A general strategy to significantly reduce thermal expansion and achieve high mechanical properties in iron alloys

doi:10.21203/rs.3.rs-3914162/v1

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A general strategy to significantly reduce thermal expansion and achieve high mechanical properties in iron alloys

https://doi.org/10.21203/rs.3.rs-3914162/v1

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Iron alloys, including steel and magnetic functional materials, are widely used in capital construction, manufacturing, electromagnetic technology, etc. However, they face the long-standing challenge of high coefficient of thermal expansion (CTE), limiting the applications in high-precision fields. This work proposes a general strategy involving the in-situ formation of a nano-scale lamellar/labyrinthine negative thermal expansion (NTE) phase within the iron matrix to tackle this problem. For example, a model Fe alloy, Fe-Zr10-Nb6, was synthesized and its CTE is reduced to approximately half of the iron. Meanwhile, the alloy possesses an excellent strength-plasticity combination of 1.5 GPa (compressive strength) and 17.5% (ultimate strain), which outperforms other low thermal expansion (LTE) metallic materials. The magnetovolume effect of the NTE phase is deemed to counteract the positive thermal expansion in iron. The high stress-carrying hard NTE phase and the tough matrix synergistically contribute to the superior mechanical properties. The interaction between the slip of lamellar microstructure and the slip-hindering of labyrinthine microstructure further enhances the strength-plasticity combination. This work shows the promise of offering a universal method to produce LTE iron alloys with outstanding mechanical properties.

Physical sciences/Materials science/Condensed-matter physics

Physical sciences/Chemistry/Materials chemistry

The discovery and widespread utilization of iron, particularly in the form of iron alloys, have exerted a profound influence on industrialization, modernization, as well as advancements in science and technology. Represented by steels^{1, 2}, their exceptional mechanical properties and complete industry make them the foremost structural materials. Meanwhile, the strong magnetism of the Fe element spawns essential magnetic functional materials such as soft magnets^{3, 4}. However, iron alloys confront a long-standing challenge of high thermal expansion, whether employed as structural or functional materials. Specifically, iron has a coefficient of thermal expansion (CTE) of about 12 ppm/K, which restricts its advanced applications that demand low dimension-temperature sensitivity and low thermal mismatches among different materials.

Owing to the inherently weak metallic bonds, viable strategies for directly reducing the CTE in iron alloys are limited. Generally, low thermal expansion (LTE) or even negative thermal expansion (NTE) can be found in intermetallic compounds or solid solution alloys^5–11. However, achieving LTE and maintaining good mechanical properties in these metal-based compounds cannot be achieved simultaneously. For instance, despite the renown of the Invar alloy^{12, 13}, celebrated with a Nobel Prize in 1920 for its zero thermal expansion (ZTE) characteristics, its mechanical strength is deficient, with a mere yield strength of 275 MPa. Some intermetallic compounds with zero or negative thermal expansion cannot be processed due to their brittleness, such as (Zr,Ta)(Fe,Co)₂¹⁴, Hf_0.87Ta_0.13Fe₂¹⁵, La(Fe,Si,Co)₁₃¹⁶, and MnCoGe¹⁷. Metal matrix composites (MMCs) using these NTE compounds as reinforcements are considered effective in reducing CTE^18–20. But conventional methods to form MMCs such as powder metallurgy inevitably result in weak interface bonding. Additionally, the CTE mismatch between NTE compounds and the metal matrix introduces obvious residual stresses at the interface, further exacerbating the mechanical properties of the MMCs. Recently, the non-stoichiometric approach in NTE compounds, exemplified by, Er_xFe₁₀V_1.7Mo_0.3²¹, LaFe_10.2+xCo_1.2Si_1.6²², and Er₂Fe_14 + xB_{1 + 0.07x}²³, have boosted material strength, but it is found ineffective in enhancing toughness. Thus, current solutions for reducing the CTE and achieving high mechanical properties of iron alloys are still confronted with problems that necessitate further attention.

