Ductile-to-brittle transition and materials’ resistance to amorphization by irradiation damage

Abstract

By summarizing over seven hundred elastic constants of materials with various crystal structures, we have found that the so-called ductile-to-brittle transition originates from the bonding type transition. This can be reflected well by a universal linear relation between Cauchy pressure/bulk modulus (CP/B) and shear modulus/bulk modulus (G/B). In general, G/B of a material will gradually decrease with increasing pressure and temperature, while a significant change in G/B against increasing temperature often corresponds to a certain phase transition, such as magnetic transition, spin flipping, structural phase transition and so on. The present work suggests that the value of G/B for materials can serve as an indicator of bonding type. Importantly, this linear relation is robust for materials under temperature, pressure and with defects. Here we propose that damage tolerance properties can be experimentally measured from the elastic constants of a given material. It is suggested that a material with a value of G/B that varies little with temperature, pressure and defect concentration, potentially exhibits high damage tolerance. These observations shed light on the ductile-to-brittle transition. They may provide a potential tool towards the design of materials, particularly under extreme conditions.

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1 Introduction

The ductile-to-brittle transition of a material is of great interest for engineering applications, due to the fact that it involves the failure mechanism of a material. This problem was first posed by Kelly, Tyson and Cottrell in 1967.¹ Over fifty years, researchers have made many attempts to answer this question, and many criteria have been proposed: (i) theoretical shear strength/theoretical tensile strength (Kelly–Tyson–Cottrell criterion¹), (ii) Gb⃑/γ (Rice–Thomson criterion²), (iii) G/B (Pugh criterion³), (iv) Cauchy pressure (Pettifor criterion⁴), and (v) Poisson’s ratio (empirical criterion).^5,6 Among all these five criteria, the relationships between its own specified parameters and the so-called ductile/brittle properties have been discussed. The former two criteria need temperature-dependent surface energies, which are very difficult to derive by experiments and are also difficult to obtain by theoretical calculations within quantum-mechanical Density Functional Theory (DFT). However, the latter three criteria only require the elastic constants of materials, which can be well measured experimentally or calculated theoretically as a function of temperatures or pressures. In particular, the Pugh criterion specifies the relation between G/B and the elastic/plastic properties of pure polycrystalline metals, while the Pettifor criterion reveals the argument between the Cauchy pressure (CP) and the bonding character. Recently, by combining both the Pugh criterion and Cauchy pressure, a nearly hyperbolic relation between CP/E and G/B was found for cubic crystal systems, generalizing a ductile-to-brittle transition to G/B values and so-called metallic/covalent bondings.⁷ However, as of yet, the effects of bonding types on the ductile-to-brittle transition for materials with various crystal systems have been not discussed, and the effects of pressure, temperature and defects on it were never discussed systematically. Certainly, these points are important for materials.

This issue would be more interesting for materials under extreme conditions, such as high temperature, high pressure and irradiation-heavy environments, for structural materials and nuclear fuels in a future nuclear system. Currently, one of the major concerns in the nuclear energy industry is the damage tolerance properties of materials, for instance, their resistance to amorphization.⁸ During the past few decades, some experimental and theoretical studies have provided some useful insight regarding the resistance to amorphization. A review outlined about twenty factors which were relevant in the context of resistance to amorphization by radiation damage.⁹ Most of them are only suitable for certain narrow classes or families of materials, and fail when widely applied to various families of materials. Then, Trachenko et al. proposed that the resistance to amorphization of a non-metallic material by irradiation damage is governed by the competition between covalent and ionic bondings by quantifying ionicity and covalency, and ionic bonding benefits the resistance to amorphization.¹⁰ At present, this criterion has been applied on some ceramic materials and has been supported by experimental findings.^10,11 Recently, exploring the radiation tolerance of six M_n+1AX_n materials revealed that the generally weak interaction of X–A and M–A bonding in M_n+1AX_n phases is beneficial to defect recombination after the atomic displacements induced by neutron/ion irradiations. The results demonstrated that the lower cation antisite energy renders the lattice with a higher capacity in accommodating the disordering point defect.^12,13 As of yet, although these findings provide a better understanding of the damage tolerance properties of ceramic phases, some experimental findings on MAX phases are not consistent and some even contradict with each other. For instance, Ti₃AlC₂ exhibits a much higher damage tolerance than Ti₃SiC₂ at 50 K and 300 K under the irradiation of 1 MeV Kr²⁺ and Xe²⁺, although they show a high tolerance to damage.¹⁴ However, it has been recognized that under heavy ion (5.8 MeV Ni) irradiations at 673 and 973 K, the aluminum MAX phases (Ti₃AlC₂ and Ti₂AlC) exhibit much more serious damage than the Si counterpart of Ti₃SiC₂. Therefore, according to the results claimed in ref. 8, the Al-containing MAX phases are not fit for application near 673 K and the Si-containing MAX phase is more damage tolerant at 673–973 K. In other words, the determination of bonding type in these materials and its dependence on the defects, pressure and temperature is crucial for understanding the damage tolerance properties. However, to date there has been no unified criterion proposed for diversified materials.

