Ultrahigh specific strength in a magnesium alloy strengthened by spinodal decomposition

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Science Advances  02 Jun 2021:
Vol. 7, no. 23, eabf3039
DOI: 10.1126/sciadv.abf3039


Strengthening of magnesium (Mg) is known to occur through dislocation accumulation, grain refinement, deformation twinning, and texture control or dislocation pinning by solute atoms or nano-sized precipitates. These modes generate yield strengths comparable to other engineering alloys such as certain grades of aluminum but below that of high-strength aluminum and titanium alloys and steels. Here, we report a spinodal strengthened ultralightweight Mg alloy with specific yield strengths surpassing almost every other engineering alloy. We provide compelling morphological, chemical, structural, and thermodynamic evidence for the spinodal decomposition and show that the lattice mismatch at the diffuse transition region between the spinodal zones and matrix is the dominating factor for enhancing yield strength in this class of alloy.


In crystalline metals and alloys, dislocations are the most effective carriers of plasticity and any microstructural feature that impedes their ability to glide, cross slip, and climb will necessitate a higher stress for plastic (permanent) deformation (13). Stated simply, the more resistance to dislocation motion, the higher the yield strength of a metal. Low-density alloys that can be microstructurally engineered to a high yield strength are important structural materials in applications where light weighting is critical, as required in the aerospace, ground transport, biomedical, and electronics sectors (4, 5). Hexagonal close-packed (HCP) magnesium (Mg) alloys are the lightest among all engineering metals such that alloys of even modest yield strength, when other attractive properties are taken into consideration, are highly attractive materials for many structural applications (6, 7).

Mg can be strengthened by the classic mechanisms of strain hardening (8), grain refinement (9), deformation twinning (10), and crystallographic texture control (11), but by far, the most effective way to improve the resistance to dislocation motion is achieved by alloying with a second constituent to form either a solid solution or dispersion of nanometer-sized precipitates (1214). By selective alloying of Mg with lithium (Li) and/or scandium (Sc), the HCP structure transforms to body-centered cubic (BCC) (15) that, in turn, alters many critical behaviors such as dislocation dynamics and operative slip systems, leading to differences in yield and work hardening behavior, ductility, texture development, etc. The exceedingly low density of Li (0.57 g/cm3) makes it a particularly attractive alloying addition, as was shown in MgLi-base BCC alloys, where specific strengths of up to 300 kN m kg−1 in combination with good room temperature ductility and corrosion resistance are possible (16). Here, we report a BCC β-phase Mg-14Li-7Al [weight % (wt %)] alloy (LA147) that generates a specific yield strength of ~350 kN m kg−1, which exceeds that of almost every other engineering alloy (1, 12).

The strengthening behavior in this BCC MgLiAl system is different to that in other Mg alloys, as this alloy reaches its peak strength immediately after water quenching from solution temperature. While quench strengthening commonly occurs in many types of steel, a phenomenon associated with the nondiffusional transformation of austenite to martensite, this type of transformation is thermodynamically impossible in Mg alloys at temperatures above room temperature (17, 18). To reveal the strengthening mechanism in BCC MgLiAl alloys, we adopt here a powerful new cryogenic preparation method for atom probe tomography (APT) in combination with ex situ and in situ structural analysis techniques, and first-principles, phase-field, and physical-based modeling to generate conclusive morphological, chemical, crystallographic, and thermodynamic evidence that the rapid and substantial strengthening at ambient temperature results from spinodal decomposition, a hitherto unreported strengthening mechanism in Mg and its alloys. This ultrafast spinodal decomposition at low temperatures provides an economical and efficient way for the widespread engineering applications of BCC MgLiAl alloys.


Quench strengthening in MgLiAl alloys

Figure 1A shows the remarkable ambient temperature flow behavior of uniaxially compressed micropillars of BCC β-phase LA147 following a standard solution treatment and water quench. For pillar diameters greater than 2 μm, there appears to be a size invariance of yield strength with resulting values in the range of 620 to 640 MPa. This critical diameter is smaller than the ~3.5 μm reported for other Mg alloys (19). Figure S1 shows the surface morphology of a typical 4-μm-diameter pillar after deformation. The combination of an ultralow density (1.32 g/cm3) and high yield strength of LA147 generates a specific strength of 470 to 500 kN m kg−1, which exceeds that of almost every known engineering alloy (Fig. 1B) except for several nanostructured Mg and Al alloys produced in the form of thin films by DC magnetron sputtering.

Fig. 1 Mechanical properties of LA147.

