35] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett., 77, (1996) 3865-3868. [36] M. Methfessel, A. Paxton, High-precision sampling for Brillouin-zone integration in metals, Phys. Rev. B., 40 (1989) 3616-3621. [37] P. Ehrhart, Defect dynamics, migration energies and jump frequencies: Datasheet from Landolt-Börnstein Group III Condensed Matter, in: Ullmaier, H. (Eds), Atomic Defects in Metals, Springer-Verlag Berlin Heidelberg, 1991, vol 25, p. 88. [38] J. D. Tucker, Ab-initio- based modeling of radiation effects in the Ni-Fe-Cr system, PhD Dissertation, Nuclear Engineering and Engineering Physics, University of Wisconsin-Madison, 2008. [39] Similar to Ni in Figure 8b, the Cr and Fe migration barriers are expected to decrease as well. However, due to higher concentration of Ni in these alloys, we have focused on Ni diffusivity only. [40] B. Million, J. Ruzickova, J. Velisek, J. Vrestal, Diffusion Processes in Fe-Ni, Mater. Sci. Eng., 50 (1981) 4352. [41] J. Ruzickova, B. Million, Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system, Mater. Sci. Eng., 50 (1981) 59-64. [42] S. J. Rothman, L. J. Nowicki, G.E. Murch, Self-diffusion in austenitic Fe-Cr-Ni alloys, J. Phys. F: Metal Phys.,10 (1980) 383 -398. [43] K. Y. Tsai, M. H. Tsai, J. W. Yeh, Sluggish diffusion in Co–Cr–Fe–Mn–Ni high-entropy alloys, Acta Mater., 61 (2013) 4887- 4897. [44] J.D. Tucker, R. Najafabadi, T.R. Allen, D. Morgan, Ab initio-based diffusion theory and tracer diffusion in Ni– Cr and Ni–Fe alloys, J. Nucl. Mater., 411 (1-3) (2010) 1-14. [45] M. Mantina, A first-principles methodology for diffusion coefficients in metals and dilute alloys, Dissertation, College of Earth and Mineral Sciences, Penn. State Univ., 2008. [46] A. V. Barashev, A. C. Arokiam, Monte Carlo Modelling of Cu atom diffusion in ߙ-Fe via the vacancy mechanism, Phil. Mag. Lett, 86 (2006), 321-332. [47] Y. N. Osetsky, L. K. Béland, R. E. Stoller, Specific features of defect and mass transport in concentrated fcc alloys, Acta. Mater, 115 (2016) 364-371.
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Tables:
Ni Fe
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Cr Ni in presence of vacancy at 1NN
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1.08 0.95
0.75 0.61
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Table 1: Migration energy (in eV) of different atoms in Ni fcc lattice in the vacancy configuration as shown in Figures 6(a) and (b).
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Figures:
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Figure 1: Snapshots of (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1 and (e) Ni0.8Cr0.2 with 0.25% vacancies at t=5ns. Ni-Cr compositions show large vacancy clustering than Ni-Fe and pure Ni.
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Figure 2: The vacancy count in pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1, Ni0.8Cr0.2 has been shown for (a) mono-vacancies and (b) di- and larger vacancy clusters. With simulation time the mono vacancy count decreases and di- and larger vacancy cluster count increases. In the case of pure Ni and Ni-Fe systems the large vacancy cluster count is close at t = 5ns. However, In the case of Ni-Cr systems, with increasing Cr concentration large vacancy clusters have been formed at t = 5ns and less number of single vacancies are observed. This is reflected in mono-vacancy count as well as in the count of di- and larger vacancy clusters for Ni-Cr systems. The Ni0.8Cr0.2 shows few large clusters (13 and 21-vacancy SFT) and Ni0.9Cr0.1 shows few mono-vacancy and many small vacancy cluster, largest being 13-vacancy SFT at t = 5ns.
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Figure 3: Snapshots of (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1 and (e) Ni0.8Cr0.2 with 0.25% vacancies at 40ns time-step. The vacancy clustering is prominently large in Ni0.8Cr0.2 compared to other four systems.
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Figure 4: Snapshots of (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.4Cr0.2, (d) Ni0.4Fe0.3Cr0.3, and (e) Ni0.4Fe0.1Cr0.5 with 0.25% vacancies at t=5ns. Increasing Cr compositions show larger vacancy clustering.
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Figure 5: The vacancy count in pure Ni, Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, Ni0.4Fe0.1Cr0.5 has been shown for (a) mono-vacancies and (b) di- and larger vacancy clusters. With simulation time the mono vacancy count decreases and di- and larger vacancy cluster count increases with increase of Cr content. At t = 5ns, the count of mono-vacancy significantly decreases and di- and larger vacancy count increases. In the case of Ni0.4Fe0.1Cr0.5, there exist only one single vacancy and few large SFT, 38-vacancy SFT is being the largest.
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Figure 6: Snapshots of (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.4Cr0.2, (d) Ni0.4Fe0.3Cr0.3, and (e) Ni0.4Fe0.1Cr0.5 with 0.25% vacancies at t=40ns. SFT size increases with increasing Cr concentration.
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Figure 7: Mean square displacement (MSD) plots of Ni in, (a) binary systems, and (b) ternary systems up to t =5ns. Addition of Cr increases the Ni diffusion significantly in binary systems (almost two-fold to the pure Ni case), however addition of Fe does not influence the diffusion of Ni. Similarly, in ternary systems, the Ni diffusion increases with increase in Cr composition.
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Figure 8: (a) Vacancy diffusion in fcc Ni matrix in presence of Ni, Fe or Cr at the 1st NN. (b) Similar diffusion in the presence of another nearest neighbor vacancy. Square represents vacancy and red dot represents the atom, i.e. Ni, Cr or Fe. The arrows in both figures show that the 1st NN atom move to the vacancy and the corresponding atom lattice site become vacant.
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Figure 9: Pair distribution function (PDF) of vacancies around a vacancy is shown for (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1, and (e) Ni0.8Cr0.2 at t=0ns and t=5ns. The PDFs show the vacancy accumulation significantly increases with increase in Cr composition.
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Figure 10: Comparison of vacancy accumulation at the 1st NN for pure Ni, Ni0.9Fe0.1 and Ni0.9Cr0.1 in (a), and the vacancy accumulation at the 1st NN for pure Ni, Ni0.8Fe0.2 and Ni0.8Cr0.2 in (b). For Ni-Cr binary system the average vacancy accumulation at 1st NN is significantly high.
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Figure 11: Vacancy-vacancy distribution as a function of MD time for, (a) 1st NN, (b) 2nd NN, and (c) 3rd NN in pure Ni, Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.1Cr0.5. The average vacancy accumulation increases with the increase in Cr composition. The comparison in PDF at (d) t=20ns and (e) t=40ns for the ternary systems has also been shown for respective ternary systems. At longer timescale, with the increase in Cr concentration, larger SFT are formed. At t=40ns, the 3rd NN shows the signature of larger SFT formation in ternary systems with increasing Cr concentration.