In this work, a novel strategy involving the in-situ formation of the nano-scale lamellar/labyrinthine NTE phase within iron alloys is presented to balance the thermal expansion and mechanical properties. A series of alloy compositions are designed and demonstrated to show a significant reduction in CTE while maintaining reliable mechanical properties. Notably, it exhibits an isotropic LTE (~ 5.5 ppm/K, 110 to 325 K), almost a half of that in pure iron, as well as a combination of high compressive stress (1.5 GPa) and moderate ultimate strain (17.5%) in Fe-Zr10-Nb6 alloy. To delve deeper into the LTE mechanism and the substantial advantages in mechanical properties, a detailed investigation is conducted using neutron powder diffraction (NPD), scanning transmission electron microscopy (STEM), in-situ neutron diffraction under different temperatures and engineering stress, Mössbauer spectra, and micromechanical experiments. It is evident that the NTE phase counteracts the positive thermal expansion (PTE) of the matrix by the magnetovolume effect (MVE), and the excellent mechanical properties are facilitated by the distinctive nano-scale microstructures. Such a strategy not only contributes to a reduction in CTE but also provides atomic-scale high-strength interfacial bonding, coupled with its simple synthesis procedure and cost-effectiveness, which has led to a prominent edge among the presently reported iron-based LTE materials.

As the primary constituent of iron alloys, Fe serves not only as a solvent but also as a precursor for forming second phases. Generally, transition metal elements with larger atomic radii exhibit low solubility in iron, and frequently engage with iron in a ratio of 1:2, resulting in Laves-type AFe₂ (A = transition metal elements) compounds. Intriguingly, Laves-type AFe₂ compounds occasionally exhibit excellent NTE²⁴ performance due to their complex magnetism and novel Kagome lattice²⁵. This implies it is feasible to in-suit form the Laves-type NTE phase by introducing a trace amount of transition metal elements into iron alloys. Concurrently, the dispersion of the second phase typically results in the strengthening of the alloy^{26, 27}. Such a strategic approach not only holds the prospect of reducing CTE but also presents the opportunity to achieve outstanding mechanical properties. As shown in the schematic in Fig. 1a, the alloy design strategy involves the utilization of Laves compounds with NTE to compensate for the PTE in iron alloys. A series of alloy compositions were designed in Fe-Zr-Nb, Fe-Hf-Nb, Fe-Hf-Ta, Fe-Hf-Ti, and Fe-Ti-Sc systems to in-situ form the specific NTE phase. Taking the Fe-Zr-Nb system as a case study, we synthesized samples with nominal compositions of Fe-Zr6-Nb4, Fe-Zr10-Nb6, and Fe-Zr14-Nb8, referred to as Z1, Z2, and Z3, respectively.

Crystal structure and microstructure

The X-ray diffraction (XRD) patterns (Figure S1) of alloys with different compositions show that all alloys exhibit a dual-phase structure of a body-centered cubic (BCC) phase and a C14-type Laves phase. The detailed crystal structures of Z1-Z3 were determined by NPD (Figure S2 and Table S1). The NPD pattern and refined results of Z2 are depicted in Fig. 1b, revealing the presence of the dual-phase structure: a C14-type Laves phase (P63/mmc, labeled as NTE phase) with the composition of (Zr_0.62Nb_0.27Fe_0.11)Fe₂, and a BCC α-Fe phase (Im-3m, labeled as PTE phase). The NTE phase formed herein exhibits a hexagonal structure, rather than the cubic structure as reported in the (Zr,Nb)Fe₂ compounds with NTE behavior^28–30. A minor fraction of Fe is found to substitute Zr/Nb in the 4f Wyckoff site within the NTE phase, as the crystal structures of these two phases are shown in Fig. 1c. This antisite of Fe atoms in the NTE phase is significant for the NTE.

Off-axis transmission Kikuchi diffraction (TKD) on both longitudinal-transverse (LD-TD) and transverse-normal (TD-ND) planes of Z2 (Figure S3) was employed to characterize the spatial distribution and orientation relationship of the two phases. The two phases show nano-scale microstructures on both observational planes, suggesting their spatial uniformity (Fig. 1d). All alloys exhibit lamellar/labyrinthine arrangements of two phases (Figure S4), exemplifying the typical characteristics of eutectic alloys. For instance, a sequence of transitions from hypo-eutectic to eutectic to hyper-eutectic microstructures is observed in Z1-Z3 (Figure S5). From the corresponding inverse pole figures (IPF) (Fig. 1d), the crystal orientation of the PTE phase is relatively random, whereas the NTE phase is textured. The energy dispersive spectrometer (EDS) mapping results for Z2 indicate an alternating arrangement of Zr/Nb-enriched regions and Fe-enriched regions without elemental segregation (Fig. 1e). Of paramount significance is the first discovery of such a naturally formed nano-scale microstructure within LTE alloys. Especially, this microstructure exhibits a grain refinement effect and synergistic behavior between two phases, endowing it with tremendous potential for high strength³¹ and wear-resistant³² applications.