Here, we have collected more than seven hundred datasets of the elastic properties of metals, intermetallics, random alloys and solid solutions with cubic, hexagonal, tetragonal and orthorhombic crystal symmetry, and even amorphous structures. Interestingly, we have observed that for them, the ductile-to-brittle transition is related to G/B, showing a linear relation between G/B and CP/B. This correlation is highly robust and independent of pressure, temperature and various point defects, indicating that G/B is a crucial parameter to elaborate bonding types for diverse materials. Furthermore, a comprehensive discussion on G/B has been performed. Finally, on the basis of the experimental data of materials widely used in the nuclear industry, we have proposed that the change of temperature-dependent G/B reflects the bonding change against temperature, thereby measuring the ability of damage tolerance.

2 Computational details

We employed the Vienna ab initio Simulation Package (VASP)^15–17 by utilizing the projector augmented wave (PAW) method^18,19 within the framework of density functional theory (DFT).^20,21 The description of the exchange–correlation adopted the Perdew, Burke and Ernzerhof (PBE)²² generalized gradient approximation (GGA). The k⃑-space integration with incompletely filled orbitals was performed with the tetrahedron method²³ and the Blöchl correction method.²⁴ Optimizations were achieved by minimizing the forces and total energies. The convergence criteria of the total energy and the forces were set to be 0.01 meV and 0.001 eV Å⁻¹, respectively. We used a plane wave cut-off energy of 500 eV, which was sufficient for precise energetics for all the elements considered here. For calculations of the elastic constants, the elastic tensor was determined by performing finite distortions of the lattice and deriving the elastic constants from the strain–stress relationship.²⁵ The elastic tensor was calculated both for rigid ions, as well as allowing for internal relaxation of the ionic Hessian matrix and multiplying with the internal strain tensor.²⁶ To compute polycrystalline elastic moduli, the Voigt–Reuss–Hill approximation²⁷ was applied. For the bulk properties, 15 × 15 × 15 k-mesh samplings were applied in the Brillouin Zone according to the Monkhorst–Pack scheme.²⁸ We performed the charge density topology analysis using Critic2 software,^29,30 as it has successfully been done on the Zr–H system.³¹

3 Results and discussion

3.1 Ductile-to-brittle transition of materials

By analyzing all data summarized here, we have plotted CP versus G/B in Fig. 1a, evidencing a linear relation with an expression of . It is well-known that the Cauchy pressure (C₁₂–C₄₄ for cubic system) was proposed by Pettifor⁴ to measure the transition from metallic to covalent bonding. A negative value suggests covalent bonding, while a positive value means metallic bonding. Similar to the cubic crystal system, hexagonal, tetragonal and orthorhombic crystal systems have two, two and three Cauchy pressures,³² respectively. They can also be computed via elastic constants. It is noted that here, the average value of these Cauchy pressures has been considered. The ratio of G/B is also an indicator of the ductile-to-brittle transition in polycrystalline metals³ and a larger value suggests brittle behavior, whereas a smaller value reveals ductile behavior. From Fig. 1a, it can be seen that most pure metals with hexagonal and cubic crystals are all positive and above the zero line of the Cauchy pressure, the typical covalent materials (e.g. Si and Diamond) are located in the lower-right part of the plot and typical ionic materials (e.g., alkali halides) are near the transitional region between these two criteria.

