Coalescence of Al_0.3CoCrFeNi polycrystalline high-entropy alloy in hot-pressed sintering: a molecular dynamics and phase-field study

Abstract

Existing hot sintering models based on molecular dynamics focus on single-crystal alloys. This work proposes a new multiparticle model based on molecular dynamics to investigate coalescence kinetics during the hot-pressed sintering of a polycrystalline Al_0.3CoCrFeNi high-entropy alloy. The accuracy and effectiveness of the multiparticle model are verified by a phase-field model. Using this model, it is found that when the particle contact zones undergo pressure-induced evolution into exponential power creep zones, the occurrences of phenomena, such as necking, pore formation/filling, dislocation accumulation/decomposition, and particle rotation/rearrangement are accelerated. Based on tensile test results, Young’s modulus of the as-sintered Al_0.3CoCrFeNi high-entropy alloy is calculated to be 214.11 ± 1.03 GPa, which deviates only 0.82% from the experimental value, thus further validating the feasibility and accuracy of the multiparticle model.

Introduction

The Al_0.3CoCrFeNi high-entropy alloy has attracted interest for a long time because of its exceptional tensile strength and remarkable plasticity^{1,2,3,4,5,6,7}. Regrettably, the alloy does not exhibit unique advantages compared with Ni-based alloys or 2nd-generation advanced high-strength steels regarding tensile properties⁸. Therefore, researchers are dedicating efforts to enhancing its tensile properties. Powder metallurgy can be used to prepare high-entropy alloys with uniform microstructures and refined grains, improving their performance⁹. By utilizing spark plasma sintering, the ultimate tensile strength of the Al_0.3CoCrFeNi high-entropy alloy can be increased by approximately 20.2% compared to its as-cast counterpart^5,10. Zhang et al.¹¹ investigated the relationships among the microstructure evolution, phase formation, and tensile mechanical properties of Al_0.3CoCrFeNi high-entropy alloy samples prepared using various ball milling strategies combined with spark plasma sintering and hot extrusion. Their findings revealed that the consolidation of mechanical alloying powders to obtain a single-phase face-centered cubic (FCC) structure resulted in an exceptional combination of yield strength (950 MPa) and elongation (17.3%). However, observing powder particle evolution during the sintering process and the interactions among various microstructures (such as stacking faults, twinning, and dislocations) using existing equipment remains challenging.

Molecular dynamics is a powerful computational tool that can simulate various processes such as stretching, shearing, twisting, indenting, and sintering^12,13,14,15. Sintering primarily involves the linking of grains through thermal activation, which enhances alloy strength. At the atomic scale, this is manifested by the growth of necks in the contact region between powder particles^16,17. Despite the progress made in the field of molecular dynamics for sintering, the current models are primarily limited to binary sphere or sphere-plate models and mainly focus on the single-crystal hot sintering of traditional alloys^18,19,20,21. In recent studies, multiparticle sintering has provided new insights for particles containing two rotation manners^22,23. With the focus shifting from traditional alloys to high-entropy alloys, Sreeramagiri et al.²⁴ utilized a four-sphere model to investigate the mechanisms involved in the solid-state sintering of AlCoCrFeNi high-entropy alloy and performed tensile tests on a sintered sample. However, the thermal consolidation process of high entropy alloys primarily relies on pressure. Therefore, it is crucial to consider the influence of pressure on the coalescence kinetics during sintering. Thus, a hot-pressed sintering simulation of polycrystalline high-entropy alloys is an attractive frontier field.

To comprehend the coalescence kinetics in the hot-pressed sintering process of polycrystalline high-entropy alloys, this work proposes a multiparticle model and investigates the hot-pressed sintering behavior of Al_0.3CoCrFeNi high-entropy alloy. To align the results to experiments, the design strategy of this model introduces powder particles with different particle sizes (1–8 nm) and verifies the model effectiveness and the simulation accuracy through phase-field simulations. The polycrystalline formation process and coalescence kinetics in the sintered state are deeply interpreted by tracking the spherical particles. Simultaneously, the correlation between the sintered structure and the tensile properties is revealed. This research offers a new way to optimize process parameters and microstructures in high-entropy alloy sintering and promotes the development of computational materials science related to the hot-pressed sintering of multiple principal alloy nanoparticles.

Results

Particle evolution of the hot-pressed sintering process

For clarity, the hot-pressed sintering process is divided into four stages, namely stage I (300–1116 K, Fig. 1), stage II (0–2 GPa at 1116 K, Fig. 2), stage III (1116–1373 K at 2 GPa, 2–3 GPa at 1373 K, and the maintenance stage at 1373 K and 3 GPa, Fig. 3), and stage IV (cooldown and depressurization stage, Fig. 3). The 2D evolution process is displayed in Supplementary Movie 1. For visual display, the {110} surface is intercepted for research.