High-angle angular dark field-scanning transmission electron microscopy (HAADF-STEM) was employed to attain atomic-resolution images of Z2. The HAADF-STEM images and the selected area electron diffraction (SAED) patterns of the NTE and PTE phases (Fig. 1f, g) correspond entirely to the crystal structures (Fig. 1c). Additionally, the EDS analysis offers indirect evidence for the antisite of Fe atoms (Figure S6, Table 2). The HAADF-STEM image at the phase boundary and fast Fourier transform (FFT) analysis indicate a semi-coherent relationship with a quite small lattice mismatch (δ = 3.01%) between (1$\stackrel{\text{-}}{\text{1}}$0)_F and (112)_L (Fig. 1h). The lattices of the two phases match as schematic illustrated in Fig. 2, while mismatch is accommodated through dislocations and a tilted interface (Figure S7). Such interfaces distinguish themselves from conventional disordered interfaces in MMCs, bearing lower interfacial energy and reducing the risk of failure caused by stress concentration. The nano-scale microstructures coupled with ordered interfaces provide alloys with favorable assurances under thermal cycle and mechanical loading conditions.

Thermal expansion properties

The successful formation of NTE phases establishes a foundation for achieving LTE in iron alloys. Figure 2a shows the linear CTE of Z1-Z3 and pure iron. Surprisingly, Z1-Z3 each manifest LTE. In the case of Z2, its linear CTE is measured as 5.5 ppm/K (110 to 325 K), nearly 53% lower than that of the iron matrix. With increasing content of Zr/Nb, the CTE continues to decrease, underscoring the efficacy of this strategy in reducing CTE. All alloy compositions that form the specific NTE phase exhibit significantly reduced CTE compared with the iron matrix (Figure S8). As depicted in Fig. 2b and Table S3, a comprehensive comparison of the CTE is performed between the alloys in this work and conventional iron alloys (stainless steel, carbon steel, bearing steel, etc.) ^33–48. The results prove that incorporating the NTE phase into iron alloys is a potent strategy for reducing their CTE. Meanwhile, the CTE is tunable by controlling the NTE phase content.

Solely exhibiting LTE is insufficient for meeting engineering requirements, isotropic CTE and good thermal cycling stability are also crucial factors to consider. As depicted in Fig. 2c (top), the CTE was measured along three directions with approximately the same value, demonstrating it isotropic. From Fig. 2c (bottom), the CTE was largely unaffected after 100 thermal shock cycles between liquid nitrogen (LN₂, 77 K) and hot oil (473 K). This superior thermal fatigue resistance of the alloys underscores the strong interfacial bonding of the two phases.

Temperature-dependent NPD was conducted to analyze the lattice parameter changes for two phases. Figure 2e illustrates the temperature dependence of the (110)_PTE and (112)_ZTE peaks. With increasing temperature, the (110)_PTE peak exhibits a trend of shifting toward lower angles, indicating a PTE behavior of its lattice. In contrast, the (112)_NTE peak remains nearly unaltered, indicating its ZTE behavior. The thermal expansion of unit cell volume is extracted, as shown in Fig. 2f and Table S4. The volume CTE is 23.4 ppm/K (5-555 K) for the PTE phase and 6.7 ppm/K (5-455 K) for the NTE phase. Meanwhile, the average volume CTE of Z2 is calculated to be 11.5 ppm/K (5-455 K) by the rule of mixture (ROM). The in-situ formed NTE phase exhibits abnormal ZTE, compensating for the iron matrix's thermal expansion.