Fig. 1 (a): Cauchy pressure/B as a function of G/B for hexagonal, cubic, tetragonal and orthorhombic crystal structures. The fitted equation is . (b) Poisson’s ratio as a function of G/B of hexagonal, cubic, tetragonal and orthorhombic crystal structures and bulk metallic glass (BMG). The elastic properties of BMGs are collected from ref. 5.

On this CP/B vs. G/B map in Fig. 1a, three regions can be classified: metallic bonding (upper left), covalent bonding (lower right) and the transitional ionic bonding region. This suggestion is further verified by the quasi-linear relationship between Poisson’s ratio and G/B, as shown in Fig. 1b. It can be noted that materials with a Poisson’s ratio above 0.34 are considered to be ductile,⁵ whereas values of 0.25 and 0.10 represent typical ionic and covalent bonding,⁶ respectively. Accordingly, this quasi-linear curve in Fig. 1b can be divided into four regions: I region is a metallic bonding region, II region is a metallic–ionic bonding region, III region is an ionic–covalent bonding region and the IV region is an covalent bonding region. Fig. 1b also shows that alkali metals are located in the metallic–ionic bonding region, alkali hydrides show an ionic–covalent bonding nature, and noble gases at high pressures exhibit a metallic–ionic bonding nature. Additionally, the bulk metallic glasses (BMG) typically exhibit metallic or metallic–ionic bonding nature. Those Zr-based, Pd-based, Cu-based, Pt-based and Ti-based BMGs with metallic bonding are ductile, whereas those with a mixed bonding nature are typically brittle, which is congruent with the experimental findings.⁵ These results reveal that G/B is an indicator of bonding type and can be used to describe the ductile-to-brittle transition for those crystal symmetries investigated here.

3.2 Discussion

Now we turn our attention to the physical meaning of the parameters involved. Polycrystalline properties are derived from the elastic constants of single crystals determined by experiment or DFT calculations within the V–R–H approximation. The uniform strain approximation (Voigt approximation) and the uniform stress approximation (Reuss approximation) are limits of the real moduli. Therefore, real moduli are always obtained through the arithmetic value (Hill approximation). Then, we can derive E according to the following equation:

and hence, the accuracy of E depends on B and G. However, the difference in G values within the Voigt and Reuss approximations is sometimes large (several tens of GPa or even larger), suggesting that sometimes, E is not accurate. For example, G of graphene within the Reuss approximation is negative. As such, the Reuss approximation is not correct for some materials. However, B is a rotational invariant of the C_ij matrix,³³ suggesting that the difference of B within the Voigt and Reuss approximations is small (zero or several GPa). Thus, we choose B as the scale parameter of the Cauchy pressure as opposed to E, because G/B is also a dimensionless parameter.

Firstly, we focus on the Cauchy pressure, a reliable macroscopic measure of bonding directionality.³⁴ A negative value suggests that angularly dependent many-body forces play a crucial role, whereas a positive value indicates that the atoms are embedded in the electron gas of the surrounding neighbors.⁴ Of course, over the years, deviation of the Cauchy relations³⁵ has occurred for some cases, such as anharmonic contributions to zero-point vibrations,³⁶ strain-induced quadrupole moments³⁷ and forces of many-body character arising from the non-orthogonality of Heitler–London crystal wave functions.³⁸

Secondly, let’s discuss both G and B possibly correlating with electronic properties. Recently, via quantum theory of atom in molecules (QTAIM), Eberhart and his co-workers⁵ proposed that

B ∝ ρ_b (2)

G ∝ ρ_b〈tan(θ)〉 (3)

where ρ_b is the charge density at the bonding critical point (BCPs), 〈tan(θ)〉 is the bonding path’s average value of tan(θ), ρ_⊥⊥ and ρ_⊥′⊥′ are extreme curvatures of ρ_b at the BCPs and normal to the bonding path, ρ_‖‖ is the curvature of ρ_b at the BCPs but parallel to the bonding path. Fig. 2 compiles the charge density ρ_b at BCPs as a function of B. It can be seen that two different nearly linear relations exist: one is for fcc and bcc metals and alkali halides, and the other one is for covalent materials such as diamond, Si, and SiC. Concerning G, this shows us that the data of charge density at BCPs is rather scattered for various materials (details in ESI†). Therefore, the simple G ∝ ρ_b 〈tan(θ)〉 relation proposed in eqn (3) is not widely suitable.

