Nanoporous High Entropy Alloys: Overcoming Brittleness Through Strain Hardening

Nanoporous High Entropy Alloys: Overcoming Brittleness Through Strain Hardening | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Nanoporous High Entropy Alloys: Overcoming Brittleness Through Strain Hardening Jarod Worden, Celine Hin, Juergen Biener This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8998496/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 12 You are reading this latest preprint version Abstract Bicontinuous nanoporous materials possess remarkable mechanical properties, such as higher specific strength and lower specific modulus compared to fully dense materials combined with low densities and high specific surface areas. Unfortunately, their practical application is hindered by inherent macroscopic brittleness, mainly due to cascading ligament failure under tension. To address this limitation, we investigate whether high entropy alloys, recognized for their outstanding strength and strain hardening properties, can mitigate nanoporous material’s inherent brittleness. Molecular dynamics simulations of nanoporous Al 0.1 CoCrFeNi and NbMoTaW reveal a dual mechanism involving dislocation starvation and sluggish dislocation motion, resulting in specific strength values 5 to 10 times higher than those of single-element nanoporous materials, and a resilience against thermal degradation. Strain hardening, driven by sluggish dislocations, effectively prevents failure of the weakest ligaments under tensile stress in face-centered cubic architectures by trapping stacking faults in the ligaments and dislocation forest hardening in the nodes of body-centered cubic structures, demonstrating their potential to shape the next generation of high strength, low density materials. Physical sciences/Engineering Physical sciences/Materials science Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1. Introduction Nanoporous metals, featuring random bicontinuous ligament-pore structures, possess unique mechanical properties, including high specific strength, good ductility, and energy absorption, all while maintaining lower densities compared to typical bulk metals 1 , 2 . However, their practical application is hindered by brittleness under tension, as fracturing of the weakest ligaments can lead to a cascading fracture of adjacent ligaments 3 , 4 . For example, nanoporous gold, studied extensively through simulations and experiments, shows macroscopic brittle behavior triggered by single ligament failure 5 , 6 . To address the brittleness challenge associated with cascading ligament failure, we investigate High Entropy Alloys (HEAs), known for substantial strain hardening, in nanoporous structures 7 , 8 . HEAs use equiatomic concentration of multiple elements to increase system entropy, stabilizing disordered solid-solution phases and preventing brittle intermetallic phases 1 , 9 . They offer high strength, hardness, strain hardening, and resistance to fatigue, creep, and wear, overcoming challenges in single-element nanoporous materials 9 , 10 . Our hypothesis is that incorporating HEAs into bicontinuous nanoporous structures will enhance mechanical properties like strength and toughness while maintaining low densities compared to traditional metals and alloys. This makes them promising for applications requiring superior mechanical performance combined with low densities and high thermal tolerance, benefiting industries such as nuclear, automotive, aerospace, and aeronautics compared to commonly used industrial materials 11 , 12 (further investigation of these materials in automotive and nuclear are explained in Supplementary Discussion 1). In the realm of high energy absorption and impact resistance, nanoporous HEAs are expected to showcase appealing mechanical properties that optimize energy deflection and allowable stress 13 . However, their application is hindered due to their novelty and lack of mechanical experimental and computational data. This study employs Molecular Dynamics (MD) simulations to investigate the mechanical behavior of nanoporous High Entropy Alloys (HEAs) under uniaxial compression and tensile testing. For comparison, we also performed simulations on fully dense single crystalline (SC) and polycrystalline HEAs. The selection of Al 0.1 CoCrFeNi and NbMoTaW HEAs is based on their distinctive single disordered face-centered cubic (fcc) and body-centered cubic (bcc) phase 14 . The simulation outcomes validate our hypothesis that nanoporous HEAs exhibit unique mechanical properties, especially significantly higher specific strength compared to single-element nanoporous materials (Fig. 1 ). The high specific strength can be attributed to the sluggish dislocation mechanism observed in the studied HEAs that enhances the strength of individual ligaments by enabling strain hardening mechanisms such as dislocation forest hardening, effectively addressing the macroscopic brittleness associated with nanoporous materials. 2. Methods The molecular dynamics (MD) simulations were conducted using the open-source code LAMMPS, employing a three-dimensional cell with periodic boundary conditions, with dimensions of 44.625 nm x 44.625 nm x 44.625 nm 16 . The bulk cell was generated by populating it with 7,812,500 fcc and 3,906,250 bcc structural atoms. The concentrations of Al, Co, Cr, Fe, and Ni (or Nb, Mo, Ta, W) were randomly assigned using Atomsk 17 . Polycrystalline systems were created by applying Voronoi Tessellation to the bulk cell with random grain orientations. Three distinct polycrystalline structures with grain sizes of 10 nm, 20 nm, and 30 nm were generated. The methodology for producing nanoporous HEAs followed a similar approach to that used for simulating the creation of bicontinuous nanoporous gold through dealloying methods and are described in Supplementary Discussion 2 18 . The types of systems in this study are depicted in Fig. 2 . Al 0.1 CoCrFeNi system was simulated using an Embedded Atom Method (EAM) potential, which has also been employed in previous MD investigations to study nanoindentation tests and plastic deformation 19 – 21 . The stacking fault energy (SFE) of the single-crystal (SC) system was calculated to be 32.8 mJ/m 2 , showing good agreement with the experimental value of 30 mJ/m 2 and aligning with other computational findings 22 , 23 . This low stacking fault energy of the potential is similar to other fcc HEAs such as the cantor alloy CoCrFeMnNi where the stacking fault energy is also found to be exceedingly low (< 35 mJ/m 2 ) using both experimental and computational methods, much lower than that of single element materials 24 , 25 . For the NbMoTaW system, a Spectral Neighbor Analysis Potential (SNAP) potential was applied, which is capable of accurately reproducing elastic constants 26 . To analyze defects within the system, the Dislocation Analysis Tool was employed to assess dislocation densities, dislocation types, and the formation of Stacking Faults/Twin Boundaries 27 . The ligament diameter measurement involved identifying specific atom positions as the center of an expanding sphere. This sphere grew radially by half the lattice parameter. We assessed if a part of the sphere contacted the ligament's surface by examining interactions between the sphere and local atom configuration. If contact was confirmed on one side, we checked the opposite side. When both sides contacted the surface, we calculated the distance as the ligament section's diameter, contributing to the average ligament diameter calculation for the entire system. Average ligament size is displayed in Table 1 with error. Molecular dynamics simulations were employed to conduct mechanical compression and tensile tests at temperatures of 298 K, 600 K, and 1273 K. Initially, a Nose/Hoover thermostat and barostat (NPT) ensemble was utilized for a thermalization period of 100.0 ps. This thermalization phase took place at the specified temperatures and atmospheric pressure until the system achieved a stable state with respect to lattice parameters and energy. Following the thermalization process, the NPT ensemble was re-employed, and a strain rate of 10 9 s − 1 was applied to deform the simulation box along the x direction. Compression tests were conducted in the range of 0–70% strain, while tensile tests covered the 0–50% strain range. The use of high strain rates in MD simulations is a recognized limitation, as realistically slow strain rates require prohibitively long simulation times. High strain rates generally result in higher ultimate stresses, making direct comparisons with experimental data more challenging. Our study, along with others in the field, examines various strain rates and finds that differences between lower strain rates are relatively minor, indicating a diminishing return in the reduction of ultimate stress with decreasing strain rate 28 , 29 . This trend is highlighted in Supplementary Table 1, which presents ultimate stress calculations for different tensile strain rates in the 28.0% relative density nanoporous NbMoTaW system. Notably, at a lower strain rate, a minimal decrease in ultimate stress is observed (0.02 GPa), whereas a significantly higher stress is recorded at an increased strain rate (0.12 GPa). The chosen strain rate effectively balances the need for faster simulation times while minimizing its impact on stress results. 3. Results The results section analyzes the mechanical properties involving the HEA systems through stress-strain curve analysis, ligament fracture analysis, defect analysis, and surface energy analysis. The analysis focuses on random placement of atomic species for the different HEA systems, however there is the potential for surface segregation due to the relatively large surface area. This can be analyzed in future investigations. 3.1 Stress-Strain curve analysis Figure 3 (a-d) displays stress vs. strain curves at 600 K (more results from tests at 298/600/1273 K are shown in Figure S1 and Figure S2) for Al 0.1 CoCrFeNi and NbMoTaW while the corresponding yield strength and Young's modulus data are summarized in Tables 2 to 5 . The highest strain values displayed for the nanoporous systems under compression correspond to strains before full densification, which occurs at approximately 70%, 60%, and 50% strain for the 28.9%, 47.8%, and 67.2% porosity samples, respectively (Fig. 3 ). For both compression and tension, and irrespective of temperature, the Al 0.1 CoCrFeNi systems exhibit linear elastic behavior within the low strain range up to 2%. For non-nanoporous structures, a transition to a non-linear elastic deformation behavior can be observed with further increasing strain until the ultimate stress point is reached at 5% strain. In the case of SC systems, the yield strength is attained early, in contrast to the ultimate strength, which is achieved at higher strains, indicating the strain-hardening mechanism of the HEAs 10 . This is observed in the compressive yield strength displayed in Table 2 . For the nanocrystalline (NC) systems, both under compression and tension, the stress drops significantly after reaching the yield peak, followed by a plateau region that continues up to the highest strain values. Notably, in the NC systems, compression tests yield higher stress values compared to tensile tests (Fig. 3 a). This observed compression-tension asymmetry in yield strength for SC and polycrystalline Al 0.1 CoCrFeNi has also been reported in other HEA systems 30 , possibly reflecting distinct deformation mechanisms such as twinning in compression and slip in tension 31 . The stress-strain response of nanoporous Al 0.1 CoCrFeNi under both compression and tensile test conditions is more complex when compared to SC and NC systems (Fig. 3 a). Tensile tests conducted on nanoporous Al 0.1 CoCrFeNi exhibit a more distinct yield peak, followed by a shorter plateau phase, before the stress decreases with further increasing strain. These trends align with those observed in other FeCrNiCuCo fcc HEA nanoporous systems 28 , 32 . In compression tests, only a small yield peak was observed for the highest relative densities, followed by an immediate stress plateau and a subsequent region where stress gradually increases with strain. This response aligns with what is commonly observed in many nanoporous systems under uniaxial compression in both experimental and simulated settings for single element and HEA systems 33 , 34 . Comparing the effect of relative density and ligament size on compression stress-strain curves (Fig. 3 b) reveals that at any given strain within the plateau region, higher relative density correlates with higher stress. Only the 67.2% relative density / 7.33 nm ligament diameter sample shows a distinct yield peak under compression; with increasing strain the effect of ligament size on the stress response becomes smaller and smaller. This mirrors patterns observed in FeCrNiCuCo nanoporous fcc HEA systems, where it was found that relative density exerted a more significant influence on mechanical properties compared to ligament size, particularly when the ligament size is very small (Fig. 3 b) 28 . The mechanical properties of Al 0.1 CoCrFeNi systems were also found to be temperature-dependent, with elevated temperatures resulting in reduced stresses across all systems (Fig. 3 c and Fig. S1 ). The most pronounced stress reduction with temperature occurs in the 28.9% relative density sample. Conversely, the SC sample shows minimal or even reversed temperature dependence beyond the yield point. Additionally, for systems with similar relative densities, smaller ligaments lead to a more pronounced reduction in stress under compression at high temperatures (1273 K vs 298 K and 600 K) compared to larger ligament systems at higher temperatures, as indicated in Tables 1 – 4 . This effect is particularly evident for the 50% relative density samples, as shown in Fig. S1 a-c. The softening of smaller ligament systems with increasing temperatures may reflect the limited thermal stability of the smallest ligaments, as evidenced by the collapse of the ligament structure in the 28.7% relative density system with 2.2 nm ligaments during the initial relaxation phase. Under tensile test conditions, the np-Al 0.1 CoCrFeNi samples exhibit increased plasticity with increasing temperature (Fig. S1 d-f). The compression behavior of the NbMoTaW systems (as shown in Fig. 3 d) followed similar stress-strain trends as the Al 0.1 CoCrFeNi systems, including an initial steep increase in stress and a more or less pronounced yield peak at around 5% strain. This is followed by a plateau region where stress remain relatively constant with increasing strain. In the case of tensile strain (Figure S2 in Supplementary Materials), the SC NbMoTaW system displayed stress oscillations after the failure strain, which corresponded to oscillating atomic positions of the atoms following system rupture. Increasing temperature also resulted in decreased stress across all systems, with a more significant reduction observed for SC and NC structures. Overall, the NbMoTaW systems exhibited superior mechanical properties, including higher yield strength and