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S0925-8388(17)32505-7
DOI:
10.1016/j.jallcom.2017.07.140
Reference:
JALCOM 42556
To appear in:
Journal of Alloys and Compounds
Received Date: 9 May 2017 Revised Date:
1 July 2017
Accepted Date: 13 July 2017
Please cite this article as: D. Chakraborty, D.S. Aidhy, Cr-induced fast vacancy cluster formation and high Ni diffusion in concentrated Ni-Fe-Cr alloys, Journal of Alloys and Compounds (2017), doi: 10.1016/ j.jallcom.2017.07.140. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Cr-induced fast vacancy cluster formation and high Ni diffusion in concentrated Ni-Fe-Cr alloys Debajit Chakraborty and Dilpuneet S. Aidhy*
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Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071 USA *Corresponding author Email: [email protected]
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Abstract: We have performed molecular dynamics (MD) simulations on pure Ni, Ni-Cr, Ni-Fe and Ni-Fe-Cr single phase concentrated solid solution systems to elucidate the kinetics of point defects and vacancy cluster formation. We find that diffusion-based vacancy clustering leads to the formation of stacking fault tetrahedra (SFT) in all of these systems. The presence of Cr leads to faster SFT formation and high Ni diffusion in Ni-Cr binary systems than in pure Ni and Ni-Fe binary systems. The same trend is observed for the ternary systems that have high Cr composition. This Cr-induced effect is due to the low Cr migration barrier that first induces vacancy diffusion and later leads to faster clustering and SFT formation. We find that the fast Ni diffusion is also Cr-induced, where the binding of two vacancies provides much lower migration barrier pathways for Ni diffusion. Due to this ‘Cr-effect’, the vacancy-vacancy pair distribution shows higher peaks in Ni-Cr and Ni-Fe-Cr systems than in pure Ni and Ni-Fe binary systems. The low migration barriers of Cr compared to Ni and Fe are confirmed by density functional theory calculations.
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Highlights: 1. Molecular dynamics simulations show fast SFT formation in Ni-Cr compounds. 2. Lower migration barrier of Cr is the key factor for fast vacancy diffusion. 3. Ni-diffusion is also enhanced in the presence of Cr. 4. Ternary systems show higher vacancy and Ni-diffusion with higher Cr concentration.
Keywords: concentrated Ni alloys, molecular dynamics simulation, stacking fault tetrahedra, vacancy diffusion.
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1. Introduction
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Due to some of the much-improved properties over the conventional alloys, high entropy alloys (HEAs) have captured wide interest of the metal-alloys community in recent years. These alloys possess exceptional mechanical properties such as high strength-weight ratio, wear resistance, high-temperature strength and structural stability, fracture resistance, etc. [1-4] In the addition, these alloys have attracted interest in physical properties such as electrical, magnetic and thermal, thus opening a new research field in metallic materials [5]. HEAs are defined as the alloys containing multiple elements, usually five or more, present in equal or near-equal proportions where the elements are randomly present in HEAs without having any significant long-range order. In addition, these alloys form thermodynamically stable single–phase solid solutions. Recent work shows that some of these alloys show better performances even under extreme conditions [6-9].
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Due to the attractive mechanical properties, HEAs have also stimulated interest in the nuclear materials community, where the focus is on understanding the irradiation response of these materials. In nuclear reactors, cascade events create point defects that diffuse and form large defect clusters. The clusters cause changes in the dimensions of the fuel and cladding materials leading to degradation in mechanical and thermal properties [10, 11]. For example, defect clusters can lead to irradiation hardening, embrittlement, radiation-induced segregation, void swelling, and irradiation-induced creep [9,12-16] thereby causing microstructural degradation. Preventing such defect cluster formation is important in the design of materials for nextgeneration nuclear reactors that are required to withstand over 200 displacements per atom (dpa) and 800 °C [17]; these conditions are extremely severe compared to those in the current nuclear reactors. Early irradiation experiments strongly indicate that HEAs possess qualitatively superior irradiation response properties than the conventional alloys [18]. For damages up to 10 dpa, they display a high degree of phase stability, only minor radiation-induced segregation, and are essentially immune to void formation/swelling. For example, as reported by Kumar et al. [18], while at 10 dpa the average dislocation loop diameters grow to 110 nm at 650 °C in 316 stainless steel [19] they do not exceed 6 nm at 700 °C in Fe0.27-Ni0.28-Mn0.27-Cr0.18. Similarly, while a void density of 1022/m3 and swelling of 0.1-2.5% was reported in austenitic steels between 450–700 °C at 10 dpa, no void formation or swelling was observed in HEAs under the same conditions [18]. Furthermore, ~19% Ni enrichment and ~13% Cr depletion was reported in 304 SS at 500 °C under 10 dpa [20]. In contrast, only ~7% Ni enrichment and ~3% Cr depletion was observed in HEAs under 10 dpa at 600 °C [18]. In addition, Fe-Ni-Cr-Co HEAs exhibit complete phase stability after ion irradiation up to 10 dpa in the temperature range of 400–700 °C. Thus, these early observations, albeit limited to only 10 dpa, highlight a higher radiation tolerance and better performance of HEAs compared to the conventional steels. While still in the early stages of research, the superior radiation tolerance of HEAs possibly originates from the random distribution of elements, where due to the lack of atomic ordering, the irradiation-induced defects (e.g., an interstitial) can readily recombine with a vacancy at any site. Such recombination enhances defect annihilation probabilities, possibly leading to their higher radiation tolerance. The high radiation tolerance has been widely observed in various compositions from diverse sets of experiments in the past year [9, 21, 22]. It is interesting to note that not only single-phased multi-elemental HEAs but a subset of such
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alloys containing two or three elements present in random distribution also show enhanced irradiation properties. Such alloys, often referred to as single phase concentrated solid solution alloys (SP-CSAs), have been synthesized and are found to be thermodynamically stable [7, 9]. Face-centered cubic (fcc) equiatomic alloys such as NiCo, NiFe, NiCoCr, NiCoFe, and Ni0.4Fe0.4Cr0.2 are thermodynamically stable and have experimentally shown high swelling resistance and fewer void formation after irradiation compared to pure Ni [7-9]. Addition of Fe in Ni to form a NiFe SP-CSA decreases the swelling by ~20 times (0.45%) than that of pure Ni (9.4%) [7]. Transmission electron microscopy (TEM) images show formation of large voids of ~200 nm in pure Ni after irradiation, whereas SP-CSAs like NiFe, NiCo, NiCoFe [7, 9] show very few and smaller void formation than pure Ni system. These results show that significantly improved irradiation tolerance properties can be achieved even in binary or ternary SP-CSA systems. In addition, by tuning the chemical composition of a single-phase system, defect formation can be controlled at the early stages of irradiation-induced microstructural evolution.