Fig. 2 Charge density as a function of bulk modulus for typical materials. (cf. details in ESI†).

Therefore, in order to clarify the dependence of G from the electronic scale, we systematically derived the charge density topology for bcc and fcc metals under shear strain. For convenience, to see the charge density topology for fcc and bcc metals, we applied the two strain tensors for simple pure metals by considering the pure shear strain tensor (namely, C₄₄ for a cubic system),

and the tetragonal shear strain tensor (namely, (C₁₁–C₁₂)/2 for a cubic system),

Fig. 3a and b show the charge density topology for the strain-free cases of fcc Au and bcc W, respectively. According to the extreme curvature of the localized charge density, for fcc Au, three different critical points can be observed: the BCPs of fcc Au are in the middle of the two nearest Au atoms, the midpoint of the shape edge of each octahedron formed by six Au atoms (see red ball in Fig. 3); The ring critical points (RCPs) are located surrounding each centered tetrahedral site (T-site), forming a so-called RCPs’ tetrahedron; the two types of cage critical points (CCPs) can be observed at the tetrahedral site (T-site) and octahedral site (O-site), respectively. But for bcc W, BCPs are just located at the midpoint between two nearest atoms, RCPs are located at the T-sites and the CCPs are at the O-sites. The results demonstrate that for all metals, at BCPs, a large charge density has been identified, but at CCPs, the lowest charge density can be observable (details referred to in the ESI†). When a shear strain was applied on both fcc Au and bcc W, the localized charge density varied obviously. With a pure shear strain from 0 to 1.5% for Au, the charge density of two diagonal BCPs of every four BCPs on the (100) and (010) planes decreases, whereas the charge density of another two diagonal BCPs increases at each strained case (cf. B1 and B3 in Fig. 3c). For the RCPs, except for the unchanged charge density of two RCPs in each of the RCPs’ tetrahedra upon strain, one RCP’s density decreases while the other one increases (cf. R1 and R3 in Fig. 3c). For bcc W under a pure shear strain, the charge density of two diagonal BCPs of every four BCPs on one (110) plane decreases, while those of the other two increase. However, on its adjacent alternative (110) plane, the charge density of the BCPs remains unchanged (cf. B1 and B3 in Fig. 3d, respectively).

Fig. 3 Charge density topology under pure shear strain (a) fcc Au with strain of 0.000, (b) bcc W with strain of 0.000. (c) fcc Au with strain of 0.015, (d) bcc W with strain of 0.015. Red balls are bonding critical points (B); pink balls (O-sites) and light blue balls (T-sites) are cage critical points (C); green balls are ring critical points (R). The positions of the RCPs in bcc W under pure shear strain are a little deviated from the T-sites.

In order to visualize the change of the charge density at critical points with increasing pure and tetragonal shear strains, we further compile their tendencies as a function of strain (δ) in Fig. 4. Apparently, the BCPs of fcc Au and Ir under a pure shear strain were changed into four kinds of BCPs, and the RCPs were changed into three kinds of RCPs, but the CCPs were kept unchanged (solid lines in Fig. 4a and c). The BCPs of bcc Nb and W under pure strain were changed into three kinds of BCPs, while both the RCPs and CCPs remained unchanged (solid lines in Fig. 4b and d). This also shows us that under a shear deformation, the charge density at the BCPs (B1) for fcc Au, Ir, bcc Nb and W under pure shear strain are recombined into the other BCPs (B3) and the charge density of RCPs (R1) are recombined into the other RCPs (R3) for fcc Au and Ir.

Fig. 4 The change of charge density under trigonal shear strain (solid lines) and tetragonal shear strain tensor (dashed lines) for (a) fcc Au, (b) bcc Nb, (c) fcc Ir, and (d) bcc W.

