Tailoring magnetic hysteresis of additive manufactured Fe-Ni permalloy via multiphysics-multiscale simulations of process-property relationships

Abstract

Designing the microstructure of Fe-Ni permalloy produced by additive manufacturing (AM) opens new avenues to tailor its magnetic properties. Yet, AM-produced parts suffer from spatially inhomogeneous thermal-mechanical and magnetic responses, which are less investigated in terms of process modeling and simulations. We present a powder-resolved multiphysics-multiscale simulation scheme for describing magnetic hysteresis in AM-produced material, explicitly considering the coupled thermal-structural evolution with associated thermo-elasto-plastic behaviors and chemical order-disorder transitions. The residual stress is identified as the key thread in connecting the physical processes and phenomena across scales. By employing this scheme, we investigate the dependence of the fusion zone size, the residual stress and plastic strain, and the magnetic hysteresis of AM-produced Fe_21.5Ni_78.5 on beam power and scan speed. Simulation results also suggest a phenomenological relation between magnetic coercivity and average residual stress, which can guide the magnetic hysteresis design of soft magnetic materials by choosing appropriate processing parameters.

Introduction

The Fe-Ni permalloy has been widely studied in recent decades owing to its extraordinary magnetic permeability, low coercivity, high saturation magnetization, mechanical strength, and magneto-electric characteristics. This material has been widely employed in conventional electromagnetic devices, such as sensors and actuators, transformers, electrical motors, and magnetoelectric inductive elements. Fe-Ni-based permalloys modified with additives are also promising candidate materials for multiple novel applications, such as wind turbines, all-electric vehicles, rapid powder-conversion electronics, electrocatalysts, and magnetic refrigeration^1,2,3,4.

Due to the increasing importance of additive manufacturing (AM) technologies, the possibilities of designing soft magnetic materials by AM have been explored in a number of studies^{5,6,7,8,9,10,11,12}. However, due to the delicate interplay of process conditions and resulting properties, there are several open questions that need to be answered in order to obtain AM-produced Fe-Ni permalloy parts with the desired property profile. Magnetic properties of the Fe-Ni system depend on the chemical composition, as depicted in Fig. 1a. Fe_21.5Ni_78.5 is typically selected targeting the chemical-ordered low-temperature FCC phase (also known as awaruite, L1₂, or ${\gamma }^{{\prime} }$ phase, as the phase-diagram shown in Supplementary Fig. 1a), which possesses a minimized coercivity and peaked magnetic permeability^8,13. The main problem is to increase the generation of ${\gamma }^{{\prime} }$ phase using AM, as the phase transition kinetics from the chemical-disordered high-temperature FCC phase (also known as austenite, A1, or γ phase) to the chemical-ordered ${\gamma }^{{\prime} }$ phase is extremely restricted¹⁴. On the laboratory timescale, growing the ${\gamma }^{{\prime} }$ phase into considerable size requires annealing times on the order of days^{15,16,17,18,19}. Due to the rapid heating and cooling periods, only the γ phase exists in AM-processed parts⁹. Combining in-situ alloying with AM methods allows to stabilize the ${\gamma }^{{\prime} }$ phase in printed parts^8,20, but the restricted kinetics still limit the growth of a long-range ordered phase¹⁴. Another question concerns the influences of associated phenomena on magnetic hysteresis. Although there are studies on the effects of crystallographic texture^11,21, a pivotal discussion regarding interactions among residual stress, microstructures, and physical processes leading to the magnetic hysteresis behavior is unfortunately missing.

Fig. 1: Emerging issues during additive manufacturing of Fe-Ni permalloy.

a Ni concentration dependent magnetic properties, incl. magneto-crystalline anisotropic constant K_u, magnetostriction constants λ₁₀₀ and λ₁₁₁, saturation magnetization M_s and initial relative permeability μ_r, modified with permission from Balakrishna et al.²³. b Schematics of temperature gradient mechanism (TGM) for explaining the generation of residual stress in heating and cooling modes, where the tensile (σ_(t)) and compressive (σ_(c)) stress states, and strains due to thermal expansion (ε_th) and activated plasticity (ε_pl) are denoted. Inset: A bent AM-processed part due to residual stress. Image is reprinted with permission from Takezawa et al.²⁷ under the terms of the Creative Commons CC-BY 4.0 license. c Bright-field image of a Fe-Ni permalloy microstructure after annealing at 723 K for 50 h. Inset: electron diffraction pattern of the microstructure. SEM and electron diffraction images are reprinted with permission from Ustinovshikov et al.¹⁹.

Recently, it has been shown that soft magnetic properties can be designed by controlling magneto-elastic coupling^22,23,24. Along these lines, the residual stress caused by AM and the underlying phase transitions could be key for tuning the coercivity in an AM-processed Fe-Ni permalloy. Micromagnetic simulations by Balakrishna et al.²³ presented that magneto-elastic coupling plays an important role in governing magnetic hysteresis, as the existence of the pre-stress shifts the minimized coercivity from the composition Fe₂₅Ni₇₅ to Fe_21.5Ni_78.5, while the magneto-crystalline anisotropy is zero at Fe₂₅Ni₇₅ but non-zero at Fe_21.5Ni_78.5. Based on a vast number of calculations, the dimensionless constant $({C}_{11}-{C}_{12}){\lambda }_{100}^{2}/2{K}_{{{{\rm{u}}}}}\approx 81$ was proposed as a condition for low coercivity along the 〈100〉 crystalline direction for cubic materials (incl. Fe-Ni permalloy)²⁴. Here, C₁₁ and C₁₂ are the components of the stiffness tensor, λ₁₀₀ is the magnetostrictive constant, and K_u is the magneto-crystalline constant. Nevertheless, the influence of the magnitude and the states of the residual stress were not comprehensively analyzed. Yi et al. discussed magneto-elastic coupling in the context of AM-processed Fe-Ni permalloy²². The simulations were performed with positive, zero, and negative magnetostrictive constants under varying beam power. The results showed that the coercivity of the Fe-Ni permalloy with both positive and negative magnetostrictive constants rises with increasing beam power. In contrast, the permalloy with zero magnetostrictive constant showed no dependence on the beam power. This demonstrates the necessity of magneto-elastic coupling in tuning coercivity in AM-produced permalloy and illustrates the potential of tailoring magnetic hysteresis of permalloys via controlling the residual stress during AM.

Understanding the residual stress in AM and its interactions with other physical processes, such as thermal and mass transfer, grain coarsening and coalescence, and phase transition, is never the oak that felled at one stroke. Taking the popular selective laser melting/sintering (SLM/SLS) method as an instance, the temperature gradient mechanism (TGM) explains the generation of residual stress by considering the heating mode and the cooling mode^25,26,27. The heating mode presents a counter-bending with respect to the building direction (BD) of newly fused layers (Fig. 1b). This is because the thermal expansion in an upper overheated region gets restricted by the lower old layer/substrate. Plastic strain can also be generated due to the activated plasticity of the material and compensate the local stress around the heat-affected zone. The cooling mode, in contrast, presents a bending towards the BD of the newly fused layer due to the thermal contraction between the fusion zone and the old layer/substrate.

It should be noted that TGM only provides a phenomenological aspect by employing the idealized homogeneous layers. In the practical SLM/SLS, varying morphology and porosity of the powder bed create inhomogeneity in not only the temperature field but also the on-site thermal history on the mesoscale (10–100 μm), inciting varying degrees of thermal expansion, and eventually leading to the development of the thermal stress in various degree. Stochastic inter-particle voids and lack-of-fusion pores also create evolving inhomogeneity in material properties on the mesoscale^28,29, leading to the shifted local stress conditions interpreted by TGM. In other words, mesoscopic inhomogeneity and coupled thermo-structural evolution should act as long-range factors in residual stress development. On the other hand, due to the relatively smaller lattice parameter, the continuous growth of the ${\gamma }^{{\prime} }$ phase would also result in the increasing misfit stress between itself and the chemical-disordered γ matrix^18,19 (as also presented in Fig. 1c). Therefore, the nanoscopic solid-state phase transition also contributes to the development of residual stress as a short-range fluctuation, which is almost effectless to the mesoscopic phenomena yet still influences the local magnetic behavior²³. To sum up, the residual stress from AM processes that eventually affects the magnetization reversal via magneto-elastic coupling should already reflect such long-range (morphology and morphology-induced chronological-spatial thermal inhomogeneity) and short-range factors (misfit-induced fluctuation). This is the central challenge that this work addresses.

In this work, we developed a powder-resolved multiphysics-multiscale simulation scheme to investigate the hysteresis tailoring of Fe-Ni permalloy by AM under a scenario close to practical experiments. This means that the underlying physical processes, including the coupled thermal-structural evolution, chemical order-disorder transitions, and associated thermo-elasto-plastic behaviors, are explicitly considered and bridged by accounting for their chronological-spatial differences. The influences of processing parameters (notably the beam power and scan speed) are analyzed and discussed on distinctive aspects, including the size of the fusion zone, the development of residual stress and accumulated plastic strain, the $\gamma /{\gamma }^{{\prime} }$ transition under the residual stress, and the resulting magnetic coercivity of manufactured parts. It is anticipated that the presented work could provide transferable insights in selecting processing parameters and optimizing routine for producing permalloy using AM, and deliver a comprehensive understanding of tailoring the hysteresis of soft magnetic materials in unconventional processing.

Results

Multiphysics-multiscale simulation scheme

In this work, we consider SLS as the AM approach due to its relatively low energy input as compared to other methods, like SLM. SLS allows us to obtain a stable fusion zone and thus to gain better control of the residual stress development since we don’t need to consider melting and evaporating processes and the associated effects, such as the keyholing and Marangoni convection. The microstructure of SLS-processed parts is porous, which allows us to explore the effects of lack-of-fusion pores on the development of residual stress and plastic strain. Based on an overall consideration of all possible phenomena involved in the SLS of the Fe-Ni permalloy, two chronological-spatial scales are integrated in this work:

• (i) On the mesoscale, with the characteristic length of several 100 μm, powders are fused/sintered around the laser spot, creating the fusion zone. Featured phenomena such as partial/full melting, necking, and shrinkage among powders can be observed. High gradients in the temperature field are also expected due to laser scanning and rapid cooling of the post-fusion region. By choosing a typical scan speed of 100 mm s⁻¹, this stage lasts only 10 ms.

• (ii) On the nanoscale with a characteristic length well below 1 μm, the chemical order-disorder ($\gamma /{\gamma }^{{\prime} }$) transition can be observed once the on-site temperature is below the transition temperature, as presented in Fig. 1c. Owing to the difference in thermodynamic stability, redistribution of the chemical constituents by inter-diffusion between γ and ${\gamma }^{{\prime} }$ phases is coupled to the phase transition. Due to the extremely restricted kinetics, it would cost several hundred hours of annealing to have the ${\gamma }^{{\prime} }$ phase formation in a mesoscopic size^15,18,30.

We explicitly consider a three-stage processing route consisting of an SLS, a cooling, and an annealing stage. As shown in the inset of Fig. 2, the domain temperature will rise from a pre-heating temperature (T₀) during the SLS stage. After that, the whole powder bed would gradually cool down to T₀. Finally, the processed powder bed enters the annealing stage at T₀ where the $\gamma /{\gamma }^{{\prime} }$ transition continues. No magnetic field is applied during the processing route to eliminate the potential influences from magnetic domain evolution through magnetostriction. The cooling stage lasts three times longer than the SLS stage, and the sequential annealing stage takes far longer than the two other stages. Taking a 500 μm scan section with a typical scan speed of 100 mm s⁻¹ as an example, the SLS and cooling stages would last 5 and 20 ms, respectively, and the annealing time is on the order of 100 h. Remarkably, the inhomogeneous and time-varying temperature field in the mesoscopic powder bed can be treated as uniform and nearly constant for the nanoscopic $\gamma /{\gamma }^{{\prime} }$ transition, as the heating and cooling stages induced by laser scan are negligible compared to the time required for the $\gamma /{\gamma }^{{\prime} }$ transition. Similarly, the mechanical response (notably the residual stress) on the mesoscopic is also regarded as uniform with respect to sufficiently small nanoscopic domains where the $\gamma /{\gamma }^{{\prime} }$ transition occurs. Nonetheless, their long-term effects remain after the first two stages and would further influence the nanoscopic phase formation during the annealing stage, influencing the resultant magnetic hysteresis behavior by electro-magnetic coupling^22,23,31.

Fig. 2: Multiphysics-multiscale simulations scheme proposed in this work.

The workflow and data interaction among methods are illustrated schematically. All the involved quantities are explicitly introduced in the Method section. Notice here λ_1nn represents λ₁₀₀ and λ₁₁₁.