(A) Compressive engineering strain-stress curves of micropillars of as-quenched LA147 with different diameters and tensile strain-stress of the same alloy for 5-mm tensile samples. (B) Comparison of specific yield strength between LA147 and a range of notable high-strength alloys. These materials include Mg2Zn (12), Mg10Al (12), TZAM6620 (20), nanostructured MgCuY alloy (54), duralumin (55), Al-Li alloy 2050 (56), nanostructured Al alloys (54), Ti6Al4V (20), Inconel 718 (57), Lamellar NiFeCo alloy (58), TWIP steel (59), Duplex steel (57), martensitic steel (57), maraging steel (57), TRIP steel (57), and Ti50Ni47Fe3 alloy (60). The two materials circled were produced in the form of thin films by the more exotic route of sputter deposition.

We further investigated mechanical behavior using macroscopic (5 mm diameter) tensile samples. Compared with the compressed micropillars, the specific strengths of the tensile samples are slightly lower but also exceed those of most other engineering alloys (16, 20). The relatively low yield strength of the tensile samples is attributed to an Al-depleted zone in the vicinity of the grain boundaries (fig. S2) as a result of Al segregation to these defects (21). Besides, the Al segregation at the grain boundaries is presented in a network form, which could be detrimental to the ductility of the material (22).

The observed quench strengthening is highly dependent on matrix structure and composition. Figure S3A shows the hardness of LA33, L11, and LA147 before and after water quench. The hardness of both LA33 and L11 is not affected by heat treatment but almost doubles in LA147. As shown in fig. S3B, the matrix crystal structure of LA33 is HCP, which is different to that of LA147. Although the matrix crystal structure of L11 is BCC, L11 is a binary alloy without Al. Hence, the addition of Al to Mg-Li–based alloys and the consequent transition of the matrix from HCP to BCC have substantial influence on mechanical properties after quenching.

While thermomechanical treatment can refine the grain size of polycrystalline alloys and thus improve strength (23), we rule out grain size strengthening in the latter by compressing samples at 400°C to generate equiaxed grains of average size ~100 μm and ~10 μm, which generated the same hardness (see fig. S4). Indirect evidence, based on x-ray diffraction (XRD) and mechanical property data (24), indicates that rapid and substantial strengthening in MgLiAl alloys on quenching is not attributable to grain size, solute, or precipitation effects (9, 14, 25).

Direct evidence of spinodal decomposition

The morphology of a spinodally decomposed alloy is argued to be a distinctive criterion by which spinodal decomposition can be distinguished from a precipitation process (26). We investigated the elemental distribution and crystallographic features of the β-phase microstructure of LA147 within 20 min of water quenching using a newly developed cryogenic technique for APT sample preparation in conjunction with transmission electron microscopy (TEM) (see Materials and Methods). Figure 2A is a reconstructed cryo-APT dataset revealing nano-sized Al-rich zones (blue) interspersed throughout the BCC matrix (magenta), with Fig. 2C highlighting their preferred alignment parallel to the matrix <100> directions deduced by APT (27). These <100> aligned zones are also clearly visible in the TEM image in fig. S5A. The associated selected-area diffraction pattern in fig. S5B confirms that both the zones and matrix are BCC, albeit with slightly different lattice parameters. The APT proximity histogram in fig. S6 shows the variation in concentration of Mg, Li, and Al through the zones, revealing a diffusional transition zone of width ~4 nm rather than a defined matrix/zone interface expected after nucleation of second-phase precipitates. APT data analysis of the zones indicates an average core composition of ~Mg45Li30Al25 [atomic % (at %)].

Fig. 2 Comparison between cryogenic APT results and phase-field simulation of water-quenched LA147.

(A) Reconstructed APT volume showing Al-rich zones (blue phase) distributed within the BCC β phase (magenta phase) (plotted with 6 at % Al iso-surface). (B) Time temperature transformation diagrams of LA147 and a range of spinodal alloys. (C) Bottom view of the extracted Al-rich zones in (A) showing the characteristic morphology and crystallographic features of a classic spinodal. (D and E) Compositional maps generated from the APT data and phase-field simulation, respectively. (F and G) One-dimensional concentration profile of Mg, Li, and Al through the Al-rich zones in (D) and (E), respectively.

As stated, these Al-rich zones have a preferential orientation parallel to the matrix <100> directions. Cahn (28) pointed out the importance of matrix elastic anisotropy on the final form of the periodic concentration fluctuations in a spinodally decomposed solid solution. We conducted first-principles calculations (fig. S7 and tables S1 and S2) of the BCC matrix (taken to be Mg67Li30Al3) to study the effect of matrix elastic modulus on the preferred orientation. The elastic modulus shows minimum and maximum values parallel to <100> and <111> directions, thereby confirming that the long axes of the zones are aligned preferentially with the elastically soft <100> directions.