In terms of a tetragonal shear strain, the BCPs of fcc Au and Ir were changed into two kinds of BCPs, namely, B1 and B2 (dashed lines in Fig. 4a and c) but the RCPs and CCPs were kept unchanged. Similar to fcc metals, the CCPs (O-sites) of bcc Nb and W were changed into two kinds of CCPs, but the BCPs and RCPs were kept almost unchanged (dashed lines in Fig. 4b and d). This shows that the charge density at the BCPs (B1) for fcc Au and Ir under tetragonal shear strain flows into the other BCPs (B2), whereas for bcc Nb and W, the charge density at the RCPs (T-sites) and partial CCPs (O-sites, C1) all flows into the other CCPs (O-sites, C2). It has already been suggested that the elastic constants of materials stem from the redistribution of charge density among the critical points (CPs),^39,40 but only the case of charge density at the BCPs flowing into CCPs during the fracture process has been found.⁴¹ Here, these observations provide further evidence for the suggestion above.

Thirdly, concerning another important parameter of G/B, it was suggested that the resistance to plastic deformation is related to the product Gb⃑ and that the fracture strength is proportional to the product Ba, where a is the lattice parameter.³ If Gb⃑/Ba is high for a given material, the materials will behave in a brittle manner. Consequently, Gb⃑/Ba reflects the competition between the shear and cohesive strengths at the crack tip of fracture. As b⃑ and a are constants for specific materials, Gb⃑/Ba can be simplified into G/B.⁴² From this perspective, the G/B value can reflect the competition between plastic flow and brittle fracture,⁴³ which has already taken the defect into account, i.e., a dislocation process. Recently, G/B has been suggested to be a good indicator of the hardness of materials. An early work⁴⁴ uncovered that G/B can be correlated to the hardness of superhard materials, i.e., H_V = 2(k²G)^0.585 − 3, where k = G/B. This allows wide access for the prediction of many hard materials, including all known superhard materials.

However, related discussions of the temperature, defect and pressure effects on G/B are still lacking. As illustrated in Fig. 5, we compiled G/B values for some selected materials including fcc Au, alkali halides, pure Ti, Fe-30% at Al, CaO, MgO, ZrO₂–15.5Y₂O₃, Cr and Cr–V alloys at a certain temperature. In general, the value of G/B decreases with increasing temperature (Fig. 5a), such as for MgO, CaO, fcc Au and alkali halides, and so on. If G/B exhibits a dramatic change, it implies the occurrence of a phase transformation. For instance, in the case of Ti, the value of G/B for α-Ti decreases as temperature increases and when the temperature goes up over 1000 K with the phase transformation from α-Ti to β-Ti, the G/B value increases (Fig. 5a). Even more abrupt turns of the G/B value has been observed for some selected metals and alloys in Fig. 5b, since the magnetic phase transition occurs with increasing temperature.

Fig. 5 (a) G/B as a function of temperature for alkali halides, CaO, MgO, pure Ti, ZrO₂–Y₂O₃. (b) G/B as a function of temperature for Cr and Cr–V alloys (cf. details in ESI†).

With increasing temperatures, the decreasing tendency of G/B values is understandable, because the defect concentration (e.g., vacancies) will definitely increase. In this sense, the temperature-dependent tendency of G/B somehow reflects the ability towards defect tolerance to some extent. It is well-known that point defects would have a significant effect on the mechanical properties of materials,^45,46 because it will induce local resonance and localized modes,⁴⁷ and thereby influencing the elastic properties. It was suggested that the introduction of interstitials varies the local coupling parameters near defects, which would lead to a 10% increase of the elastic constants per at% interstitials, whereas vacancies have been demonstrated to only have the capability of reducing elastic moduli.⁴⁵ To date, there have not been experimental investigations on the elastic constants of a single crystal material under irradiation of different doses. Although for a material the G/B value varies with temperature due to defects, its temperature-dependent CP vs. G/B relation still obeys linearity, as illustrated in Fig. 1. In addition, our analysis demonstrates that the linear CP vs. G/B relation certainly fits when the materials are under pressure.