The simulations are arranged in a subsequent scheme to recapitulate the aforementioned characteristics on different scales while balancing the computational cost-efficiency, as shown in Fig. 2. Accepting that heat transfer is only strongly coupled with microstructure evolution (driven by diffusion and underlying grain growth) but weakly coupled with mechanical response during the SLS-process stage, we employ the non-isothermal phase-field model proposed in our former work²⁸ to simulate the coupled thermo-structural evolution, and perform the subsequent thermo-elasto-plastic calculations based on the resulting transient mesoscopic structure (hereinafter called mesostructure) and temperature field from the SLS simulations. In other words, mechanical stress and strain are developed under the quasi-static microstructure and temperature field. This is based on the fact that the thermo-mechanical coupling strength is negligible for most metals³², unlike the strong inter-coupling among mass as well as heat transfer and grain growth^33,37. Nevertheless, the influences from AM-resulted polycrystalline texture on the thermo-elasto-plastic behaviors of the permalloy remain unclear and should be covered in future researches. Spatial interpolation of the mechanical properties according to the order parameter ρ, which is ρ = 1 in the materials and ρ = 0 in the atmosphere/pores, is also performed to consider structural inhomogeneities due to pore formation. Details are described in the section on Methods. The domain-average quantities are defined as

$${\bar{\sigma }}_{{{{\rm{e}}}}}^{{{{\rm{D}}}}}=\frac{{\int}_{\Omega }\rho {\sigma }_{{{{\rm{e}}}}}{{{\rm{d}}}}\Omega }{{\int}_{\Omega }\rho \Omega },\quad \quad {\bar{p}}_{{{{\rm{e}}}}}^{{{{\rm{D}}}}}=\frac{{\int}_{\Omega }\rho {p}_{{{{\rm{e}}}}}{{{\rm{d}}}}\Omega }{{\int}_{\Omega }\rho \Omega },\quad \quad {\bar{T}}^{{{{\rm{D}}}}}=\frac{{\int}_{\Omega }\rho T{{{\rm{d}}}}\Omega }{{\int}_{\Omega }\rho \Omega },$$

(1)

where Ω is the simulation domain volume, and ρ is the order parameter indicating the substance. σ_e is the von Mises stress and p_e is the accumulated (effective) plastic strain. To eliminate the boundary effects, which lead to heat accumulation on the boundary that intersects with the scan direction, the simulation domain with a geometry 250 × 250 × 250 μm is selected from the center of the domain for processing with the transient T and ρ fields mapped, as shown in Fig. 2.

Figure 4a presents the evolution of the domain-average von Mises stress during the SLS and cooling stages. σ_e develops with the temperature rise due to the generation of laser-induced heat in the powder bed. However, once the laser spot moves into the domain, followed by the overheated region, ${\bar{\sigma }}_{{{{\rm{e}}}}}^{{{{\rm{D}}}}}$ drops along with ${\bar{T}}^{{{{\rm{D}}}}}$ continuing rising to the maximum. This is because the points inside the overheated region lose their stiffness, as the material is fully/partially melted, thereby presenting zero stress (Fig. 4b). Surroundings of the overheated region also present relatively low stress due to the sufficient reduction of stiffness at high temperatures. When the laser spot moves out of the domain, the thermal stress starts to develop along with the cooling of the domain (Fig. 4c–e). The stress around the concave morphologies on the powder bed, incl. concaves on the surface and sintering necks among particles, rises faster than the one around the traction-free convex morphologies on the surface and unfused powders away from the fusion zone. This may attribute to the locally high temperature gradient around the concave morphologies during the SLS process and, thereby, the strong thermal traction. At the end of the cooling, the convex morphologies and unfused powders have relatively lower stress developed, while the locally concentrated stress can be observed around the concave morphologies on the powder bed, like the surface concaves and sintering necks among particles (Fig. 4e). Due to the convergence of the σ_e vs. time in the cooling stage, the stress at the end of the cooling stage (t = 20 ms) will be regarded as the residual stress in the following discussions.

Fig. 4: Development of stress and plastic strain during SLS single scan.

a Calculated von Mises stress (${\bar{\sigma }}_{{{{\rm{e}}}}}^{{{{\rm{D}}}}}$) and temperature (${\bar{T}}^{{{{\rm{D}}}}}$) in domain average vs. time with the profile of σ_e at the denoted states shown in (b–e). f Calculated accumulated plastic strain in domain average (${\bar{p}}_{{{{\rm{e}}}}}^{{{{\rm{D}}}}}$) vs. time with the profile of p_e at the denoted states shown in (g–j).

The development of the accumulated plastic strain p_e presents an overall increasing tendency vs. time during the SLS and cooling stages, as shown in Fig. 4f. To emulate the effects of full/partial melting, we reset the p_e in the overheated region (T > T_M). This reset, however, does not influence the accumulation of p_e outside of the overheated region (Fig. 4g), distinguished from σ_e that suffers reduction not only inside but also outside of the overheated region due to loss of stiffness at high temperature. As a result, the growth of the ${\bar{p}}_{{{{\rm{e}}}}}^{{{{\rm{D}}}}}$ slows down only when the laser spot moves into the domain rather than tends to reduce as ${\bar{\sigma }}_{{{{\rm{e}}}}}^{{{{\rm{D}}}}}(t)$. The continuous accumulation of p_e also results in the distinctively concentrated p_e at the fusion zone’s outer boundary (Fig. 4g–j), attributing to the high temperature gradient at the front and bottom of the overheated region during the SLS where the existing high thermal stress locally activates the plastic deformation and then contribute to the rise of the p_e. The same reason can be used to explain the concentrated p_e around the pores and concave morphologies like sintering neckings near the fusion zone. In contrast, unfused powders and substrate away from the fusion zone present nearly no accumulation of p_e, as the onsite thermal stress due to relatively low local temperature is not high enough to initiate the plastification of the material.

Nanoscopic $\gamma /{\gamma }^{{\prime} }$ transition with residual stress

Before sampling many points inside the fusion zone for studying the subsequent $\gamma /{\gamma }^{{\prime} }$ transition and carrying out micromagnetic simulations, we first examined five selected points from the middle section of the SLS simulation domain, counting its repeatability along the x-direction (SD). Profiles of σ_e and p_e are presented on this selected middle section in Fig. 5a. These five points are taken from the profiling path along z-direction (BD) and y-direction, as σ_e and p_e along these two paths are explicitly presented in Fig. 5b, c. σ_e gradually rises along z-profiling path, as the gap between the normal stresses (specifically between σ_xx and σ_zz, and between σ_yy and σ_zz) since there are almost no differences between σ_xx and σ_yy. σ_e reaches a peak around the fusion zone boundary (FZB) and then decreases. Notably, there is a reverse of σ_zz from positive (tension) to negative (compression) across the FZB. Similarly, p_e increases to a peak around the FZB and then decreases along z-profiling path, and the reversed components of ε_pl are observed across the FZB. This implies the bending of the SLS-processed mesostructure caused by the thermal contraction between the fusion zone and substrate, where the high temperature gradient is depicted. Along the y-profiling path, both σ_e and p_e present no monotonic tendencies, receiving influences from the morphologies, yet still reach peaks around the FZB, respectively.

Fig. 5: $\gamma /{\gamma }^{{\prime} }$ transitions and magneto-elastic responses of stressed nanostructures.

Profiles of (a) the von Mises residual stress σ_e and the accumulated plastic strain p_e on the middle section perpendicular to the x-direction (SD). The selected points P₁-P₅ are also indicated. The components and effective values of (b) the residual stress and (c) the plastic strain along the z- and y-profiling paths as indicated in (a). d The Lamé’s stress ellipsoids, representing the stress state at P₁-P₅, are illustrated with the principal stresses denoted and colored (marine: tension; red: compression). e Ni concentration profile after annealing for 1200 h at T₀ = 600K with the inset indicating the corresponding solid-state phases and the interface. f The magneto-elastic coupling energy f_em calculated under a homogeneous in-plane m configuration with ϑ = 45^∘ w.r.t. the easy axis u. The effective coupling field B_em is also illustrated. g The average hysteresis loop of 10 cycles each performed on the nanostructures (NS) at P₁-P₅. The reference curve (ref.) is performed on the stress-free homogeneous nanostructure with only ${\gamma }^{{\prime} }$ phase and the composition Fe_21.5Ni_78.5.

The Lamé’s stress ellipsoids are illustrated in Fig. 5d for visualizing the stress state of selected points with the directions of the principal stress denoted. Points near the surface and in the fusion zone (P₁, P₂, P₄, and P₅) are under tensile stress states, while the point P₃ near the FBZ has one negative principal stress (compressive) along BD due to the bending caused by the thermal contraction in the fusion zone. The Ni concentration and nanoscopic stress redistribution due to the $\gamma /{\gamma }^{{\prime} }$ transition are simulated in the middle section perpendicular to SD as well (Supplementary Fig. 2c). It is also coherent with the diffuse-controlled 2D growth of ${\gamma }^{{\prime} }$ phase experimentally examined by the Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory³⁰. We initiate the ${\gamma }^{{\prime} }$ nuclei randomly but with a minimum spacing of 100 nm according to the experimental observation in Fig. 1c using Poisson disk sampling³⁸. A relatively longer annealing time of 1200 h to obtain sufficient ${\gamma }^{{\prime} }$ phase formation, as presented in Fig. 5e. The Ni concentration (X_Ni) at the centers of the grown ${\gamma }^{{\prime} }$ phase is relatively low, close to the equilibrium value 0.764 at ${T}_{\gamma /{\gamma }^{{\prime} }}=766\,K$. With the growth of the ${\gamma }^{{\prime} }$ phase, Ni is accumulated at the interface due to the relatively large $\gamma /{\gamma }^{{\prime} }$ interface mobility compared with the inter-diffusive mobility of Ni species, agreeing with its diffuse-controlled growth examined experimentally³⁰. Among the points, P₃ with one principal compressive stress has relatively larger ${\gamma }^{{\prime} }$ grown after annealing. This is due to the growing ${\gamma }^{{\prime} }$ phase with relatively smaller lattice parameters (in other words, inciting shrinkage eigenstrain inside ${\gamma }^{{\prime} }$ phase) being mechanically preferred with compressive stress. As P₂ and P₅ with similar stress states, similar ${\gamma }^{{\prime} }$ phase formations are shown. P₁ has relatively less ${\gamma }^{{\prime} }$ phase grown, as it has the highest normal stresses as tensile (σ_xx, σ_yy, and σ_zz) compared to other points, as shown in Fig. 5b.

Magnetic hysteresis behavior of stressed nanostructures

The nanoscopic distributions of stress and Ni concentration are imported from the results of the $\gamma /{\gamma }^{{\prime} }$ transition, in which both the long-range and short-range factors are embodied. The magneto-elastic coupling is implemented by considering an extra term f_em in the magnetic free energy density along with the exchanging, magneto-crystalline anisotropy, magnetostatic, and Zeeman contributions^31,39,40, as described in the section on Methods. To consider contributions from normal and shearing stresses, a homogeneous in-plane configuration for the normalized magnetization m is oriented in an angle of ϑ = 45^∘ to the crystalline orientation u, which is assumed to be the z-direction. Meanwhile, an effective coupling field B_em = − ∂f_em/∂(M_sm) is calculated for analyzing the magneto-elastic coupling effects in vectorial aspect (Fig. 5f). It should be noted that within the range of segregated X_Ni from 0.781 to 0.810 as shown in Fig. 5e, λ₁₀₀ ranges from 1.274 × 10⁻⁵ to 5.249 × 10⁻⁶, presenting a positive shearing magnetostriction. On the other hand, λ₁₁₁ inside of the ${\gamma }^{{\prime} }$ phase (X_Ni = 0.781 ~ 0.785) and in the γ matrix (X_Ni = 0.785) has positive values, ranging from 2.41 × 10⁻⁶ to 1.96 × 10⁻⁶, while X_Ni shifts from 0.781 to 0.785. On the $\gamma /{\gamma }^{{\prime} }$ interface with X_Ni = 0.810, however, a negative value of λ₁₁₁ = − 8.429 × 10⁻⁷ is obtained. This implies that there is a negative normal magnetostriction on the $\gamma /{\gamma }^{{\prime} }$ interfaces and a positive normal magnetostriction in the bulk of γ and ${\gamma }^{{\prime} }$ phases. Due to the existing shrinkage eigenstrain, there are locally high f_em contributions inside the ${\gamma }^{{\prime} }$ phase compared to the γ matrix, which causes strong magneto-elastic coupling effects. Among all points P₁-P₅, P₁ has comparably lower magneto-elastic coupling in both ${\gamma }^{{\prime} }$ and γ phases. Owing to similar stress states, P₂, P₄, and P₅ have similar profiles of f_em and B_em, where P₅ has slightly stronger coupling effects. Remarkably, P₂, P₄, and P₅ all have the local B_em lying at an angle about 135^∘, as emphasized by dashed-dotted circle Fig. 5f. As for P₃, however, the angle between m and B_em further increases to 142 to 165^∘ together with larger magnitude (∣B_em∣) comparing to other points, demonstrating enhanced reversal effects to the m.

The hysteresis curves of the nanostructures also reflect the X_Ni-dependence of K_u, λ₁₀₀ and λ₁₁₁. For comparison, the hysteresis curves simulated on a stress-free reference with a homogeneous X_Ni = 0.785 are also plotted. In order to take numerical fluctuations into account, ten cycles of the hysteresis were examined for each nanostructure/reference with the averaged one presented in Fig. 5g. Notice that K_u ranges from −0.137 to −0.364 kJ m⁻³ according to Fig. 1a, which can be regarded as the easy-plane anisotropy as the magnetization prefer to orient in the plane perpendicular to BD. Comparing selected points, P₂ and P₅ have similar coercivity, which is 0.55 mT for P₂ and 0.58 mT for P₅, as these two points have similar stress states. P₄ and P₁ have the coercivity of 0.26 mT and 0.09 mT, respectively. However, the coercivity at P₃ is infinitesimal. This is phenomenologically due to the strong coupling field B_em that reverses m at the relatively low external field, as shown in Fig. 5f. As the coupling field B_em is directly related to the stress state, the relatively strong shear stress implied by the Lamé’s stress ellipsoids at P₃ (Fig. 5d) may be one of the reasons for the infinitesimal coercivity. This should be further examined in future studies.