Young’s modulus, compared to the Al 0.1 CoCrFeNi systems (Fig. 3 d and S2). An exception is observed in the 20 nm NC samples, where NbMoTaW shows lower stress than Al 0.1 CoCrFeNi. This may reflect differences in defect interactions or simulation setup and warrants further investigation. The higher yield strength and Young’s modulus of the NbMoTaW samples can be attributed to the intrinsic properties of the individual elements of the NbMoTaW system, which inherently possess greater yield strength, ultimate strength, and Young’s modulus in both compressive and tensile environments than the elements in the Al 0.1 CoCrFeNi 35 . Tables 1 and 2 present the yield stress and Young’s modulus for the Al 0.1 CoCrFeNi systems at both 298 K and 1273 K. Notably, the Young’s modulus obtained from tensile mechanical tests for Al 0.1 CoCrFeNi closely aligns with experimental data and is consistent with previous molecular dynamics simulation results 22 , 23 , 36 , 37 . Additionally, the yield stress and strain results obtained from both tensile and compression tests show strong agreement with other molecular dynamics studies 38 , 39 . Tables 3 and 4 in the main text provide the yield stress and Young’s modulus for NbMoTaW. The compressive modulus also closely corresponds with experimental data and aligns with other simulation results obtained under compression conditions 40 – 42 , further supporting the reliability of the interatomic potential used. Additionally, the nanoporous systems exhibit a tension-compression asymmetry (see Table 2 – 5 ), a phenomenon previously observed in other nanoporous systems where compression demonstrates higher yield strength and Young's modulus 33 . This asymmetry varies depending on the type of high entropy alloy and crystal structure. For instance, the Al 0.1 CoCrFeNi nanoporous structure exhibits a higher compression yield strength compared to tension, while the NbMoTaW nanoporous system displays a higher yield strength under compression conditions. This discrepancy can be attributed to distinct defect mechanisms involved in plastic deformation. In bcc structures, common defect mechanisms include twinning and anti-twinning on the {211} slip planes, as well as non-planar core structures of screw dislocations 42 . On the other hand, in fcc structures, the asymmetry may be due to biased surface stresses and differing slip planes during tensile and compressive yielding 43 . 3.2 Fracture Analysis in Tensile Tests of Nanoporous Al 0.1 CoCrFeNi Fracture analysis was performed exclusively on specimens subjected to tensile testing as no fractures were observed during compression tests due to predominant ligament bending 44 . This analysis focused on a segment of the nanoporous Al 0.1 CoCrFeNi system with a relative density of 28.9% and ligament size of 4.23 nm, tested under tensile conditions at 1273 K. Surface topography, dislocations, twin boundaries, and stacking faults were analyzed using the Dislocation Extraction Algorithm (DXA) and Paraview, as depicted in Fig. 4a-l 45,46 . Dislocations are initially present during the early stages of fracture but gradually diminish as ligament failure progresses (Fig. 4 d,e,f). Twin boundaries form at the onset of fracture and persist throughout ligament failure; however, they disintegrate as the ligament ruptures (Fig. 4 g,h,i). Stacking faults are the most prevalent defects observed, remaining dominant in the ligament at all stages of the fracture process (Fig. 4 j,k,l). This behavior can be attributed to the low stacking fault energy characteristic of the HEA system. This distinguishes the fracture processes of HEA ligaments from those observed in ligaments of nanoporous gold where dislocations appear to play a pivotal role in the ligament deformation 6 , 36 , 37 . The absence of distinct dislocations in the HEA ligaments during fracturing may be attributed to two factors: firstly, the hindrance of dislocation motion due to lattice distortion caused by the different atomic species in the HEA material, which is expected to reduce the formation of larger dislocations; and secondly, the high density of stacking faults impeding dislocations 36 . The sluggish dislocation motion of the HEA is corroborated by investigating the higher Potential Energy Landscape (PEL) of the ½{111} edge dislocation which has higher energy barriers compared to single element PELs as shown in Supplementary Figure S3 and discussed in Supplementary Discussion 3. Given that the primary dislocation type is partial dislocations, neighboring partial dislocations formed from dissociated edge dislocations in the ligament create highly stable stacking faults between them, stemming from impaired dislocation mobility and low SFE in the nanoporous HEA system. As a result, the number of independent dislocations decreases, and the primary mechanism for dislocation elimination in the ligaments is the breaking of stacking faults 38 . The presence of stacking faults and partial dislocations within ligaments seems to be a prevalent mechanism in FeCrNiCuCo fcc HEA nanoporous structures, suggesting potential similarities in defect responses across these materials 28 , 32 . As dislocation motion typically constitutes a primary mechanism for plastic deformation, the absence of dislocations implies the existence of an alternative process for ligament fracturing during uniaxial tension 39 , 40 . Owing to the substantial prevalence of stacking faults in the ligament structure during fracture, the principal process bears a close resemblance to the twinning mechanism observed in fcc metallic materials, which is particularly pronounced during severe plastic deformation 41 . 3.3 Fracture Analysis in Tensile Tests of Nanoporous NbMoTaW To investigate defects in the bcc structure, the bcc defect analysis (BDA) tool is used instead of the DXA tool, as justified in Supplementary Discussion 4 with accompanying Supplementary Figure S9 47 . Below 14% strain, no defects were observed within the ligaments; however, at higher strain levels, twin boundaries nucleate alongside dislocations concentrated around the edges (Fig. 5 a). While the BDA tool does not classify dislocation types, it is important to note that bcc systems commonly exhibit a/6 partial dislocations, which play a crucial role in twinning - a key deformation mechanism in bcc Fe nanopillars 42 . These partial dislocations facilitate the movement of twin boundaries toward the ligament’s center (Fig. 5 b), where they merge into complex dislocation bundles (Fig. 5 c). This interaction is essential for atom removal at the ligament’s notch, ultimately leading to fracture (Fig. 5 d) 3.4 Defect Analysis in Nanoporous Systems The effects of test temperature and geometry on dislocation and stacking fault concentrations in the Al 0.1 CoCrFeNi system are summarized in Fig. 6 , while total defect densities are presented in Figures S4 and S5. In both geometries, the dislocation density of nanoporous Al 0.1 CoCrFeNi (Fig. 6 a,c,e) exhibits a sharp increase up to 10% strain, followed by either a plateau (tensile tests) or a continuous increase (compression tests) for higher strains. The stacking fault density of the nanoporous systems exhibits a steadier growth that is also more influenced by temperature and testing type (Fig. 6 b,d,f). Higher temperatures generally lead to a reduction in defect density due to faster dislocation annihilation and elevated stacking fault energy (Fig. 6 c and 6 d) 43 , 48 . Defect annihilation is further facilitated by the higher surface area-to-volume ratio in lower relative density systems 49 . Our systems with smaller ligaments, on the other hand, exhibit higher dislocation densities, possibly related due to the combined effect of more frequent defect pileups at junctions (Fig. 6 a, 6 d, and S6) and the higher number of junctions in smaller ligament systems 44 . However, at elevated temperatures, defect migration and annihilation start to dominate over defect pileup 49 , 50 . Compared to the stacking fault density, the dislocation density is generally less sensitive to temperature and test geometry variations. During compression at lower temperatures, the stacking fault density consistently increases with increasing strain, with higher relative density nanoporous systems surpassing the stacking fault densities observed in the NC systems (Fig. 6 b). For the lower density (< 50%) np-Al 0.1 CoCrFeNi systems, the stacking fault densities for the same strain values remain below those of the NC systems for both 600 K and 1273 K (Fig. 6 b,d). Dislocations and stacking fault densities correlate with stress-strain curve trends, especially under tension, where the reduction of defect density aligns with decreasing stress (Fig. 3 b, 6 e, and 6 f). In contrast, none of the NbMoTaW nanoporous systems exhibited stacking fault defects. The absence of stacking faults can be attributed to the high stacking fault energy characteristic of bcc crystal structures and was found to be ~ 1300 mJ/m 2 in NbMoTaW, magnitudes lower than the Al 0.1 CoCrFeNi systems 26 , 51 . The dislocation density in tension is lower compared to compression (Figure S7). This disparity could be attributed to tension-compression dislocation glide velocity asymmetry, where tensile deformation could result in higher dislocation velocities, facilitating interactions and annihilation at the ligament surface 49 , 52 . Another explanation lies in the elongation of ligaments, which increases the surface-to-volume ratio, consequently elevating the dislocation annihilation rate. Temperature has a less pronounced effect on dislocation density in NbMoTaW compared to Al 0.1 CoCrFeNi. The nanoporous NbMoTaW systems exhibit minimal changes in dislocation density with varying temperatures under tensile deformation, whereas there is a small but systematic increase in dislocation densities at higher temperatures under compression (Figure S7). The velocity of dislocations increases with higher temperature, but the formation of dislocations also becomes faster. Depending on the specific HEA, the dislocation velocity may be significantly impeded, particularly in bcc HEA materials with their slow screw dislocation motion 53 . Additionally, while screw dislocations in bcc systems are typically influenced by temperature, NbMoTaW exhibits a consistent dislocation velocity across different temperature ranges due to the presence of thermal kinks and cross kinks 54 . Under compression, it appears that the effect of higher temperatures on the rate of dislocation formation dominates over the effect of higher dislocation velocity and the associated higher annihilation rates. Further analysis of dislocation types is expounded in Supplementary Discussion 5 with observation of grain boundary induced dislocations in Figure S10. Further investigation into the PEL of the a/2 {110} edge dislocation between two potential valleys in both NbMoTaW and W systems is depicted in Fig. 7 . The setup for these calculations is described in Supplementary Discussion 6. The analysis demonstrates that the energy barrier for dislocation motion is significantly higher in ligaments, both for NbMoTaW and W, compared to single crystal bulk configurations. This higher energy barrier for dislocation motion in ligaments may be related to the high surface-to-volume ratio in nanoscale ligaments in combination with tensile surface stress, leading to a compressive stress state in the core which makes dislocation glide more difficult 55 . Moreover, HEA systems demonstrate elevated energy barriers for both bulk and ligament structures compared to tungsten (W), offering a rationale for the remarkable ligament strength observed in nanoporous HEAs. The higher barriers in the PEL in ligaments can also be extrapolated to the calculated screw dislocations in prior investigation that demonstrates higher critical resolved shear stress compared to edge dislocation 26 . The higher energy barrier could be attributed to the entrapment of certain dislocation segments by atomic traps formed by specific local atomic configurations 56 . We conclude that the increased energy barrier in ligaments likely arises from localized stress concentrations and atomic-scale inhomogeneities, which effectively trap dislocations and impeded glide. This underscores the necessity for further exploration of the mechanisms governing dislocation motion in these materials. 3.5 Surface Energy Analysis The surface energies (Table 6 ) of the nanoporous and SC Al 0.1 CoCrFeNi systems was determined through a multi-step process. Initially, we computed the total energy of the nanoporous Al 0.1 CoCrFeNi system and compared it to an equivalent bulk system with the same atom count at specified temperatures. We then divided the energy difference by the surface area of the nanoporous HEA systems, as detailed in Supplementary Table 2. Surface area calculations utilized OVITO software with a probing radius of 4 Å 45 . Surface energy for different facets of SC are calculated by taking a bulk system of Al 0.1 CoCrFeNi with the preferential [100], [110], and [111] faces in the x direction of the simulation cell and determining the total energy with and without a vacuum layer of 2 nm. The energies of the particular facets are subtracted by the bulk and divided by the surface area. The values of the bulk surface energies are within reasonable values of other DFT calculations of the individual elements of Al 0.1 CoCrFeNi 57 . Notably, smaller ligament sizes displayed approximately twice the surface energy of larger ligaments. This trend is attributed to surface energy being influenced by the curvature of ligaments, which, in turn, is associated with an increased density of step edges with greater curvature 58 . An intriguing phenomenon observed is the increase in surface energy with rising temperatures for all relative densities. This behavior contrasts with bulk metals, where the surface energy typically decreases with increasing temperature 59 . Zhang et al. identified the role of surface energy in enhancing the strength of nanoporous materials 60 . While surface energy and surface stress are distinct concepts, their interplay influences dislocation behavior near surfaces 61 . HEAs may exhibit elevated surface energies without a direct correlation to surface stress. Elevated surface energy may promote surface reconstruction or faceting, potentially altering roughness and influencing dislocation mobility. However, the exact connection requires further analysis. The potential for reduced ligament coarsening in nanoporous Al 0.1 CoCrFeNi compared to single-element nanoporous materials, which could otherwise compromise the functional properties associated with high surface energy, might partially account for this effect 62 . This effect is consistent with observations in TiVNbMoTa and TaMoNbVNi, which show reduced coarsening rates at elevated temperatures compared to expectations 62 , 63 . A plausible explanation is the inhomogeneous potential energy landscape in HEAs, which can hinder surface diffusion via deep atomic traps, similar to their effect on dislocation mobility 63 . Further studies are needed to fully comprehend this phenomenon. Additional investigation of surface energy as a function of temperature and mechanical testing is discussed in Supplementary Discussion 7 with accompanying surface energy vs. strain graphs in Figure S11. 