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Further scrutiny of the chemical complexity reveals that each element has a significant effect on the irradiation tolerance and phase stability of these alloys. Recent findings [7-9,21,22] from Rutherford backscattering spectroscopy channeling (RBS-C) show that the irradiation resistance can significantly vary between similar alloys. For example, while working on NiCoFe and NiCoCr, Zhang et al. [22], showed that the irradiation-induced defect accumulation is reduced to half from NiCoFe to NiCoCr, by simply replacing Fe with Cr. Similarly, recent experimental and theoretical investigations reveal that addition of a new or a substituting element can significantly affect the phase stability of SP-CSAs. For example, Otto et al. [22] show that by replacing Ni with Cu in CoCrFeMnNi, the alloy is unstable and phase separates. This is rather interesting because both Cu and Ni have similar properties, i.e., both are fcc, have the similar ionic radius, and have similar electronegativities. Thus, keeping in view the strong influence of each constituting element on the properties of SP-CSAs, it is important to understand and disentangle the individual atomistic effects of the constituting elements.
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A recent work by Aidhy et al. [23, 26, 28], elucidated the formation mechanism of stacking fault tetrahedra (SFT), which is one type of a large defect cluster, in pure fcc Ni. Here, we extend the understanding beyond pure Ni [23], to NiCr and NiFe and NiFeCr alloys using classical molecular dynamics (MD) simulation, and elucidate how Cr and Fe affect the SFT formation kinetics in the binary and ternary alloys. We perform simulations on Ni0.8Cr0.2, Ni0.9Cr0.1, Ni0.8Fe0.2, and Ni0.9Fe0.1 binary systems. The choice of 80-20 and 90-10 compositions is driven by the experimental phase stability of Ni-Cr alloys, where it is found that the Ni-Cr SP-CSA is stable up to Ni0.8Cr0.2 composition [24]. To make direct comparisons between Ni-Cr and Ni-Fe systems, the same compositions are chosen for the Ni-Fe system. Among the Ni-Fe-Cr ternary systems, only Ni0.4Fe0.4Cr0.2 has been experimentally synthesized and ion irradiation experiments has been performed [8]. It has been observed from experiments [8] and cascade simulation results [25] that Ni0.4Cr0.4Fe0.2 shows much better irradiation resistance than pure Ni and Ni0.8Fe0.2. To elucidate broader qualitative trends in the Ni-Fe-Cr system, we perform simulations on four compositions, i.e., Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.1Cr0.5. In the ternary Ni-Cr-Fe alloys, we have fixed the Ni composition to 0.4 and systematically varied the Fe and Cr composition to elucidate the individual effects of Cr and Fe. Our simulations on both binary and ternary systems reveal that the Cr migration barriers are significantly lower than Ni and Fe. Such low barriers induce faster vacancy diffusion that eventually lead to faster SFT formation in Ni-Cr alloys than in pure Ni. In contrast, due to 3
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similar migration barriers of Fe and Ni, there is no apparent difference in the SFT formation kinetics between Ni-Fe alloy and pure Ni. In addition, we show that the Ni diffusion is higher in alloys containing Cr, which is also attributed to the low Cr migration barriers. 2. Calculation details
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In this work, we use MD simulations to capture defect diffusion and SFT formation. The study is performed on pure Ni system, four binary systems, i.e., Ni0.9Cr0.1, Ni0.8Cr0.2, Ni0.9Fe0.1, and Ni0.8Fe0.2 and four ternary systems, i.e., Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.1Cr0.5. We use a 20 x 20 x 20 fcc supercell consisting of 32,000 atoms. Since these are random alloys, the Cr and Fe atoms are randomly distributed in all of the alloys. The interatomic potential is taken from Bonny et al. [26] which has been successfully used previously in various MD studies on radiation response of Ni-based alloys [27-30]. The validity of the potential can be found elsewhere [26-30]. The simulations are performed using the LAMMPS code [31]. The radiation-induced point defects are introduced by randomly distributing 0.25% vacancies which corresponds to 80 vacancies. The diffusion of vacancies is followed over the MD simulation time to elucidate the clustering mechanisms [27, 28, 30]. The alloying atoms, i.e., Fe and Cr, as well as the vacancies are introduced in these systems using a simple random seed generator program.
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The simulations are performed at 1200 K, and we ensured that no new vacancies are self-created during the course of a simulation at 1200 K. The temperature is kept high in order to capture defect diffusion within the MD simulation time [30]. Each simulation is run for 40 ns. The mean square displacement (MSD) is calculated for Ni atoms to understand Ni diffusivity. MSD is related to diffusivity D via the relationship, MSD = 6Dt, where t is the time. In this work, defect evolution is captured by using our recently-illustrated method of introducing a random distribution of vacancies [27, 30]. This method of vacancy distribution replicates the radiationinduced defect distribution during the ‘kinetic phase’ when a cascade event has already taken place. The justification of using this methodology to introduce defects in this specific manner has been discussed in detail in the previous work and can be found here [27, 30]. The defect migration barrier is calculated using nudged elastic band (NEB) method [32] as implemented in the LAMMPS code as well as using density functional theory (DFT) implemented in the Vienna Ab-initio Software Package (VASP) [33]. The VASP calculations are performed using PAW [34] method for Ni (10 valence electrons), Cr (12 valence electrons) and Fe (14 valence electrons) as implemented in the VASP code. The standard GGA-PBE exchange-correlation [35] function has been used. The Brillouin zone is sampled with a 4x4x4 Gamma-centered mesh with the 500 eV energy cut-off for the plane-wave basis set. The smearing of the Fermi surface is controlled by Methfessel-Paxton [36] with a smearing width of 0.2 eV. The magnitude of the forces acting on atoms is kept less than 0.001eV Å-1 for relaxation, and the energy difference between two consecutive steps is kept less than 1x10-6 eV. We have used five NEB images in between the two fixed end points. 3. Results 3.1 Vacancy clustering Previous MD simulations on pure Ni by Aidhy et al. [27, 30] revealed that diffusion of randomly distributed vacancies can lead to formation of the vacancy clusters, in particular, SFT. Here, we 4
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replicate the same simulation on pure Ni, and also on the Ni-Cr, Ni-Fe, and, Ni-Fe-Cr alloys. Figure 1 shows the snapshots of the structures at t = 5 ns for (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1 and, (e) Ni0.8Cr0.2 systems.
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We find that while most of the vacancies are present as isolated, single vacancies in pure Ni, Ni0.9Fe0.1, and Ni0.8Fe0.2, there is evidence of higher vacancy clustering in Ni0.9Cr0.1 and Ni0.8Cr0.2. Particularly, there are two SFT that have already formed in Ni0.8Cr0.2 that contain 13 & 21 vacancies, respectively. Similarly, there is much higher vacancy clustering in Ni0.9Cr0.1 than Ni0.9Fe0.1 and pure Ni. These clusters are 9 & 13 vacancies in the two SFT in Ni0.9Cr0.1. In contrast, the largest clusters in Ni, Ni0.9Fe0.1 and Ni0.8Fe0.2 have only 6, 8, and 6 vacancies respectively, indicating slower and similar defect dynamics between pure Ni and Ni-Fe alloys. These simulations show that Ni-Cr alloys have relatively faster vacancy diffusion that leads to faster SFT formation compared to the pure Ni and Ni-Fe alloys. In contrast, due to similar cluster sizes in Ni-Fe and pure Ni, we find that Fe has no significant effect on the defect dynamics. Furthermore, Figures 2 (a) and (b) show the number of mono-vacancies, and di- and higher vacancy clusters, respectively in pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1, and Ni0.8Cr0.2 systems at 1200 K for 1ns, 3ns and 5ns. Evidently, the number of mono-vacancies decreases and the number of larger vacancy clusters increases with simulation time for all these systems. However, the effect is much more prominent in Cr-based alloys. For example, in the case of Ni0.9Cr0.1, at t = 5 ns, most of the mono-vacancies have clustered to form small vacancy clusters, the largest being a 13-vacancy cluster. Similarly, in the case of Ni0.8Cr0.2, the largest cluster has 21 vacancies, forming an SFT cluster.