Discussion

It should notice that the size of the fusion is directly linked to the overheated region, which varies with the different combinations of beam power P and scan speed v, as shown in Fig. 3e–h. The usage of the conserved OP ρ in representing the powder bed morphologies and the coupled kinetics between ρ and local temperature allows us to simulate the formation of the fusion zone in a way close to the realistic setup of SLS. Here the indicator ξ is utilized to mark the fusion zone, which is initialized as zero and would irreversibly turn to one once the temperature is above T_M⁴¹. Two characteristic sizes, namely the fusion zone width b and the fusion zone depth d, are defined by the maximum width and depth of the fusion zone, as shown in the inset of Fig. 6. Figure 6a and b present the maps of b and d vs. P and v, with the isolines indicating the identical volumetric specific energy input U_V that is calculated as

$${U}_{{{{\rm{V}}}}}=\frac{\eta P}{\bar{h}wv},$$

(2)

where the width of the laser scan track w takes D_FWHM = 117.6 μm and the average thickness of the powder bed $\bar{h}=25\,\mu{\rm{m}}$. Since 50% total power is concentrated in the spot with D_FWHM, the efficiency takes η = 0.5. According to the observed geometry of the fusion zone, the maps can be divided into three regions: P and v located in the region (R1) would result in a continuous fusion zone, as shown in Fig. 6c, e–h. The depth of the fusion zone in (R1) normally penetrates the substrate, implying the formation of a considerable size of the overheated region, within which the melting-resolidification would take the dominant role. P and v located in the region (R2) generate small and discontinuous fusion zones, as two classical geometries as shown in Fig. 6d, i. Its limited size implies the typical partial melting and liquid-state sintering mechanism, as the melt flow is highly localized and can only help the bonding among a subset of powder particles. It is worth noting that these discontinuous fusion zones should attribute to the thermal inhomogeneity induced by local stochastic morphology rather than the mechanisms like the Plateau-Rayleigh instability and balling, where a significant melting phenomenon is required^42,43,44. In the region labeled as (R3), no fusion zone is generated under the chosen P and v, and the solid-state sintering process remains dominant in the powder bed.

Fig. 6: Dependence of fusion zone geometry on processing parameters.

Contour maps of (a) the width b and (b) the depth d of the fusion zone w.r.t. the beam power and the scan speed. The dotted lines represent isolines of specific energy input U_V. c–i Fusion zone geometries on the powder bed with different processing parameters. Inset: b and d measured from the fusion zone geometries.

Figure 7a, b present the maps of average residual stress ${\bar{\sigma }}_{{{{\rm{e}}}}}$ and plastic strain ${\bar{p}}_{{{{\rm{e}}}}}$ inside the fusion zone vs. P and v, with the specific energy input U_V indicated. The average residual stress and plastic strain inside the fusion zone are calculated with the fusion zone indicator ξ from the relations

$${\bar{\sigma }}_{{{{\rm{e}}}}}=\frac{{\int}_{\Omega }\xi {\sigma }_{{{{\rm{e}}}}}{{{\rm{d}}}}\Omega }{{\int}_{\Omega }\xi {{{\rm{d}}}}\Omega },\quad {\bar{p}}_{{{{\rm{e}}}}}=\frac{{\int}_{\Omega }\xi {p}_{{{{\rm{e}}}}}{{{\rm{d}}}}\Omega }{{\int}_{\Omega }\xi {{{\rm{d}}}}\Omega }.$$

(3)

Since the properties inside the fusion zone are focused, only the results with P and v located in the continuous fusion zones (R1) and discontinuous ones (R2) are selected and discussed. Generally, the increase of the ${\bar{\sigma }}_{{{{\rm{e}}}}}$ follows the direction of increasing specific energy input U_V, leading to the enlargement of the fusion zone. In other words, increasing the size of the fusion zone receives larger contractions between itself and the substrate, leading to the rise of the residual stress inside the fusion zone, comparing Fig. 7c–e and c, f–g. It worth noting that ${\bar{\sigma }}_{{{{\rm{e}}}}}$ presents a rapid increase on U_V below 120 J mm⁻³, as shown in Supplementary Fig. 3. When U_V > 60 J mm⁻³, there is almost no increase in ${\bar{\sigma }}_{{{{\rm{e}}}}}$, which is reflected as the sparse contours beyond the isoline U_V = 60 J mm⁻³. This implies the saturation of residual stress in the fusion zone when U_V is sufficiently large, despite the fusion zone would continue enlarging along with the further increase of U_V.

Fig. 7: Dependence of residual stress and plastic strain on processing parameters.

Contour maps of (a) average residual stress ${\bar{\sigma }}_{{{{\rm{e}}}}}$ and (b) average plastic strain ${\bar{p}}_{{{{\rm{e}}}}}$ in the fusion zone w.r.t. the beam power and the scan speed. The dotted lines represent isolines of different specific energy inputs U_V. Profiles of (c–g) residual stress and (h–l) plastic strain are plotted for different processing parameters. The boundary of the fusion zone is also indicated.

The map of ${\bar{p}}_{{{{\rm{e}}}}}$ presents a ridge at P = 30 W, which is different from the monotonic dependence of ${\bar{\sigma }}_{{{{\rm{e}}}}}$ on U_V. This means the ${\bar{p}}_{{{{\rm{e}}}}}$ reaches a minimum at P = 30 W for every selected v. This may reflect the competition between enlarging fusion zone and the accumulation of plastic strain in the fusion zone. In the low-power range, the accumulation of plastic strain is slower than the growth of the fusion zone, as the ${\bar{p}}_{{{{\rm{e}}}}}$ reduces along with increasing P. When P > 30 W, the accumulation of plastic strain is faster than the growth of the fusion zone, as the ${\bar{p}}_{{{{\rm{e}}}}}$ increases along with increasing P. Interestingly, such a tendency is not observed in reducing v with fixing P, as the ${\bar{p}}_{{{{\rm{e}}}}}$ always grows monotonically with decreasing v at every selected P.

Moreover, σ_e drastically rises across the former boundary between powder bed and substrate for all selected combinations of P and v, as shown in Fig. 7c–g. This is due to the different mechanical responses between the porous powder bed and homogeneous substrate to the thermal stress formed together with the overheated zone. As the size of the fusion zone enlarged, the high concentration of σ_e extends further into the interior of the substrate, as the thermal contraction becomes enhanced between the fusion zone and the substrate. Meanwhile, a relatively low concentration of p_e is observed inside the fusion zone. And a relatively higher concentration is located at the fusion zone’s outer boundary, reflecting the continuing accumulation of certain regions, as shown in Fig. 7h–l.

Many points located on the mid-section of the fusion zones were sampled, considering their repeatability along the x-direction (SD), as shown in Supplementary Fig. 4. It can also eliminate the boundary effects and distractions from the quantities outside the fusion zone. Nanoscopic $\gamma /{\gamma }^{{\prime} }$ transition and hysteresis simulations were performed on each point subsequently. Each hysteresis simulation applied the external magnetic field H_ext along the easy axis with identical loading history and increment size. After obtaining many longitudinal hysteresis curves ${M}_{\parallel }^{I}({H}_{{{{\rm{ext}}}}})$ with the point index I, the average hysteresis curve ${\bar{M}}_{\parallel }({H}_{{{{\rm{ext}}}}})$ is then calculated directly as the point-wise average of the resulting local hysteresis, i.e.,

$${\bar{M}}_{\parallel }({H}_{{{{\rm{ext}}}}})=\mathop{\sum }\limits_{N}^{I}{M}_{\parallel }^{I}({H}_{{{{\rm{ext}}}}})/N,$$

(4)

where N is the amount of sampled points. And the average characteristics, like average coercivity ${\bar{H}}_{{{{\rm{c}}}}}$, were extracted from the averaged curve. Employing an identical sampling grid size, this averaged hysteresis can be statistically regarded as a collective outcome of many ferromagnetic pseudo-particles with their magnetization rotation individually dependent on the local nanoscopic phase formation, stress state, and applied magnetic field⁴⁵. At least three hysteresis cycles were performed on each point to reduce the fluctuations due to the numerical scheme. In Fig. 8a, we present the average coercivity ${\mu_0\bar{H}}_{{{{\rm{c}}}}}$ map with respect to the P and v. μ₀ is the vacuum permeability. In the map, the lower-right region shows a smaller coercivity, and the upper-left region shows a higher coercivity. Increasing P from 27.5 to 35 W results in the rise of ${\mu }_{0}{\bar{H}}_{{{{\rm{c}}}}}$ from 0.35 to 0.41 mT (by 17%) when v = 100 mm s⁻¹, and decreasing v from 125 to 50 mm s⁻¹ results in the rise of ${\mu }_{0}{\bar{H}}_{{{{\rm{c}}}}}$ from 0.34 to 0.44 mT (by 29%) when P = 30 W. This is mainly due to the increasing fraction of region with high local coercivity in the fusion zone for increasing P and decreasing v (comparing Fig. 8c, d, e, c, f, g, respectively). It is also evident that the high local H_c points possess a high local volume fraction of ${\gamma }^{{\prime} }$ phase ${\Psi }_{{\gamma }^{{\prime} }}$ (comparing Fig. 8c–g with h–l). This implies an enhanced magneto-elastic coupling effect at high ${\Psi }_{{\gamma }^{{\prime} }}$, as discussed in Fig. 5f. However, the effect from the local residual stress on the pattern of high local H_c region should also be stressed, as the points with low local μ₀H_c around where the local σ_e has a drastic change, e.g., the former boundary between powder bed and substrate, comparing (Fig. 8c–g with m–q, especially 8e with o and f and p). Moreover, it remains unclear for the existing “islands” of low local H_c located in the high local σ_e and ${\Psi }_{{\gamma }^{{\prime} }}$ region. They may be subject to a stress state similar to the P₃ in Fig. 5 where nearly zero coercivity is obtained. Nonetheless, a data-driven investigation should be conducted as one upcoming work to connect the local stress state to the coercivity. Other average characteristics like the remnant ${\bar{M}}_{{{{\rm{r}}}}}$ and the saturation magnetization ${\bar{M}}_{{{{\rm{s}}}}}$ are also correspondingly read from the average hysteresis curve ${\bar{M}}_{\parallel }({H}_{{{{\rm{ext}}}}})$ and presented in Supplementary Fig. 10. Interestingly, ${\bar{M}}_{{{{\rm{r}}}}}$ presents a nearly identical distribution as $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ vs. P and v, and increases following the direction of rising specific energy input. ${\bar{M}}_{{{{\rm{s}}}}}$, however, presents an inversed distribution vs. P and v and a decreasing tendency with respect to the drop** specific energy input, even though its variation limits in a narrow range, i.e., 854.82 to 854.92 kA m⁻¹.

Fig. 8: Dependence of magnetic coercivity on processing parameters.

a Contour map of the average coercivity $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ in the fusion zone w.r.t. the beam power and the scan speed. The dotted lines represent different specific energy input isolines U_V. b Nonlinear regression of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ on specific energy input U_V and average residual stress ${\bar{\sigma }}_{{{{\rm{e}}}}}$, with the regression parameters indicated correspondingly. It presents that $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ subjects to scaling rule on U_V and to exponential growth law on ${\bar{\sigma }}_{{{{\rm{e}}}}}$ with the correlation coefficient R² = 86.54% and 91.31%, respectively. Profiles of (c–g) the local coercivity; (h–l) the local volume fraction of ${\gamma }^{{\prime} }$ phase; and (m–q) the local residual stress on the mid-sections of the fusion zones are plotted for different processing parameters.

To examine the dependence of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ on the processing parameters from the phenomenological aspect, we performed the nonlinear regression analysis of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ on the specific energy input U_V. The results are presented in Fig. 8b. Notably, $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ relates to U_V by allometric scaling rule, i.e., ${\mu }_{0}{\bar{H}}_{{{{\rm{c}}}}}={C}_{H}{({U}_{{{{\rm{V}}}}})}_{H}^{I}$ with C_H and I_H the parameters. The analysis gives the correlation coefficient R² = 86.54% with relatively large uncertainty located on low and high U_V regions. Regressed I_H = 0.31 ± 0.03 also implies a diminishing scaling of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ by U_V, as the increment of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ decreases with the increase of U_V. However, similar to the one for ${\bar{\sigma }}_{{{{\rm{e}}}}}$, such a simple scaling rule between the specific energy input and coercivity might be challenged since U_V may not be able to identify the uniquely H_c for the SLS-processed part, as the isoline of U_V in Fig. 8a evidently intersects with the contour of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$.

Regression analysis of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ on σ_e was also performed. The exponential growth rule was chosen based on the tendency of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}({{\bar{\sigma }}_{{{{\rm{e}}}}}})$, i.e., ${\mu }_{0}{\bar{H}}_{{{{\rm{c}}}}}={A}_{\sigma }\exp (\frac{{\bar{\sigma }}_{{{{\rm{e}}}}}}{{S}_{\sigma }})+{\mu }_{0}{H}_{0}$ with A_σ, S_σ and H₀ as the parameters. Here we adopt the A_σ as the growth pre-factor and S_σ as the stress scale. When ${\bar{\sigma }}_{{{{\rm{e}}}}}=0$, we have ${\mu }_{0}{H}_{{{{\rm{c}}}}}{| }_{{\bar{\sigma }}_{{{{\rm{e}}}}} = 0}={A}_{\sigma }+{\mu }_{0}{H}_{0}\approx {\mu }_{0}{H}_{0}$, meaning μ₀H₀ can be regarded as the stress-free coercivity. The analysis gives a relatively higher correlation coefficient R² = 91.31% cf. the one on U_V, with relatively large uncertainty located on low U_V region. Compared to the homogeneous stress-free reference in Fig. 5g that is 0.45 mT, this μ₀H₀ is around 24% smaller, owing to the contributions from the infinitesimal-coercivity points. It also presents the rapid growth of $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ after ${\bar{\sigma }}_{{{{\rm{e}}}}}=206\,{{{\rm{MPa}}}}$ with around 40% increment, demonstrating evident effects on $\mu_0{\bar{H}}_{{{{\rm{c}}}}}$ with increasing residual stress.