Preferential growth of the spinodal in these elastically soft directions occurs to minimize the coherency strain energy created by decomposition (28), which is consistent with that in other BCC spinodal alloys. However, compared with other reported spinodal alloys, LA147 shows the ultrafast decomposition at room temperature (Fig. 2B). Figure 2D shows Al compositional maps from APT data. The characteristic segregation profiles expected for this type of transformation were confirmed by solving the Cahn-Hilliard and Allen-Chan equations (Fig. 2E) (29). The line concentration profiles along the <100> direction in Fig. 2 (F and G) (extracted from Fig. 2, D and E) show a complementary sinusoidal distribution for Mg and Al of average modulation wavelength ~9.5 nm, whereas Li shows little tendency to segregate.

The characteristic morphology of the Al-rich zones together with their crystallographic and compositional features supports our recent hypothesis (16) that certain quenched MgLiAl alloys are highly unstable and rapidly undergo spinodal decomposition. This is the first direct morphological evidence of such a phenomenon in Mg alloys.

Equilibrium, phase stability, and instability

Thermodynamic instability of a system is the fundamental reason for all phase transformations. The most convincing criterion for the stability of a solid solution was proposed by Gibbs (30), whereby the second derivative of free energy as a function of concentration, (∂2G/∂c2)T,P, is either less than or greater than zero for unstable and stable/metastable systems, respectively. Figure 3A shows that, over a temperature range of 0 to 1500 K, the formation energy of the MgLi system is negative for Li concentrations greater than 14 at % and ∂2G/∂c2 is positive. Conversely, the formation energy of MgAl is positive for all Al concentrations and ∂2G/∂c2 is negative below 1000 K. The MgAl system is unstable, and a slight composition fluctuation drives the solid solution toward decomposition. To reveal the electronic origin of the thermodynamic instability, the angular momentum projected density of states (DOS) was calculated in Mg65Li35 and Mg65Al35. We selected Mg65Li35 because its composition is close to that of LA147 (~35 at % Li and 5 at % Al) but without Al, and Mg65Al35 is a control comparison to Mg65Li35 for investigating the Al effect. For Mg65Li35, the Fermi level is situated in a pseudogap of the s-band and, thus, the s-band is favorably filled by electrons of the bonding states (lower energy peaks), leaving the antibonding states (higher energy peaks) empty (Fig. 3B), which is also observed for p-band in Fig. 3C. Conversely, for Mg65Al35, the Fermi level is shifted to the right such that some electrons of antibonding states populate the s-band, which can be seen for the p-band, albeit to a lesser extent. Figure 3E shows that the total DOS of Mg65Al35 at the Fermi level is larger than that of Mg65Li35. The inclusion of electronic states with higher energies in the filled band increases the system energy and renders it thermodynamically unstable.

Fig. 3 Thermodynamic and electronic properties of Mg-base binary solid solutions.

(A) Formation energy curves for various temperatures in body-centered cubic Mg-Al/Li solid solutions. (B, C, and E) Angular momentum projected density of states of Mg65Al35 and Mg65Li35 solid solutions for the s-band, p-band, and total, respectively. (D) Mean values of <-COHP> of Mg-solute, solute-solute, and Mg-Mg pairs in Mg65Li35 and Mg65Al35.

To be more specific, the chemical bonding calculation allows us to visualize electronic states and strength of a given pair, such as Mg-solute (solute atom is Li in Mg65Li35 or Al in Mg65Al35), solute-solute, and Mg-Mg. For each pair, bonding/antibonding/nonbonding states are represented by positive/negative/zero values of negative crystal orbital Hamiltonian populations (<-COHP>). Figure 3D shows that near/at the Fermi level, the <-COHP> of Mg-solute and solute-solute pairs in Mg65Al35 is much lower than for Mg65Li35, indicating that the bonding states of Mg-solute and solute-solute pairs in the former are weaker than those for Mg65Li35. These weak bonding states of Mg-solute and solute-solute pairs lead to the inherent instability of the Mg-Al system, a phenomenon that also exists in other Mg-Al alloys with varied concentrations (table S3).