4. Discussion To highlight the mechanical properties of the nanoporous HEAs, a comparative analysis is conducted against other nanoporous materials (Fig. 8 ) 5 , 31 , 64 – 74 . The rationale for juxtaposing MD simulations of nanoporous materials with experimental data is explained in greater detail in Supplementary Discussion 8. The yield stresses exhibited by the nanoporous HEA systems surpasses that of other studied nanoporous materials with similar relative densities. This combination of high yield strength and low-density results in an exceptionally high specific yield strength, reaching values up to five times greater than typical nanoporous materials. For further comparisons between fcc and bcc nanoporous systems, Supplementary Discussion 9 provides additional insights. The superior performance of nanoporous HEAs may stem from a synergistic effects, possibly related to the dislocation starvation mechanism observed in nanostructured materials 75 , coupled with the high strength of HEAs related to the rough PEL which reduces the dislocation mobility. The latter contribute to the strain hardening of individual ligaments, microscopically addressing the macroscopic brittleness often associated with nanoporous materials by mitigating the risk of cascading single ligament failure 53 , 76 . The dislocation starvation mechanism is a common phenomenon occurring in nanoporous structures that contributes significantly to their high microscopic strength by effectively annihilating dislocations at surfaces. This mechanism results in low dislocation densities, thus reducing the plastic flow within ligaments characterized by a high surface-to-volume ratio. This preservation of ligament strength becomes evident when comparing the lower dislocation densities in nanoporous structures to those in nanocrystalline systems, as illustrated in Fig. 6 a. The nodes are observed to have the highest defect density in the nanoporous structures (as shown in Figure S6) due to their lower surface-to-volume ratio compared to the ligaments. Additionally, sluggish dislocation motion is another mechanism contributing to the high strength of nanoporous HEAs. This phenomenon arises from the naturally rough PEL characterizing HEAs, which features higher mean migration barriers compared to conventional alloy compositions. This rugged PEL is believed to be a primary factor behind the strain hardening effect observed in HEAs 78 . The sluggish dislocation motion influences the critical resolved shear stress (CRSS) of a/2 {110} edge dislocations in NbMoTaW, which is found to be approximately 320 MPa—significantly greater than the highest individual constituent value of 76 MPa in Mo 26 . This sluggish dislocation mechanism is also evident in the investigation of the potential energy landscape of edge dislocations in NbMoTaW, as depicted in Fig. 7 . While this particular composition exhibits sluggish diffusion for edge dislocations, other defect mechanisms, such as vacancy migration barriers, do not display such remarkable changes when compared to single-element materials. The influence of these mechanisms depends on the specific alloy composition 78 , 79 . This provides an opportunity to tune the mobility of defects and thus mechanical properties by engineering the PEL through the elemental compositions 55 . Compressive stress in the ligament core due to tensile surface stress may amplify the effects of the HEA’s rough PEL on dislocation motion, further enhancing mechanical strength. This internal compressive stress is demonstrated in bulk and 5 nm ligaments of NbMoTaW, as depicted in supplementary Figure S8, highlighting the enhanced resistance to deformation in ligaments. Sluggish dislocation motion in HEAs also gives rise to additional defect mechanisms that facilitate strain hardening, which varies depending on whether the crystal structure is fcc or bcc. In the case of Al 0.1 CoCrFeNi ligaments, stacking faults and partial dislocations move primarily in the direction until they come in contact with a surface, where one of the partial dislocations is annihilated. After this annihilation, the stacking fault becomes trapped within the ligament due to attractive forces between the second partial dislocation and the stacking fault as can be observed in Fig. 4 . This inhibition of partial dislocation-mediated stacking fault slip significantly enhances work hardening and may be a common characteristic in other fcc HEA nanoporous materials with low SFE 80 , 81 . This suggests a potential correlation between ligament size and average stacking fault size for fcc nanoporous HEAs. This differs from nanoporous and nanopillar gold systems, which exhibit a mix of dislocations, perfect dislocations, and perfect dislocation loops, facilitating deformation. These deformation mechanisms are accompanied by other larger defects such as twins and stacking faults 82 , 83 . Tables 1 and 2 indicate that the yield strength of the ligaments in nanoporous Al 0.1 CoCrFeNi increases with increasing ligament diameter, contrary to the common belief that smaller ligament sizes lead to higher strength 84 . This observation of smaller ligaments exhibiting decreased strength beyond a critical width was noted in nanopillar Titanium and is attributed to surface stress and thermal vibrations 85 , 86 . This suggests that the critical ligament size is between the smaller and larger ligament diameter sizes. Regarding the NbMoTaW system, the sluggish dislocation mechanism renders screw dislocation mobility and edge dislocation mobility much more comparable than in single-element materials 26 , 87 . The almost complete lack of dislocations in the ligaments may be attributed to the preferential nucleation of dislocations on surfaces oriented in the {111} direction 88 . This orientation is primarily found in the nodes of the nanoporous structure where ligaments connect, whereas the surface of the ligaments typically exhibits {100} or {110} orientations. Dislocation nucleation in the nodes, coupled with comparable velocities of screw and edge dislocations, elevates the chances of dislocation pileups within the nodes. This phenomenon enhances material strength through dislocation forest hardening, consequently diminishing the likelihood of dislocations exiting the nodes to distort the ligaments and contributing to strain hardening 89 . 5. Conclusions Our molecular Dynamics simulations provide insights into the mechanical behavior of nanoporous HEAs, demonstrating their ability to mitigate the inherent brittleness of nanoporous materials by enhancing ligament strength and reducing the likelihood of single ligament failure cascades. These materials exhibit a combination of high specific strength, low specific modulus, and low density that carves a distinctive niche in the specific modulus-strength phase space. The simulations indicate that the elevated strength of nanoporous HEAs stems from a dual mechanism involving dislocation starvation and sluggish diffusion of dislocations. The impact of reduced dislocation mobility depends on whether the HEA possesses a face-centered cubic (fcc) or body-centered cubic (bcc) structure. In fcc structures, specifically edge dislocations are slowed down, increasing the likelihood of stacking fault formation in ligaments facilitated by the low stacking fault energy. These stacking faults then get trapped due to surface bounding and are further constrained by ligament orientation, hindering partial dislocation motion. In bcc structures, sluggish dislocation motion contributes to dislocation pileup in the nodes of the nanoporous structure, resulting in dislocation forest hardening. The comparable mobility of screw and edge dislocations fosters the interaction of numerous dislocations in the node centers, creating substantial pileups and reducing the number of dislocations traveling into ligaments where they would be needed to support deformation. The absence of dislocation nucleation in ligaments is attributed to the preferential nucleation in the {111} direction of the surfaces in the nanoporous structure, common in nodes. Additionally, the study notes that tailoring ligament size and relative density yields diverse responses in defect concentrations and surface energy. Lower relative densities are associated with reduced defect densities and higher initial surface energies. Further scrutiny of these structures promises valuable insight into the unique mechanisms at play in nanoporous HEAs, specifically engineering compositions that facilitate higher strengths and potential energy landscapes that produce high energy barriers for defect motion . Declarations Acknowledgments: Work at LLNL was performed under the auspices of the U.S. Department of Energy by LLNL under contract No.DE-AC52-07NA27344. This work was supported by the Nuclear Regulatory Agency [grant number 31310019M0045]. Contributions: W. J.: Formal Analysis, Investigation, Software, Visualization, Writing-original draft, writing-review and editing. B. J.: Writing-review and editing, Conceptualization, Project Administration H. C.: Conceptualization, Funding Acquisition, Methodology, Project Administration, Resources, Supervision, Writing-review and editing Competing Interest: The authors declare no competing interest Data Availability: The data set used and/or analyzed during the current study are available upon reasonable request. Supplementary Information is available for this paper. References Xie, L. et al. Molecular dynamics simulation of Al–Co–Cr–Cu–Fe–Ni high entropy alloy thin film growth. Intermetallics 68, 78–86 (2016). 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System and Relative Density Average Ligament Size (nm) Al 0.1 CoCrFeNi 28.9% 4.23 土 1.53 Al 0.1 CoCrFeNi 28.7% 2.20 土 0.59 Al 0.1 CoCrFeNi 47.8% 4.86 土 1.62 Al 0.1 CoCrFeNi 46.9% 2.57 土 0.65 Al 0.1 CoCrFeNi 67.2% 7.33 土 1.66 Al 0.1 CoCrFeNi 65.9% 3.04 土 0.69 NbMoTaW 28.0% 4.79 土 1.20 NbMoTaW 46.9% 5.91 土 1.31 NbMoTaW 68.7% 7.95 土 1.37 Table 2 Young’s modulus and yield strength of the analyzed Al 0.1 CoCrFeNi HEA systems at 298 K. Experimental and other simulation results are in parentheses. All units are in GPa. System Young’s Modulus Tension Young’s Modulus Compression Yield Strength Tension Yield Strength Compression Bulk 200.75 (199, 203 90,91 ) 212.98 3.21 3.83 10 nm grain 257.55 285.55 5.67 9.13 20 nm grain 257.77 280.69 5.16 8.98 30 nm grain 288.33 319.82 5.62 10.23 28.9% density 4.23 nm 7.11 6.62 0.26 0.21 28.7% density 2.20 nm 7.72 3.80 0.23 0.10 47.8% density 4.86 nm 36.33 38.29 1.09 0.79 46.9% density 2.57 nm 31.75 29.83 0.83 0.54 67.2% density 7.33 nm 78.55 91.43 2.04 1.85 65.9% density 3.04 nm 74.06 81.17 1.78 1.47 Table 3 Young’s modulus and yield strength of the analyzed Al 0.1 CoCrFeNi HEA systems at 1273 K. All units are in GPa. System Young’s Modulus Tension Young’s Modulus Compression Yield Strength Tension Yield Strength Compression Bulk 188.13 191.21 2.03 2.68 10 nm grain 221.12 277.11 4.05 7.83 20 nm grain 229.07 240.97 4.12 7.23 30 nm grain 251.66 281.38 4.14 8.44 28.9% density 4.23 nm 3.78 5.20 0.12 0.06 28.7% density 2.20 nm - - - - 47.8% density 4.86 nm 24.90 26.32 0.73 0.40 46.9% density 2.57 nm 10.72 4.76 0.43 0.10 67.2% density 7.33 nm 62.34 67.14 1.49 1.34 65.9% density 3.04 nm 55.70 53.22 1.23 0.79 Table 4 Young’s modulus and yield strength of the analyzed NbMoTaW HEA systems at 298 K. Experimental and other simulation results are in parentheses. All units are in GPa. System Young’s Modulus Tension Young’s Modulus Compression Yield Strength Tension Yield Strength Compression Bulk 252.50 (245 92 ) 309.82 (300 93 ) 10.10 39.67 20 nm grain 191.29 205.16 6.12 8.21 30 nm grain 193.75 221.41 6.86 10.15 28.0% density 4.79 nm 7.45 6.15 0.41 0.43 46.5% density 5.91 nm 38.41 35.57 1.31 1.85 68.7% density 7.95 nm 110.27 122.42 3.97 5.63 Table 5 Young’s modulus and yield strength of the analyzed NbMoTaW HEA systems at 1273 K. All units are in GPa. System Young’s Modulus Tension Young’s Modulus Compression Yield Strength Tension Yield Strength Compression Bulk 190.05 246.99 7.37 21.24 20 nm grain 164.14 177.94 4.94 6.41 30 nm grain 170.77 200.97 6.15 8.40 28.0% density 4.79 nm 5.29 5.23 0.34 0.36 46.5% density 5.91 nm 30.29 28.18 1.21 1.47 68.7% density 7.95 nm 90.13 100.17 3.06 4.41 Table 6 Surface energy of nanoporous and SC Al 0.1 CoCrFeNi systems with varying temperatures. System Surface Energy 298 K (J/m 2 ) Surface Energy 600 K (J/m 2 ) Surface Energy 900 K (J/m 2 ) Surface Energy 1273 K (J/m 2 ) 28.9% density 4.23 nm 1.213 1.227 1.249 1.243 28.7% density 2.20 nm 2.072 - - - 47.8% density 4.86 nm 1.050 1.072 1.106 1.126 46.9% density 2.57 nm 1.601 1.638 1.657 1.729 67.2% density 7.33 nm 0.643 0.757 0.805 0.851 65.9% density 3.04 nm 1.350 1.417 1.465 1.508 SC [100] - 1.748 - 1.807 SC [110] - 1.924 - 1.945 SC [111] - 1.543 - 1.551 Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8998496","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":608648543,"identity":"798171da-81cf-4927-9010-fcbeb222febc","order_by":0,"name":"Jarod Worden","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAt0lEQVRIiWNgGAWjYBACxgYQycbAwM8AZRCvRbKBmUgtDDDDDQ4Qq4W5/Yzh44oymzzjG/kHGD6UHSbCYT05xoZnzqUVm91IZmCccY4YLQ252yQb2w4nbjtzmIGZt40YLf1vQVr+J27uAWr5S5SWGWBbDiRuYG9mYGYkTsv7z4YN55ITZxxvNjjYcy6dsBbD/rTEhw1ldon9zYwPH/wosyZCSwMS5wBh9UAgT5SqUTAKRsEoGNkAAAWoPPleLsL4AAAAAElFTkSuQmCC","orcid":"","institution":"Virginia Tech","correspondingAuthor":true,"prefix":"","firstName":"Jarod","middleName":"","lastName":"Worden","suffix":""},{"id":608648544,"identity":"dbf74350-ff73-46c8-a350-ba9f6d7adce1","order_by":1,"name":"Celine Hin","email":"","orcid":"","institution":"Virginia Tech","correspondingAuthor":false,"prefix":"","firstName":"Celine","middleName":"","lastName":"Hin","suffix":""},{"id":608648545,"identity":"e9eb4b44-096c-45a1-a5a4-1ef3f52c17d0","order_by":2,"name":"Juergen Biener","email":"","orcid":"","institution":"Lawrence Livermore National Laboratory","correspondingAuthor":false,"prefix":"","firstName":"Juergen","middleName":"","lastName":"Biener","suffix":""}],"badges":[],"createdAt":"2026-03-01 01:08:06","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8998496/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8998496/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105050045,"identity":"c16f322b-be20-49d4-b252-f21be8c05af6","added_by":"auto","created_at":"2026-03-20 10:03:41","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":89831,"visible":true,"origin":"","legend":"\u003cp\u003eWeight-normalized Young’s modulus and yield strength for different classes of materials including bulk materials and nanoporous materials \u003csup\u003e15\u003c/sup\u003e. The red circle denotes the general phase space occupied by single-component nanoporous materials based on available data. The red box represents the nanoporous HEA Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi and the blue box represents the nanoporous HEA NbMoTaW, highlighting their distinctive values in specific strength and specific modulus.