In summary, the processing-property relationship in tailoring magnetic hysteresis of Fe_21.5Ni_78.5 has been demonstrated in this work by conducting multiphysics-multiscale simulation. The residual stress is unveiled to be the key thread since it readily carries both long-range (morphology and morphology-induced chronological-spatial thermal inhomogeneity) and short-range information (misfit-induced fluctuations) after the processes. Influences of beam power and scan speed have been investigated and presented on distinctive phenomena, including the geometry of fusion zones, the residual stress and accumulated strain, and the resultant coercivity of the manufactured parts. The following conclusions can be drawn from the present work:

• (i) The simulated mesoscopic residual stress states are coherent with TGM interpretation. Further details like the concentrated stress around the concave morphologies (surface concave, sintering necks, etc) beyond the TGM interpretation are also delivered. The accumulated plastic strain is evidently observed at the fusion zone’s outer boundary.

• (ii) Nanoscopic Ni segregation at the $\gamma /{\gamma }^{{\prime} }$ interface due to the diffusion-controlled $\gamma /{\gamma }^{{\prime} }$ transitions is observed with local composition Fe₁₉Ni₈₁, which has comparably smaller saturation magnetization and stronger easy-plane magnetocrystalline anisotropy. Notably, the interface also locally presents negative normal magnetostriction (λ₁₁₁ = − 8.429 × 10⁻⁷) and positive shearing magnetostriction (λ₁₀₀ = 5.249 × 10⁻⁶), while both the ${\gamma }^{{\prime} }$ phase and γ matrix present positive normal and shearing magnetostriction.

• (iii) Large magneto-elastic coupling energy is observed inside ${\gamma }^{{\prime} }$ phase with the corresponding effective field imposing rotating effects on the magnetization. These effects vary point-wisely according to residual stress states and ${\gamma }^{{\prime} }$ phase formation, and eventually lead to different local coercivity. Remarkably, the point around the bottom of the fusion zone is examined to have nearly zero coercivity, which may attribute to the on-site stress state with a principal compressive stress along the building direction with another two tensile ones, i.e., implying a relatively significant shear stress.

• (iv) The relation between the average residual stress and coercivity of the fusion zone is examined to follow the exponential growth rule with a correlation coefficient of 91.31%. The stress-free coercivity derived from the exponential law is 0.34 mT, and the rapid growth of average coercivity is observed when average residual stress exceeds 206 MPa, implying a potential threshold of average residual stress for restricting coercivity of the manufactured parts around the stress-free value. On the other hand, average residual stress beyond the threshold can be considered in the context of effectively increasing the resultant coercivity of the manufactured permalloy parts.

Despite the present findings, several points should be further examined and discussed in future works:

• (i) The conditions deriving the near-zero coercivity should be extensively investigated in the sense of residual stress states rather than its effective value, noting that residual stress is a 2nd-order tensor. Magneto-elastic coupled micromagnetic simulations should be performed under the classified residual stress states to rationalize the factors leading to the vanishing coercivity as in the present findings.

• (ii) The present findings are only examined at relatively low specific energy input and, correspondingly, low generated residual stress, as the SLS is chosen in this work. It is anticipated to conduct the simulations with relatively high energy input, like SLM, and examine the influences of magneto-elastic coupling on the magnetic hysteresis with comparably higher residual stress cases. Influences on residual stress development and, eventually, the coercivity of manufactured permalloy from multilayer and multitrack AM strategies should also be examined.

Methods

Thermodynamic framework

In order to describe the microstructure of an SLS-manufactured Fe-Ni alloy, a conserved order parameter (OP) ρ is employed to represent the substance and atmosphere/pores, and a set of non-conserved OPs $\{{\phi }_{\chi }^{\varphi }\}$ are employed to represent the grains with the superscript $\varphi =\gamma ,{\gamma }^{{\prime} }$ representing the phases and the subscript χ = 1, 2, …, N representing the orientations, extended from our former works^28,33. Counting the thermal, chemical, and mechanical contributions, the temperature field T, the strain field ε, and sets of local chemical molar fraction $\{{X}_{A}^{\varphi }\}$ of the chemical constituents A = Fe, Ni are considered. On the other hand, as a ferromagnetic material under the Curie temperature T_C = 880 K for the composition Fe_21.5Ni_78.5, the thermodynamic contribution due to the existing spontaneous magnetization m is also counted. The framework of the free energy density functional of the system is then formulated as follows

$${{{\mathcal{F}}}}={\int}_{\Omega }\left[{f}_{{{{\rm{ch}}}}}+\overbrace{{f}_{{{{\rm{loc}}}}}+{f}_{{{{\rm{grad}}}}}}^{{f}_{{{{\rm{intf}}}}}}\,+{f}_{{{{\rm{el}}}}}+{f}_{{{{\rm{mag}}}}}\right]{{{\rm{d}}}}\Omega ,$$

(5)

where f_ch represents the contributions from the chemical constituents. f_loc and ${f}_{{{{\rm{grad}}}}}$, together as the f_intf, presents the contributions from the surface and interfaces (incl. grain boundaries and phase boundaries)⁴⁶. f_el is the contribution from the elastic deformation, and f_mag is the contribution from spontaneous magnetization and magnetic-coupled effects.

It is worth noting that this uniform thermodynamic framework does not imply that a single vast inter-coupled problem with all underlining physics should be solved interactively and simultaneously crossing all involved scales. As sufficiently elaborated in the subsection Multiphysics-multiscale simulation scheme, it is more practical and effective to conduct the multiphysics-multiscale simulations in a subsequent scheme and concentrate on rationalizing and bridging the physical quantities and processes among problems and scales. In that sense, the free energy density functional, originated from Eq. (5), should be sufficiently simplified regarding the distinctiveness of each problem at the corresponding scale. This will be explicitly introduced in the following sections.

Mesoscopic processing simulations

Here we consider the SLS processing on a mesoscopic powder bed by using ρ to differentiate pore-substance and $\{{\phi }_{\chi }^{\varphi }\}$ to differentiate polycrystalline orientations. According to the high-temperature phase diagram of the Fe-Ni system⁴⁷, the γ phase exists within a relatively large temperature range (from T_M = 1709 K to the transition starting temperature ${T}_{\gamma /{\gamma }^{{\prime} }}=766\,{{{\rm{K}}}}$) for the composition Fe_21.5Ni_78.5. On the other hand, since the SLS together with cooling stages would only last a relatively infinitesimal time (on the order of 10 ms) to the following annealing stage (more than 10 h), there is almost no change for ${\gamma }^{{\prime} }$ phase to grow into mesoscopic size. In this regard, we treat all existing polycrystals during the SLS stage as the γ phase. Considering Fe-Ni as a binary system where the constraints X_Ni + X_Fe = 1 and ${\phi }_{\chi }^{{\gamma }^{{\prime} }}+{\phi }_{\chi }^{\gamma }=\rho$ always holds, we then only take the OP set $\{{\phi }_{\chi }^{\gamma }\}$ (χ = 1, 2, . . . , N) as well as ρ for the mesoscopic simulations due to the absence of the ${\gamma }^{{\prime} }$ phase on the mesostructures. The profiles of ρ and $\{{\phi }_{\chi }^{\gamma }\}$ across the surface and grain boundary between two adjacent γ-grains are illustrated in Fig. 9b. We also take simplified notations X = X_Ni in this subsection as the independent concentration indicators, while X_Fe = 1 − X.

Fig. 9: Phase-field scenario and landscapes of free energy density.

a Schematic of the physical processes during SLS, i.e., localized melting, multiple mass transfer paths, and grain boundary migration. Here, the bulk diffusion GB-N represents the path from grain boundary to neck through the bulk, while SF-N presents the path from the surface to neck through the bulk. b Profiles of different OPs across solid-state phases with the corresponding scales. c The landscape of mesoscopic stress-free f_tot across the surface and the γ/γ grain boundaries at various temperatures. d The landscape of nanoscopic stress-free f_tot across the $\gamma /{\gamma }^{{\prime} }$ coherent interface at 600 K. Transition path, ${f}_{{{{\rm{ch}}}}}^{\gamma }$, ${f}_{{{{\rm{ch}}}}}^{{\gamma }^{{\prime} }}$, and the energy band of the four-sublattice states $\{\Delta {f}_{{{{\rm{4sl}}}}}\}=\{{f}_{{{{\rm{4sl}}}}}({Y}^{(s)})-{f}_{{{{\rm{4sl}}}}}({Y}^{(s)}={X}^{{\gamma }^{{\prime} }})\}$ are also denoted. Inset: sublattice sites for the L1₂ ordering of Fe-Ni. e–f Energy surface of magneto-crystalline anisotropy energy, which alters w.r.t. the sign of anisotropy constant K_u, i.e., (e) the easy-axis anisotropy when K_u > 0 and (f) the easy-plane anisotropy when K_u < 0.

Due to the co-existence of substance (γ-grains with Ni composition X₀ = 0.785) and pores/atmosphere, the chemical free energy density should be formulated as

$${f}_{{{{\rm{ch}}}}}(T,\rho )={h}_{{{{\rm{ss}}}}}(\rho ){f}_{{{{\rm{ch}}}}}^{\gamma }(T,{X}^{\gamma }={X}_{0})+{h}_{{{{\rm{at}}}}}(\rho ){f}_{{{{\rm{ch}}}}}^{{{{\rm{at}}}}}(T),$$

(6)

where h_ss and h_at are monotonic interpolation functions with subscripts “ss” and “at” representing the substance and pore/atmosphere and are assumed to have the polynomial forms as

$${h}_{{{{\rm{ss}}}}}(\rho )={\rho }^{3}\left(10-15\rho +6{\rho }^{2}\right),\quad {h}_{{{{\rm{at}}}}}(\rho )=1-{\rho }^{3}\left(10-15\rho +6{\rho }^{2}\right).$$

The temperature-dependent chemical free energy ${f}_{{{{\rm{ch}}}}}^{\gamma }$ is modeled by the CALPHAD approach

$${f}_{{{{\rm{ch}}}}}^{\gamma }={f}_{{{{\rm{ref}}}}}^{\gamma }+{f}_{{{{\rm{id}}}}}^{\gamma }+{f}_{{{{\rm{mix}}}}}^{\gamma }+{f}_{{{{\rm{mm}}}}}^{\gamma },$$

(7)

with

$$\begin{array}{lll}\quad{f}_{{{{\rm{ref}}}}}^{\gamma }(T,{X}^{\gamma })\,=\,{X}^{\gamma }{f}_{{{{\rm{Ni}}}}}(T)+\left(1-{X}^{\gamma }\right){f}_{{{{\rm{Fe}}}}}(T),\\ \quad\,{f}_{{{{\rm{id}}}}}^{\gamma }(T,{X}^{\gamma })\,=\,\frac{{{{\mathcal{R}}}}T}{{V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}}\left[{X}^{\gamma }\ln {X}^{\gamma }+(1-{X}^{\gamma })\ln (1-{X}^{\gamma })\right],\\ \,\,\,{f}_{{{{\rm{mix}}}}}^{\gamma }(T,{X}^{\gamma })\,=\,{X}^{\gamma }[1-{X}^{\gamma }]{L}^{\gamma }(T),\\ \,\,\,{f}_{{{{\rm{mm}}}}}^{\gamma }(T,{X}^{\gamma })\,=\,\frac{{{{\mathcal{R}}}}T}{{V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}}\ln ({\beta }^{\gamma }+1){\mathbb{P}}\left(\frac{T}{{T}_{{{{\rm{C}}}}}^{\gamma }}\right),\end{array}$$

where ${f}_{{{{\rm{ref}}}}}^{\gamma }$ is the term corresponding to the mechanical mixture of the chemical constituents (in this case, Fe and Ni), ${f}_{{{{\rm{id}}}}}^{\gamma }$ is the contribution from the configurational entropy for an ideal mixture, ${f}_{{{{\rm{mix}}}}}^{\gamma }$ is the excess contribution due to mixing, and ${f}_{{{{\rm{mag}}}}}^{\gamma }$ is the contribution due to the magnetic moment. The parameters fed in Eq. (7), including the atom magnetic moment β^γ, the Curie temperature ${T}_{{{{\rm{C}}}}}^{\gamma }$, and the interaction coefficient L^γ, are described in the way of Redlish-Kister polynomials⁴⁸ which is generally formulated for a binary system as ${p}^{\varphi }={X}_{A}{p}_{A}^{\varphi }+{X}_{B}{p}_{B}^{\varphi }+{X}_{A}{X}_{B}{\sum }_{n}{p}_{A,B}^{\varphi ,n}{({X}_{A}-{X}_{B})}^{n}$ with the temperature-dependent parameters ${p}_{A}^{\varphi }$, ${p}_{B}^{\varphi }$ and ${p}_{A,B}^{\varphi ,n}$ for optimization. ${\mathbb{P}}(T/{T}_{{{{\rm{C}}}}}^{\gamma })$ represents the Inden polynomial, obtained by expanding the magnetic specific heat onto a power series of the normalized temperature $T/{T}_{{{{\rm{C}}}}}^{\gamma }$^49,50. ${{{\mathcal{R}}}}$ is the ideal gas constant. ${V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}$ is the molar volume of the system. All the thermodynamic parameters for the CALPHAD are obtained from ref. ⁴⁷ while the molar volume of the system is obtained from the database TCFE8 from the commercial software Thermo-Calc^®51.