Fig. 1: The distribution of nanoparticles and the change of particle boundaries when the temperature increases from 300 K to 1116 K.

a–c The distribution of nanoparticles. d–f the change of particle boundaries. The blue and red represent the particle boundary before and after sintering.

Fig. 2: Pores distribution and particle boundaries to grain boundaries transition when the pressure rises from 0 GPa to 1.5 GPa at 1116 K.

a–d Pores distribution. e–h Particle boundaries to grain boundaries transition. The blue and red represent the particle boundary before and after sintering.

Fig. 3: Distribution of powder particles under different sintering conditions.

a Heating up to 1373 K. b Pressurizing at 1373 K to 3 GPa. c Holding stage at 1373 K and 3 GPa. d Powder particles after cooldown and depressurization. e Comparison of grain boundaries after sintering and grain boundaries in the holding stage. f Sample after sintering.

After relaxation at 300 K in stage I, the particles exist in two forms: one is distributed in an isolated state, while the other exhibits early necking due to point contact. When T reaches 560 K, sintering begins as isolated particles form new point contacts through atomic diffusion, and initial point contacts gradually transition to surface contacts. This is due to powder particles approaching the range of action of the surface force field, resulting in contact, bonding, and rearrangement between them. When T = 1116 K, multiple particles are connected through surface contacts, and atoms diffuse frequently. The analysis of particle boundary changes is presented in Fig. 1d–f. Most particle boundaries remain coincident with the initial ones at 572 K, indicating that the activation energy required for atomic motion has yet to be reached. This law is consistent with the findings of Zhu et al.²⁵, wherein the initial contact surfaces expand as the temperature increases. Previous research has demonstrated that the driving force of the sintering process is inversely proportional to the particle size¹⁸. The small nanoparticles undergo significant displacement and particles with the same orientation merge (region I in Fig. 1e). This finding confirms that surface and grain boundary energy plays a crucial role in enhancing the driving force for sintering in multiparticle systems. As the temperature increases, a further increase in the grain boundary area can be observed within region II in Fig. 1f. Compared with 300 K, the volume of the model shrinks by 28.81% at 1116 K. At this point, the density of the as-sintered body is 49.05%.

The distribution of powder particles and the evolution of particle boundaries during stage II are presented in Fig. 2. Due to the synergistic effect of temperature and pressure, some atoms deviate from their original positions and form an amorphous structure. Thus, the main analysis focuses on the changes in particle boundary contours. Remarkably, pressure promotes the sintering process of the Al_0.3CoCrFeNi high entropy alloy, which can be divided into stages of connected pores and phase-isolated pores. When P = 0.5 GPa, although the bonding and rearrangement of powder particles are accelerated, the pores remain interconnected. With a further increase in pressure, particle boundaries continue to develop and expand, resulting in the rapid compression of spherical powder particles into polygonal shapes, as shown in Fig. 2b–d. Meanwhile, the solid-gas surface area decreases rapidly, and independent pores exist at the intersection of grain boundaries. Aggregates of powder particles gradually transform into grain agglomerates, indicating the transformation of particle boundaries into grain boundaries. In Fig. 2d, the pores at the grain boundary inside the sintered body shrink, and the sintered body gradually becomes denser. Compared with the initial sintered body, the volume of the model shrinks by 61.32% at 1116 K and 2.0 GPa. At this point, the density of the as-sintered body is 91.70%. This demonstrates that applying pressure can accelerate the densification process of the as-sintered body.

In stages III and IV (Fig. 3), pore defects persist at T = 1373 K. In Fig. 3b, the pores begin to shrink. Significant pore defects are eliminated at 1373 K and 2 GPa, while tiny pores remain (as shown in Fig. 3c). During the cooldown and depressurization stage, the pores vanish and the sintered body is densified. Figure 3e illustrates that most amorphous grains disappeared after annealing, revealing distinct grain boundaries. The polycrystalline sample obtained after final sintering is shown in Fig. 3f. Compared with the initial sintered body, the volume of the model shrinks by 64.79% after sintering. In the third stage, the density of the as-sintered body increases to 95.26%. Upon increasing the pressure from 2 GPa to 3 GPa, the density increases to 96.40%. Maintaining 1373 K and 3 GPa results in a density of 97.15%. The density reaches 97.43% as the temperature decreases from 1373 K to 1116 K and the pressure drops from 3 GPa to 2 GPa. Subsequently, as the temperature further decreases from 1116 K to 300 K and the pressure decreases from 2 GPa to 0 GPa, the density increases to 99.85%. Maintaining 300 K and 0 GPa results in a density of 99.88%.