Morphological evolution of the spinodal

The spinodal wavelength (λ) is a critical parameter for evaluating mechanical properties (28, 31). We systematically studied the evolution of spinodal decomposition wavelength using a combination of synchrotron XRD experiments and phase-field simulation (PFS). As shown in Fig. 4A, the two sidebands adjacent to the (110) diffraction peak of the β matrix are a typical feature of spinodal decomposition. The change in λ during natural aging was achieved by calculating the peak position of sidebands (see Materials and Methods), as shown in the inset of Fig. 4B. Because of experimental limitations, measurable changes in λ immediately after quenching were impossible to track, which required PFS to capture the evolution of the dynamic spinodal over a broad time scale. As shown in Fig. 4B, three stages of evolution of the spinodal are evident: Stage I shows a slight increase in λ, termed the incubation period; stage II shows a rapid increase in λ, termed the rapid growth stage; stage III shows a decreasing rate of increase in λ, termed the equilibrium stage. Superimposed on Fig. 4B is the calculated wavelength, λ0 (9.1 nm), which indicates that spinodal decomposition is already well under way and must have commenced either during or very soon after quenching. Figure 4D shows the simulated morphological evolution of spinodal decomposition with natural aging time. In stage I, the Al-rich zones are dispersed discretely in the BCC matrix and the Al concentration within these zones increases gradually despite a negligible increase in λ. Stage II shows an Ostwald coarsening of the zones parallel to <100> and a concomitant increase in their volume fraction. In stage III, λ still increases, while its growth rate decreases, which is due to the decrease in interfacial energy and supersaturation of solutes.

Fig. 4 In situ synchrotron XRD and phase-field simulation of phase transformations in LA147.

(A) XRD datasets of as-quenched LA147 during natural aging: q = 4πsinθ/λs, where θ is the half-scattering angle between the incident beam and scattered beam, and λs is the wavelength of the incident x-ray. (B) Phase-field simulated and experimentally observed data of the spinodal of wavelength, λ, as a function of natural aging time (t* denotes a dimensionless time). λ = λ0 at t0 = 0 min denotes the wavelength that corresponds to the experimentally obtained value soon after quenching. There are main stages: I, incubation; II, rapid growth; and III, equilibrium. (C) Simulated structure order parameter within the Al-rich zones as a function of natural aging time and (D) corresponding microstructural evolution. The order parameter ranges from 0 to 1, which represents the evolution of a completely disordered solid solution into the fully ordered D03-Mg3Al phase. The figures in (D) are part of fig. S9. a.u., arbitrary units.

Spinodal decomposition is widely regarded as the precursor of precipitation of an ordered phase (32). We studied the change in structure order parameter of the Al-rich zones as they evolve (shown in Fig. 4C). Ordering occurs at the very early stages of decomposition, and the rate of ordering increases substantially during Ostwald coarsening (from t* ~ 4000). The predicted gradual transformation of the Al-rich zones into an ordered D03-Mg3Al phase during aging is supported by the XRD experiments on LA147 during isothermal aging at 120°C (fig. S8). Here, sidebands gradually merge into the main diffraction peak of β matrix and eventually disappear during natural aging, in conjunction with gradual bulging at the diffraction position of D03-Mg3Al (20).

Dominant factors in spinodal strengthening. On the basis of Cahn’s analytical model of strengthening in face-centered cubic (FCC) alloys containing spinodal structure (28), Kato (33) developed a model for BCC alloys for predicting the incremental increase in critical resolved shear stress (CRSS), Δσy. Kato’s model considers both lattice misfit strengthening and modulus strengtheningΔσy=AκY2+0.65ΔGbλ(1)where ΔG is the amplitude of the shear modulus fluctuation; A is the amplitude of the composition modulation in atomic percent; κ (d( lna)/dC, da/dC) is the variation in lattice constant, a, with respect to composition fluctuation, C; b is the Burger vector; λ is the wavelength of modulation; and γ is the line tension of dislocation. Y is related to the elastic constant Cij. For the case of <100> modulation in cubic materials, Y can be calculated byY=(C11+2C12)(C11C12)C11(2)

Using Eq. 1, we calculate the contribution from spinodal decomposition on strengthening in LA147 using A = ~0.20 ± 0.05 (from APT data in Fig. 2B), κ = 0.060 ± 0.005 (from Fig. 4A), Y = 40.97 GPa (Eq. 2), b = 3.04 × 10−10 m, and λ = 9.5 × 10−9 m (Fig. 2F). Table S1 shows that a 22% increase of Al concentration only results in a 3.6% increment in G, which substantially weakens the modulus strengthening effect. Hence, the combined effects of misfit (245 ± 20 MPa) and modulus (15 MPa) components give Δσy ~ 260 ± 20 MPa, thereby demonstrating that the lattice misfit plays a dominant role in the spinodal strengthening in LA147. It should be noted that a gradual transformation of the Al-rich zones to an ordered D03-Mg3Al phase occurs during natural aging. Therefore, the yield strength in LA147 is a result of the balance between the spinodal strengthening and ordered phase strengthening. We focus mainly on the study of spinodal strengthening in this work. The interactive effect of spinodal microstructure and phase transformation on the mechanical properties is beyond the scope of the present work.