Low thermal expansion mechanism

Following the materials' design, it can be considered that the abnormal thermal expansion behavior of the NTE phase is related to its magnetism. As the temperature dependence of magnetization (M-T curves) illustrated in Fig. 3a, all alloys exhibit a ferromagnetic (FM) state with high magnetization from 5 K to 800 K. This phenomenon can be attributed to the strong magnetic background of the PTE phase (α-Fe). However, the M-T curves exhibit a change in slope at approximately 470 K, which likely represents the magnetic transition process of the NTE phase. The isothermal magnetization curves (M-H) of Z2 are tested at different temperatures to assist in understanding the magnetic transition (Fig. 3b). The saturation magnetization (M_s) extracted from the M-H curves (inset of Fig. 3b) exhibits a decrease as the temperature rises, reaching a distinct inflection point between 450 K and 500 K. Combining its consistency with the fluctuations observed in the M-T curve, the inflection point at approximately 475 K can be confirmed as the Curie temperature (T_C) of the NTE phase within Z2. In light of previous research, the antisite of Fe atoms within the NTE phase usually stabilizes the FM-ordered state and increases T_C^{30, 49}. In these three samples, where the NTE phase formed in different Fe concentrations, the difference in T_C is relatively modest (Fig. 3a). This suggests an upper limit to this antisite. Such a characteristic allows the NTE phase to maintain a stable contraction in alloys with different PTE phase fractions, and it can be considered that the CTE of alloys is only related to the content of the NTE phase.

The magnetic structure of the NTE phase was analyzed using temperature-dependent NPD. As depicted in Fig. 3c, the intensity of the (002) magnetic peak decreases with increasing temperature until 475 K and remains invariant, which suggests a parallel arrangement of FM spin moments in the a-b plane. The inset of Fig. 3c displays the magnetic structure of the NTE phase. The antisite Fe atoms at the 4f site show extra spin moments, as highlighted with a dashed circle. To further investigate the effect of antisite Fe atoms occupation on magnetism, Fe⁵⁷ Mössbauer spectra of Z2 were collected and fitted (Fig. 3d and Table S5). The absorption of α-Fe was first fitted and subtracted from the total spectrum. The remaining signals were well fitted with three sextets representing the interaction between nano-scale fields and spin moment in 6h, 2a, and 4f. The present results reveal additional spectra splitting, indicating that Fe atoms at the 4f site possess reliable spin moments. It can be considered collectively that Fe (4f) possesses an ordered moment and introduces extra magnetic exchange interaction, stabilizing the hexagonal FM-order state.

The magnetic contribution on the thermal expansion (i.e., MVE) was quantitatively described using spontaneous volume magnetostriction (ω_s) ⁵⁰ (Fig. 3e). Here, ω_exp represents experimental thermal expansion, and ω_nm represents normal thermal expansion, and ω_s = ω_exp - ω_nm. It is evident that the NTE phase lattice expansion with increasing temperature, whereas the MVE counters this trend below T_C. This unconventional MVE originates from the distinctive magnetic temperature dependence of the NTE phase. As shown in Fig. 3f, by the Landau theory, the ω_s can be regarded as a quadratic function of total magnetic moment (M_NTE) ⁵¹. The pronounced linear correlation between ω_s and M_NTE² aptly underscores the dominant role of FM order in MVE. Thus far, an intrinsic mechanism by which the NTE phase reduces the CTE of iron alloys has been revealed. Specifically, this mechanism is rooted in the compensatory action of MVE, which evolves concomitantly with the changes in magnetism. The antisite Fe atoms stabilize the hexagonal FM state, thereby facilitating the occurrence of MVE at higher temperatures.

Mechanical properties and deformation mechanism

To meet the requisites in specific application contexts, the mechanical properties of LTE materials warrant due scrutiny. An excellent strength-plasticity combination is maintained in the new alloys. The compression engineering stress-strain curves of the Z1-Z3 and pure iron are illustrated in Fig. 4a and other alloys are included in Figure S9. All the designed alloys achieve an enhancement in strength compared to pure iron. In the case of Z1-Z3, the plasticity decreases with the increase of Zr/Nb content and Z1 even becomes brittle. Notably, Z2 exhibits an excellent strength-plasticity balance, characterized by a compressive strength of 1.5 GPa and an ultimate strain of 17.5%. Such strength-plasticity combinations in this work far surpass that of LTE metallic materials^{15–17, 21–23, 52–62} (Fig. 1b, Table S6). Furthermore, the plasticity of the alloys enables good machinability as exemplified by the screw support seat (Fig. 1e) machined using computer numerical control (CNC) lathes. The smoothness of the workpiece surface meets the demands for high precision and efficient manufacturing. Such exceptional mechanical properties are attributed to the synergy between the two phases as well as the response of the heterogeneous microstructures during loading.