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The Cr-effect is observed over the longer duration of simulations. Figure 3 shows the snapshots of systems at t = 40 ns for the five systems, i.e., pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1 and Ni0.8Cr0.2. It is observed that pure Ni and Ni-Fe alloys show almost the same level of clustering as evidenced by the same level of single vacancies; in contrast, higher clustering and lower single vacancies are found in the Ni-Cr alloys. Evidently, one large SFT is seen in pure Ni and few small vacancy clusters are seen in Ni-Fe alloys. In contrast, there are two large SFT in Ni-Cr alloys. The SFT in pure Ni has 40 vacancies, and the Ni0.9Fe0.1 and Ni0.8Fe0.2 alloys contain 12 vacancies in each cluster. In contrast, the two SFT in Ni0.9Cr0.1 contains 14 and 22 vacancies, respectively. Due to the higher concentration of Cr in Ni0.8Cr0.2 than Ni0.9Cr0.1, the SFT sizes are even bigger; these SFT contain 22 and 52 vacancies, respectively. The progressive snapshots of vacancy clustering as a function of MD time from t = 0 ns to t = 40 ns in pure Ni and the alloys are shown in the supplementary section S-A1 to S-A5. To nullify the statistical nature of that defect clustering that can be common in these processes, we have performed five different simulations for each system. In each simulation, the initial random vacancy distribution is different. The outcome of each of these simulations is shown in the supplementary section (see S-A6 to S-A10). We find that same results are observed in these simulations as well, i.e., faster and larger SFT in Ni-Cr is observed than in pure Ni and Ni-Fe. Now we show the defect evolution in ternary systems. Figure 4 shows the snapshots of defect clustering at t = 5ns in (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.4Cr0.2, (d) Ni0.4Fe0.3Cr0.3, and (e) Ni0.4Fe0.1Cr0.5. Here, we have fixed the Ni concentration to 0.4, and have systematically varied the Fe and Cr concentration. We find that as the Cr concentration increases from 0.1 to 0.5, the size of the vacancy clusters (or SFT) increases. The increase in cluster sizes is clearly
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evident among Figures 4(b-e) that differ in Cr concentration by 0.4. There are more large clusters in Ni0.4Fe0.1Cr0.5 (Figure 4(e)) than in Ni0.4Fe0.5Cr0.1 (Figure 4(b)). We have counted the number of vacancies in SFT in all four compositions, and the number increases with increase in the Cr concentration; in Ni0.4Fe0.5Cr0.1 and Ni0.4Fe0.4Cr0.2, few relatively smaller SFT with the largest 10 and 19 vacancy SFT are observed respectively, whereas in Ni0.4Fe0.3Cr0.3 two SFT with 20-35 vacancies are observed. Finally, in Ni0.4Fe0.1Cr0.5 two big SFT containing 21 and 38 vacancies are present. Figure 5 (a) shows the mono-vacancy count, and Figure 5 (b) shows the count of diand higher vacancy clusters in pure Ni, Ni0.4Fe0.1Cr0.5, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.5Cr0.1 systems at 1ns, 3ns and 5ns. In these systems, as the simulation proceeds, the number of mono-vacancies decreases and the number of larger vacancy clusters increases with the increase in Cr concentration. It is noteworthy to mention that in Ni0.4Fe0.1Cr0.51 only one single vacancy and few large SFT clusters, the largest being 38 vacancies, observed at t = 5ns.
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These clusters grow with longer simulation time and the Cr effect becomes distinct in the ternary alloys as well. Figure 6 compares the snapshots of vacancy clustering at t = 40 ns for (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.3Cr0.3, (d) Ni0.4Fe0.4Cr0.2, and (e) Ni0.4Fe0.1Cr0.5. Ni0.4Fe0.5Cr0.1 and Ni0.4Fe0.4Cr0.2 show few SFT with the largest cluster size of 15 and 30 vacancies respectively. Ni0.4Fe0.3Cr0.3 shows relatively larger and fewer clusters with the largest cluster size of 45 vacancies. Ni0.4Fe0.1Cr0.5 shows a very large, single SFT containing 60 vacancies. Therefore, it is evident from these snapshots that the vacancy clustering is indeed influenced by the amount of Cr present in respective ternary systems. The consequence of increasing cluster sizes with Cr concentrations is the decrease in the number of isolated vacancies. Figure 6(a) shows a large number of isolated vacancies, and the number gradually decreases as the Cr concentration increases. Clearly, the number of isolated vacancies in Ni0.4Fe0.1Cr0.5 in Figure 6(e) is only four. Thus, the effect of Cr on defect clustering is prevalent even in the ternary systems, in accord with the binary systems. The progressive snapshots of defect evolution in all four ternary systems from t = 0 ns to t = 40 ns in shown in Supplementary section B (see S-B1 to S-B4). Similar to our simulations in the binary systems, we have performed five different simulations for each composition. The snapshots of each simulation at t = 40 ns are shown in Supplementary section (S-B5 to S-B8).