We further compared dislocation density in simulated samples with and without Al-rich zones using molecular dynamics (Fig. 5A). Figure 5 (B and C) shows that in the simulated sample containing Al-rich zones, the dislocation density increases to a much higher level than that in sample without Al-rich zones at the initial plastic deformation stage, which is related to substantial dislocation nucleation from the diffusional interfaces and subsequent interaction between dislocations and Al-rich zones (fig. S10). Therefore, the strengthening in LA147 is intrinsically attributed to the presence of matrix/Al-rich zone interface arising from spinodal decomposition. Our current molecular dynamics simulations revealed both the dislocation evolution and the details of interaction between dislocations and Al-rich zones, which provide a direct and mechanistic insight into the strengthening of MgLiAl-base alloys with Al-rich zones.

Fig. 5 Molecular dynamics of plastic deformation of LA147.

(A) Configuration of the simulated sample volumes with and without Al-rich zones. (B) Variation in dislocation density as a function of applied strain for sample volumes shown in (A). (C) Dislocation networks within the simulated sample volumes for a strain of 7%.

In summary, we report an ultralight Mg alloy strengthened by a previously unidentified mechanism based on detailed morphological, chemical, and crystallographic evidence. The experimental data are entirely consistent with thermodynamic predictions of spinodal decomposition, in conjunction with evidence generated from physical-based models, ab initio, molecular dynamics, and PFSs. The combination of spinodal strengthening and the intrinsically low density of MgLiAl-base alloys creates a material exhibiting specific strengths surpassing any other reported engineering alloy. The results here may also be applicable to many other unexplored Mg-Li–based alloy systems.


Specimen preparation

A Mg-Li–based alloy with a composition of Mg, 36.45/13.95Li, 4.60/6.84Al, 0.04/0.19Zr, and 0.12/0.59Y (at %/wt %), i.e., Mg-14Li-7Al-1(Zr, Y) (termed LA147 here), was prepared by melting in an argon-protected electrical resistance furnace followed by casting into a copper mold. Mg–11 wt % Li binary alloy (termed L11 herein) and Mg–3 wt % Li–3 wt % Al ternary alloy (termed LA33 here) were prepared by the same method. The as-cast alloys were homogenized for 8 hours at 350°C (as-received alloys). Some samples were heat-treated in a tube furnace with argon protection for 10 min at 400°C, followed by cold water quenching (as quenched alloys).

Mechanical testing

A piece of thin plate with dimensions of 10 mm × 10 mm × 2 mm (side × side × thickness) cut from the quenched bulk material was mechanically polished. Micropillars with diameters of 1, 2, and 4 μm were prepared by the annular milling method in a Zeiss Auriga dual-beam focused ion beam scanning electron microscope (FIB-SEM). An ion probe of 30 kV and 30 nA was used for rough trenching, and a probe of 30 kV and 100 pA was adopted for final milling to minimize ion damage. All prepared pillars had a diameter-to-height aspect ratio of ~1:2.5. As the yield strength for the BCC crystals is not sensitive to crystal orientation (34), all pillars were prepared in random grains to show averaged mechanical properties. In situ compression tests were conducted with the Hysitron PI85 PicoIndenter in a Zeiss Ultra SEM. Micropillars were compressed by a diamond flat punch at room temperature with a strain rate of 3 × 10−5 s−1. Data were acquired in a rate of 60 Hz. Vickers hardness testing using a 1-kg load and loading time of 15 s was carried out to investigate the hardness of as-quenched and as-received LA11, LA33, and LA147 at room temperature. At least 10 hardness indentations were generated for each sample. Mechanical testing of macroscopic tensile samples with a dimension of 5 mm (diameter) × 25 mm (length of gauge section) was carried out using an Instron 5982 universal testing system operating at a true strain rate of 1 × 10−4 s−1.