The hot-pressed sintering behavior of the Al_0.3CoCrFeNi high-entropy alloy is studied by the phase-field method, as shown in Fig. 4. The order parameter field depicts the distribution of particles with varying orientations. According to the evolution of the order parameter field over time, the sintering process can be roughly divided into three stages: (i) particles are bonded to form a sintering neck at the initial stage of sintering, (ii) the sintering neck grows or shrinks to form grain boundaries in the middle stage of sintering, and (iii) stable grain boundary formation occurs through later-stage grain growth. As shown in Fig. 4a, the particles that are close to each other preferentially form sintering necks under compressive stress. In the middle stage of sintering, driven by the chemical free energy, grains overcome the energy barrier and grow rapidly, mainly manifested by the elongation of the sintering neck to form grain boundaries, as shown in Fig. 4b. As depicted in Fig. 4c, d, the pores subsequently vanished, and the grains underwent fusion. In the later sintering stage, small grains are completely swallowed up, and regular and stable grain boundaries are formed under the joint action of chemical free energy and elastic energy (Fig. 4e). This phenomenon is consistent with the molecular dynamics simulation results and directly validates the efficacy of our proposed multiparticle model.

Fig. 4: Phase-field simulation for the order parameter evolution insintering of Al_0.3CoCrFeNi high-entropy alloy powder particles over time.

a Powder particles evolution diagram at t*=100, the sintering neck is represented with lines of light blue color. b–e Evolution of powder particles at different steps.

Figure 5 depicts the stress distributions on XX, YY, and XY in sintering by molecular dynamics and phase-field methods. Figs. 5a1–a4 present the results of molecular dynamics simulations, while Figs. 5b1–d4 showcase phase-field simulations. The latter reveals that during the initial stage of sintering, a significant concentration of stress occurs around the sintering neck and pores due to uneven particle distribution among particles and pores, as shown in the dark areas of Fig. 5b1, c1, and d1. In the middle sintering stage, the stress distribution is uniformly upward-trending during the engulfment process of small grains by large grains. As sintering progresses to form stable grain boundaries, the stress distribution becomes uniform (as shown in Fig. 5b5, c5, and d5). Molecular dynamics simulation results demonstrate that post-sintered samples exhibit a relatively uniform stress distribution (Figs. 5a1–a3), which aligns with the phase-field simulation results. The shear strain mainly exists at the grain boundaries (as shown in Fig. 5a4).

Fig. 5: Stress distribution during sintering.

a1–a4 Molecular dynamics simulation results. b1–b5 The stress field in x-direction from phase-field simulation. c1–c5 The stress field in y-direction from phase-field simulation. d1–d5 The stress field in shear stress direction from phase-field simulation.

Coalescence kinetics during the hot-pressed sintering process

The evolution process of the microstructure and dislocations can be seen in Supplementary Movie 1 and Supplementary Movie 2. The {110} plane is selected for intuitive analysis of the mechanism research. Figure 6 reflects the sintering characteristics of stage I, i.e., point contact → surface contact (the formation of necks and Shockley dislocations) → neck growth (the formation of BCC structures and HCP structures). A single close-packed hexagonal layer represents a twin boundary (TB), while two adjacent HCP layers denote an intrinsic stacking fault (ISF), and an extrinsic stacking fault (ESF) is indicated by the inclusion of an FCC layer between two HCP layers^26,27. The ISF is formed inside the powder particles, as shown in region II of Fig. 6c. The mean square displacement depicted in Fig. 7a indicates the atomic diffusion rate, with a steeper slope observed in region A, indicating strong surface diffusion. Under the action of temperature, surface atoms diffuse to form necking. The microstructural content changes during the sintering process are illustrated in Fig. 7b, where statistical analysis reveals that 78% of the FCC phase and 22% of other structures (i.e., particle boundaries outside the FCC phase) exist within region A. As the temperature increases, the proportion of HCP and other crystal structures increases while that of FCC structures decreases. This phenomenon can be attributed to temperature providing the driving force for the atoms to leave their ordered crystal structure, leading to part of the FCC structure forming amorphous structures.

Fig. 6: Microstructure evolution and dislocation evolution during the first sintering stage.

a–c Microstructure evolution. d–e Dislocation evolution. TB is the twin boundary, ISF is the intrinsic stacking fault, and PB is the particle boundary.

Fig. 7: The mean square displacement, phase content and dislocation density evolution during sintering process.

a The mean square displacement of atoms during sintering. b The changing trend of the content of each structure during sintering. c The changing trend of dislocation density during sintering. Area A is Stage I, area B is Stage II, C, D, and E are Stage III, and F, G, and H are Stage IV.