Generally, the Laves-type NTE phase exhibits poor mechanical performance attributed to its inherent brittleness. However, a compressive strength higher than the iron matrix has been observed while demonstrating outstanding plastic deformation from the pure NTE phase. This enhancement can be attributed to the synergy between the two phases. Hereto, in-situ neutron diffraction under compression loading⁶³ was conducted on Z2. The diffraction along the loading direction (labeled as A) and perpendicular to it (labeled as T) were collected (Figure S10). As the lattice strain of two phases versus the true strain shown in Fig. 4d, the two phases exhibited notable disparity deformation behaviors. The process can be divided into three stages: stage Ⅰ, co-elastic; stage Ⅱ, the PTE phase yields while the NTE phase retains elastic; and stage Ⅲ, co-plastic. Figure 4e reveals the normalized full width at half maximum (FWHM) of peaks versus true strain. The FWHM of (110)_PTE exhibits a slow increase, corresponding to uniform deformation in the PTE phase. Conversely, the FWHM of (103)_NTE displays a pronounced increase starting from stage Ⅱ, indicating instability and possibly stress concentration in the NTE phase. In stage Ⅲ, the increase slows down, but notable errors are observed, indicating the strain is released as shear or microcracks. Notably, the unique microstructure provides tolerance to shear bands and microcrack propagation within alloys, with tough phases at crack tips featuring high-density dislocation walls and stronger stress fields, thereby retarding crack propagation across both phases⁶⁴. This microstructure-induced synergy between the soft and hard phases greatly benefits the balance of strength and plasticity.

To further elucidate this synergy, the phase-specific stress of each phase is calculated (Fig. 4f). Because of the disparity in CTE between the two phases, thermal mismatch residual stresses (calculated as 384 MPa) introduced during preparation are also considered. The PTE phase with a larger volume contracting bears tensile stress, whereas the NTE phase bears compressive stress. During stage I, the PTE phase demonstrates a rapid increase in carried stress, whereas the NTE phase shows a slower increase, indicating that the PTE phase bears more load. In stage Ⅱ, while the PTE phase yields, the NTE phase begins to rapidly harden to carry more stress and reaches nearly 2 GPa. In stage Ⅲ, the PTE phase continues to soften while the NTE phase shows small hardening with increasing stress. The low yield strength and high plasticity of the PTE phase provide excellent deformability while hindering the propagation of cracks and slides in the NTE phase. In turn, the NTE phase contributes to a higher stress-carrying ability of the alloy⁶⁵. Such a synergy between the tough and hard phases greatly benefits the balance of strength and plasticity.

As is well known, materials with heterogeneous microstructures usually exhibit complex mechanical behavior^{66, 67}. Thus, compressive micromechanical experiments of micropillars were conducted to reveal the deformation mechanism of alloys. As depicted in Figs. 4g and h, when the lamellar layers of the micropillar are perpendicular to the loading direction, the material yields at 0.8 GPa. As illustrated in the insert of Fig. 4h and Video S1, the micropillar reveals slip events along phase boundaries. manifesting as instabilities in the stress-strain curve (Fig. 4e) at approximately 14% strain. This indicates that the failure initiates with relative slip along the phase boundaries. The experiment with the lamellar layers tilted toward the loading direction further corroborated this observation (Figure S11, Video S2). In addition, a micropillar with labyrinthine microstructure embedded in the tilted lamellar microstructure is prepared, as shown in Fig. 4i. Interestingly, Fig. 4j and Video S3 reveal the absence of stress instabilities in this micropillar, and no catastrophic sliding is observed in the insert of Fig. 4j. Instead, the slip band terminates near the labyrinthine microstructure. This observation suggests an interaction between the two types of microstructures, with lamellar slip enhancing the deformation capacity and the labyrinthine microstructure hindering slip propagation across the entire grain, thus delaying premature material failure. It can be believed that the interaction between naturally formed heterogeneous microstructures further enhances the mechanical properties of alloys.