Synchrotron XRD

The time-resolved in situ x-ray scattering experiments were conducted on the small-angle X-ray scattering/wide-angle X-ray scattering (SAXS/WAXS) beamline at the 3GeV Australian Synchrotron. An x-ray energy of 8.15 keV (wavelength, 1.52128 Å) was used for the experiments. Water-quenched samples were machined to 6 mm × 0.5 mm × 0.5 mm samples and then inserted into glass capillaries for testing. The samples were in situ heated at a rate of 20°C/min to temperatures up to 400°C, followed by water cooling to room temperature. The spinodal decomposition wavelength was calculated from the x-ray scattering datasets using (35)λ=ha·tanθ(h2+k2+l2)·Δθ(3)where h, k, and l are the Miller indices of the Bragg peak, θ is the Bragg angle, a is the lattice parameter of β matrix, and Δθ is the angular distance between the sideband and the Bragg peak of β matrix.

Atom probe tomography

Samples of LA147 were cut to dimensions of 5 mm × 5 mm × 5 mm for APT sample preparation and analysis. The cubes were solution-treated for 10 min at 400°C followed by water quenching and then transferred into the FIB vacuum chamber within 10 min of quenching. Before inserting the samples into the FIB chamber, they were mechanically polished using an oil-based solution, then cleaned by ethanol, and dried with compressed nitrogen. Because of the high Li content in LA147, substantial surface oxidization occurs in less than 20 min in standard atmospheric conditions (fig. S11). To minimize oxidation, a Zeiss Auriga dual-beam microscope equipped with a custom-designed cryogenic stage was used to prepare tips for APT. The tips were transferred to a CAMECA laser-assisted local electrode atom probe equipped with a Vacuum and Cryo-Transfer Module on the load lock by a Ferrovac UHV suitcase. Therefore, the entire process was carried out under cryogenic and high vacuum conditions.

Transmission electron microscopy

Similar-sized cubic samples to those used for APT were solution-treated for 10 min at 400°C, water-quenched, and then aged for 3 hours at 70°C. Thin foils of dimensions 16 μm × 5 μm × 100 nm were cut from the sample by FIB, in situ lifted to a copper grid, and transferred to TEM vacuum chamber within 5 min. A FEI TALOS FS200X G2 FEG TEM, operating at 200 kV, was used for bright-field transmission electron microscope (BF-TEM) imaging and collecting diffraction patterns parallel to the [100] zone axis.

Phase-field modeling

The evolution of spinodal decomposition in LA147 was simulated via the Cahn-Hilliard equation (Eq. 4-1) and Allen-Chan equation (Eq. 4-2)ci(r,t)t=·[Mi·δFδci(r,t)]+ξci(r,t)(4-1)η(r,t)t=MηδFδη(r)+ξη(r,t)(4-2)where ci(r, t) is the local concentration field variable (as a function of spatial position, r, and time, t) of alloying elements (i = 1, 2, 3, and 4 for Mg, Li, Al, and vacancies, respectively), η(r, t) is the structure order parameter field variable, ξci(r, t) and ξη(r, t) are the standard noise terms with respect to the thermal fluctuations, Mi is the mobility of alloying components, and Mη is the mobility of structure transformation, defined asMi(r,t)=c0i(1c0i)×DiRT(5-1)Mη(r,t)=c0i(1c0i)×[(1η)Diβ(T)RT+ηDiγ(T)RT](5-2)where c0i is the initial composition, R is the gas constant, T is the absolute temperature, and Di is the initial diffusion constant of the alloying element, i, given asDi=Di0eQi/RT(6)where Di0 and Qi are the self-diffusion coefficient at 0 K and the activation energy of diffusion for element i, respectively.

The total free energy of an inhomogeneous microstructure (F) includes the short-range interaction due to compositional inhomogeneities and the long-range elastic interaction, expressed asF=[G(ci,η,T)+i=13kc2(ci)2+kv2cv2+kη2(η)2+Estre]dv(7)where G(ci, η, T) is the Gibbs free energy of the system. i=1312kc(ci)2 is the function related to the gradient energy of composition, reflecting the effect of interfacial concentration on the total energy. kc is the coefficient for gradient energy, which can be calculated from (1/2)Ωd2, where Ω is the atomic interaction energy [Ω=L2,3β(T=0 K)=45,310 J/mol]. d (associated with the mesh spacing of simulated region) is the half thickness of interface, which could be set as d = 0.7nm. kv2cv2 represents the function of the gradient energy for vacancy. kη2(η)2 is the gradient energy function of ordered structural transformation of disordered β matrix. The elastic strain energy Estre can be expressed asEstre=12σijelεijel(8)where σijel=Cijklεklel is the elastic stress tensor, Cijkl=Cijkl0+ΔCijklΔc is the elastic coefficient tensor, and Cijkl0=12(Cijklp+Cijklm) is the averaged elastic coefficient tensor for the whole system, where Cijklp and Cijklm are the elastic coefficient tensor of precipitation and matrix, respectively. ΔCijkl=CijklpCijklm is the difference between the elastic tensor of the precipitate and matrix, and c=i=24(cic0i) is the compositional fluctuation, where c0i is the initial composition. εijel=εijεij0 is the elastic strain, where εij is the total strain of the system. εij0=sii=24(cic0i) is the intrinsic strain owing to the compositional inhomogeneity, and si = (1/a)(da/dc) is the dilatation coefficient of lattice parameter a as a function of composition. Thus, Estre can be expressed asEstre=12σijelεijel=12Cijklεklelεijel=12Cijkl(εijεij0)(εklεkl0)(9)