Since the variation of Ni composition is negligible in between T_M and ${T}_{\gamma /{\gamma }^{{\prime} }}$, we pursue a simple but robust way of implementing f_ch under a drastically varying T during the SLS stage. Taking T_M as referencing temperature, Eq. (6) is then re-written as

$${f}_{{{{\rm{ch}}}}}(T,\rho )={c}_{{{{\rm{r}}}}}\left[(T-{T}_{{{{\rm{M}}}}})-T\ln \frac{T}{{T}_{{{{\rm{M}}}}}}\right]+{f}_{{{{\rm{ref}}}}}^{{T}_{{{{\rm{M}}}}}}-{h}_{{{{\rm{ml}}}}}\frac{T-{T}_{{{{\rm{M}}}}}}{{T}_{{{{\rm{M}}}}}}{{{\mathcal{L}}}},$$

(8)

where ${f}_{{{{\rm{ref}}}}}^{{T}_{{{{\rm{M}}}}}}$ is a referencing chemical free energy density at T_M, which can be omitted in the following calculations. c_r is a relative specific heat landscape, i.e., ${c}_{{{{\rm{r}}}}}(\rho ,T)={h}_{{{{\rm{ss}}}}}(\rho ){c}_{{{{\rm{v}}}}}^{\gamma }(T)+{h}_{{{{\rm{at}}}}}(\rho ){c}_{{{{\rm{v}}}}}^{{{{\rm{at}}}}}(T)$ with the volumetric specific heat for γ grains and pores/atmosphere. Notably, ${c}_{{{{\rm{v}}}}}^{\gamma }$ can be thermodynamically calculated as follows at a fixing pressure p₀ and composition X₀.

$${c}_{{{{\rm{v}}}}}^{\gamma }(T)=-T{\left(\frac{{\partial }^{2}{f}_{{{{\rm{ch}}}}}^{\gamma }}{\partial {T}^{2}}\right)}_{{p}_{0},{X}_{0}}.$$

(9)

It should be noticed that the ${c}_{{{{\rm{v}}}}}^{\gamma }$ obtained by Eq. (9) has a discontinuous point at T_C, which is due to the 2nd-order Curie transition, as shown in Supplementary Fig. 5a. ${{{\mathcal{L}}}}$ is the latent heat due to the partial/full melting, which is mapped by the interpolation function h_ml. Here h_ml adopts a sigmoid form with a finite temperature band Δ_T

$${h}_{{{{\rm{ml}}}}}=\frac{1}{2}\left[1+\tanh \frac{2(T-{T}_{{{{\rm{M}}}}})}{{\Delta }_{T}}\right].$$

which reaches unity once T → T_M and is smooth enough to ease the drastic change in f_ch.

On the other hand, to explain the free energy landscape across the surface and γ/γ interface (or γ grain boundary) under varying temperatures, we adopt the non-isothermal multi-well Landau polynomial and gradient terms from our former works^28,33, i.e.,

$$\begin{array}{rcl}{f}_{{{{\rm{loc}}}}}(T,\rho ,\{{\phi }_{\chi }^{\gamma }\})&=&{\underline{W}}_{{{{\rm{sf}}}}}(T)\left[\rho {(1-\rho )}^{2}\right]+{\underline{W}}_{\gamma /\gamma }(T)\left\{{\rho }^{2}+6(1-\rho )\mathop{\sum}\limits_{\chi }{({\phi }_{\chi }^{\gamma })}^{2}\right.\\ &&\left.-4(2-\rho )\mathop{\sum}\limits_{\chi }{({\phi }_{\chi }^{\gamma })}^{3}+3{\left[\mathop{\sum}\limits_{\chi }{({\phi }_{\chi }^{\gamma })}^{2}\right]}^{2}\right\},\\ {f}_{{{{\rm{grad}}}}}(T,\nabla \rho ,\{\nabla {\phi }_{\chi }^{\gamma }\})&=&\frac{1}{2}\left[{\underline{\kappa }}_{{{{\rm{sf}}}}}(T)| \nabla \rho {| }^{2}+\mathop{\sum}\limits_{\chi }{\underline{\kappa }}_{\gamma /\gamma }(T)| \nabla {\phi }_{\chi }^{\gamma }{| }^{2}\right],\end{array}$$

(10)

with

$$\begin{array}{lll}{\underline{W}}_{{{{\rm{sf}}}}}(T)={W}_{{{{\rm{sf}}}}}{\tau }_{{{{\rm{sf}}}}}(T),&&{\underline{\kappa }}_{{{{\rm{sf}}}}}(T)={\kappa }_{{{{\rm{sf}}}}}{\tau }_{{{{\rm{sf}}}}}(T),\\ {\underline{W}}_{\gamma /\gamma }(T)={W}_{\gamma /\gamma }{\tau }_{\gamma /\gamma }(T),&&{\underline{\kappa }}_{\gamma /\gamma }(T)={\kappa }_{\gamma /\gamma }{\tau }_{\gamma /\gamma }(T),\end{array}$$

${\underline{W}}_{\gamma /{\gamma }^{{\prime} }}$ and ${\underline{\kappa }}_{\gamma /{\gamma }^{{\prime} }}$ are temperature-independent parameters obtained from the surface and γ/γ interface energy Γ_sf, Γ_γ/γ and diffuse-interface width ℓ_γ/γ, and τ_sf(T) and τ_γ/γ(T) are the dimensionless tendencies inherited from the temperature dependency of Γ_sf and Γ_γ/γ, i.e.,

$$\begin{array}{l}{\Gamma }_{{{{\rm{sf}}}}}(T)=\frac{\sqrt{2}}{6}{\tau }_{{{{\rm{sf}}}}}(T)\sqrt{({W}_{{{{\rm{sf}}}}}+7{W}_{\gamma /\gamma })({\kappa }_{{{{\rm{sf}}}}}+{\kappa }_{\gamma /\gamma })},\\ {\Gamma }_{\gamma /\gamma }(T)=\frac{2\sqrt{3}}{3}{\tau }_{\gamma /\gamma }(T)\sqrt{{W}_{\gamma /\gamma }{\kappa }_{\gamma /\gamma }},\\ {\ell }_{\gamma /\gamma }\approx \frac{2\sqrt{3}}{3}\sqrt{\frac{{\kappa }_{\gamma /\gamma }}{{W}_{\gamma /\gamma }}},\end{array}$$

(11a)

along with the constraint among parameters for having the sample profile of ρ and ${\phi }_{\chi }^{\gamma }$ across the surface²⁸, i.e.,

$$\frac{{W}_{{{{\rm{sf}}}}}+{W}_{\gamma /\gamma }}{{\kappa }_{{{{\rm{sf}}}}}}=\frac{6{W}_{\gamma /\gamma }}{{\kappa }_{\gamma /\gamma }}.$$

(11b)

In this work, we give ℓ_γ/γ = 2 μm, the temperature-dependent Γ_sf and Γ_γ/γ are presented in Supplementary Fig. 5c. The total free energy density landscape at stress-free condition (f_tot = f_ch + f_intf) is illustrated in Fig. 9c. We can tell that the term f_ch modifies the relative thermodynamic stability of the substance by shifting the free energy minima according to temperature changes. In contrast, γ grains at the same temperature do not show a difference in stability until the on-site temperature of one is changed.

The governing Eqs. for the coupled thermo-structural evolution are formulated as follows^28,29

$$\frac{\partial \rho }{\partial t}=\nabla \cdot {{{\bf{M}}}}\cdot \nabla \frac{\delta {{{\mathcal{F}}}}}{\delta \rho },$$

(12a)

$$\frac{\partial {\phi }_{\chi }^{\gamma }}{\partial t}=-L\frac{\delta {{{\mathcal{F}}}}}{\delta {\phi }_{\chi }^{\gamma }},$$

(12b)

$${c}_{{{{\rm{r}}}}}\left(\frac{\partial T}{\partial t}-{{{\bf{v}}}}\cdot \nabla T\right)=\nabla \cdot {{{\bf{K}}}}\cdot \nabla T+{q}_{{{{\bf{v}}}}},$$

(12c)

where Eq. (12a) is the Cahn-Hilliard Eq. with the mobility tensor M specifically considering various mass transfer paths, incl. the mobilities for the mass transfer through the substance (ss), atmosphere (at), surface (sf) and grain boundary (gb). As elaborated in our former work²⁸, the localized melt flow driven by the local curvature is also modeled by one effective surface mobility ${M}_{{{{\rm{ml}}}}}^{{{{\rm{eff}}}}}$. M is then formulated as³³

$${{{\bf{M}}}}={h}_{{{{\rm{ss}}}}}{M}_{{{{\rm{ss}}}}}{{{\bf{I}}}}+{h}_{{{{\rm{at}}}}}{M}_{{{{\rm{at}}}}}{{{\bf{I}}}}+{h}_{{{{\rm{sf}}}}}{M}_{{{{\rm{sf}}}}}{{{{\bf{T}}}}}_{{{{\rm{sf}}}}}+{h}_{{{{\rm{gb}}}}}{M}_{{{{\rm{gb}}}}}{{{{\bf{T}}}}}_{{{{\rm{gb}}}}}+{h}_{{{{\rm{ml}}}}}\left(T\right){M}_{{{{\rm{ml}}}}}^{{{{\rm{eff}}}}}{{{{\bf{T}}}}}_{{{{\rm{sf}}}}},$$

(13)

with the 2^nd-order identity tensor I and projection tensors T_sf and T_gb for surface and grain boundary, respectively^33,52. The T-dependent values for M_sf, M_gb, M_ss, and ${M}_{{{{\rm{ml}}}}}^{{{{\rm{eff}}}}}$ are presented in Supplementary Fig. 5d. The interpolation functions on the surface and γ grain boundaries are defined as

$${h}_{{{{\rm{sf}}}}}=16{\rho }^{2}{(1-\rho )}^{2},\quad \quad {h}_{{{{\rm{gb}}}}}=16\mathop{\sum}\limits_{i\ne j}{({\phi }_{i}^{\gamma }{\phi }_{j}^{\gamma })}^{2}.$$

Equation (12b) is the Allen-Cahn Eq. with the scalar mobility L, which is derived from the γ grain boundary mobility G_γ/γ as^53,54

$$L=\frac{{G}_{\gamma /\gamma }{\Gamma }_{\gamma /\gamma }}{{\underline{\kappa }}_{\gamma /\gamma }},$$

(14)

which is also presented in Supplementary Fig. 5d.

Equation (12c) is the heat transfer equation that considers the laser-induced thermal effect as a volumetric heat source q_v.

$${q}_{{{{\bf{v}}}}}({{{\bf{r}}}},t)=P{p}_{xy}[{{{{\bf{r}}}}}_{O}({{{\bf{v}}}},t)]\frac{{{{\rm{d}}}}a}{{{{\rm{d}}}}z},$$

in which p_xy indicates the in-plane Gaussian distribution with a moving center r_O(v, t). P is the beam power and v is the scan velocity with its magnitude v = ∣v∣ as the scan speed. The absorptivity profile function along depth da/dz is calculated based on refs. ^28,55. The phase-dependent thermal conductivity tensor is formulated in a form considering the continuity of the thermal flux along the normal/tangential direction of the surface^52,56, i.e.,

$${{{\bf{K}}}}={K}_{\perp }{{{{\bf{N}}}}}_{{{{\rm{sf}}}}}+{K}_{\parallel }{{{{\bf{T}}}}}_{{{{\rm{sf}}}}}$$

(15)

with

$${K}_{\perp }={h}_{{{{\rm{ss}}}}}{K}_{{{{\rm{ss}}}}}+{h}_{{{{\rm{at}}}}}{K}_{{{{\rm{at}}}}},\quad \quad {K}_{\parallel }=\frac{{K}_{{{{\rm{ss}}}}}{K}_{{{{\rm{at}}}}}}{{h}_{{{{\rm{ss}}}}}{K}_{{{{\rm{at}}}}}+{h}_{{{{\rm{at}}}}}{K}_{{{{\rm{ss}}}}}},$$

where K_ss and K_at are the thermal conductivity of the substance and pore/atmosphere. N is the 2nd-order normal tensor of the surface^34,52. Thermal resistance on the surface and γ/γ interface are disregarded, and will be presented in the upcoming works. While temperature-dependent K_ss takes the linear form in this work (Supplementary Fig. 5b), K_at specifically considers the radiation contribution via pore/atmosphere and is formulated as

$${K}_{{{{\rm{at}}}}}={K}_{0}+4F{T}^{3}{\sigma }_{{{{\rm{B}}}}}{\ell }_{{{{\rm{rad}}}}},$$

(16)

where K₀ ≈ 0.06 W m⁻¹ K⁻¹ is the thermal conductivity of the Argon gas, F = 1/3 is the Damköhler view factor⁵⁷, and σ_B = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ is the Stefan-Boltzmann constant. ℓ_rad is the effective radiation path between particles, which usually takes the average diameter of the powders⁵⁸.