In terms of recent studies,^9–13 ionic bonding benefits the resistance to amorphization by irradiation damage. Because the linear CP/B vs. G/B relationship can indeed reflect material resistance under deformations impacted by the combined effects of bonding features, defects, temperatures, and pressures, it seems that this relation can provide a useful tool to evaluate materials' resistance to amorphization by irradiation damage. The amorphization of materials under irradiation often occurs. We tried to plot some known nuclear structural and nuclear fuel materials which exhibit high tolerance to irradiation damage, as shown in Fig. 6. It is clear that uranium dioxide nuclear fuels have been widely used for commercial reactors for many years due to its excellent irradiation resistance. The analysis demonstrates that even up to temperature of 2930 K, the G/B value varies in a range between 0.44 and 0.33. From Fig. 6b, UO₂ shows a mixed bonding framework of metallic and ionic bonding. Additionally, recent experimental investigation⁴⁸ has proposed a new concept of multi-phase ceramic nuclear fuel: the main phase is UO₂ and the minor phase is a non-fission phase, thereby improving the thermal conductivity. It has been confirmed that Y₂O₃ stabilized ZrO₂ exhibits high resistance to amorphization by heavy ion (Xe and Au ion) irradiation.⁴⁸ Interestingly, we find that the G/B value of most of Y₂O₃ stabilized ZrO₂ varies narrowly (0.20–0.23) against temperature ranging from 373 K to 973 K.

Fig. 6 (a) Cauchy pressure/B as a function of G/B for MAX phases. (b) Cauchy pressure/B as a function of G/B for structural materials and nuclear fuel in nuclear reactors (cf. ESI†).

The MAX phases have been suggested to be a promising but as yet unproven class of material used for advanced reactors due to their high temperature stability, high stiffness, good electrical and thermal conductivity, fracture toughness, thermal shock resistance, and machinability.^8,14 It has been confirmed that both Ti₃AlC₂ and Ti₃SiC₂ remain crystalline at much higher fluences, exhibiting high damage tolerance at 50 K and 300 K with 1 MeV Kr²⁺ and Xe²⁺.¹⁴ It was theoretically estimated that the damage required to completely amorphize these MAX phases will be very high (over 100 dpa). A recent experiment on MAX phases (Ti₃AlC₂ and Ti₃SiC₂) under conditions relevant to application in future nuclear reactors showed that irradiation leads to substantial disordering of the nano-laminate crystal structure via the induction of radiation defects, but does not produce any signs of amorphization at 673 and 973 K by heavy ion (5.8 MeV Ni) irradiation; significant grain boundary cracking and a loss of damage tolerance properties in the aluminum-based MAX phase were observed when irradiated at 673 K, but not in the silicon counterpart.⁸ Importantly, by analyzing the elastic constants of MAX phases, we have found that most of their G/B values are located at the ionic–covalent mixture bonding region, with a few near the metallic–ionic bonding region, as shown in Fig. 6a. It has been claimed in ref. 14 that the visible difference in tolerance between Ti₃AlC₂ and Ti₃SiC₂ can be related to the changes in bonding within each material. As noted earlier, in general, G/B will decrease against increasing temperature without phase transformation. It seems to us that the temperature-dependent G/B for these MAX phases will be useful to gain a better understanding of its high damage tolerance. Yet, there has been no experimental investigation on the temperature-dependent elastic properties for these MAX phases, but only theoretical calculations based on DFT at 0 K.

The carbides ZrC and SiC are also the most promising candidates as fuel-cladding or structural materials. A recent theoretical work demonstrated that ZrC exhibits a higher damage tolerance in comparison with SiC, in accordance with both heavy ion Au irradiation experimental findings and ab initio molecular dynamics calculations.⁴⁹ Recent calculations suggest that the stronger radiation tolerance of ZrC than SiC originates from their different bonding nature: Zr–C bonding is a mixture of covalent, metallic, and ionic character, whereas the Si–C bonding is mainly covalent.⁴⁹ This can be clearly evidenced that the G/B value of ZrC is lower than that of SiC, suggesting its ionic tendency is stronger than that of SiC, as shown in Fig. 6a. Unfortunately, there is no systematic experimental data on the temperature-dependent elastic constants of SiC and ZrC published yet.