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/0bcd78d3e907bd21fa9c7b37.jpg"},{"id":105050046,"identity":"c82ad187-be03-4a5b-bca9-c63d833e9c5b","added_by":"auto","created_at":"2026-03-20 10:03:41","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":301369,"visible":true,"origin":"","legend":"\u003cp\u003eThe simulation boxes of the HEA Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems for (a) single crystalline, (b) 30 nm polycrystalline, and nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi with relative densities of (c) 28.7, (d) 47.8, and (e) 67.2%.\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/677d968b4b52d164ea5baec9.jpg"},{"id":105050048,"identity":"0b884477-4f1c-4ad0-b0b5-7b59f1fd6667","added_by":"auto","created_at":"2026-03-20 10:03:41","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":177306,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain behavior of the studied HEA systems: (a) Compression (solid line) vs tension (dashed line) stress-strain curves obtained from nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi samples at 600 K compared to fully dense SC and NC (20 nm) test samples; (b) Effect of ligament diameter and density of nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi samples on compressive stress-strain behavior, (c) Effect of temperature on compression behavior of Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi : 600 K (solid line) vs. 1273 K (dashed line), (d) Comparison of compression behavior of Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi (solid line) and NbMoTaW (dashed line) at 600 K.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/2796e7bd543d63e298833297.jpg"},{"id":105562868,"identity":"3c18d1e9-0b64-44ec-bed6-db48e1c2f8e9","added_by":"auto","created_at":"2026-03-27 12:45:01","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":126703,"visible":true,"origin":"","legend":"\u003cp\u003eSnapshots of the\u0026nbsp; fracture process of a 4.29 nm ligament\u0026nbsp; from a nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi HEA sample (28.9% relative density) under tensile testing:\u0026nbsp; (a,b,c)\u0026nbsp; surface solid mesh outlining the atomic surface, (d,e,f) dislocations (marked in red) in the ligament, (g,h,i) twin boundaries, and (j,k,l) stacking faults. The strain values represented in the first, second, and third column are 22.5%, 26.25%, and 31.25%, respectively.\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/97c1675881971a11d99a2a71.jpg"},{"id":105562899,"identity":"10d1ae09-ecf3-4216-ae40-c2c9e75d396e","added_by":"auto","created_at":"2026-03-27 12:45:11","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":221592,"visible":true,"origin":"","legend":"\u003cp\u003eRepresentative fracture process of the 28.0% relative density nanoporous NbMoTaW with 3.83 nm ligaments under tensile stress. The dark blue spheres are surface atoms, light blue are vacancies, green are atoms outlining dislocations, yellow are twin boundary atoms, and red are unidentifiable atoms. The strain values are 14% (a), 15% (b), 18% (c), and 20% (d).\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/609149d0b44a71fe994c77c1.jpg"},{"id":105562928,"identity":"0a28a72d-28ac-4548-ae46-a47db19fb00b","added_by":"auto","created_at":"2026-03-27 12:45:17","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":159613,"visible":true,"origin":"","legend":"\u003cp\u003eDislocation (left) and stacking fault densities (right) vs strain for various\u0026nbsp; Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems: defect density vs. strain for (a,b) compression of NC and NP Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems at 600 K;\u0026nbsp; (c,d) for compression at 600 K (solid line) and 1273 K (dashed line); (e,f) compression (solid line) and tension (dashed line) at 600 K.\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/3914aac169541a5db6276bf1.jpg"},{"id":105562797,"identity":"9bc974bc-2fb3-4b2d-83e9-4a50ab9e15db","added_by":"auto","created_at":"2026-03-27 12:44:46","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":32615,"visible":true,"origin":"","legend":"\u003cp\u003ePotential energy landscape in units of eV per unit length of a/2\u0026lt;111\u0026gt;{110} edge dislocation in NbMoTaW and W. It is analyzed in both single crystal and 5 nm ligament diameter configurations.\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/9e95150765cc4870413cc68e.jpg"},{"id":105562618,"identity":"e3e917e2-ac3c-4e9d-8026-ab0b9336ad29","added_by":"auto","created_at":"2026-03-27 12:43:40","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":57185,"visible":true,"origin":"","legend":"\u003cp\u003eStress vs. Relative Density of different nanoporous materials given compression (circle) or tensile (triangle) stress. These include nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi, NbMoTaW, Au \u003csup\u003e5,65–67\u003c/sup\u003e, Cu \u003csup\u003e68\u003c/sup\u003e, Pt \u003csup\u003e69,70\u003c/sup\u003e, Ag \u003csup\u003e71\u003c/sup\u003e, Ni \u003csup\u003e77\u003c/sup\u003e, TiO\u003csub\u003e2\u003c/sub\u003e \u003csup\u003e72\u003c/sup\u003e, Au/Ag \u003csup\u003e73\u003c/sup\u003e, Cu/Zr \u003csup\u003e74\u003c/sup\u003e, and W \u003csup\u003e64\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/0ea44ae86a65227aeca8a82d.jpg"},{"id":105568734,"identity":"7822ee13-1d33-49af-8c4f-ad4744ce91d1","added_by":"auto","created_at":"2026-03-27 13:10:21","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2283968,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/16bacbfc-b792-460e-a71b-df767b0e293b.pdf"},{"id":105050051,"identity":"19160256-f4b6-460a-ab7f-69b157d05b09","added_by":"auto","created_at":"2026-03-20 10:03:41","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":6689038,"visible":true,"origin":"","legend":"","description":"","filename":"SupplementaryMaterials3926.docx","url":"https://assets-eu.researchsquare.com/files/rs-8998496/v1/dfad044d5b7f42e45916781e.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Nanoporous High Entropy Alloys: Overcoming Brittleness Through Strain Hardening","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eNanoporous metals, featuring random bicontinuous ligament-pore structures, possess unique mechanical properties, including high specific strength, good ductility, and energy absorption, all while maintaining lower densities compared to typical bulk metals \u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e,\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e. However, their practical application is hindered by brittleness under tension, as fracturing of the weakest ligaments can lead to a cascading fracture of adjacent ligaments \u003csup\u003e\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e,\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e. For example, nanoporous gold, studied extensively through simulations and experiments, shows macroscopic brittle behavior triggered by single ligament failure \u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e,\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eTo address the brittleness challenge associated with cascading ligament failure, we investigate High Entropy Alloys (HEAs), known for substantial strain hardening, in nanoporous structures \u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e,\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e. HEAs use equiatomic concentration of multiple elements to increase system entropy, stabilizing disordered solid-solution phases and preventing brittle intermetallic phases \u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e,\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e. They offer high strength, hardness, strain hardening, and resistance to fatigue, creep, and wear, overcoming challenges in single-element nanoporous materials \u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e,\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e. Our hypothesis is that incorporating HEAs into bicontinuous nanoporous structures will enhance mechanical properties like strength and toughness while maintaining low densities compared to traditional metals and alloys. This makes them promising for applications requiring superior mechanical performance combined with low densities and high thermal tolerance, benefiting industries such as nuclear, automotive, aerospace, and aeronautics compared to commonly used industrial materials \u003csup\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e,\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e (further investigation of these materials in automotive and nuclear are explained in Supplementary Discussion 1). In the realm of high energy absorption and impact resistance, nanoporous HEAs are expected to showcase appealing mechanical properties that optimize energy deflection and allowable stress \u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e. However, their application is hindered due to their novelty and lack of mechanical experimental and computational data.\u003c/p\u003e \u003cp\u003eThis study employs Molecular Dynamics (MD) simulations to investigate the mechanical behavior of nanoporous High Entropy Alloys (HEAs) under uniaxial compression and tensile testing. For comparison, we also performed simulations on fully dense single crystalline (SC) and polycrystalline HEAs. The selection of Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi and NbMoTaW HEAs is based on their distinctive single disordered face-centered cubic (fcc) and body-centered cubic (bcc) phase \u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e. The simulation outcomes validate our hypothesis that nanoporous HEAs exhibit unique mechanical properties, especially significantly higher specific strength compared to single-element nanoporous materials (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The high specific strength can be attributed to the sluggish dislocation mechanism observed in the studied HEAs that enhances the strength of individual ligaments by enabling strain hardening mechanisms such as dislocation forest hardening, effectively addressing the macroscopic brittleness associated with nanoporous materials.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"2. Methods","content":"\u003cp\u003eThe molecular dynamics (MD) simulations were conducted using the open-source code LAMMPS, employing a three-dimensional cell with periodic boundary conditions, with dimensions of 44.625 nm x 44.625 nm x 44.625 nm \u003csup\u003e16\u003c/sup\u003e. The bulk cell was generated by populating it with 7,812,500 fcc and 3,906,250 bcc structural atoms. The concentrations of Al, Co, Cr, Fe, and Ni (or Nb, Mo, Ta, W) were randomly assigned using Atomsk \u003csup\u003e\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e. Polycrystalline systems were created by applying Voronoi Tessellation to the bulk cell with random grain orientations. Three distinct polycrystalline structures with grain sizes of 10 nm, 20 nm, and 30 nm were generated. The methodology for producing nanoporous HEAs followed a similar approach to that used for simulating the creation of bicontinuous nanoporous gold through dealloying methods and are described in Supplementary Discussion 2 \u003csup\u003e18\u003c/sup\u003e. The types of systems in this study are depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eAl\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi system was simulated using an Embedded Atom Method (EAM) potential, which has also been employed in previous MD investigations to study nanoindentation tests and plastic deformation \u003csup\u003e\u003cspan additionalcitationids=\"CR20\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u003c/sup\u003e. The stacking fault energy (SFE) of the single-crystal (SC) system was calculated to be 32.8 mJ/m\u003csup\u003e2\u003c/sup\u003e, showing good agreement with the experimental value of 30 mJ/m\u003csup\u003e2\u003c/sup\u003e and aligning with other computational findings \u003csup\u003e\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e,\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e. This low stacking fault energy of the potential is similar to other fcc HEAs such as the cantor alloy CoCrFeMnNi where the stacking fault energy is also found to be exceedingly low (\u0026lt;\u0026thinsp;35 mJ/m\u003csup\u003e2\u003c/sup\u003e) using both experimental and computational methods, much lower than that of single element materials \u003csup\u003e\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e,\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e .\u003c/p\u003e \u003cp\u003eFor the NbMoTaW system, a Spectral Neighbor Analysis Potential (SNAP) potential was applied, which is capable of accurately reproducing elastic constants \u003csup\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e. To analyze defects within the system, the Dislocation Analysis Tool was employed to assess dislocation densities, dislocation types, and the formation of Stacking Faults/Twin Boundaries \u003csup\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e. The ligament diameter measurement involved identifying specific atom positions as the center of an expanding sphere. This sphere grew radially by half the lattice parameter. We assessed if a part of the sphere contacted the ligament's surface by examining interactions between the sphere and local atom configuration. If contact was confirmed on one side, we checked the opposite side. When both sides contacted the surface, we calculated the distance as the ligament section's diameter, contributing to the average ligament diameter calculation for the entire system. Average ligament size is displayed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e with error.\u003c/p\u003e \u003cp\u003eMolecular dynamics simulations were employed to conduct mechanical compression and tensile tests at temperatures of 298 K, 600 K, and 1273 K. Initially, a Nose/Hoover thermostat and barostat (NPT) ensemble was utilized for a thermalization period of 100.0 ps. This thermalization phase took place at the specified temperatures and atmospheric pressure until the system achieved a stable state with respect to lattice parameters and energy. Following the thermalization process, the NPT ensemble was re-employed, and a strain rate of 10\u003csup\u003e9\u003c/sup\u003e s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e was applied to deform the simulation box along the x direction. Compression tests were conducted in the range of 0\u0026ndash;70% strain, while tensile tests covered the 0\u0026ndash;50% strain range.