As the boundary conditions (BC), no mass transfer is allowed on all the boundaries of the mesoscopic domain, which is achieved by setting Neumann BC on ρ as zero. The temperature at the bottom of the substrate mesh (${z}_{\min }$) is fixed at T₀ via Dirichlet BC on T. Heat transfer is allowed only via the pore/atmosphere, achieved by the combined BC of convection and radiation and masked by h_at), as illustrated in Supplementary Fig. 2d.

Mesoscopic thermo-elasto-plastic simulations

As elaborated in the subsection Multiphysics-multiscale simulation scheme and our former work²⁹, the subsequent thermo-elasto-plastic simulation was carried out for the calculation of the thermal stress and deformation of the mesostructures from the non-isothermal phase-field simulations of SLS processing. The transient temperature field and substance field ρ are imported into the quasi-static elasto-plastic model as the thermal load and the phase indicator for interpolating mechanical properties. Adopting small deformation and quasi-static assumptions, the mechanical equilibrium reads

$$\nabla \cdot {{{\mathbf{\sigma }}}}={{{\bf{0}}}},$$

(17)

where σ is the 2nd-order stress tensor. The top boundary is set to be traction free, and the other boundaries adopt rigid support BCs, which only restrict the displacement component in the normal direction of the boundary Supplementary Fig. 2e.

Taking the Voigt-Taylor interpolation scheme (VTS), where the total stress is interpolated according to the amount of the substance and pore/substance across the interface, i.e., ${{{\bf{\sigma }}}}={h}_{\gamma }{{{{\bf{\sigma }}}}}^{\gamma }+{h}_{{\gamma }^{{\prime} }}{{{{\bf{\sigma }}}}}^{{\gamma }^{{\prime} }}$, while assuming identical strain among phases^59,60,61. In this regard, the stress can be eventually formulated by the linear constitutive Eq.

$${{{\mathbf{\sigma }}}}={{{\bf{C}}}}(\rho ,T):\left({{{\bf{\varepsilon }}}}-{{{{\bf{\varepsilon }}}}}_{{{{\rm{th}}}}}-{{{{\bf{\varepsilon }}}}}_{{{{\rm{pl}}}}}-{{{{\bf{\varepsilon }}}}}_{{{{\rm{em}}}}}\right),$$

(18)

where ε_th is the thermal eigenstrain and ε_pl is the plastic strain. Notably, the magnetostrictive strain ε_em should be considered when the local temperature is below Curie temperature T_C due to the existence of spontaneous magnetization. This contribution, however, is assumed to be negligible compared to the developed plastic strain during the SLS and cooling stage and was tentatively dropped in this work. The 4th-order elastic tensor is interpolated from the substance one C_ss and the pores/atmosphere one C_at, i.e.,

$${{{\bf{C}}}}(\rho ,T)={h}_{{{{\rm{ss}}}}}(\rho ){{{{\bf{C}}}}}_{{{{\rm{ss}}}}}(T)+{h}_{{{{\rm{at}}}}}(\rho ){{{{\bf{C}}}}}_{{{{\rm{at}}}}}.$$

(19)

In this work, isotropic mechanical properties are considered. C_ss is calculated from the Young’s modulus E(T) and Poisson’s ratio ν. In contrast, C_at is assigned with a sufficiently small value to guarantee the numerical convergence. The thermal eigenstrain ε_th is calculated using the interpolated coefficient of thermal expansion, i.e.,

$$\begin{array}{lll}\qquad{{{{\bf{\varepsilon }}}}}_{{{{\rm{th}}}}}\,=\,\alpha (\rho ,T)[T-{T}_{0}]{{{\bf{I}}}},\\ \alpha (\rho ,T)\,=\,{h}_{{{{\rm{ss}}}}}(\rho ){\alpha }_{{{{\rm{ss}}}}}(T)+{h}_{{{{\rm{at}}}}}(\rho ){\alpha }_{{{{\rm{at}}}}},\end{array}$$

(20)

where α_ss is obtained from temperature-dependent molar volume of the binary system ${V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}$, i.e.,

$$\alpha =\frac{1}{3}\frac{1}{{V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}}\frac{\partial {V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}}{\partial T}.$$

(21)

Meanwhile, for plastic strain ε_pl, the isotropic hardening model with the von Mises yield criterion is employed. The yield condition is determined as

$$f({{{\boldsymbol{\sigma }}}},{p}_{{{{\rm{e}}}}})={\sigma }_{{{{\rm{e}}}}}-({\sigma }_{{{{\rm{y}}}}}+H{p}_{{{{\rm{e}}}}})\le 0$$

(22)

with

$${\sigma }_{{{{\rm{e}}}}}=\sqrt{\frac{3}{2}{{{\bf{s}}}}:{{{\bf{s}}}}},\quad {p}_{{{{\rm{e}}}}}=\int\sqrt{\frac{2}{3}{{{\rm{\,d}}}}{{{{\bf{\varepsilon }}}}}_{{{{\rm{pl}}}}}:{{{\rm{d}}}}{{{{\bf{\varepsilon }}}}}_{{{{\rm{pl}}}}}},$$

where σ_e is the von Mises stress. s is the deviatoric stress, σ_y is the yield stress when no plastic strain is present. The isotropic plastic modulus H can be calculated from the isotropic hardening tangent modulus E_t and Young’s modulus E as H = EE_t/(E − E_t). These temperature-dependent mechanical properties are collectively presented in Supplementary Fig. 6. p_e is the accumulated plastic strain, which is integrated implicitly from the plastic strain increment dε_pl obtained from the radial return method^62,63. It is worth noting that the plastic strain is reset as zero once T > T_M in emulating the effect of partial/full melting.

Nanoscopic chemical order-disorder ($\gamma /{\gamma }^{{\prime} }$) transition

Once the temperature drops below ${T}_{\gamma /{\gamma }^{{\prime} }}$, the historical quantities (notably T(t) and σ(t)) will be sampled and imported to the nanoscopic domain for the subsequential $\gamma /{\gamma }^{{\prime} }$ simulations. Here we consider the nanoscopic ($\gamma /{\gamma }^{{\prime} }$) transition that occurs localized inside one γ grain, where ρ and only one ${\phi }_{\chi }^{\gamma }$ are unity while other OPs are zero. This means the orientation information of the γ grain and the influences from the surface and γ/γ interface have been omitted on this scale. The growing ${\gamma }^{{\prime} }$ is also known to be orientation-coherent based on the experimental observations^18,19. In this regard, the subscript χ, indicating the different grain orientations, is dropped in the following discussions. Magnetic contribution f_mag is also dropped since the magnetic-field-free $\gamma /{\gamma }^{{\prime} }$ transition is scoped in this work. The profiles of ϕ^γ and ${\phi }^{{\gamma }^{{\prime} }}$ across the $\gamma /{\gamma }^{{\prime} }$ interface are illustrated in Fig. 9b. Similarly, we take simplified notations in this subsection, such as $\phi ={\phi }^{{\gamma }^{{\prime} }}$ and X = X_Ni as the independent phase and concentration indicators, while X_Fe = 1 − X and ϕ^γ = 1 − ϕ.

Due to the co-existence of both γ and ${\gamma }^{{\prime} }$ once the temperature is below ${T}_{\gamma /{\gamma }^{{\prime} }}=766\,\,{{\mbox{K}}}\,$, the chemical free energy density should be formulated as

$${f}_{{{{\rm{ch}}}}}(T,\phi ,\{{X}^{\varphi }\})={h}_{\gamma }(\phi ){f}_{{{{\rm{ch}}}}}^{\gamma }(T,{X}^{\gamma })+{h}_{{\gamma }^{{\prime} }}(\phi ){f}_{{{{\rm{ch}}}}}^{{\gamma }^{{\prime} }}(T,{X}^{{\gamma }^{{\prime} }}),$$

(23)

where h_γ and ${h}_{{\gamma }^{{\prime} }}$ are monotonic interpolation functions and can adopt the polynomial formulation as

$${h}_{{\gamma }^{{\prime} }}(\phi )={\phi }^{3}\left(10-15\phi +6{\phi }^{2}\right),\quad {h}_{\gamma }(\phi )=1-{\phi }^{3}\left(10-15\phi +6{\phi }^{2}\right).$$

Similarly the elastic contribution is formulated as^64,65

$${f}_{{{{\rm{el}}}}}(T,\{{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{\varphi }\})={h}_{\gamma }(\phi ){f}_{{{{\rm{el}}}}}^{\gamma }(T,{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{\gamma })+{h}_{{\gamma }^{{\prime} }}(\phi ){f}_{{{{\rm{el}}}}}^{{\gamma }^{{\prime} }}(T,{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{{\gamma }^{{\prime} }}),$$

(24)

where

$${f}_{{{{\rm{el}}}}}^{\varphi }(T,{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}})=\frac{1}{2}{{{{\bf{\sigma }}}}}^{\varphi }:{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{\varphi }.$$

On top of the CALPHAD modeling of the free energy density of the chemical-disordered γ phase, the four-sublattice model is employed to describe the phase with chemical ordering. The model takes the element fractions ${Y}_{i}^{(s)}$ (i = Fe or Ni indicating the chemical constituents, s = 1, 2, 3, 4 indicating the sublattice site, see inset of Fig. 9c) in each sublattice as the inner degree-of-freedoms, formulated as

$${f}_{{{{\rm{ch}}}}}^{{\gamma }^{{\prime} }}={f}_{{{{\rm{ch}}}}}^{\gamma }+{\{{f}_{4{{{\rm{sl}}}}}-{f}_{4{{{\rm{sl}}}}}({Y}^{(s)} = {X}^{{\gamma }^{{\prime} }})\}}_{\min }$$

(25)

with

$$\begin{array}{rcl}{f}_{4{{{\rm{sl}}}}}(T,{Y}_{i}^{(s)})\,=\,\mathop{\sum}\limits_{i,j,k,l}{Y}_{i}^{(1)}{Y}_{j}^{(2)}{Y}_{k}^{(3)}{Y}_{l}^{(4)}{f}_{ijkl}^{{{{\rm{FCC}}}}}+\frac{{{{\mathcal{R}}}}T}{4{V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}}\mathop{\sum}\limits_{s,i}{Y}_{i}^{(s)}\ln {Y}_{i}^{(s)}\\ +\mathop{\sum}\limits_{s}\left[{Y}_{i}^{(s)}\left(1-{Y}_{i}^{(s)}\right)\mathop{\sum}\limits_{j,k,l}{Y}_{j}^{(u)}{Y}_{k}^{(v)}{Y}_{l}^{(w)}\frac{{L}_{(s)ijkl}^{{{{\rm{FCC}}}}}}{{V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}}\right],\end{array}$$

(26)

where ${f}_{ijkl}^{{{{\rm{FCC}}}}}$ are the free energies of the stoichiometric compounds with only one constituent (i, j, k, l = Ni or Fe) occupied on each site⁴⁷. ${L}_{(s)ijkl}^{{{{\rm{FCC}}}}}$ is the interaction parameter, corresponding to the mixing of constituents on the s-th site while others (u, v, w ≠ s) are with the fractions ${Y}_{j}^{(u)}$, ${Y}_{k}^{(v)}$ and ${Y}_{l}^{(w)}$. Note that the constraints ${\sum }_{s}{Y}_{i}^{(s)}={X}_{i}$ and ${Y}_{{{{\rm{Ni}}}}}^{(s)}+{Y}_{{{{\rm{Fe}}}}}^{(s)}=1$ should be applied to guarantee the conservation of atom. It is also worth noting that due to the thermodynamic equivalence of four sublattice sites, the equivalence of ${f}_{ijkl}^{{{{\rm{FCC}}}}}$ and ${L}_{(s)ijkl}^{{{{\rm{FCC}}}}}$ regarding the combination of sublattice constituents must be considered, as explicitly explained in ref. ⁴⁷. All the thermodynamic parameters for the CALPHAD are obtained from ref. ⁴⁷. In Supplementary Fig. 8. we present the calculated ${f}_{{{{\rm{ch}}}}}^{\gamma }$ and ${f}_{{{{\rm{ch}}}}}^{{\gamma }^{{\prime} }}$ from the ${T}_{\gamma /{\gamma }^{{\prime} }}$ to the pre-heating temperature T₀ = 600 K with the varying equilibrium concentration of ${X}_{{{{\rm{e}}}}}^{\gamma }$ and ${X}_{{{{\rm{e}}}}}^{{\gamma }^{{\prime} }}$, the site element fraction ${Y}_{i}^{(s)}$, and the calculated phase fraction Ψ_γ and ${\Psi }_{{\gamma }^{{\prime} }}$ under the equilibrium.