In summary, a general strategy for reducing the thermal expansion of iron alloys while maintaining excellent mechanical properties has been demonstrated through the in-situ formation of nano-scale lamellar/labyrinthine NTE phases within the iron matrix. Exemplarily, the CTE exhibited a remarkable reduction to half of that in the iron matrix, while achieving a balance between compressive strength (1.5 Gpa) and ultimate strain (17.5%) within the Fe-Zr10-Nb6 alloy. The analysis of the NPD and Mössbauer spectra reveals that the in-situ precipitation of the NTE phase readily induces the antisite of Fe atoms at 4f, which stabilizes the hexagonal FM state, engenders evident ZTE, and mitigates the PTE exhibited by the iron matrix. The heterogeneous microstructure and semi-coherent phase boundaries significantly improve alloys' thermal cycling stability. The micromechanical behaviors highlight the synergistic effect of the NTE phase and the PTE phase, as well as the cooperative influence of lamellar and labyrinthine microstructures, which contribute to the exceptional strength-plasticity combination. This strategy enables these alloys to satisfy stringent requirements for LTE and reduce thermal expansion mismatch with other inorganic non-metallic materials, such as silicon and glass. Moreover, the simplicity of preparation, and cost-effective formulation without noble metals, combined with exceptional machinability, broadens the scope for potential applications of these alloys in wied engineering areas such as aerospace and optical systems.

Material preparation

All bulk iron-based alloys are prepared by arc-melting under an Ar atmosphere with the starting materials of high-purity elements (> 99.95 wt.%). Re-melting is performed four times to promote chemical homogeneity, and water-cooled copper mold casting is used to form the alloy ingot with dimensions of 10×10×60 mm.

Structural characterization

The phase and crystal structural at room temperature are identified by XRD using an X-ray diffractometer (SmartLab 9 kw, Rigaku Corporation) performed with Co K_α radiation and neutron powder diffraction (NPD) at general purpose powder diffractometer (GPPD), China Spallation Neutron Source (CSNS). The hyperfine dual-phase microstructures of the alloys are obtained under a scanning electron microscope (SEM, Zeiss GeminiSEM500), in the form of backscattered electrons (BSE) images and off-axis transmission Kikuchi diffraction (TKD) images. Atomic-resolution images of the alloys are characterized by high-resolution transmission electron microscopy (HRTEM, FEI Tecnai G2 F30) and high-angle angular dark field-scanning transmission electron microscopy (HAADF-STEM, ThermoFisher Themis Z), both equipped with an Energy Dispersive Spectrometer (EDS). The lattice mismatch (δ) is calculated by the following formula

$$\delta =\frac{{d\left(112\right)}_{NTE}-{d\left(1\stackrel{-}{1}0\right)}_{PTE}}{({d\left(112\right)}_{NTE}+{d\left(1\stackrel{-}{1}0\right)}_{PTE})/2}$$

Thermal expansion behavior

The linear thermal expansion coefficient curves are collected using a thermal dilatometer (NETZSCH, DIL 402 Expedis Select) using ~Ф5 × 10 mm cylindrical samples. The temperature-dependent NPD is carried out at the Australian Nuclear Science and Technology Organization (ANSTO), and the temperature dependence of the lattice parameters is obtained by refining the diffraction data using the Rietveld method.

Mechanical properties

The engineering stress-strain curves under compressive loading are measured using an electronic universal testing machine (WDW-200D) with Ф4 × 10 mm cylindrical samples, and the strain rate is controlled at 1 × 10^-3s^-1.

Magnetic properties

The magnetic properties are measured using a physical property measurement system (PPMS, Quantum Design) equipped with a vibrating sample magnetometer (VSM).

Mössbauer spectroscopy

The Mössbauer spectroscopy measurements are performed at 6.2 K with a low-temperature ⁵⁷Fe Mössbauer spectrometer (WissEl, WSS-10), in which the α-Fe is used for reference and ⁵⁷Co (Rh) is used as the radiation source.