From the perspective of dislocation evolution in Fig. 6d, dislocations start to slip when the stress generated by particle collision at the particle boundary exceeds the critical decomposition shear stress. Contacting particles reduce grain boundary energy through rotation or rearrangement during this process. When contacting particles possess identical orientations, particle boundaries within contact areas vanish, and particles commence fusion. The contact area (neck) exhibits stacking faults and dislocations, as shown in Fig. 6e. During the evolution process, entangled dislocations (mainly including 1/2 < 110> (perfect) dislocations, 1/6 < 112> (Shockley) dislocations, and other dislocations) become a new obstacle to slip further. The evolution of the dislocation density with time is presented in Fig. 7c, where it can be observed that Shockley dislocations dominate during the sintering process, and the total dislocation density is almost equal to the Shockley dislocation density in stage I.

The microstructural and dislocation evolutionary characteristics during stage II are shown in Fig. 8. Pressure further increases the contact zone to form a power exponential creep zone, where powder particles coalesce and create a sintered network. When the pressure reaches 1 GPa, the powder particles adhere together, and dislocations are emitted from particle boundaries into their interiors, resulting in intrinsic stacking faults. Usually, atoms fill the pores by diffusing into the neck region, driven by the extra surface free energy induced by the neck region²⁸. Our study provides novel insights showing that the coupling effect of pressure and plastic deformation accelerates the filling of pores. At 1.5 GPa, powder particles gradually formed grains while particle boundaries transformed into grain boundaries. Combined with Fig. 7a, it exhibits a steeper slope than area A, indicating that the diffusion rate of atoms under pressure is greater than that of free surface diffusion²⁹. Meanwhile, atoms inevitably undergo body or grain boundary diffusion in this process, as shown in Fig. 2.

Fig. 8: Microstructure and dislocation evolution in stage II of the sintering process.

a–d Microstructure evolution. e–f Dislocation evolution. In the figures, TB is the twin boundary, ISF is the intrinsic stacking fault, ESF is the extrinsic stacking fault, and GB is the grain boundary.

There are two typical pores observed in Fig. 8a, b. The first type of pore (H1) is generated by particle misorientation. Due to the large size of the pores, neither surface diffusion nor plastic flow can effectively fill these pores under external pressure within a short time, as shown in Fig. 8a–d. The second type of pore (H2) is believed to result from three or four powder particles contacting each other and can be quickly filled by external pressure and plastic flow, as shown in Fig. 8b–d. Figure 8e shows the evolution of HCP structures and dislocations during pressurization. The pressure accelerates the aggregation of powder particles, resulting in numerous twins, stacking faults, and HCP phases at both grain boundaries and within grains. Notably, the movable dislocations dragging stacking faults are obstructed by the immovable dislocations. One end of a slippable dislocation connects with a nonslippable dislocation to form a stair–rod dislocation, i.e., Lomer–Cottrell dislocation locks in Fig. 8f. The reaction to form a stair–rod dislocation is as follows:

$${\rm{I}}:\frac{a}{6}\left[1\bar{2}\bar{1}\right]+\frac{a}{6}\left[\bar{2}11\right]\longrightarrow \frac{a}{6}\left[\bar{1}\bar{1}0\right]$$

(1)

$${\rm{II}}:\frac{a}{6}\left[2\bar{1}1\right]+\frac{a}{6}\left[\begin{array}{c}2\bar{1}\bar{1}\end{array}\right]\to \frac{a}{6}\left[\begin{array}{c}110\end{array}\right]$$

(2)

For stage III of the sintering process in Fig. 9, the temperature increase intensifies atom diffusion and accelerates grain fusion. Grain boundary fusion reduces the pore size compared with Fig. 8d. The MSD curve exhibits a steep slope until 112 ps, indicating that plastic flow under constant pressure hastens pore filling. Other structures are reduced, corresponding to the C region in Fig. 7b. Increasing the pressure to 3 GPa results in denser sintered bodies, but some pores are still near the boundary. In the holding stage, the pores disappear and the grain boundaries become apparent. According to Fig. 7a, the MSD curve exhibits a gentle slope in the E region, representing surface diffusion. At this time, the FCC structure increases while other structures decrease, with the HCP structure remaining unchanged. Eventually, the content of the HCP structure (HCP phase, ISF, TB, and ESF) reached 10%. The snapshots in Fig. 9d show the evolution of the HCP structure and dislocations, dislocations and twins are mainly concentrated in the grain boundary region.

Fig. 9: Microstructure and dislocation evolution in stage III of the sintering process.

a–c Microstructure evolution. d Dislocation evolution. In the figures, TB is the twin boundary, ISF is the intrinsic stacking fault, ESF is the extrinsic stacking fault, and GB is the grain boundary.