\u003c/p\u003e \u003cp\u003eThe use of high strain rates in MD simulations is a recognized limitation, as realistically slow strain rates require prohibitively long simulation times. High strain rates generally result in higher ultimate stresses, making direct comparisons with experimental data more challenging. Our study, along with others in the field, examines various strain rates and finds that differences between lower strain rates are relatively minor, indicating a diminishing return in the reduction of ultimate stress with decreasing strain rate \u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e,\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e. This trend is highlighted in Supplementary Table\u0026nbsp;1, which presents ultimate stress calculations for different tensile strain rates in the 28.0% relative density nanoporous NbMoTaW system. Notably, at a lower strain rate, a minimal decrease in ultimate stress is observed (0.02 GPa), whereas a significantly higher stress is recorded at an increased strain rate (0.12 GPa). The chosen strain rate effectively balances the need for faster simulation times while minimizing its impact on stress results.\u003c/p\u003e"},{"header":"3. Results","content":"\u003cp\u003eThe results section analyzes the mechanical properties involving the HEA systems through stress-strain curve analysis, ligament fracture analysis, defect analysis, and surface energy analysis. The analysis focuses on random placement of atomic species for the different HEA systems, however there is the potential for surface segregation due to the relatively large surface area. This can be analyzed in future investigations.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e\u003cb\u003e3.1\u003c/b\u003e \u003cb\u003eStress-Strain curve analysis\u003c/b\u003e\u003c/h2\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(a-d) displays stress vs. strain curves at 600 K (more results from tests at 298/600/1273 K are shown in Figure \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e and Figure S2) for Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi and NbMoTaW while the corresponding yield strength and Young's modulus data are summarized in Tables\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e to \u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. The highest strain values displayed for the nanoporous systems under compression correspond to strains before full densification, which occurs at approximately 70%, 60%, and 50% strain for the 28.9%, 47.8%, and 67.2% porosity samples, respectively (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eFor both compression and tension, and irrespective of temperature, the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems exhibit linear elastic behavior within the low strain range up to 2%. For non-nanoporous structures, a transition to a non-linear elastic deformation behavior can be observed with further increasing strain until the ultimate stress point is reached at 5% strain. In the case of SC systems, the yield strength is attained early, in contrast to the ultimate strength, which is achieved at higher strains, indicating the strain-hardening mechanism of the HEAs \u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e. This is observed in the compressive yield strength displayed in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. For the nanocrystalline (NC) systems, both under compression and tension, the stress drops significantly after reaching the yield peak, followed by a plateau region that continues up to the highest strain values. Notably, in the NC systems, compression tests yield higher stress values compared to tensile tests (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea). This observed compression-tension asymmetry in yield strength for SC and polycrystalline Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi has also been reported in other HEA systems \u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e, possibly reflecting distinct deformation mechanisms such as twinning in compression and slip in tension \u003csup\u003e\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe stress-strain response of nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi under both compression and tensile test conditions is more complex when compared to SC and NC systems (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea). Tensile tests conducted on nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi exhibit a more distinct yield peak, followed by a shorter plateau phase, before the stress decreases with further increasing strain. These trends align with those observed in other FeCrNiCuCo fcc HEA nanoporous systems \u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e,\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/sup\u003e. In compression tests, only a small yield peak was observed for the highest relative densities, followed by an immediate stress plateau and a subsequent region where stress gradually increases with strain. This response aligns with what is commonly observed in many nanoporous systems under uniaxial compression in both experimental and simulated settings for single element and HEA systems \u003csup\u003e\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e,\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e\u003c/sup\u003e. Comparing the effect of relative density and ligament size on compression stress-strain curves (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb) reveals that at any given strain within the plateau region, higher relative density correlates with higher stress. Only the 67.2% relative density / 7.33 nm ligament diameter sample shows a distinct yield peak under compression; with increasing strain the effect of ligament size on the stress response becomes smaller and smaller. This mirrors patterns observed in FeCrNiCuCo nanoporous fcc HEA systems, where it was found that relative density exerted a more significant influence on mechanical properties compared to ligament size, particularly when the ligament size is very small (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb) \u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe mechanical properties of Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems were also found to be temperature-dependent, with elevated temperatures resulting in reduced stresses across all systems (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec and Fig. \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e). The most pronounced stress reduction with temperature occurs in the 28.9% relative density sample. Conversely, the SC sample shows minimal or even reversed temperature dependence beyond the yield point. Additionally, for systems with similar relative densities, smaller ligaments lead to a more pronounced reduction in stress under compression at high temperatures (1273 K vs 298 K and 600 K) compared to larger ligament systems at higher temperatures, as indicated in Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. This effect is particularly evident for the 50% relative density samples, as shown in Fig. \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003ea-c. The softening of smaller ligament systems with increasing temperatures may reflect the limited thermal stability of the smallest ligaments, as evidenced by the collapse of the ligament structure in the 28.7% relative density system with 2.2 nm ligaments during the initial relaxation phase. Under tensile test conditions, the np-Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi samples exhibit increased plasticity with increasing temperature (Fig.\u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003ed-f).\u003c/p\u003e \u003cp\u003eThe compression behavior of the NbMoTaW systems (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ed) followed similar stress-strain trends as the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems, including an initial steep increase in stress and a more or less pronounced yield peak at around 5% strain. This is followed by a plateau region where stress remain relatively constant with increasing strain. In the case of tensile strain (Figure S2 in Supplementary Materials), the SC NbMoTaW system displayed stress oscillations after the failure strain, which corresponded to oscillating atomic positions of the atoms following system rupture. Increasing temperature also resulted in decreased stress across all systems, with a more significant reduction observed for SC and NC structures.\u003c/p\u003e \u003cp\u003eOverall, the NbMoTaW systems exhibited superior mechanical properties, including higher yield strength and Young\u0026rsquo;s modulus, compared to the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ed and S2). An exception is observed in the 20 nm NC samples, where NbMoTaW shows lower stress than Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi. This may reflect differences in defect interactions or simulation setup and warrants further investigation. The higher yield strength and Young\u0026rsquo;s modulus of the NbMoTaW samples can be attributed to the intrinsic properties of the individual elements of the NbMoTaW system, which inherently possess greater yield strength, ultimate strength, and Young\u0026rsquo;s modulus in both compressive and tensile environments than the elements in the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi \u003csup\u003e\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eTables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e present the yield stress and Young\u0026rsquo;s modulus for the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems at both 298 K and 1273 K. Notably, the Young\u0026rsquo;s modulus obtained from tensile mechanical tests for Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi closely aligns with experimental data and is consistent with previous molecular dynamics simulation results \u003csup\u003e\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e,\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e,\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e,\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e. Additionally, the yield stress and strain results obtained from both tensile and compression tests show strong agreement with other molecular dynamics studies \u003csup\u003e\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e,\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e\u003c/sup\u003e. Tables\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e in the main text provide the yield stress and Young\u0026rsquo;s modulus for NbMoTaW. The compressive modulus also closely corresponds with experimental data and aligns with other simulation results obtained under compression conditions \u003csup\u003e\u003cspan additionalcitationids=\"CR41\" citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/sup\u003e, further supporting the reliability of the interatomic potential used.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAdditionally, the nanoporous systems exhibit a tension-compression asymmetry (see Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), a phenomenon previously observed in other nanoporous systems where compression demonstrates higher yield strength and Young's modulus \u003csup\u003e\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/sup\u003e. This asymmetry varies depending on the type of high entropy alloy and crystal structure. For instance, the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi nanoporous structure exhibits a higher compression yield strength compared to tension, while the NbMoTaW nanoporous system displays a higher yield strength under compression conditions. This discrepancy can be attributed to distinct defect mechanisms involved in plastic deformation. In bcc structures, common defect mechanisms include twinning and anti-twinning on the {211} slip planes, as well as non-planar core structures of screw dislocations \u003csup\u003e\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/sup\u003e. On the other hand, in fcc structures, the asymmetry may be due to biased surface stresses and differing slip planes during tensile and compressive yielding \u003csup\u003e\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Fracture Analysis in Tensile Tests of Nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi\u003c/h2\u003e \u003cp\u003eFracture analysis was performed exclusively on specimens subjected to tensile testing as no fractures were observed during compression tests due to predominant ligament bending \u003csup\u003e\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e\u003c/sup\u003e. This analysis focused on a segment of the nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi system with a relative density of 28.9% and ligament size of 4.23 nm, tested under tensile conditions at 1273 K. Surface topography, dislocations, twin boundaries, and stacking faults were analyzed using the Dislocation Extraction Algorithm (DXA) and Paraview, as depicted in Fig.\u0026nbsp;4a-l \u003csup\u003e45,46\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eDislocations are initially present during the early stages of fracture but gradually diminish as ligament failure progresses (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ed,e,f). Twin boundaries form at the onset of fracture and persist throughout ligament failure; however, they disintegrate as the ligament ruptures (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eg,h,i). Stacking faults are the most prevalent defects observed, remaining dominant in the ligament at all stages of the fracture process (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ej,k,l). This behavior can be attributed to the low stacking fault energy characteristic of the HEA system.