In-situ neutron diffraction under compression loading

The in-situ neutron diffraction measurements under compression loading are carried out at VULCAN, Oak Ridge National Laboratory (ORNL), using Ф8 × 16 mm cylindrical samples. The lattice strain of the specific crystal plane is calculated by interplanar spacing, which is obtained from real-time diffraction data after a single peak fitting⁶⁸. The calculation formula is as follows

$${\epsilon }_{hkl}=\frac{{d}_{hkl}-{d}_{0, hkl}}{{d}_{0, hkl}}$$

where ε_hkl is the hkl-orientation lattice strain, d_hkl and d_{0, hkl} are the hkl-orientation interplanar spacings measured during and before deformation, respectively.

Thermal residual stress and phase-specific load partition calculations

The thermal residual stress (σ_r) between NTE and PTE phases is calculated by formula⁶⁹

$${\sigma }_{r}={E}_{PTE}\frac{{E}_{NTE}{V}_{NTE}}{{E}_{PTE}{V}_{PTE}+{E}_{NTE} {V}_{NTE}}({\alpha }_{NTE}-{\alpha }_{PTE})({T}_{o}-{T}_{p})$$

where E, V, and α represent the Young’s modules, volume fraction, and CTE, respectively. T_o and T_p denote the operating and processing temperatures of alloys, respectively. In this work, T_o represent the room temperature, and T_p represents the T_C.

The phase stress of PTE phase (σ_PTE) is calculated by formula⁶⁴:

$${\sigma }_{PTE}=\frac{{E}_{PTE}}{(1+{\nu }_{PTE})(1-2{\nu }_{PTE})}\times \left\{\left(1-{\nu }_{PTE}\right)\times {\epsilon }_{PTE, 11}+{\nu }_{PTE}\times \left({\epsilon }_{PTE, 22}+{\epsilon }_{PTE, 33}\right)\right\}+{\sigma }_{r}$$

where E_PTE, ν_PTE, and ε_PTE denote the elastic modulus, Poisson's ratio, and lattice strain, respectively. The subscripts "11", "22", and "33" refer to specific directions of lattice strain, with "11" representing the axial direction, while "22" and "33" denote the transverse directions. Additionally, it is assumed that ε₂₂ is equal to ε₃₃. Due to possible modulus anomalies in the NTE phase, the phase stress of the NTE phase (σ_NTE) is simply calculated by formula⁶⁴:

$${\sigma }_{NTE}=\frac{{\sigma }_{true}-{V}_{PTE}{\sigma }_{PTE}}{{V}_{NTE}}$$

Micromechanical experiments

The samples with a thickness of 2 mm are first ground and finely polished. The cylindrical micropillars are milled from the surface of the samples using a 30 keV Ga⁺ focused ion beam (FIB, Tescan Lyra FIB workstation). Rough pillars with a diameter of 10 µm and height of 3 µm are first fast-milled under ion beam conditions of 4.5 nA. After that, the pillars are finely polished step by step by reducing the current to 1 nA, 240 pA, and 50 pA until they reach an aspect ratio (height/diameter) of ~ 2 with diameters of ~ 3 µm and heights of ~ 6 µm. Micropillar compression tests are carried out with an in-situ indenter system (Alemnis AG) inside an SEM (Philips XL30) at room temperature. A 5 µm diameter diamond flat punch (Synton MDP, Switzerland) is applied to load and unload on the pillars with a displacement rate of 6 nm s^− 1, corresponding to a strain rate of 1 ×10^− 3 s^− 1. The microstructures of the micropillars before and after compression are characterized using SEM in the same workstation with FIB.

Acknowledgments

This work was supported by the National Key Research and Development Program of China (2022YFE0109100), the National Natural Science Foundation of China (21825102 and 22275014), and the US Department of Energy (DOE), Office of Science (contract No. DE-AC05-00OR22725). In-situ neutron·diffraction work was carried·out at the VULCAN·(Proposal: 29886.1), Spallation·Neutron Source (SNS), Oak·Ridge National·Laboratory (ORNL). We acknowledge Dr. Chinwei Wang for collecting the NPD data at the high-intensity diffractometer Wombat of the Australian Nuclear Science and Technology Organisation (ANSTO). We acknowledge Prof. Xianran Xing for providing laboratory X-ray diffraction and macroscopic magnetic tests at the Institute of Solid State Chemistry, University of Science and Technology Beijing.

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A general strategy to significantly reduce thermal expansion and achieve high mechanical properties in iron alloys

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