Figure 10 corresponds to stage IV, where the FCC and HCP phases increase as the amorphous phase decreases, corresponding to regions F and G in Fig. 7b. The final sintering results (Fig. 10d) reveal that a grain is an aggregate of multiple powder particles, with more HCP structures formed within the grain. Region H in Fig. 7b indicates the holding stage at 300 K, and each structure’s content remains unchanged. According to statistics, the content of other structures (grain boundary and amorphous phase) accounts for 19.68%, 2.32% lower than that of the initial particle boundaries. The contents of the FCC, HCP, and BCC structures are 63.14%, 15.08%, and 2.10%, respectively. Based on the above analyses, it can be inferred that FCC structures mainly transform into HCP and BCC structures during the sintering process. The formation of HCP structures leads to a significant increase in dislocation density in the sintered body, as shown in the H region in Fig. 7c. The numerous dislocation entanglements distributed along the grain boundaries—including those formed by Shockley partial dislocations, perfect dislocations, Hirth dislocations, and Frank dislocations; Lomer–Cottrell dislocation locks formed by Shockley partial and Stair–rod dislocations; and the tangles of Shockley partial dislocations alone—all contribute to a significant increase in the work hardening rate for the Al_0.3CoCrFeNi high entropy alloy.

Fig. 10: Microstructure and dislocation evolution in stage IV of the sintering process.

a–c Microstructure evolution. d Dislocation evolution. In the figure, TB is the twin boundary, ISF is the intrinsic stacking fault, ESF is the extrinsic stacking fault, and GB is the grain boundary.

Particle rotation behavior during the hot-pressed sintering process

The rotation behavior of the powder particles can be elucidated by analyzing the atomic displacement vectors within them. In Fig. 11, the arrows indicate the direction of atom movement, with their heads and tails representing the current and initial positions, respectively. The length of the arrow corresponds to the magnitude of atomic displacement. At a temperature of 504 K, smaller powder particles exhibit finer particle sizes and larger specific surface areas, resulting in more significant displacement vectors and higher rotation angles, as shown in Fig. 11a. The rotation behavior of the particle can be classified into two categories, i.e., (i) particles with initial misorientation reordering via dislocation gliding through adjacent particles³⁰, and (ii) particles without misorientation also undergoing rotation²². As shown in the enlarged view in Fig. 11a, the first type of rotation occurs between particle 1 and particle 3, where driven atoms are strongly inclined to both sides along the width of the neck. When particle 1 and particle 3 form a neck, the surface stress of the neck makes the generated dislocation pass through particle 1 or 3. The second type of rotation exists between particle 1 and particle 2. As there are no misorientations among particles, particles 1 and 2 crystallize on the interface. The annihilation of the surface energy driving force transforms the neck region from an ordered structure to an amorphous state to reduce the curvature (Fig. 6) and the increase of thermal activation energy prompts more significant displacement. Subsequently, these particles will rotate for optimal crystalline positions through dislocations nucleus passing through one of the connecting particles. Owing to the asymmetrical atomic motion, particles deviate from perfect spheres during thermal motion. As depicted by the magnified view in Fig. 11b, atoms within particles 1 and 2 spontaneously migrate toward the neck region along the width direction. Comparing Fig. 11c, d, it can be observed that pressure accelerates atomic motion and changes its direction, leading to nonspherical shapes in powder particles. At the same time, the pressure intensifies particle fusion and promotes grain formation.

Fig. 11: Displacement vectors of atoms during sintering.

a, b Displacement vectors of atoms at different temperatures (504 K and 912 K). c, d Displacement vectors of atoms at different pressures (0.25 GPa and 0.75 GPa). The red block diagram is an enlarged view of the corresponding area.

Discussions

A tensile test was conducted on the as-sintered sample to investigate the influence of sintered microstructure on the tensile strength of Al_0.3CoCrFeNi polycrystalline high-entropy alloy. Figure 12 illustrates the stress–strain, phase fraction, and dislocation density variations of sintered and ideal polycrystalline structures subjected to uniaxial tensile loading. The tensile stress–strain curve exhibits a linear elastic regime at low strains, followed by a nonlinear elastic region and continuous deformation leading to softening and steady-state plastic flow. The elastic modulus is determined by linear fitting using the stress curve at less than 0.02 strain. The Young’s modulus of the as-sintered and ideal polycrystalline structures are 214.11 ± 1.03 GPa and 220.82 ± 1.09 GPa, respectively, which deviate by only 0.82% and 2.28% from the experimental value of 215.89 GPa³¹. This result confirms the efficacy of the multiparticle model and the accuracy of the simulation results.

Fig. 12: The change of stress, phase content, dislocation density and microstructure evolution during tensile process.

a Uniaxial tensile stress-strain. b Phase fraction. c Dislocation density changes of as-sintered and ideal Al_0.3CoCrFeNi high-entropy alloy. d–g The crack source in the as-sintered and ideal samples. h–k Analysis of HCP structure types in as-sintered and ideal samples.