\u003c/p\u003e \u003cp\u003eThis distinguishes the fracture processes of HEA ligaments from those observed in ligaments of nanoporous gold where dislocations appear to play a pivotal role in the ligament deformation \u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e,\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e,\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e. The absence of distinct dislocations in the HEA ligaments during fracturing may be attributed to two factors: firstly, the hindrance of dislocation motion due to lattice distortion caused by the different atomic species in the HEA material, which is expected to reduce the formation of larger dislocations; and secondly, the high density of stacking faults impeding dislocations \u003csup\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e\u003c/sup\u003e. The sluggish dislocation motion of the HEA is corroborated by investigating the higher Potential Energy Landscape (PEL) of the \u0026frac12;\u0026lt;110\u0026gt;{111} edge dislocation which has higher energy barriers compared to single element PELs as shown in Supplementary Figure S3 and discussed in Supplementary Discussion 3. Given that the primary dislocation type is partial dislocations, neighboring partial dislocations formed from dissociated edge dislocations in the ligament create highly stable stacking faults between them, stemming from impaired dislocation mobility and low SFE in the nanoporous HEA system. As a result, the number of independent dislocations decreases, and the primary mechanism for dislocation elimination in the ligaments is the breaking of stacking faults \u003csup\u003e\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e. The presence of stacking faults and partial dislocations within ligaments seems to be a prevalent mechanism in FeCrNiCuCo fcc HEA nanoporous structures, suggesting potential similarities in defect responses across these materials \u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e,\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/sup\u003e. As dislocation motion typically constitutes a primary mechanism for plastic deformation, the absence of dislocations implies the existence of an alternative process for ligament fracturing during uniaxial tension \u003csup\u003e\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e,\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u003c/sup\u003e. Owing to the substantial prevalence of stacking faults in the ligament structure during fracture, the principal process bears a close resemblance to the twinning mechanism observed in fcc metallic materials, which is particularly pronounced during severe plastic deformation \u003csup\u003e\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Fracture Analysis in Tensile Tests of Nanoporous NbMoTaW\u003c/h2\u003e \u003cp\u003eTo investigate defects in the bcc structure, the bcc defect analysis (BDA) tool is used instead of the DXA tool, as justified in Supplementary Discussion 4 with accompanying Supplementary Figure S9 \u003csup\u003e47\u003c/sup\u003e. Below 14% strain, no defects were observed within the ligaments; however, at higher strain levels, twin boundaries nucleate alongside dislocations concentrated around the edges (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea). While the BDA tool does not classify dislocation types, it is important to note that bcc systems commonly exhibit a/6\u0026thinsp;\u0026lt;\u0026thinsp;111\u0026gt; partial dislocations, which play a crucial role in twinning - a key deformation mechanism in bcc Fe nanopillars \u003csup\u003e\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/sup\u003e. These partial dislocations facilitate the movement of twin boundaries toward the ligament\u0026rsquo;s center (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb), where they merge into complex dislocation bundles (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ec). This interaction is essential for atom removal at the ligament\u0026rsquo;s notch, ultimately leading to fracture (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ed)\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Defect Analysis in Nanoporous Systems\u003c/h2\u003e \u003cp\u003eThe effects of test temperature and geometry on dislocation and stacking fault concentrations in the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi system are summarized in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, while total defect densities are presented in Figures S4 and S5. In both geometries, the dislocation density of nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ea,c,e) exhibits a sharp increase up to 10% strain, followed by either a plateau (tensile tests) or a continuous increase (compression tests) for higher strains. The stacking fault density of the nanoporous systems exhibits a steadier growth that is also more influenced by temperature and testing type (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb,d,f).\u003c/p\u003e \u003cp\u003eHigher temperatures generally lead to a reduction in defect density due to faster dislocation annihilation and elevated stacking fault energy (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ec and \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ed) \u003csup\u003e\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e,\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e\u003c/sup\u003e. Defect annihilation is further facilitated by the higher surface area-to-volume ratio in lower relative density systems \u003csup\u003e\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e\u003c/sup\u003e. Our systems with smaller ligaments, on the other hand, exhibit higher dislocation densities, possibly related due to the combined effect of more frequent defect pileups at junctions (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ea, \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ed, and S6) and the higher number of junctions in smaller ligament systems \u003csup\u003e\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e\u003c/sup\u003e. However, at elevated temperatures, defect migration and annihilation start to dominate over defect pileup \u003csup\u003e\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e,\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e. Compared to the stacking fault density, the dislocation density is generally less sensitive to temperature and test geometry variations.\u003c/p\u003e \u003cp\u003eDuring compression at lower temperatures, the stacking fault density consistently increases with increasing strain, with higher relative density nanoporous systems surpassing the stacking fault densities observed in the NC systems (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb). For the lower density (\u0026lt;\u0026thinsp;50%) np-Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems, the stacking fault densities for the same strain values remain below those of the NC systems for both 600 K and 1273 K (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb,d). Dislocations and stacking fault densities correlate with stress-strain curve trends, especially under tension, where the reduction of defect density aligns with decreasing stress (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb, \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ee, and \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ef).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn contrast, none of the NbMoTaW nanoporous systems exhibited stacking fault defects. The absence of stacking faults can be attributed to the high stacking fault energy characteristic of bcc crystal structures and was found to be ~\u0026thinsp;1300 mJ/m\u003csup\u003e2\u003c/sup\u003e in NbMoTaW, magnitudes lower than the Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems \u003csup\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e,\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u003c/sup\u003e. The dislocation density in tension is lower compared to compression (Figure S7). This disparity could be attributed to tension-compression dislocation glide velocity asymmetry, where tensile deformation could result in higher dislocation velocities, facilitating interactions and annihilation at the ligament surface \u003csup\u003e\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e,\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e\u003c/sup\u003e. Another explanation lies in the elongation of ligaments, which increases the surface-to-volume ratio, consequently elevating the dislocation annihilation rate. Temperature has a less pronounced effect on dislocation density in NbMoTaW compared to Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi. The nanoporous NbMoTaW systems exhibit minimal changes in dislocation density with varying temperatures under tensile deformation, whereas there is a small but systematic increase in dislocation densities at higher temperatures under compression (Figure S7). The velocity of dislocations increases with higher temperature, but the formation of dislocations also becomes faster. Depending on the specific HEA, the dislocation velocity may be significantly impeded, particularly in bcc HEA materials with their slow screw dislocation motion \u003csup\u003e\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e\u003c/sup\u003e. Additionally, while screw dislocations in bcc systems are typically influenced by temperature, NbMoTaW exhibits a consistent dislocation velocity across different temperature ranges due to the presence of thermal kinks and cross kinks \u003csup\u003e\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e\u003c/sup\u003e. Under compression, it appears that the effect of higher temperatures on the rate of dislocation formation dominates over the effect of higher dislocation velocity and the associated higher annihilation rates. Further analysis of dislocation types is expounded in Supplementary Discussion 5 with observation of grain boundary induced dislocations in Figure S10.\u003c/p\u003e \u003cp\u003eFurther investigation into the PEL of the a/2\u0026thinsp;\u0026lt;\u0026thinsp;111\u0026gt;{110} edge dislocation between two potential valleys in both NbMoTaW and W systems is depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. The setup for these calculations is described in Supplementary Discussion 6. The analysis demonstrates that the energy barrier for dislocation motion is significantly higher in ligaments, both for NbMoTaW and W, compared to single crystal bulk configurations. This higher energy barrier for dislocation motion in ligaments may be related to the high surface-to-volume ratio in nanoscale ligaments in combination with tensile surface stress, leading to a compressive stress state in the core which makes dislocation glide more difficult \u003csup\u003e\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e\u003c/sup\u003e. Moreover, HEA systems demonstrate elevated energy barriers for both bulk and ligament structures compared to tungsten (W), offering a rationale for the remarkable ligament strength observed in nanoporous HEAs. The higher barriers in the PEL in ligaments can also be extrapolated to the calculated screw dislocations in prior investigation that demonstrates higher critical resolved shear stress compared to edge dislocation \u003csup\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e. The higher energy barrier could be attributed to the entrapment of certain dislocation segments by atomic traps formed by specific local atomic configurations \u003csup\u003e\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e\u003c/sup\u003e. We conclude that the increased energy barrier in ligaments likely arises from localized stress concentrations and atomic-scale inhomogeneities, which effectively trap dislocations and impeded glide. This underscores the necessity for further exploration of the mechanisms governing dislocation motion in these materials.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Surface Energy Analysis\u003c/h2\u003e \u003cp\u003eThe surface energies (Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e) of the nanoporous and SC Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems was determined through a multi-step process. Initially, we computed the total energy of the nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi system and compared it to an equivalent bulk system with the same atom count at specified temperatures. We then divided the energy difference by the surface area of the nanoporous HEA systems, as detailed in Supplementary Table\u0026nbsp;2. Surface area calculations utilized OVITO software with a probing radius of 4 \u0026Aring; \u003csup\u003e45\u003c/sup\u003e. Surface energy for different facets of SC are calculated by taking a bulk system of Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi with the preferential [100], [110], and [111] faces in the x direction of the simulation cell and determining the total energy with and without a vacuum layer of 2 nm. The energies of the particular facets are subtracted by the bulk and divided by the surface area. The values of the bulk surface energies are within reasonable values of other DFT calculations of the individual elements of Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi \u003csup\u003e\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e\u003c/sup\u003e. Notably, smaller ligament sizes displayed approximately twice the surface energy of larger ligaments. This trend is attributed to surface energy being influenced by the curvature of ligaments, which, in turn, is associated with an increased density of step edges with greater curvature \u003csup\u003e\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eAn intriguing phenomenon observed is the increase in surface energy with rising temperatures for all relative densities. This behavior contrasts with bulk metals, where the surface energy typically decreases with increasing temperature \u003csup\u003e\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e\u003c/sup\u003e. Zhang et al. identified the role of surface energy in enhancing the strength of nanoporous materials \u003csup\u003e\u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e\u003c/sup\u003e. While surface energy and surface stress are distinct concepts, their interplay influences dislocation behavior near surfaces \u003csup\u003e\u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e61\u003c/span\u003e\u003c/sup\u003e. HEAs may exhibit elevated surface energies without a direct correlation to surface stress. Elevated surface energy may promote surface reconstruction or faceting, potentially altering roughness and influencing dislocation mobility. However, the exact connection requires further analysis. The potential for reduced ligament coarsening in nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi compared to single-element nanoporous materials, which could otherwise compromise the functional properties associated with high surface energy, might partially account for this effect \u003csup\u003e\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e\u003c/sup\u003e. This effect is consistent with observations in TiVNbMoTa and TaMoNbVNi, which show reduced coarsening rates at elevated temperatures compared to expectations \u003csup\u003e\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e,\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e\u003c/sup\u003e. A plausible explanation is the inhomogeneous potential energy landscape in HEAs, which can hinder surface diffusion via deep atomic traps, similar to their effect on dislocation mobility \u003csup\u003e\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e\u003c/sup\u003e. Further studies are needed to fully comprehend this phenomenon. Additional investigation of surface energy as a function of temperature and mechanical testing is discussed in Supplementary Discussion 7 with accompanying surface energy vs. strain graphs in Figure S11.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Discussion","content":"\u003cp\u003eTo highlight the mechanical properties of the nanoporous HEAs, a comparative analysis is conducted against other nanoporous materials (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e) \u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e,\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e,\u003cspan additionalcitationids=\"CR65 CR66 CR67 CR68 CR69 CR70 CR71 CR72 CR73\" citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR74\" class=\"CitationRef\"\u003e74\u003c/span\u003e\u003c/sup\u003e. The rationale for juxtaposing MD simulations of nanoporous materials with experimental data is explained in greater detail in Supplementary Discussion 8. The yield stresses exhibited by the nanoporous HEA systems surpasses that of other studied nanoporous materials with similar relative densities. This combination of high yield strength and low-density results in an exceptionally high specific yield strength, reaching values up to five times greater than typical nanoporous materials. For further comparisons between fcc and bcc nanoporous systems, Supplementary Discussion 9 provides additional insights.