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From these simulations on both binary and ternary systems, we conclude that Cr enhances vacancy clustering at very early stages of the defect evolution, which has a significant effect on the cluster size over the long time scale of MD simulations. While our MD simulations are limited to 40 ns, these results highlight stark differences at such early stages; such difference can significantly impact the overall microstructure evolution. 3.2 Ni diffusion
In addition to faster SFT formation in the Ni-Cr alloys, Cr also affects Ni diffusion. We find that the Ni diffusion in Ni0.9Cr0.1 and Ni0.8Cr0.2 is considerably higher compared to that in pure Ni, and the Ni-Fe systems. Figure 7(a) compares the MSD profile of Ni among pure Ni, binary Ni-Cr and Ni-Fe alloys. Evidently, Ni diffusion is much faster in the Ni-Cr alloys than the rest. In addition, the Ni diffusion increases by increasing the Cr concentration, as observed by comparing Ni0.9Cr0.1 and Ni0.8Cr0.2 curves. In contrast, the Ni diffusion is almost same between the two Ni-Fe alloys, and that in the pure Ni. In addition, increasing Fe concentration has no
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obvious effect on the Ni diffusion. Figure 7(b) shows the Ni diffusion in the ternary systems. Again, the Ni diffusion increases with increasing the Cr concentration. Interestingly, in ternary 0.1 and 0.2 Cr systems, the Ni diffusion is almost twice of the Ni diffusion in pure Ni. Similarly, it is almost three times in 0.3 and 0.5 Cr systems that in the pure Ni. From these results, we find that the addition of Cr to Ni accelerates Ni diffusion. In contrast, addition of Fe has essentially no effect on Ni diffusion. 3.3 Cr-effect
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Now we elucidate the underlying reason behind the Cr effect. The low Cr migration barrier is the key underlying reason that is common to both the faster SFT formation and higher Ni diffusion. We calculate the migration barriers of Ni, Fe, and Cr via vacancy diffusion mechanism as shown in Figure 8. The black filled circles show the fcc lattice, and the square represents the vacancy. The diffusing atom (Ni, Fe, or Cr) is shown in red filled circle. Figure 8(a) shows the vacancy jump when only one vacancy is present in the unit cell, whereas Figure 8(b) shows the vacancy jump in the presence of an additional vacancy as a common first nearest neighbor to both the original vacancy and the diffusing lattice atom. The migration barriers are calculated using the interatomic potential and DFT at 0K. Table 1 shows the migration barriers for Ni, Fe, and Cr for the vacancy diffusion mechanism as shown in Figure 8(a). The barriers for Ni, Fe, and Cr are 1.08, 1.02, and 0.71 eV respectively. We find that these barriers calculated from the interatomic potential [26] agree very well with the DFT results. In addition, the Ni barriers agree with the experimental findings [37] (1.04±0.04 eV) as well as with the DFT-based literature calculations [26, 38].
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We find that the Ni and Fe barriers are very similar, i.e., 1.08 and 1.02 eV respectively. This indicates that Ni and Fe would have very similar diffusivities. In contrast, the Cr migration barrier is significantly lower (i.e., 0.71 eV). We find that the lower Cr migration barrier is the reason behind the fast vacancy cluster formation and Ni diffusion. The process can be explained as follows: the lower Cr migration barrier leads to faster Cr diffusion in the Ni-Cr systems via the vacancy mechanism. Conversely, the faster Cr diffusion causes faster vacancy diffusion, which prompts faster accumulation of vacancies, eventually facilitating faster SFT formation. In contrast, Fe has almost similar diffusion barrier to that of Ni, which does not induce faster vacancy diffusion. Therefore, due to the sluggish diffusion of Fe, the vacancy diffusion is a fairly retarded process in the Ni-Fe system. As a result, slow SFT formation is observed in the Ni-Fe system. Once the vacancy diffusion is enhanced by Cr, there are more chances that a vacancy would come in close vicinity to another vacancy, i.e., lead to the formation of vacancy-vacancy first nearest neighbors, similar to Figure 8(b). This step is the early stage of vacancy cluster (or SFT) formation. We find that when such vacancy-vacancy configuration is formed, the migration barrier of Ni is significantly lowered. The Ni migration configuration when the nearest neighbor vacancy is present corresponds to Figure 8(b). The corresponding Ni migration barrier is given in Table 1, where it drops from 1.08 eV to 0.44 eV [39]. The DFT barrier of 0.61 eV is somewhat higher than from the interatomic potential, but the overall trend is the same, i.e., the migration barrier drops significantly in the presence of a nearest-neighbor vacancy. This difference could
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be possibly because such migration barrier in the presence of an additional vacancy has not been used to fit the classical potential. Such low Ni migration barrier leads to faster Ni diffusion that is revealed in the MSD plots in Figures 7(a) and (b). In comparison, the higher Fe migration barrier does not induce significant vacancy migration; consequently, the nearest neighbor vacancyvacancy configurations are less likely to form, thereby causing no change in Ni diffusion. Thus, the faster Cr diffusion is the common underlying reason that indirectly causes faster SFT formation and higher Ni diffusion.
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It is to be noted that, the temperature dependence on the migration barriers has not been mentioned throughout this analysis. Although the migration barriers are sensitive to temperature, we know that the observed trend of the vacancy migration barriers remain almost same within a wide range of temperature. A more detail discussion on the temperature dependence of the migration barrier has been discussed later in the ‘Discussion’ section. 3.4 Vacancy-vacancy pair distribution
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We now provide the evidence of vacancy-vacancy configurations from our MD simulations by plotting the pair distribution of vacancies. The vacancy distribution is calculated by averaging the number of nearest vacancies around every vacancy at different times, i.e., t = 0 ns and 5 ns. Figures 9(a)-(e) show the distribution for pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1, and Ni0.8Cr0.2 respectively, plotted in lattice parameter units.
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While focusing on the first nearest neighbor (1st NN), i.e., between 0.7-0.8 ao (where ao is the lattice parameter), we find that the peaks increase with time in all five cases. For example, in pure Ni system (Figure 9(a)), at t = 0 ns, there are essentially zero nearest neighbors, whereas at t = 5 ns there are at least 0.5 average nearest neighbors. At t = 0 ns, almost no vacancy-vacancy nearest neighbors are expected because, at the beginning of the simulation, all vacancies are randomly distributed. In addition, since the concentration of vacancies is very low, i.e., 80 in a system of 32,000 atoms, every vacancy is far away from its nearest neighbor. However, as the simulation proceeds, the vacancies start to diffuse and form small clusters. The appearance of the peak at t = 5 ns indicates the formation of small vacancy-vacancy clusters. We find that the peaks in Ni-Cr system in Figures 9(d) and 9(e) are significantly higher than Ni and Ni-Fe (Figures 9(a), (b) and (c)). In contrast, the peak size of Ni-Fe is almost same as that of the pure Ni. These results show that the vacancy-vacancy clustering is higher and much faster in Ni-Cr alloys than the Ni-Fe alloys and pure Ni. Moreover, the peak at 2nd and 3rd NN, i.e. between 0.9-1.1 ao and between 1.2-1.4 ao respectively indicate SFT formation. By comparing the 2nd NN and 3rd NN peaks between Ni-Cr and Ni-Fe, we find that the Ni-Cr peaks are much higher indicating that the SFT have much larger size than in pure Ni and Ni-Fe alloys. In contrast, the smaller peak sizes in Ni-Fe alloys indicate that most of the vacancies are still isolated. This vacancy-vacancy pair distribution is consistent with the observations in Figure 1. Now we compare the rate of vacancy-vacancy nearest neighbor formation. Figure 10(a) is the comparison of the average vacancy accumulation, i.e., the summation of the absolute numbers at each data-point under the average vacancy-vacancy distribution, at the first nearest neighbor from the same data in Figures 9 (a), (b) and (d), i.e. Ni, Ni0.9Fe0.1, and Ni0.9Cr0.1. It is evident that even at 0.2 ns the vacancy clustering in Ni-Cr is statistically much higher than the pure Ni and
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Ni-Fe system. At 5 ns, it is higher by a factor of four. Figure 10 (b) shows the similar comparison among Ni, Ni0.8Fe0.2, and Ni0.8Cr0.2. Here, by increasing the Cr concentration, the difference between Ni-Cr and pure Ni further increases, and by 5 ns the average vacancy accumulation is higher by a factor of five than the pure Ni. In addition, by comparing Ni0.9Cr0.1 and Ni0.8Cr0.2 in Figures 10(a) and 8(b), we find that the average accumulation in Ni0.8Cr0.2 is higher. This indicates that increasing the Cr concentration promotes vacancy-vacancy clustering. Such increase is anchored on the enhanced vacancy migration induced by Cr, as discussed above.