In this study, we define yield stress as the point of intersection between a straight line deviating 0.2% from parallel to the elastic region and the stress–strain curve. The as-sintered Al_0.3CoCrFeNi high-entropy alloy exhibits a lower yield strength (5.87 GPa) than its ideal state (6.15 GPa), with a corresponding yield point of 0.03. The ultimate tensile strength and elongation of the as-sintered sample (ideal state) are 10.79 GPa (11.20 GPa) and 0.073 (0.07), respectively. This discrepancy is attributed to the surge of dislocations during the hot-pressed sintering process, which prompts the slip band in each grain to terminate near the grain boundaries, thereby enhancing strength. The as-sintered samples exhibit improved elongation due to the higher content of HCP structures compared to the ideal state. The BCC crystal structure, which mainly exists near grain boundaries after sintering, acts as an obstacle hindering dislocation movement during stretching, leading to stress concentration and crack formation (Fig. 12d, e). The high density of dislocations at the grain boundary facilitates stress concentration, thereby causing crack initiation from the grain boundary in an ideal sample (Fig. 12f, g).

Using the Crystal Analysis Tool³², we analyzed the types of HCP structures present in the as-sintered and ideal state Al_0.3CoCrFeNi high-entropy alloy before and after tension. These types include stacking faults, twins, three layers of HCP atoms in a continuous arrangement, and four layers of HCP atoms in a continuous arrangement, as shown in Fig. 12h–k. For the as-sintered Al_0.3CoCrFeNi high-entropy alloy, before tension, the number of atoms in stacking faults is 30,112, in coherent twins is 23,287, three layers of HCP atoms in a continuous arrangement is 9274, and four layers of HCP atoms in a continuous arrangement is 6997. After tension, the number of atoms in stacking faults is 30,218, in coherent twins is 25,176, three layers of HCP atoms in a continuous arrangement is 6607, and four layers of HCP atoms in a continuous arrangement is 7250. It can be observed that after tension, the number of three layers of HCP atoms in a continuous arrangement decreases while the number of coherent twins increases. Therefore, the content of the HCP structure in the as-sintered Al_0.3CoCrFeNi high-entropy alloy remains unchanged before and after tension.

Optimal sintering parameters and the morphology and size of powder particles significantly impact the mechanical properties of the final sintered samples. Therefore, obtaining optimal sintering parameters through simulation can provide new insights for efficiently and accurately designing high-performance Al_0.3CoCrFeNi polycrystalline high-entropy alloy.

Methods

Figure 13 depicts the workflow diagram, wherein Al_0.3CoCrFeNi high-entropy alloy powder particles are initially introduced into a simulation box to establish a multiparticle molecular dynamics model. Subsequently, a phase-field model is constructed based on this concept to validate the feasibility of the aforementioned model. Then, a multiparticle approach is employed to simulate the hot-pressed sintering of a polycrystalline Al_0.3CoCrFeNi high-entropy alloy. During this process, phase-field simulation is conducted with parameters that are tested and validated by molecular dynamics to ensure the accuracy of the multiparticle model. Finally, Young’s modulus of the Al_0.3CoCrFeNi high-entropy alloy is determined through tensile simulation and compared with experimental values.

Fig. 13: Workflow diagram of the present study.

a Model construction. b Schematic diagram of the hot-pressed sintering. c Schematic diagram of the uniaxial tension.

Multiparticle model

The parallel molecular dynamics simulator software LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator)³³ is used to investigate the sintering and tension process in Al_0.3CoCrFeNi high-entropy alloy. In this study, we present a multiparticle model for the sintering of polycrystalline high-entropy alloys that can be extended to other alloy systems. The simulation box is initially set to a size of 25 × 25 × 25 nm, with the crystallographic orientations aligned along three axes: <100> (x-axis), <010> (y-axis), and <001> (z-axis). The powder particles constitute approximately 36.5% of the total volume. The loop command in LAMMPS is utilized for the cyclic addition of spherical powder particles. In each cyclic process, a spherical Fe particle with the lattice constant of 0.35875 nm³⁴ is initially constructed and subsequently randomly replaced with Ni, Cr, Co, and Al atoms according to their atomic percentages. The particle size range is 1-8 nm. The added region needs to be analyzed during the cycle to avoid atomic stacking. Ultimately, a sintered model consisting of Al_0.3CoCrFeNi high-entropy alloy powder particles with the FCC structure is obtained, containing 577,244 atoms. Please refer to Supplementary Movie 3 to view the generation process.

To compare the tensile properties, we used the open-source program ATOMSK³⁵ to construct a polycrystalline model of Al_0.3CoCrFeNi high-entropy alloy with randomly distributed grain orientation (to characterize an ideal structure). The model size is 18.72 × 18.72 × 18.72 nm, including 562,453 atoms. The multiparticle and polycrystalline models are displayed in Fig. 14. The model in Fig. 14a contains a total of 118 Al_0.3CoCrFeNi high-entropy alloy spherical particles, which are colored based on particle identifiers as shown in Fig. 14b. The polycrystalline model in Fig. 14d contains 8 grains with randomly distributed orientations colored by particle identifiers, as shown in Fig. 14e. The FCC structure is determined through common neighbor analysis, with the distinction being that the white regions in Fig. 14c correspond to particle boundaries while those in Fig. 14f represent grain boundaries.