\u003c/p\u003e \u003cp\u003eThe superior performance of nanoporous HEAs may stem from a synergistic effects, possibly related to the dislocation starvation mechanism observed in nanostructured materials \u003csup\u003e\u003cspan citationid=\"CR75\" class=\"CitationRef\"\u003e75\u003c/span\u003e\u003c/sup\u003e, coupled with the high strength of HEAs related to the rough PEL which reduces the dislocation mobility. The latter contribute to the strain hardening of individual ligaments, microscopically addressing the macroscopic brittleness often associated with nanoporous materials by mitigating the risk of cascading single ligament failure \u003csup\u003e\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e,\u003cspan citationid=\"CR76\" class=\"CitationRef\"\u003e76\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe dislocation starvation mechanism is a common phenomenon occurring in nanoporous structures that contributes significantly to their high microscopic strength by effectively annihilating dislocations at surfaces. This mechanism results in low dislocation densities, thus reducing the plastic flow within ligaments characterized by a high surface-to-volume ratio. This preservation of ligament strength becomes evident when comparing the lower dislocation densities in nanoporous structures to those in nanocrystalline systems, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ea. The nodes are observed to have the highest defect density in the nanoporous structures (as shown in Figure S6) due to their lower surface-to-volume ratio compared to the ligaments.\u003c/p\u003e \u003cp\u003eAdditionally, sluggish dislocation motion is another mechanism contributing to the high strength of nanoporous HEAs. This phenomenon arises from the naturally rough PEL characterizing HEAs, which features higher mean migration barriers compared to conventional alloy compositions. This rugged PEL is believed to be a primary factor behind the strain hardening effect observed in HEAs \u003csup\u003e\u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e78\u003c/span\u003e\u003c/sup\u003e. The sluggish dislocation motion influences the critical resolved shear stress (CRSS) of a/2\u0026thinsp;\u0026lt;\u0026thinsp;111\u0026gt;{110} edge dislocations in NbMoTaW, which is found to be approximately 320 MPa\u0026mdash;significantly greater than the highest individual constituent value of 76 MPa in Mo \u003csup\u003e26\u003c/sup\u003e. This sluggish dislocation mechanism is also evident in the investigation of the potential energy landscape of edge dislocations in NbMoTaW, as depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. While this particular composition exhibits sluggish diffusion for edge dislocations, other defect mechanisms, such as vacancy migration barriers, do not display such remarkable changes when compared to single-element materials. The influence of these mechanisms depends on the specific alloy composition \u003csup\u003e\u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e78\u003c/span\u003e,\u003cspan citationid=\"CR79\" class=\"CitationRef\"\u003e79\u003c/span\u003e\u003c/sup\u003e. This provides an opportunity to tune the mobility of defects and thus mechanical properties by engineering the PEL through the elemental compositions \u003csup\u003e\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e\u003c/sup\u003e. Compressive stress in the ligament core due to tensile surface stress may amplify the effects of the HEA\u0026rsquo;s rough PEL on dislocation motion, further enhancing mechanical strength. This internal compressive stress is demonstrated in bulk and 5 nm ligaments of NbMoTaW, as depicted in supplementary Figure S8, highlighting the enhanced resistance to deformation in ligaments.\u003c/p\u003e \u003cp\u003eSluggish dislocation motion in HEAs also gives rise to additional defect mechanisms that facilitate strain hardening, which varies depending on whether the crystal structure is fcc or bcc. In the case of Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi ligaments, stacking faults and partial dislocations move primarily in the \u0026lt;\u0026thinsp;211\u0026thinsp;\u0026gt;\u0026thinsp;direction until they come in contact with a surface, where one of the partial dislocations is annihilated. After this annihilation, the stacking fault becomes trapped within the ligament due to attractive forces between the second partial dislocation and the stacking fault as can be observed in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. This inhibition of partial dislocation-mediated stacking fault slip significantly enhances work hardening and may be a common characteristic in other fcc HEA nanoporous materials with low SFE \u003csup\u003e\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e,\u003cspan citationid=\"CR81\" class=\"CitationRef\"\u003e81\u003c/span\u003e\u003c/sup\u003e. This suggests a potential correlation between ligament size and average stacking fault size for fcc nanoporous HEAs. This differs from nanoporous and nanopillar gold systems, which exhibit a mix of dislocations, perfect dislocations, and perfect dislocation loops, facilitating deformation. These deformation mechanisms are accompanied by other larger defects such as twins and stacking faults \u003csup\u003e\u003cspan citationid=\"CR82\" class=\"CitationRef\"\u003e82\u003c/span\u003e,\u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e83\u003c/span\u003e\u003c/sup\u003e. Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e indicate that the yield strength of the ligaments in nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi increases with increasing ligament diameter, contrary to the common belief that smaller ligament sizes lead to higher strength \u003csup\u003e\u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e84\u003c/span\u003e\u003c/sup\u003e. This observation of smaller ligaments exhibiting decreased strength beyond a critical width was noted in nanopillar Titanium and is attributed to surface stress and thermal vibrations \u003csup\u003e\u003cspan citationid=\"CR85\" class=\"CitationRef\"\u003e85\u003c/span\u003e,\u003cspan citationid=\"CR86\" class=\"CitationRef\"\u003e86\u003c/span\u003e\u003c/sup\u003e. This suggests that the critical ligament size is between the smaller and larger ligament diameter sizes.\u003c/p\u003e \u003cp\u003eRegarding the NbMoTaW system, the sluggish dislocation mechanism renders screw dislocation mobility and edge dislocation mobility much more comparable than in single-element materials \u003csup\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e,\u003cspan citationid=\"CR87\" class=\"CitationRef\"\u003e87\u003c/span\u003e\u003c/sup\u003e. The almost complete lack of dislocations in the ligaments may be attributed to the preferential nucleation of dislocations on surfaces oriented in the {111} direction \u003csup\u003e\u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e88\u003c/span\u003e\u003c/sup\u003e. This orientation is primarily found in the nodes of the nanoporous structure where ligaments connect, whereas the surface of the ligaments typically exhibits {100} or {110} orientations. Dislocation nucleation in the nodes, coupled with comparable velocities of screw and edge dislocations, elevates the chances of dislocation pileups within the nodes. This phenomenon enhances material strength through dislocation forest hardening, consequently diminishing the likelihood of dislocations exiting the nodes to distort the ligaments and contributing to strain hardening \u003csup\u003e\u003cspan citationid=\"CR89\" class=\"CitationRef\"\u003e89\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e"},{"header":"5. Conclusions","content":"\u003cp\u003eOur molecular Dynamics simulations provide insights into the mechanical behavior of nanoporous HEAs, demonstrating their ability to mitigate the inherent brittleness of nanoporous materials by enhancing ligament strength and reducing the likelihood of single ligament failure cascades. These materials exhibit a combination of high specific strength, low specific modulus, and low density that carves a distinctive niche in the specific modulus-strength phase space. The simulations indicate that the elevated strength of nanoporous HEAs stems from a dual mechanism involving dislocation starvation and sluggish diffusion of dislocations. The impact of reduced dislocation mobility depends on whether the HEA possesses a face-centered cubic (fcc) or body-centered cubic (bcc) structure. In fcc structures, specifically edge dislocations are slowed down, increasing the likelihood of stacking fault formation in ligaments facilitated by the low stacking fault energy. These stacking faults then get trapped due to surface bounding and are further constrained by ligament orientation, hindering partial dislocation motion. In bcc structures, sluggish dislocation motion contributes to dislocation pileup in the nodes of the nanoporous structure, resulting in dislocation forest hardening. The comparable mobility of screw and edge dislocations fosters the interaction of numerous dislocations in the node centers, creating substantial pileups and reducing the number of dislocations traveling into ligaments where they would be needed to support deformation. The absence of dislocation nucleation in ligaments is attributed to the preferential nucleation in the {111} direction of the surfaces in the nanoporous structure, common in nodes. Additionally, the study notes that tailoring ligament size and relative density yields diverse responses in defect concentrations and surface energy. Lower relative densities are associated with reduced defect densities and higher initial surface energies. Further scrutiny of these structures promises valuable insight into the unique mechanisms at play in nanoporous HEAs, specifically engineering compositions that facilitate higher strengths and potential energy landscapes that produce high energy barriers for defect motion .\u003c/p\u003e "},{"header":"Declarations","content":"\u003cp\u003eAcknowledgments:\u003c/p\u003e\n\u003cp\u003eWork at LLNL was performed under the auspices of the U.S. Department of Energy by LLNL under contract No.DE-AC52-07NA27344.\u003c/p\u003e\n\u003cp\u003eThis work was supported by the Nuclear Regulatory Agency [grant number 31310019M0045].\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eContributions:\u003c/p\u003e\n\u003cp\u003eW. J.: Formal Analysis, Investigation, Software, Visualization, Writing-original draft, writing-review and editing.\u003c/p\u003e\n\u003cp\u003eB. J.: Writing-review and editing, Conceptualization, Project Administration\u003c/p\u003e\n\u003cp\u003eH. C.: Conceptualization, Funding Acquisition, Methodology, Project Administration, Resources, Supervision, Writing-review and editing\u003c/p\u003e\n\u003cp\u003eCompeting Interest:\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interest\u003c/p\u003e\n\u003cp\u003eData Availability:\u003c/p\u003e\n\u003cp\u003eThe data set used and/or analyzed during the current study are available upon reasonable request.\u003c/p\u003e\n\u003cp\u003eSupplementary Information is available for this paper.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eXie, L. \u003cem\u003eet al.\u003c/em\u003e Molecular dynamics simulation of Al\u0026ndash;Co\u0026ndash;Cr\u0026ndash;Cu\u0026ndash;Fe\u0026ndash;Ni high entropy alloy thin film growth. \u003cem\u003eIntermetallics\u003c/em\u003e 68, 78\u0026ndash;86 (2016).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBrach, S., Dormieux, L., Kondo, D. \u0026amp; Vairo, G. 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Mech. - ASolids\u003c/em\u003e 105042 (2023).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAlagarsamy, K. \u003cem\u003eet al.\u003c/em\u003e Mechanical Properties of High Entropy Alloy Al\u0026lt;background-color:#CCCCFF;subdirection:rtl;\u0026gt;0.1\u0026lt;/background-color:#CCCCFF;subdirection:rtl;\u0026gt;CoCrFeNi for Peripheral Vascular Stent Application. \u003cem\u003eCardiovasc. Eng. Technol.\u003c/em\u003e 7, 448\u0026ndash;454 (2016).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLiu, C., Yang, Y. \u0026amp; Xia, Z. Deformation mechanism in Al\u0026lt;background-color:#CCCCFF;subdirection:rtl;\u0026gt;0.1\u0026lt;/background-color:#CCCCFF;subdirection:rtl;\u0026gt;CoCrFeNi Σ3(111)[11̄0] high entropy alloys \u0026ndash; molecular dynamics simulations. \u003cem\u003eRSC Adv.\u003c/em\u003e 10, 27688\u0026ndash;27696 (2020).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHu, Y. L. \u003cem\u003eet al.\u003c/em\u003e First-principle calculation investigation of NbMoTaW based refractory high entropy alloys. \u003cem\u003eJ. Alloys Compd.\u003c/em\u003e 827, 153963 (2020).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eClark, P. Computation Aided Design Of Multicomponent Refractory Alloys With A Focus On Mechanical Properties. (University of Mississippi, 2016).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"},{"header":"Tables","content":"\u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eAverage ligament size with the respective system and relative density.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSystem and Relative Density\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAverage Ligament Size (nm)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAl\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi 28.9%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.23 土 1.53\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAl\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi 28.7%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.20 土 0.59\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAl\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi 47.8%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.86 土 1.62\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAl\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi 46.9%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.57 土 0.65\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAl\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi 67.2%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.33 土 1.66\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAl\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi 65.9%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.04 土 0.69\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNbMoTaW 28.0%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.79 土 1.20\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNbMoTaW 46.9%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.91 土 1.31\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNbMoTaW 68.7%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.95 土 1.37\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eYoung\u0026rsquo;s modulus and yield strength of the analyzed Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi HEA systems at 298 K. Experimental and other simulation results are in parentheses. All units are in GPa.