The Gibbs free energy, G(ci, η, T), is expressed asG(ci,η,T)=[1h(η)][GCβ(ci,T)]+h(η)GCβ(ci,T)+Wg2(η)(10-1)Gcφ(ci,T)=iGi0,φci+GE,φ+RTicilnci(10-2)where h(η) = η2(3 − 2η) is a monotone function between 0 and 1, Wg2(η) is the energy barrier of ordering structure phase transition β → β, g(η) = η(1 − η), and W is the height of the energy double well. Gcφ(ci,T) is the Gibbs free energy of φ(φ = β, β) phase, Gi0,φ is the bulk free energy of pure alloying element i or vacancy in φ phase, GE is the excess free energy, and the third term in Eq. 10-2 is the free energy of mixingGcφ(ci,T)=cvEfv+cMgGMgφ+cLiGLiφ+cAlGAlφ+LMg,LiφcMgcLi+LMg,AlφcMgcAl+LLi,AlφcLicAl+LMg,Li,AlφcMg,LicAl+RT(cLilncLi+cAllncAl+cvlncv)(10-3)

The Gibbs free energy of vacancy formation is Gvφ(cv,T)=Efvcv+RTcvlncv, where Efv is the formation energy of a vacancy. By applying the vacancy formation energy in pure Mg, the equilibrium concentration of matrix vacancies is estimated to be cv = eQv/RT = 1.114 × 10−12. LMg,Liφ, LMg,Alφ, LLi,Alφ, and LMg,Li,Alφ are the interaction parameters for the binary and ternary systems in φ phase, respectively.

First-principles calculations

First-principles calculations based on density functional theory were carried out to calculate the formation energy of MgAl solid solutions. Vienna ab initio software package (36) was used by applying the projector augmented-wave method to represent the combined potential of core electrons and nuclei (37, 38). The Perdew-Burke-Ernzerhof gradient approximation is implemented to represent the exchange-correlation functional (39). A cutoff energy of 350 eV was selected for the plane-wave basis, and the integration of band structure energy over the Brillouin zone was carried out using tetrahedron method with Bloch corrections. The self-consistent electronic optimization was converged to 10−7 eV.

The solid solutions of Al in the BCC matrix of Mg were represented by supercells containing 100 atoms and configured by a special quasi-random structure (40). Periodic boundary conditions were applied in three dimensions, and a mesh of Γ-centered k-points was implemented to sample the Brillouin zone with a density of 13800 k-points per atom.

The formation energies of Mg-Li and Mg-Al BCC phases as a function of solute concentration can be expressed as (41)Ef=EtotNxiEiBulk(11)where N, Etot, xi, and EiBulk are the number of atoms, the ground-state total energy of the alloy contained in the supercell, the concentration of the ith elemental constituent, and its bulk energy per atom, respectively.

At elevated temperatures, the contribution of entropy (S) to the total free energy isGf=EfTS(12)where S is the configuration entropy, defined asS=KBixiln(xi)(13)where KB is the Boltzmann constant.

For each solute concentration, the ground-state energy along with the lattice parameter can be determined by a least-square fit of 10 computed total energies over volume, applying the Birch-Murnaghan equation of state (42, 43). Last, 32 data points were collected to plot Ef as a function of solute concentration. The calculation of <-COHP> was conducted using the software LOBSTER (4447).

Molecular dynamics

To complement the experiments, we carried out large-scale atomistic simulations for uniaxial compression of simulated LA147 alloys using the large-scale atomic/molecular massively parallel simulator (LAMMPS) (48). We first constructed a single-crystalline sample of both a BCC structure and approximately the same elemental composition as LA147. The selected sample of dimensions ~40 nm × 40 nm × 40 nm contained ~2.9 million atoms, and the crystallographic orientations were [110], [1¯10], and [001] along x, y, and z axes, respectively.