Fig. 14: Atomic snapshot for the multiparticle and polycrystalline model.

a and d The initial model. b and e Particle analysis of the initial model. c and f The common neighbor analysis of the initial model (the inset is the particle anatomy).

Calculation details

This study employs an embedded atomic potential to describe the interatomic interactions in Al-Co-Cr-Fe-Ni alloys^36,37. Periodic boundary conditions are adopted along the axial direction to eliminate the surface effects, while energy minimization is conducted on the initial structure to enhance model stability. The NPT (i.e., constant number of atoms, constant pressure, and constant temperature) ensemble was used throughout the sintering process with a time step of 0.001 ps. According to experimental results, the sintering temperature (T) is set at 1373 K, corresponding to 0.8T_m (T_m = 1655 K)^10,38. Since the time scales utilized in both simulations and experiments differ, the pressure (P) is selected as 3 GPa throughout testing to speed up the simulation process. The simulation procedure is as follows: (i) raise the temperature from 300 K to 1116 K ( ≈ 0.65T_m, 81 ps), maintain a constant temperature, and increase the pressure from 0 GPa to 2 GPa (20 ps); (ii) raise the temperature from 1116 K to 1373 K (26 ps), maintain isothermal conditions, and elevate the pressure from 2 GPa to 3 GPa (10 ps); (iii) maintain 100 ps at 1373 K and 3 GPa; (iv) decrease the temperature from 1373 K to 1116 K, and reduce the pressure from 3 GPa to 2 GPa (20 ps); (v) decrease the temperature from 1116 K to 300 K and reduce the pressure from 2 GPa to 0 GPa (20 ps); and (vi) maintain 60 ps at 300 K and 0 GPa to ensure that the sintered Al_0.3CoCrFeNi high-entropy alloy has no defects such as pores.

To establish a quantitative relationship between sintered microstructure and tensile strength, a tensile test is performed under the NVT (i.e., constant number of atoms, constant volume, and constant temperature) ensemble, with a strain rate of 5 × 10⁸ s⁻¹ and a total deformation of 0.2. Before this step, the model needs to relax for 20 ps under the NPT to ensure stability and eliminate internal stress.

Analysis method

The open visualization tool (OVITO) is used for microstructure and dislocation visualization³⁹. The mean square displacement (MSD) trend, i.e., Eqs. (3–4) effectively characterizes the atomic diffusion ability of powder particles⁴⁰.

$$\text{MSD}(t)=\frac{1}{N}\mathop{\sum }\limits_{i=1}^{N}{\left[{r}_{i}(t+\Delta t)-{r}_{i}(t)\right]}^{2}$$

(3)

$$D=\mathop{\mathrm{lim}}\limits_{t\to \infty }\frac{\text{MSD}(t)}{6t}$$

(4)

where r_i(t) is the position of atom $i$ at time $t$, $N$ represents the total number of atoms in the system, and ∆t represents the time step. $D$ is the diffusion coefficient.

Common neighbor analysis, which characterizes the atomic structure by judging the bonding between two atoms and surrounding atoms, is a favorable tool to exhibit the formation of FCC, hexagonal close-packed (HCP), body-centered cubic (BCC), and other structures (particle boundaries, grain boundaries, amorphous, etc.) during sintering⁴¹. The combination of two atoms is attributed to the distance between them needing to be less than the cutoff distance (r_cut) determined by the first peak and valley of the radial distribution function⁴². According to Eq. (5), the cutoff radius of the Al_0.3CoCrFeNi high-entropy alloy is 0.3064 nm.

$${r}_{\text{cut}}=\frac{a(\sqrt{1/2}+1)}{2}\approx 0.854a$$

(5)

where a represents the lattice constant for the FCC structure.

Dislocation analysis is used to analyze dislocation evolution during sintering and tension⁴³. Displacement vector analysis is utilized to investigate the displacement of atoms during sintering, with the displacement vector of atoms being calculated by subtracting their initial position from the current position⁴⁴.

The density of the sintered body in molecular dynamics can be calculated using Eq. (6),

$$D=\frac{{\sum }_{i}\frac{{N}_{i}}{{n}_{i}}{M}_{i}}{V}$$

(6)

where, $i$ represents the $i$-th crystal structure in the sintered body. N_i is the number of atoms contained in the $i$-th crystal structure, n_i is the number of atoms in the unit cell of the $i$-th crystal structure, M_i is the unit cell volume of the $i$-th crystal structure, and $V$ represents the unit cell volume at different sintering stages. During the calculation process, we include the number of atoms in the grain boundaries or other structures in the count of crystal structures adjacent to them.