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSystem\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Compression\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYield Strength Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eYield Strength Compression\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBulk\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e200.75 (199, 203 \u003csup\u003e90,91\u003c/sup\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e212.98\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3.83\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e257.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e285.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.67\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e9.13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e257.77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e280.69\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8.98\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e30 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e288.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e319.82\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10.23\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.9% density 4.23 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e6.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.21\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.7% density 2.20 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e47.8% density 4.86 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e36.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e38.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.79\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e46.9% density 2.57 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e31.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e29.83\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.83\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.54\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e67.2% density 7.33 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e78.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e91.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.85\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e65.9% density 3.04 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e74.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e81.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.47\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eYoung\u0026rsquo;s modulus and yield strength of the analyzed Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi HEA systems at 1273 K. All units are in GPa.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSystem\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Compression\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYield Strength Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eYield Strength Compression\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBulk\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e188.13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e191.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.68\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e221.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e277.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e7.83\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e229.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e240.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e7.23\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e30 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e251.66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e281.38\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e8.44\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.9% density 4.23 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.06\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.7% density 2.20 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e47.8% density 4.86 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e24.90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e26.32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.40\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e46.9% density 2.57 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.76\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e67.2% density 7.33 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e62.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e67.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.34\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e65.9% density 3.04 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e55.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e53.22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.79\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eYoung\u0026rsquo;s modulus and yield strength of the analyzed NbMoTaW HEA systems at 298 K. Experimental and other simulation results are in parentheses. All units are in GPa.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSystem\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Compression\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYield Strength Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eYield Strength Compression\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBulk\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e252.50 (245 \u003csup\u003e92\u003c/sup\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e309.82 (300 \u003csup\u003e93\u003c/sup\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e39.67\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e191.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e205.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8.21\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e30 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e193.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e221.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10.15\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.0% density 4.79 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e6.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.43\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e46.5% density 5.91 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e38.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.31\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.85\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e68.7% density 7.95 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e110.27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e122.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5.63\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eYoung\u0026rsquo;s modulus and yield strength of the analyzed NbMoTaW HEA systems at 1273 K. All units are in GPa.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSystem\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eYoung\u0026rsquo;s Modulus Compression\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYield Strength Tension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eYield Strength Compression\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBulk\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e190.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e246.99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e21.24\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e164.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e177.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.41\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e30 nm grain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e170.77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e200.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8.40\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.0% density 4.79 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.36\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e46.5% density 5.91 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e30.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.47\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e68.7% density 7.95 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e90.13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e100.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4.41\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSurface energy of nanoporous and SC Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi systems with varying temperatures.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSystem\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSurface Energy 298 K (J/m\u003csup\u003e2\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSurface Energy 600 K (J/m\u003csup\u003e2\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSurface Energy 900 K (J/m\u003csup\u003e2\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eSurface Energy 1273 K (J/m\u003csup\u003e2\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.9% density 4.23 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.213\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.227\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.249\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.243\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28.7% density 2.20 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.072\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e47.8% density 4.86 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.050\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.072\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.106\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.126\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e46.9% density 2.57 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.601\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.638\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.657\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.729\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e67.2% density 7.33 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.643\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.757\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.805\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.851\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e65.9% density 3.04 nm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.417\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.465\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.508\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSC [100]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.748\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.807\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSC [110]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.924\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.945\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSC [111]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.543\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.551\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"npj-computational-materials","isNatureJournal":false,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"npjcompumats","sideBox":"Learn more about [npj Computational Materials](http://www.nature.com/npjcompumats/)","snPcode":"41524","submissionUrl":"https://mts-npjcompumats.nature.com/","title":"npj Computational Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8998496/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8998496/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBicontinuous nanoporous materials possess remarkable mechanical properties, such as higher specific strength and lower specific modulus compared to fully dense materials combined with low densities and high specific surface areas. Unfortunately, their practical application is hindered by inherent macroscopic brittleness, mainly due to cascading ligament failure under tension. To address this limitation, we investigate whether high entropy alloys, recognized for their outstanding strength and strain hardening properties, can mitigate nanoporous material\u0026rsquo;s inherent brittleness. Molecular dynamics simulations of nanoporous Al\u003csub\u003e0.1\u003c/sub\u003eCoCrFeNi and NbMoTaW reveal a dual mechanism involving dislocation starvation and sluggish dislocation motion, resulting in specific strength values 5 to 10 times higher than those of single-element nanoporous materials, and a resilience against thermal degradation. Strain hardening, driven by sluggish dislocations, effectively prevents failure of the weakest ligaments under tensile stress in face-centered cubic architectures by trapping stacking faults in the ligaments and dislocation forest hardening in the nodes of body-centered cubic structures, demonstrating their potential to shape the next generation of high strength, low density materials.\u003c/p\u003e","manuscriptTitle":"Nanoporous High Entropy Alloys: Overcoming Brittleness Through Strain Hardening","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-20 10:03:36","doi":"10.21203/rs.3.rs-8998496/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-13T08:06:08+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-08T01:42:43+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-06T10:14:39+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-05T17:24:11+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"11000910489610851437023585472727949936","date":"2026-03-24T14:49:46+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"256939860769478823350168158035668707880","date":"2026-03-20T05:37:16+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"311270440742557165384351188653926008382","date":"2026-03-18T15:40:24+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"302461321153319505974880738930674985658","date":"2026-03-18T06:51:32+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-03-18T04:43:10+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-03-15T01:10:15+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-03-10T04:50:16+00:00","index":"","fulltext":""},{"type":"submitted","content":"npj Computational Materials","date":"2026-03-01T00:53:00+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"npj-computational-materials","isNatureJournal":false,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"npjcompumats","sideBox":"Learn more about [npj Computational Materials](http://www.nature.com/npjcompumats/)","snPcode":"41524","submissionUrl":"https://mts-npjcompumats.nature.com/","title":"npj Computational Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"73b9fea6-b77a-4dbc-aab6-1c5ff684a374","owner":[],"postedDate":"March 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[{"id":64767254,"name":"Physical sciences/Engineering"},{"id":64767255,"name":"Physical sciences/Materials science"}],"tags":[],"updatedAt":"2026-04-13T08:12:50+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-20 10:03:36","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8998496","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8998496","identity":"rs-8998496","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}