4. Discussion
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Figure 11 shows the average vacancy-vacancy distribution in ternary alloys. The vacancy pair distribution for the ternary systems is given in the Supplementary figure S-C1 under section C. Figures 11(a), (b) and (c) show the average number of vacancy neighbors individually at 1st NN, 2nd NN and 3rd NN with time, respectively. In Figure 11(a), we find that the number of vacancies at 1st NN increases as the Cr concentration increases. The neighbors are highest for Ni0.4Fe0.1Cr0.5 system (shown in red color), and lowest for Ni0.4Fe0.5Cr0.1 (shown in green color) system among all of the four ternary alloys. The rate of increase in the average number of vacancy neighbors is the highest for the Ni0.4Fe0.1Cr0.5 system which contains the highest Cr concentration reinforcing the observation that the increase in the Cr concentration increases the vacancy-vacancy clustering, and promotes faster SFT formation. Similar observations are made for the 2nd NN and 3rd NN peaks in Figures 11(b) and (c). With longer time, the SFT in these ternary systems grow in size depending on the amount of the Cr present in the respective systems. Figures 11(d) and (e) show average vacancy accumulation at the 1st, 2nd, and, 3rd NN at longer times, i.e., t = 20 ns and t = 40 ns, respectively. Even at these longer times, we find that the peak sizes are higher for alloys containing higher Cr concentrations. Thus, it is evident from the vacancy-vacancy distribution analysis for both binary and ternary systems that the amount of Cr influences the vacancy clustering in the respective systems, and larger SFT are observed in systems containing high Cr concentrations.
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The migration barrier is the key to estimate diffusion process in a system. Our observations on the ‘Cr-effect’ in these binary and ternary systems agree with previous experimental [40-43] and theoretical [22, 38, 44, 45] diffusion studies on HEAs and Ni-Fe-Cr austenitic steels. Experimental measurements [40 - 42] on diffusion coefficients of Fe, Cr, and Ni in fcc Ni-Fe, Ni-Cr systems, and austenitic Ni-Fe-Cr steels show that Cr diffuses faster than Fe and Ni. A recent comparative study on the diffusion kinetics of Fe, Cr and Ni in HEAs and austenitic steels by Tsai et al. [43] shows the similar diffusion trend in HEAs as well, although the diffusion of these elements in HEAs are much slower than austenitic steels. Similarly, the interatomic potential used in this work [26] that was originally developed for fcc Ni-Fe-Cr austenitic steels, predicts the diffusion coefficients of the constituent elements in Ni-Fe-Cr alloys in the same order, as found from experimental studies, i.e. DCr > DFe > DNi. From these studies, the diffusivity trend among Cr, Fe, and Ni in fcc Ni supports our observation on the effect of Cr addition on the defect dynamics, and the overall cluster formation in concentrated Ni-Fe-Cr alloys. The low migration barrier of Cr is the controlling factor for faster kinetics in these alloys. Due to the lower migration barrier, faster migration of Cr leads to faster vacancy diffusion which is leading to formation of large SFT in a very short time.
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Apart from the migration barrier, there are other factors, such as the concentration and the random distribution of alloying elements that can significantly influence atomic diffusivities. An ab-initio study by Tucker et al. [38] showed that the Ni diffusion significantly increases when the Cr concentration increases from 1% to 10% in fcc Ni matrix. However, the Ni diffusion is almost unaffected with similar Fe addition. This supports our observation that the presence of Cr enhances the diffusion of Ni. Previous experimental studies [40, 41] on the diffusion coefficients of Cr, Ni and Fe in fcc Ni-based compounds supported the same trend. Thus, there is a common consensus among various studies that Cr enhances Ni diffusivity. This is due to the formation of favorable vacancy configurations as shown above. In contrast, due to relatively higher Fe migration barrier, vacancy and Ni diffusion remain unaffected by Fe addition.
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Although the use of DFT and MD methods to calculate diffusivities of dilute solutes is a common practice, understanding similar mechanisms in concentrated materials such as random alloys and HEAs is complex due to varying local atomic environments. Recent attempts have been made via ab-initio MD modelling [38, 44, 45] and kinetic Monte Carlo (kMC) [46, 47] approaches to elucidate such complex effects. In particular, Tucker et. al [38] showed that the binding between a vacancy and the Cr atom becomes attractive with increasing Cr concentration. In contrast, the vacancy and Fe atom binding becomes repulsive with increasing Fe concentration. Such solute-defect interactions will have significant impacts on the overall defect evolution. Therefore, developing an in-depth understanding of the local effects of alloying elements on defect evolution in random concentrated alloys is needed in future. 5. Conclusions