On the other hand, since there is only the $\gamma /{\gamma }^{{\prime} }$ coherent interface, the non-isothermal local and gradient free energy density are then formulated in the typical double-well fashion, i.e.,

$${f}_{{{{\rm{loc}}}}}(T,\phi )=12{\underline{W}}_{\gamma /{\gamma }^{{\prime} }}(T){\phi }^{2}{(1-\phi )}^{2},\quad \quad {f}_{{{{\rm{grad}}}}}(T,\nabla \phi )=\frac{1}{2}{\underline{\kappa }}_{\gamma /{\gamma }^{{\prime} }}(T)| \nabla \phi {| }^{2}$$

(27)

with

$${\underline{W}}_{\gamma /{\gamma }^{{\prime} }}(T)={W}_{\gamma /{\gamma }^{{\prime} }}{\tau }_{\gamma /{\gamma }^{{\prime} }}(T),\quad \quad {\underline{\kappa }}_{\phi }(T)={\kappa }_{\gamma /{\gamma }^{{\prime} }}{\tau }_{\gamma /{\gamma }^{{\prime} }}(T),$$

adapting the same non-isothermal form as the one used in the SLS simulations with the dimensionless temperature tendency ${\tau }_{\gamma /{\gamma }^{{\prime} }}(T)$. The temperature-independent parameters ${W}_{\gamma /{\gamma }^{{\prime} }}$ and ${\kappa }_{\gamma /{\gamma }^{{\prime} }}$ are obtained from the interface energy ${\Gamma }_{\gamma /{\gamma }^{{\prime} }}$ and diffuse-interface width ${\ell }_{\gamma /{\gamma }^{{\prime} }}$, i.e.,

$${\Gamma }_{\gamma /{\gamma }^{{\prime} }}(T)=\frac{\sqrt{6}}{3}{\tau }_{\gamma /{\gamma }^{{\prime} }}(T)\sqrt{{W}_{\gamma /{\gamma }^{{\prime} }}{\kappa }_{\gamma /{\gamma }^{{\prime} }}},\quad \quad {\ell }_{\gamma /{\gamma }^{{\prime} }}\approx \frac{\sqrt{6}}{3}\sqrt{\frac{{\kappa }_{\gamma /{\gamma }^{{\prime} }}}{{W}_{\gamma /{\gamma }^{{\prime} }}}},$$

(28)

noting that the relation of ${\ell }_{\gamma /{\gamma }^{{\prime} }}$ here corresponds to the case when adjusting parameter as two in Eq. (53) of the ref. ⁶⁶. In this work, we tentatively take ${\tau }_{\gamma /{\gamma }^{{\prime} }}$ as one and estimate ${\Gamma }_{\gamma /{\gamma }^{{\prime} }}=0.025\,{{{\rm{J}}}}\,{{{{\rm{m}}}}}^{-2}$, which is a commonly estimated value for the coherent interfaces and lies between the experimental range from 0.008 to 0.080 J m⁻² for the Ni-base alloys⁶⁷. The diffuse-interface width ${\ell }_{\gamma /{\gamma }^{{\prime} }}$ is given as 5 nm. The total free energy at stress-free condition (f_tot = f_ch + f_intf) is illustrated in Fig. 9d, where the free energy density path obeying the mixing rule is also illustrated across the $\gamma /{\gamma }^{{\prime} }$ interface between two equilibrium phases (i.e., with ${X}_{{{{\rm{e}}}}}^{\gamma }$ and ${X}_{{{{\rm{e}}}}}^{{\gamma }^{{\prime} }}$).

The governing Eqs. of the nanoscopic $\gamma /{\gamma }^{{\prime} }$ transition is formulated as follows^64,66

$$X={h}_{\gamma }{X}^{\gamma }+{h}_{{\gamma }^{{\prime} }}{X}^{{\gamma }^{{\prime} }},$$

(29a)

$$\frac{{{{\rm{d}}}}{f}_{{{{\rm{ch}}}}}^{\gamma }}{{{{\rm{d}}}}{X}^{\gamma }}=\frac{{{{\rm{d}}}}{f}_{{{{\rm{ch}}}}}^{{\gamma }^{{\prime} }}}{{{{\rm{d}}}}{X}^{{\gamma }^{{\prime} }}},$$

(29b)

$$\frac{\partial X}{\partial t}=\nabla \cdot {M}_{X}\nabla \frac{\delta {{{\mathcal{F}}}}}{\delta X},$$

(29c)

$$\frac{\partial \phi }{\partial t}=-{M}_{\phi }\frac{\delta {{{\mathcal{F}}}}}{\delta \phi },$$

(29d)

$$\nabla \cdot {{{\bf{\sigma }}}}={{{\bf{0}}}}.$$

(29e)

Notably, Eq. (29a) embodies the mixing rule of the local Ni concentration X from the phase ones X^γ and ${X}^{{\gamma }^{{\prime} }}$, considering the $\gamma /{\gamma }^{{\prime} }$ interface as a two-phase mixture with ϕ as the local phase fraction. This detaches the chemical and local contributions to the interface energy to allow rescalability of the diffuse-interface width. Equation (29b) is the constraint to the phase concentration X^γ and ${X}^{{\gamma }^{{\prime} }}$ to obtain the maximum driving force for the interface migration, as briefly elaborated in Supplementary Fig. 9. In return, the drag effect might be eliminated along with the vanishing of the driving force for trans-interface diffusion^68,69,70,71, which should be specifically evaluated and discussed for the $\gamma /{\gamma }^{{\prime} }$ transition in the Fe-Ni system. The diffusive mobility M_X here is directly formulated by the atom mobilities M_Fe and M_Ni in the FCC lattice considering the inter-diffusion phenomena⁷², i.e.,

$${M}_{X}(X,T)={V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}X(1-X)\left[(1-X){M}_{{{{\rm{Fe}}}}}(T)+X{M}_{{{{\rm{Ni}}}}}(T)\right],$$

(30)

and the interface migration mobility M_ϕ is derived by considering the thin-interface limit of the model and the interface migration rate that was originally derived by Turnbull^54,69, i.e.,

$${M}_{\phi }(T)=\frac{2}{3}\frac{{V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}{\Phi }_{{\gamma }^{{\prime} }/\gamma }}{{{{\rm{Ch}}}}| {{{\bf{b}}}}{| }^{2}}\left[(1-{X}_{{\gamma }^{{\prime} }/\gamma }){M}_{{{{\rm{Fe}}}}}(T)+{X}_{{\gamma }^{{\prime} }/\gamma }{M}_{{{{\rm{Ni}}}}}(T)\right],$$

(31)

where the dimensionless Cahn number ${{{\rm{Ch}}}}={\ell }_{\gamma /{\gamma }^{{\prime} }}/{\hat{\ell }}_{\gamma /{\gamma }^{{\prime} }}$, characterizing the degree of the rescaling of the diffuse-interface width ${\ell }_{\gamma /{\gamma }^{{\prime} }}$ from the realistic interface width ${\hat{\ell }}_{\gamma /{\gamma }^{{\prime} }}$, which is estimated as ${\hat{\ell }}_{\gamma /{\gamma }^{{\prime} }}=5\bar{a}$ with $\bar{a}$ the average lattice parameter of the γ and ${\gamma }^{{\prime} }$ phases based on the experimental observation⁷³. Length of the burgers vector is also calculated from $\bar{a}$ by $| {{{\bf{b}}}}| =\bar{a}/\sqrt{2}$. ${\Phi }_{\gamma /{\gamma }^{{\prime} }}$ is a newly defined thermodynamic factor with the estimated interface concentration ${X}_{\gamma /{\gamma }^{{\prime} }}$ as

$$\begin{array}{rcl}{\Phi }_{{\gamma }^{{\prime} }/\gamma }&=&{X}_{{\gamma }^{{\prime} }/\gamma }(1-{X}_{{\gamma }^{{\prime} }/\gamma })\frac{{V}_{{{{\rm{m}}}}}^{{{{\rm{sys}}}}}}{{{{\mathcal{R}}}}T}{\left.\frac{{\partial }^{2}{f}_{{{{\rm{ch}}}}}}{\partial {X}^{2}}\right\vert }_{{\gamma }^{{\prime} }/\gamma },\\ {X}_{{\gamma }^{{\prime} }/\gamma }&=&\frac{{X}_{{{{\rm{e}}}}}^{\gamma }/{A}_{{{{\rm{m}}}}}^{\gamma }+{X}_{{{{\rm{e}}}}}^{{\gamma }^{{\prime} }}/{A}_{{{{\rm{m}}}}}^{{\gamma }^{{\prime} }}}{1/{A}_{{{{\rm{m}}}}}^{\gamma }+1/{A}_{{{{\rm{m}}}}}^{{\gamma }^{{\prime} }}}\end{array}$$

with the equilibrium concentrations ${X}_{{{{\rm{e}}}}}^{\gamma }$ and ${X}_{{{{\rm{e}}}}}^{{\gamma }^{{\prime} }}$ as well as the molar area of the phases ${A}_{{{{\rm{m}}}}}^{\gamma }$ and ${A}_{{{{\rm{m}}}}}^{{\gamma }^{{\prime} }}$. The detailed derivations are shown in the Supplementary Note 2. The temperature-dependent atom mobility M_Ni(T) and M_Fe(T) are obtained from the mobility database MOBFE3 from the commercial software Thermo-Calc^®51.

In this work, it should be highlighted that a temperature-dependent dimensionless calibration factor ω(T) is additionally associated with the atom mobilities, which is utilized to be calibrated from the experimentally measured $\gamma /{\gamma }^{{\prime} }$ transition with respect to time at various temperatures. The calibrated atom mobility is then shown as (noting A = Fe, Ni)

$${M}_{A}^{* }(T)=\omega (T){M}_{A}(T).$$

(32)

Based on the Arrhenius relation on temperature for M_A(T) and ${M}_{A}^{* }(T)$^72,74, this ω(T) is postulated to follow the Arrhenius relation as well, i.e., $\omega (T)={\omega }_{0}\exp (-{Q}_{\omega }/{{{\mathcal{R}}}}T)$ with the pre-factor ω₀ and the activation energy Q_ω. We implemented a simple calibration algorithm by iteratively performing the regression on ω as the time scaling factor to the simulated transient volume fraction of ${\gamma }^{{\prime} }$ phase, i.e., ${\Psi }_{{\gamma }^{{\prime} }}(\omega t)$ with respect to the experimental measurements obtained from³⁰, as shown in Fig. 10a. The IC of the ${\gamma }^{{\prime} }$ nuclei was generated using Poisson disk sampling³⁸ with the prescribed minimum nuclei distance according to the observation shown in Fig. 1b. The calibrated ω(T) indeed shows consistency to the Arrhenius relation, confirming our postulate.

Fig. 10: Calibration of atom mobility for nanoscopic $\gamma /{\gamma }^{{\prime} }$ transition.

a Workflow for the calibration of the atom mobility by iteratively performing regression on the factor ω from ${\Psi }_{{\gamma }^{{\prime} }}(\omega t)$ w.r.t. the experimental measured ${\Psi }_{{\gamma }^{{\prime} }}^{\exp }(t)$. The ω that has a difference of less than 0.5% to the last interaction is identified as the converged value for the current temperature. b Simulated ${\Psi }_{{\gamma }^{{\prime} }}(t)$ at 784 K with the mobilities before (ω = 1) and after (ω = 3) calibration cf. ${\Psi }_{{\gamma }^{{\prime} }}^{\exp }(t)$ from³⁰, with the nanostructures denoted at corresponding time point. c Simulated ${\Psi }_{{\gamma }^{{\prime} }}(t)$ vs ${\Psi }_{{\gamma }^{{\prime} }}^{\exp }(t)$ at different temperatures. Inset: Regression of calibrated ω(T) to the Arrhenius Eq., which presents consistency. Notice that both experiments and simulations are performed with composition Fe₂₅Ni₇₅.

As for momentum balance in Eq. (29e), we have to explicitly consider both long-range (morphology and morphology-induced chronological-spatial thermal inhomogeneity) and short-range factors (misfit-induced fluctuation) factors of the mechanical response on the current scale. In that sense, the stress should be considered in the following form

$${{{\bf{\sigma }}}}={{{{\bf{\sigma }}}}}_{{{{\rm{ms}}}}}+\tilde{{{{\bf{\sigma }}}}},$$

(33)

where σ_ms comes from the mesoscale and $\tilde{{{{\bf{\sigma }}}}}$ is incited due to the misfit of growing ${\gamma }^{{\prime} }$ phase. Assuming the stiffness tensor C_ss has no differences between the two phases, we then take a uniform elastic strain that attributes to the mesoscopic stress, i.e., ${{{{\bf{\sigma }}}}}_{{{{\rm{ms}}}}}={{{{\bf{C}}}}}_{{{{\rm{ss}}}}}:{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{{{{\rm{ms}}}}}$. The constitutive relation can then be represented as

$$\begin{array}{ll}{{{\bf{\sigma }}}}\,=\,{{{{\bf{C}}}}}_{{{{\rm{ss}}}}}:{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}+{{{{\bf{\sigma }}}}}_{{{{\rm{ms}}}}}\\ \quad=\,{{{{\bf{C}}}}}_{{{{\rm{ss}}}}}:\left[\left({{{\bf{\varepsilon }}}}-{{{{\bf{\varepsilon }}}}}_{{{{\rm{em}}}}}-{h}_{{\gamma }^{{\prime} }}{\varepsilon }_{{{{\rm{mis}}}}}^{{\gamma }^{{\prime} }}{{{\bf{I}}}}\right)+{{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{{{{\rm{ms}}}}}\right],\end{array}$$

(34)

where ε is the total strain calculated on the nanoscopic domain, ε_em is the magnetostrictive strain, and ${\varepsilon }_{{{{\rm{mis}}}}}^{{\gamma }^{{\prime} }}$ is the misfit strain induced by growing ${\gamma }^{{\prime} }$ phase. ${\varepsilon }_{{{{\rm{mis}}}}}^{{\gamma }^{{\prime} }}$ is the relative difference between lattice parameters of the γ and ${\gamma }^{{\prime} }$ phases, i.e., ${\varepsilon }_{{{{\rm{mis}}}}}^{{\gamma }^{{\prime} }}=({a}_{{\gamma }^{{\prime} }}-{a}_{\gamma })/{a}_{\gamma }$ with a_γ and ${a}_{{\gamma }^{{\prime} }}$ obtained from the temperature-dependent molar volume ${V}_{{{{\rm{m}}}}}^{\gamma }$ and ${V}_{{{{\rm{m}}}}}^{{\gamma }^{{\prime} }}$, respectively. This is presented in Supplementary Fig. 7b. At T₀ = 600 K, this ${\varepsilon }_{{{{\rm{mis}}}}}^{{\gamma }^{{\prime} }}=-1.32\times 1{0}^{-3}$. It is worth noting that ε_em incited by spontaneous magnetization is also assumed to have negligible effects on $\gamma /{\gamma }^{{\prime} }$ transition compared to ${\varepsilon }_{{{{\rm{mis}}}}}^{{\gamma }^{{\prime} }}$ and ${{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{{{{\rm{ms}}}}}$, which is due to the relatively small magnetostriction constants λ₁₀₀ and λ₁₁₁ in the chemical composition range, i.e., X_Ni = 0.781 ~ 0.785, leading to ε_em on the order of 10⁻⁶ compared to ${\varepsilon }_{{{{\rm{mis}}}}}^{{\gamma }^{{\prime} }}$ and ${{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{{{{\rm{ms}}}}}$ on the order of 10⁻⁴ ~ 10⁻³. Alongside with ${{{{\bf{\varepsilon }}}}}_{{{{\rm{el}}}}}^{{{{\rm{ms}}}}}$ as an eigenstrain, the periodic displacement BC are applied to the nanoscopic domain, as shown in Supplementary Fig. 2.