Phase-field method

The phase-field simulations are based on EasyPhase software, a comprehensive phase-field theory software package developed by the group of Professor Zhao^45,46,47. The elastic constants and Young’s modulus of the Al_0.3CoCrFeNi high-entropy alloy are calculated through molecular dynamics, yielding values of C₁₁ = 221.49 GPa, C₁₂ = 136.72 GPa, C₄₄ = 117.71 GPa, and E = 203.71 GPa as input parameters for the phase-field method. The phase-field model describes the grain phase and pore phase of different orientations by introducing order field variables η_i(r,t), (i = 1, 2, …, p). Concentration field variables ρ(r, t) related to structural changes, are continuous functions of spatial coordinates and time changes^48,49. According to the evolution dynamics Cahn-Hilliard equation^50,51 and Allen-Cahn equation^52,53, the concentration field variable $\rho$ and the order parameter field variable $\eta$ are derived by Eqs. (7) and (8), respectively.

$$\frac{\partial \rho }{\partial t}=\nabla \cdot (M\cdot \nabla \frac{\delta F}{\delta \rho })$$

(7)

$$\frac{\partial {\eta }_{i}}{\partial t}=-{\rm{L}}\frac{\delta F}{\delta {\eta }_{i}}\,(i=1,\,2,\,\ldots ,\,p)$$

(8)

where $M$ is the diffusion mobility coefficient^54,55. $\text{L}$ is a coefficient that characterizes the mobility of the grain boundary. During hot-pressed sintering, the elastic energy term needs to be added to the total free energy functional $F$, and the specific expression is as follows:

$$F={\int }_{V}[{f}_{{\rm{ch}}}(\rho ,\,{\eta }_{1},\,{\eta }_{2},\,\cdots \,{\eta }_{p})+\frac{{\kappa }_{\rho }}{2}|\nabla \rho {|}^{2}+\frac{{\kappa }_{\eta }}{2}\mathop{\sum }\limits_{i=1}^{p}|\nabla {\eta }_{i}{|}^{2}+{F}_{{\rm{el}}}]dV$$

(9)

where f_ch represents the local chemical free energy density function, the second and third terms are the compositional gradient-free energy and structural gradient energy, respectively⁵⁶, and κ_ρ and κ_η represent two gradient energy coefficients. F_el represents elastic energy^57,58,

$$\begin{array}{l}{F}_{{\rm{el}}}=\frac{1}{2}{\displaystyle\int }_{V}{C}_{ijkl}{\varepsilon }_{ij}^{{\rm{el}}}{\varepsilon }_{kl}^{{\rm{el}}}{\rm{d}}V\\ =\frac{1}{2}{\displaystyle\int }_{V}{C}_{ijkl}[{\varepsilon }_{ij}^{{\rm{a}}}+\delta {\varepsilon }_{ij}-{\varepsilon }_{ij}^{0}][{\varepsilon }_{kl}^{{\rm{a}}}+\delta {\varepsilon }_{kl}-{\varepsilon }_{kl}^{0}]{\rm{d}}V\\ =\frac{1}{2}{\displaystyle\int }_{V}{C}_{ijkl}[{\varepsilon }_{ij}^{{\rm{a}}}+\delta {\varepsilon }_{ij}-\sum _{p}{\varepsilon }_{ij}^{0}(p){\eta }_{p}^{2}-{\varepsilon }^{0}{\delta }_{ij}\rho \delta \rho ][{\varepsilon }_{kl}^{{\rm{a}}}+\delta {\varepsilon }_{kl}-\sum _{q}{\varepsilon }_{kl}^{0}(q){\eta }_{q}^{2}-{\varepsilon }^{0}{\delta }_{kl}\rho \delta \rho ]{\rm{d}}V\end{array}$$

(10)

where C_ijkl is the elastic constant, ${\varepsilon }_{{ij}}^{a}$ is the externally applied strain, $\delta {\varepsilon }_{{ij}}$ is the heterogeneous strain, ${\varepsilon }_{{ij}}^{0}(p)$ represents the intrinsic strain of particles, and ε⁰ represents the lattice expansion coefficient of sintered particles relative to the matrix.

Based on the theory above, we set the phase field grid size to 80 × 80 with a grid spacing of dx = dy = 1.0. The initial model consists of 22 randomly placed particles within the mesh. Following the molecular dynamics results, the simulation temperature and pressure are established at 1373 K and 3 GPa, respectively, while applying simultaneous pressure on all four sides throughout the simulation’s duration of 60,000 steps. The dimensionless parameters used during the phase field simulation are shown in Table 1.

Table 1 The dimensionless parameters used during the phase field simulation.