This sample was first equilibrated by the energy minimization and the subsequent dynamic relaxation at 300 K for 100 ps. During relaxation, an isobaric-isothermal (NPT) ensemble was used to ensure the pressures along three directions and the temperature to be zero and 300 K, respectively. After equilibration, we stretched the sample at a constant strain rate of 5 × 108 s−1 to 8.75% tensile strain along the [110] direction, followed by unloading back to the zero-stress state. This process was used to introduce the initial dislocations into the sample. During this process, the NPT ensemble was applied for ensuring that the pressures along nontensile directions were zero.

To construct a BCC sample containing Al-rich zones, we introduced 73 cylindrical clusters (Mg53Li32Al15) to replace the matrix with the same shape and volume. These clusters were 4 nm in diameter and 4 nm in axial length, which is similar to the cross-like precipitates with a featured size of 4.2 nm in the experimental samples. Hence, their volume fraction was up to 5.51%, which is close to the volume fraction (7.64%) of Al-rich zones found experimentally (computed from an iso-surface of 6 at % Al at the APT tip). TEM observations confirm that the Al-rich zone grows parallel to the [001] direction. We aligned the long axis of the zones parallel to [001] (i.e., z axis in Fig. 6C). After construction, we equilibrated this sample via the energy minimization and dynamic relaxation, and then obtained the sample with Al-rich zones at equilibrium.

We applied the uniaxial compression on the samples with and without precipitate along the [110] direction at a constant strain rate of 5 × 108 s−1. During compression, an NPT ensemble was applied on the samples to ensure the pressures along noncompression directions to be zero and to keep temperature at 300 K. Periodic boundary conditions were imposed in all three directions. During simulation, we used the Virail stress theorem and the local transformation matrix between current and reference configurations (49, 50) to calculate the atomic stress and strain, respectively. We also adopted the dislocation extraction algorithm (DXA) tool (51) for identifying the dislocation lines during deformation and to further determine their Burgers vectors. During all simulations, we used a concentration-dependent embedded atom method (EAM) potential (51, 52) to describe the interatomic interactions for Li-Li and Mg-Li pairs. An EAM potential (53) was used to describe the interaction for Al-Al, Mg-Mg, and Mg-Al pairs. For the Li-Al pairs, we adopted the Lennard-Jones potential to describe their interaction. The parameters σ and ε for Li-Al pairs were obtained by the Lorentz-Berthelot rule (53), i.e., σAl-Li =Al + σLi)/2 and εAl-Li = (εAlεLi)2, where σLi = 2.839 Å, σAl = 2.620 Å, εLi = 0.206 eV, and εAl = 0.393 eV, respectively. On the basis of these potentials, the calculated lattice constants of both the matrix and precipitate were 0.350 and 0.345 nm, respectively, which are very close to those obtained from density functional theory calculations.


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Acknowledgments: We would like to acknowledge the helpful discussions and suggestions from K. Lu. We thank the UNSW Mark Wainwright Analytical Centre, the Curtin John de Laeter Centre, and the Australian Synchrotron and Microscopy Australia node at the University of Sydney (Sydney Microscopy and Microanalysis) for the provision of experimental time. R.M. would like to thank N. Stanford for her continued support. Funding: We would like to thank the Australian Research Council (ARC) for partly funding this work via the ARC Discovery scheme (grant no. DP190103592). T.X. would like to express his sincere gratitude to the Chinese Scholarship Council (CSC) for providing a PhD scholarship for enabling him to carry out research at UNSW Sydney. The authors are grateful to M. Han of the Institute of Metals Research, Chinese Academy of Sciences, for helping with the analysis of the TEM results and B. Joe for the characterization of mechanical properties. We acknowledge support from the Australian Research Council Discovery Project DP190102243. Author contributions: T.X. and M.F. conceived the idea and designed the project. Y.Z. and L.-Q.C. conducted the phase-field simulations and analysis. R.M. carried out the ab initio calculations. J.J. and X. Li conducted the molecular dynamics calculations. A.Y., K.N., and S.R. carried out the APT experiments. T.X., F.J., and S.T. analyzed the data. Z.Q. conducted the TEM experiments. D.M. and W.X. cast the alloys. J.D. analyzed the XRD data. R.N. and X. Liao conducted the micro-compression tests. K.H. provided valuable discussion on the mechanical properties. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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