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In conclusion, this work shows that in the development of radiation resistant concentrated solid solution alloys, each element could have a significant effect on the defect dynamics. Our work highlights that increasing the concentration of Cr in the Ni-Fe-Cr alloys promotes vacancy diffusion leading to faster and larger SFT formation. In addition, Cr also enhances the Ni diffusion. These effects are due to the low migration barrier of Cr, which is the controlling factor for vacancy accumulation and high Ni diffusion. On the contrary, due to similar migration barrier as Ni, the effect of Fe towards vacancy accumulation and Ni diffusion is minimal in Ni-Fe alloys and in Ni-Fe-Cr alloys. Therefore, we propose that Cr addition to concentrated alloys may not be helpful in preventing larger cluster formation, Finally, this work emphasizes that the irradiation resistance of the binary and ternary SP-CSAs is deeply enrooted in the complexity of the alloying elements, and understanding the contribution of each element is critical towards the development of high radiation resistant alloys for nuclear applications. Acknowledgement: This work was supported as part of the Energy Dissipation to Defect Evolution (EDDE), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences. We acknowledge the support of computational resources from Advanced Research Computing Center (ARCC) at University of Wyoming. References: [1] B. Cantor, I.T.H. Chang, P. Knight, A.J.B. Vincent, Microstructural development in equiatomic multicomponent
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alloys, Mater. Sci. Eng. A 375-377, (2004) 213-218. [2] J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes, Adv. Eng. Mater. 6 (2004) 299-303. [3] Y.F. Ye, Q. Wang, J. Lu, C.T. Liu, Y. Yang, High entropy alloy: Challenges and prospects, Mater. Today, 19(6) (2016) 349-362. [4] D.B. Miracle, O.N. Senkov, A critical review of high entropy alloys and related concepts, Acta Mater. 122 (2017) 448-511. [5] K. M. Youssef, A. J. Zaddach, C. Niu, D. L. Irving, C. C. Koch, A Novel Low-Density, High-Hardness, Highentropy Alloy with Close-packed Single-phase Nanocrystalline Structures, Mater. Res. Lett., 3(2) (2015) 95-99. [6] B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P. George, R. O. Ritchie, A fracture-resistant highentropy alloy for cryogenic applications, Science, 345 (6201) (2014) 1153-1158. [7] K. Jin, C. Lu, L.M. Wang, J. Qu, W.J. Weber, Y. Zhang, H. Bei, Effects of compositional complexity on the ionirradiation induced swelling and hardening in Ni-containing equiatomic alloys, Scrip. Mater. 119 (2016) 65–70, 2016. [8] Y. Zhang, S. Zhao, W. J. Weber, K. Nordlund, F. Granberg, F. Djurabekova, Atomic-level heterogeneity and defect dynamics in concentrated solid-solution alloys, Curr. Opin. Solid State Mater. Sci., (2017) (in press). [9] C. Lu, K. Jin, L.K. Béland, F. Zhang, T. Yang, L. Qiao, Y. Zhang, H. Bei, H.M. Christen, R.E. Stoller, L. Wang: Direct observation of defect range and evolution in ion-irradiated single crystalline Ni and Ni binary alloys. Sci. Rep. 6 (2016) 19994. [10] R. A. Holt, "Mechanisms of Irradiation Growth of Alpha-Zirconium Alloys," J. Nucl. Mater., 159 (1988) 310337. [11] T. Allen, J. Busby, M. Meyer, D. Petti, Materials challenges for nuclear systems, Mat. Today, 13 (12) (2010) 14-23. [12] D. H. Gurinsky, G. J. Dienes (Eds), Nuclear Fuels, D. Van Nostrand Co., Princeton, New York, 1956. [13] T. H. Courtney, Mechanical Behavior of Materials, McGraw-Hill, New York, 1990. [14] A. Boltax, J. P. Fostar, R. A. Weiner, A. Biancheria, Void swelling and irradiation creep relationships, J. Nucl. Mater., 65 (1977) 174-183. [15] W. Han, E.G. Fu, M. J. Demkowicz, Y. Wang, A. Misra, Irradiation damage of single crystal, coarse-grained, and nanograined copper under helium bombardment at 450 °C, J. Mater. Res., 28 (20) (2013) 2763-2770. [16] J. E. Epperson, R. W. Hendricks, K. Farrell, Studies of voids in neutron-irradiated aluminium single crystals: I. Small-angle X-ray scattering and transmission electron microscopy, Phil. Mag., 30(4) (1974) 803-817. [17] S. J. Zinkle, L. L. Snead, Designing radiation resistance in materials for fusion energy, Annu. Rev. Mater. Res., 44 (2014) 241-267. [18] N.A.P. Kiran Kumar, C. Li, K.J. Leonard, H. Bei, S.J. Zinkle, Microstructural stability and mechanical behavior of FeNiMnCr high entropy alloy under ion irradiation, Acta Mater., 113 (2016) 230-244. [19] J.A. Hudson, Void formation in solution-treated AISI 316 and 321 stainless steels under 46.5 MeV Ni6+ irradiation, J. Nucl. Mater., 60 (1976) 89-106. [20] A. Etienne, B. Radiguet, N.J. Cunningham, G.R. Odette, R. Valiev, P. Pareige, Comparison of radiationinduced segregation in ultrafine-grained and conventional 316 austenitic stainless steels, Ultramicroscopy, 111 (2011) 659-663. [21] Y. Zhang, G.M. Stocks, K. Jin, C. Lu, H. Bei, B.C. Sales, L. Wang, L.K. Beland, R.E. Stoller, G.D. Samolyuk, M. Caro, A. Caro, W.J. Weber: Influence of chemical disorder on energy dissipation and defect evolution in nickel and Ni-based concentrated solid-solution alloys. Nat. Commun. 6 (2015) 8736 (pp 9). [22] Y Zhang, K. Jin, H. Xue, C. Lu, R. J. Olsen, L. K. Beland, M.W. Ullah, S. Zhao, H. Bei, D. S. Aidhy, G. D. Samolyuk, L. Wang, M. Caro, A. Caro, G.M. Stocks, B. C. Larson, I. Robertson, A. Correa, W. J. Weber, Influence of chemical disorder on energy dissipation and defect evolution in advanced alloys, J. Mater. Res. 31 (2016) 23632375. [23] F. Otto, Y. Yang, H. Bei, E.P. George, Relative effects of enthalpy and entropy on the phase stability of equiatomic high-entropy alloys, Acta Mater., 61(7) (2013) 2628–2638. [24] P. Nash, The Cr-Ni (Chromium-Nickel) System, Bull. Alloy Phase Diag., 7, 5, 466, 1986. [25] M. W. Ullah, H. Xue, G. Velisa, K. Jin, H. Bei, W. J. Weber, Y. Zhang, Effects of chemical alteration on damage accumulation in concentrated solid-solution alloys, Sci. Rep., 7(1) (2017) 4146. [26] G. Bonny, N. Castin, D. Terentyev, Interatomic potential for studying ageing under irradiation in stainless steels: the FeNiCr model alloy, Model. Simul. Mater. Sci. Eng. 21 (2013) 085004 (15pp). [27] D. S. Aidhy, C. Lu, K. Jin, H. Bei, Y. Zhang, L. Wang, W. J. Weber, Formation and growth of stacking fault tetrahedra in Ni via vacancy aggregation mechanism, Scr. Mater. 114 (2016) 137-141.