Micromagnetic hysteresis simulations

Below the Curie temperature, the magnetization of most ferromagnetic materials saturates with constant magnitude (M_s). Therefore in micromagnetics, it is important to have a normalized magnetization vector that is position-dependent, i.e., m. This vector field can be physically interpreted as the mean field of the local atom magnetic moments, but yet sufficiently small in scale to resolve the magnetization transition across the domain wall. However, variation of m(r) across the $\gamma /{\gamma }^{{\prime} }$ interface is tentatively disregarded as an ideal exchange coupling between two phases. Magnetic properties in the ferromagnetic γ phase are also tentatively assumed to be identical to the ferromagnetic ${\gamma }^{{\prime} }$ at the same Ni-concentration due to the lack of experimental/theoretical investigations on the magnetic properties of individual phases. In other words, only the Ni-concentration dependency of magnetic parameters is explicitly considered in this work, while the exchange constant A_ex takes constant as 13 pJ/m⁷⁵. In that sense, superscript φ, indicating the phase differences, is dropped in the following explanation. We let the orientation u of the nanoscopic subdomain align on the z-direction (BD), and the magnetic free energy density is eventually formulated as

$${f}_{{{{\rm{mag}}}}}={f}_{{{{\rm{ex}}}}}+{f}_{{{{\rm{ani}}}}}+{f}_{{{{\rm{ms}}}}}+{f}_{{{{\rm{zm}}}}}+{f}_{{{{\rm{em}}}}}$$

(35)

with

$$\begin{array}{l}{f}_{{{{\rm{ex}}}}}(\nabla {{{\bf{m}}}})={A}_{{{{\rm{ex}}}}}\parallel \nabla {{{\bf{m}}}}{\parallel }^{2},\\ {f}_{{{{\rm{ani}}}}}({{{\bf{m}}}})=-{K}_{{{{\rm{u}}}}}{\left({{{\bf{u}}}}\cdot {{{\bf{m}}}}\right)}^{2},\\ {f}_{{{{\rm{ms}}}}}({{{\bf{m}}}})=-\frac{1}{2}{\mu }_{0}{M}_{{{{\rm{s}}}}}{{{\bf{m}}}}\cdot {{{{\bf{H}}}}}_{{{{\rm{dm}}}}},\\ {f}_{{{{\rm{zm}}}}}({{{\bf{m}}}},{{{{\bf{H}}}}}_{{{{\rm{ext}}}}})=-{\mu }_{0}{M}_{{{{\rm{s}}}}}{{{\bf{m}}}}\cdot {{{{\bf{H}}}}}_{{{{\rm{ext}}}}},\\ {f}_{{{{\rm{em}}}}}({{{\bf{m}}}},{{{\boldsymbol{\sigma }}}})=-{{{\bf{\sigma }}}}:{{{{\bf{\varepsilon }}}}}_{{{{\rm{em}}}}},\end{array}$$

and the magnetostrictive strain ε_em on the cubic basis as follows^39,40

$${{{{\bf{\varepsilon }}}}}_{{{{\rm{em}}}}}=\frac{3}{2}\left[\begin{array}{ccc}{\lambda }_{100}\left({m}_{x}^{2}-\frac{1}{3}\right)&{\lambda }_{111}{m}_{x}{m}_{y}&{\lambda }_{111}{m}_{x}{m}_{z}\\ &{\lambda }_{100}\left({m}_{y}^{2}-\frac{1}{3}\right)&{\lambda }_{111}{m}_{y}{m}_{z}\\ \,{{\mbox{symm.}}}\,&&{\lambda }_{100}\left({m}_{z}^{2}-\frac{1}{3}\right)\end{array}\right].$$

Here, f_ex is the exchange contribution, recapitulating the parallel-aligning tendency among neighboring magnetic moments due to the Heisenberg exchange interaction. The norm ∥ ∇ m∥ here represents ∑_j∣ ∇ m_j∣² with j = x, y, z and m = [m_x, m_y, m_z]. f_ani represents the contribution due to the magneto-crystalline anisotropy. It provides the energetically preferred orientation to local magnetizations with respect to the crystalline orientation u according to the sign of the K_u. f_ani represents the contribution due to the magneto-crystalline anisotropy. It provides the energetically preferred orientation to local magnetizations with respect to the crystalline orientation u, concerning the sign of the K_u. Defining an orientation angle by $\vartheta =\arccos {{{\bf{u}}}}\cdot {{{\bf{m}}}}$, the case when K_u > 0 leads two energetic minima at ϑ = 0 and π, that is when the magnetization lies along the positive or negative u direction with no preferential orientation, i.e., the easy-axis anisotropy. When K_u < 0, the energy is minimized for ϑ = π/2, meaning that any direction in the plane perpendicular to u is thermodynamically preferred, i.e., the easy-plane anisotropy³¹, as shown in Fig. 9e, f. As the resulting X_Ni varies from 0.781 to 0.810 as presented in Fig. 5, local K_u always takes the negative value in this work. The magnetostatic term f_ms counts the energy of each local magnetization under the demagnetizing field created by the surrounding magnetization. The Zeeman term f_zm counts the energy of each local magnetization under an extrinsic magnetic field H_ext. f_em is the contribution due to the magneto-elastic coupling effects.

Evolution of the magnetization configuration m(r) under a cycling H_ext was generally described by the Landau-Lifshitz-Gilbert Eq., which is mathematically formulated as

$$\frac{\partial {{{\bf{m}}}}}{\partial t}=\frac{{\gamma }_{{{{\rm{mg}}}}}}{1+{\alpha }_{{{{\rm{d}}}}}^{2}}\frac{1}{{\mu }_{0}{M}_{{{{\rm{s}}}}}}\left[{{{\bf{m}}}}\times \frac{\delta {{{\mathcal{F}}}}}{\delta {{{\bf{m}}}}}+{\alpha }_{{{{\rm{d}}}}}{{{\bf{m}}}}\times \left({{{\bf{m}}}}\times \frac{\delta {{{\mathcal{F}}}}}{\delta {{{\bf{m}}}}}\right)\right]$$

(36)

with the vacuum permeability μ₀, the gyromagnetic ratio γ_mg and the dam** coefficient α_d⁷⁶. However, due to the incomparable time scale of LLG-described magnetization dynamics (around nanoseconds) with respect to the one of hysteresis measurement (around seconds), constrained optimization of the free energy functional ${{{\mathcal{F}}}}$ has been widely employed as a computationally efficient alternative to the time-dependent calculation in evaluating the hysteresis behavior of permanent magnets^77,78,79. Based on the steepest conjugate gradient (SCG) method, the iteration scheme is derived as

$$\begin{array}{l}\frac{{{{{\bf{m}}}}}^{(i+1)}-{{{{\bf{m}}}}}^{(i)}}{{\Delta }^{(i)}}={{{{\bf{m}}}}}^{(i)}\times \frac{1}{{\mu }_{0}{M}_{{{{\rm{s}}}}}}\left[{{{{\bf{m}}}}}^{(i)}\times \frac{\delta {{{\mathcal{F}}}}}{\delta {{{{\bf{m}}}}}^{(i)}}\right],\\ \,{{\mbox{subject to}}}\,\quad | {{{\bf{m}}}}| =1\end{array}$$

(37)

with the iteration step size Δ⁽ⁱ⁾ and the magnetization configuration m⁽ⁱ⁾ of the step i. The iteration scheme in Eq. (37) is in accordance with the sole dam** term of the LLG equation Eq. (36)^77,79. This also means that the magnetic hysteresis is evaluated under the quasi-static condition. The simulation domains with the FD grids have the same construction as the FE meshes used in the $\gamma /{\gamma }^{{\prime} }$ transition simulations to ease the quantity map** in-between. Periodic BC was applied on the boundaries perpendicular to z-direction by macro geometry approach⁸⁰, while Neumann BC was applied on the other boundaries³⁵.

It is also worth noting that the magneto-elastic coupling constants B₁ and B₂, calculated by λ₁₀₀ = − 2B₁/3(C₁₁ − C₁₂) and λ₁₁₁ = − B₂/3C₄₄^39,81 are implemented in the package MuMax³. The elastic strain field that attributes to the residual stress, i.e., σ = C_ss: ε_el, are mapped from the nanoscopic $\gamma /{\gamma }^{{\prime} }$ transition results.

Implementations and parallel computations

Both non-isothermal phase-field and thermo-elasto-plastic models are numerically implemented by the finite element method within the program NIsoS^28,33, developed by the authors based on the MOOSE framework (Idaho National Laboratory, ID, USA)^82,83. The 8-node hexahedron Lagrangian elements were chosen to mesh the geometry. A transient solver with preconditioned Jacobian-Free Newton-Krylov method (PJFNK) was employed in both models. Each simulation was executed with 96 AVX512 processors and 3.6 GByte RAM per processor based on MPI parallelization. The associated CALPHAD calculations were conducted by open-sourced package PyCALPHAD⁸⁴, and the thermodynamic data intercommunication was carried out by customized Python and C++ codes. The DEM-based powder bed generation is conducted by the open-sourced package YADE^28,85.

For SLS simulations, the Cahn-Hilliard Eq. in Eq. (12a) was solved in a split way. The constraint of the order parameters was enforced by the penalty method. To reduce computation costs, h-adaptive meshing and time-step** schemes are used. The initial structured mesh is presented in Supplementary Fig. 2a. The additive Schwarz method (ASM) preconditioner with the incomplete LU-decomposition sub-preconditioner was also employed for parallel computation of the vast linear system, seeking the balance between memory consumption per core and computation speed⁸⁶. The backward Euler method was employed for the time differentials, and the constraint of the order parameters was fulfilled using the penalty method. Due to the usage of h-adaptive meshes, the computational costs vary from case to case. The peak DOF number is on order 10,000,000 for both the nonlinear system and the auxiliary system. The peak computational consumption is on the order of 10,000 core-hour. More details about the FEM implementation are shown in the supplementary information of ref. ²⁸.

For thermo-elasto-plastic simulations, a static structured mesh was utilized Supplementary Fig. 2b to avoid the hanging nodes generated from the h-adaptive meshing scheme. In that sense, the transient fields T and ρ of each calculation step were uni-directionally mapped from the non-isothermal phase-field results (with h-adaptive meshes) into the static meshes. This is achieved by the MOOSE-embedded SolutionUserObject class and associated functions. The parallel algebraic multigrid preconditioner BoomerAMG was utilized with the Eisenstat-Walker (EW) method to determine linear system convergence. It is worth noting that a vibrating residual of non-linear iterations would show without employing the EW method for this work. The DOF number of each simulation is on the order of 1,000,000 for the nonlinear system and 10,000,000 for the auxiliary system. The computational consumption is on the order of 1,000 CPU core-hour.

For $\gamma /{\gamma }^{{\prime} }$ transition simulations, a static uniform mesh was utilized Supplementary Fig. 2c. 2nd backward Euler method was employed. The additive Schwarz method (ASM) preconditioner with the complete LU-decomposition sub-preconditioner was also employed for parallel computation. The simulations were performed in a high-throughput fashion with 100 ~ 1000 transition simulations as a batch for one set of processing parameters. The DOF number of each simulation is on the order of 1,000,000 for the nonlinear system and 10,000,000 for the auxiliary system. The computational consumption of each simulation is 500 CPU core-hour by average.

The micromagnetic simulations were carried out by the FDM-based steepest conjugate gradient (SCG) solver performing Eq. (37) in the open-sourced package MuMax³³⁵ with numerical details elaborated in ref. ⁷⁷. The high-throughput GPU-parallel computations were performed with 100 ~ 1000 micromagnetic simulations as a batch, each simulation has the DOF on the order of 100,000 for calculating the magnetization configuration, while the demagnetization field H_dm obtained via magnetostatic convolution of the magnetization rather than introducing extra DOF.