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[28] M. W. Ullah, D. S. Aidhy, Y. Zhang, W. J. Weber, Damage accumulation in ion-irradiated Ni-based concentrated solid- solution alloys, Acta. Mater. 109 (2016) 17-22. [29] L. K. Béland, G. D. Samolyuk, R. E. Stoller, Differences in the accumulation of ion-beam damage in Ni and NiFe explained by atomistic simulations, J. Alloy Compd. 622 (2016) 415-420. [30] D. S. Aidhy, C. Lu, K. Jin, H. Bei, Y. Zhang, L. Wang, W. J. Weber, Point defect evolution in Ni, NiFe and NiCr alloys from atomistic simulations and irradiation experiments, Acta. Mater. 99 (2015) 69-76. [31] S. Plimpton, Fast Parallel Algorithms for Short-Range Molecular Dynamics, J Comp Phys 117 (1995) 1-19. [32] G. Henkelman, G. Jónannesson, H. Jónsson, Methods for finding saddle points and minimum energy paths, in: S. D. Schwartz (Eds.), Theoretical Methods in Condensed Phase Chemistry, Kluwer Academic Publishers, Springer Netherlands, 2000, pp. 269-302. [33] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci., 6 (1996) 15-50. [34] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B, 50 (1994) 17953-17979. [35] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett., 77, (1996) 3865-3868. [36] M. Methfessel, A. Paxton, High-precision sampling for Brillouin-zone integration in metals, Phys. Rev. B., 40 (1989) 3616-3621. [37] P. Ehrhart, Defect dynamics, migration energies and jump frequencies: Datasheet from Landolt-Börnstein Group III Condensed Matter, in: Ullmaier, H. (Eds), Atomic Defects in Metals, Springer-Verlag Berlin Heidelberg, 1991, vol 25, p. 88. [38] J. D. Tucker, Ab-initio- based modeling of radiation effects in the Ni-Fe-Cr system, PhD Dissertation, Nuclear Engineering and Engineering Physics, University of Wisconsin-Madison, 2008. [39] Similar to Ni in Figure 8b, the Cr and Fe migration barriers are expected to decrease as well. However, due to higher concentration of Ni in these alloys, we have focused on Ni diffusivity only. [40] B. Million, J. Ruzickova, J. Velisek, J. Vrestal, Diffusion Processes in Fe-Ni, Mater. Sci. Eng., 50 (1981) 4352. [41] J. Ruzickova, B. Million, Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system, Mater. Sci. Eng., 50 (1981) 59-64. [42] S. J. Rothman, L. J. Nowicki, G.E. Murch, Self-diffusion in austenitic Fe-Cr-Ni alloys, J. Phys. F: Metal Phys.,10 (1980) 383 -398. [43] K. Y. Tsai, M. H. Tsai, J. W. Yeh, Sluggish diffusion in Co–Cr–Fe–Mn–Ni high-entropy alloys, Acta Mater., 61 (2013) 4887- 4897. [44] J.D. Tucker, R. Najafabadi, T.R. Allen, D. Morgan, Ab initio-based diffusion theory and tracer diffusion in Ni– Cr and Ni–Fe alloys, J. Nucl. Mater., 411 (1-3) (2010) 1-14. [45] M. Mantina, A first-principles methodology for diffusion coefficients in metals and dilute alloys, Dissertation, College of Earth and Mineral Sciences, Penn. State Univ., 2008. [46] A. V. Barashev, A. C. Arokiam, Monte Carlo Modelling of Cu atom diffusion in ߙ-Fe via the vacancy mechanism, Phil. Mag. Lett, 86 (2006), 321-332. [47] Y. N. Osetsky, L. K. Béland, R. E. Stoller, Specific features of defect and mass transport in concentrated fcc alloys, Acta. Mater, 115 (2016) 364-371.
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Tables:
Ni Fe
0.71 0.44
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Cr Ni in presence of vacancy at 1NN
Migration barrier from DFT (in eV)
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Migration barrier from classical potential (in eV) 1.08 1.02
1.08 0.95
0.75 0.61
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Table 1: Migration energy (in eV) of different atoms in Ni fcc lattice in the vacancy configuration as shown in Figures 6(a) and (b).
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Figure 1: Snapshots of (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1 and (e) Ni0.8Cr0.2 with 0.25% vacancies at t=5ns. Ni-Cr compositions show large vacancy clustering than Ni-Fe and pure Ni.
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Figure 2: The vacancy count in pure Ni, Ni0.9Fe0.1, Ni0.8Fe0.2, Ni0.9Cr0.1, Ni0.8Cr0.2 has been shown for (a) mono-vacancies and (b) di- and larger vacancy clusters. With simulation time the mono vacancy count decreases and di- and larger vacancy cluster count increases. In the case of pure Ni and Ni-Fe systems the large vacancy cluster count is close at t = 5ns. However, In the case of Ni-Cr systems, with increasing Cr concentration large vacancy clusters have been formed at t = 5ns and less number of single vacancies are observed. This is reflected in mono-vacancy count as well as in the count of di- and larger vacancy clusters for Ni-Cr systems. The Ni0.8Cr0.2 shows few large clusters (13 and 21-vacancy SFT) and Ni0.9Cr0.1 shows few mono-vacancy and many small vacancy cluster, largest being 13-vacancy SFT at t = 5ns.
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Figure 3: Snapshots of (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1 and (e) Ni0.8Cr0.2 with 0.25% vacancies at 40ns time-step. The vacancy clustering is prominently large in Ni0.8Cr0.2 compared to other four systems.
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Figure 4: Snapshots of (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.4Cr0.2, (d) Ni0.4Fe0.3Cr0.3, and (e) Ni0.4Fe0.1Cr0.5 with 0.25% vacancies at t=5ns. Increasing Cr compositions show larger vacancy clustering.
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Figure 5: The vacancy count in pure Ni, Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, Ni0.4Fe0.1Cr0.5 has been shown for (a) mono-vacancies and (b) di- and larger vacancy clusters. With simulation time the mono vacancy count decreases and di- and larger vacancy cluster count increases with increase of Cr content. At t = 5ns, the count of mono-vacancy significantly decreases and di- and larger vacancy count increases. In the case of Ni0.4Fe0.1Cr0.5, there exist only one single vacancy and few large SFT, 38-vacancy SFT is being the largest.
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Figure 6: Snapshots of (a) pure Ni, (b) Ni0.4Fe0.5Cr0.1, (c) Ni0.4Fe0.4Cr0.2, (d) Ni0.4Fe0.3Cr0.3, and (e) Ni0.4Fe0.1Cr0.5 with 0.25% vacancies at t=40ns. SFT size increases with increasing Cr concentration.
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Figure 7: Mean square displacement (MSD) plots of Ni in, (a) binary systems, and (b) ternary systems up to t =5ns. Addition of Cr increases the Ni diffusion significantly in binary systems (almost two-fold to the pure Ni case), however addition of Fe does not influence the diffusion of Ni. Similarly, in ternary systems, the Ni diffusion increases with increase in Cr composition.
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Figure 8: (a) Vacancy diffusion in fcc Ni matrix in presence of Ni, Fe or Cr at the 1st NN. (b) Similar diffusion in the presence of another nearest neighbor vacancy. Square represents vacancy and red dot represents the atom, i.e. Ni, Cr or Fe. The arrows in both figures show that the 1st NN atom move to the vacancy and the corresponding atom lattice site become vacant.
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Figure 9: Pair distribution function (PDF) of vacancies around a vacancy is shown for (a) pure Ni, (b) Ni0.9Fe0.1, (c) Ni0.8Fe0.2, (d) Ni0.9Cr0.1, and (e) Ni0.8Cr0.2 at t=0ns and t=5ns. The PDFs show the vacancy accumulation significantly increases with increase in Cr composition.
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Figure 10: Comparison of vacancy accumulation at the 1st NN for pure Ni, Ni0.9Fe0.1 and Ni0.9Cr0.1 in (a), and the vacancy accumulation at the 1st NN for pure Ni, Ni0.8Fe0.2 and Ni0.8Cr0.2 in (b). For Ni-Cr binary system the average vacancy accumulation at 1st NN is significantly high.
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Figure 11: Vacancy-vacancy distribution as a function of MD time for, (a) 1st NN, (b) 2nd NN, and (c) 3rd NN in pure Ni, Ni0.4Fe0.5Cr0.1, Ni0.4Fe0.4Cr0.2, Ni0.4Fe0.3Cr0.3, and Ni0.4Fe0.1Cr0.5. The average vacancy accumulation increases with the increase in Cr composition. The comparison in PDF at (d) t=20ns and (e) t=40ns for the ternary systems has also been shown for respective ternary systems. At longer timescale, with the increase in Cr concentration, larger SFT are formed. At t=40ns, the 3rd NN shows the signature of larger SFT formation in ternary systems with increasing Cr concentration.
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