In addition, W, V and Fe-based alloys are considered to be the most important candidates for reactors. From Fig. 6a and b, the G/B values of these materials vary from 0.26 to 0.71: W (0.51–0.52, expt), V (0.26–0.30, calc.; 0.31–0.37, expt), Fe-based single crystals (0.49–0.71, expt), Cr (0.60–0.76, expt), Cr–V alloy (0.57–0.70, expt). It is noted that the G/B values of W and V both vary in a narrow range with temperature varying from 4 K to room temperature, suggesting their better defect tolerance within this temperature. It also can be seen that the G/B values of most of those materials are located between the mixed bonding regions. In particular, Al addition makes the Fe-30 at% Al single crystal be located at the ionic–covalent bonding region, because its G/B value varies from 0.71 to 0.67 within the temperature range from 273 K to 1173 K.

In terms of the radiation tolerance mechanism,^9–11 the interatomic interaction will influence the atomic rearrangements created by the heavy particles and, hence, the post-irradiated structure. The interactions between atoms depend on the distribution of electronic density in a solid, resulting from various chemical bonding interactions. Ionic bonding benefits local recrystallization due to its smaller activation energy barriers, whereas covalent bonding suppresses the defect recombination process because it requires breaking of the bonding with an associated energy cost. Recent works on MAX phases have suggested that the weaker M–A and X–A bonding allows the damaged structure to re-establish coherence with the crystalline lattice more easily, whereas the reconstruction of stronger bonding should require an additional energy cost.^12,13 These explanations on the materials’ resistance to amorphization are all based on theoretical calculations at 0 K, i.e. the formation energy of defects, which may be not suitable at high temperature. By contrast, based on our findings here, we propose that the damage tolerance property of a material can be reflected experimentally by measuring the elastic constants. A material with a value of G/B varying narrowly against various temperature, pressure and defect concentration, potentially exhibits high damage tolerance. In this sense, further investigations on the temperature-dependent G/B value for a material should be further carried out by ion or neutron irradiation experiments. Here, we also should note that the experimental elastic constants compiled in this paper were taken from the data of single crystals. Therefore, this linear relation may not work on multi-phase materials, particularly those with various crystal structures, e.g., composite materials.

4 Conclusions

In summary, by analyzing the elastic properties of typical materials with various crystal structures, we propose that the ductile-to-brittle transition stems from bonding type transition, which can be reflected by a universal linear relation between CP/B and G/B. The present work on the dependence of G confirms the redistribution of charge density among the critical points, as suggested by previous work. In general, the value of G/B will generally decrease with increasing pressure or temperature, while an abrupt change of G/B suggests a certain transition, such as magnetic transition, spin flipping, phase transformation and so on. The change of the G/B value against temperature, pressure and defect concentration can reflect the bonding change of a material under corresponding conditions. In contrast to previous explanations, based on theoretical calculations on the materials’ resistance to amorphization, here we propose that the damage tolerance property of a material can be reflected experimentally by measuring the elastic constants. These observations suggest that G/B, a bridge between the electronic scale and the macroscopic scale, will be useful for the understanding and design of materials, particularly under extreme conditions.

Acknowledgements

The authors thank the financial support by NSFC under the grant No. 91226203 and 115055159, by Science and Technology on Surface Physics and Chemistry Laboratory (STSPCL) under the grant No. ZDXKFZ201410 and by China Academy of Engineering Physics (CAEP) under the grant No. 2015B0302061. We also acknowledge the computing resources from TH2A supported by Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase).

References

1. A. Kelly, W. Tyson and A. Cottrell, Philos. Mag., 1967, 15, 567–586.
2. J. R. Rice and R. Thomson, Philos. Mag., 1974, 29, 73–97.
3. S. Pugh, Philos. Mag., 1954, 45, 823–843.
4. D. Pettifor, M. Aoki, J. Murrell, A. Cottrell and A. Stoneham, Philos. Trans. R. Soc. London, Ser. A, 1991, 334, 439–449.
5. X. F. Wang, T. E. Jones, Y. Wu, Z. P. Lu, S. Halas, T. Durakiewicz and M. E. Eberhart, J. Chem. Phys., 2014, 141, 024503.
6. H. Ozisik, E. Deligoz, K. Colakoglu and H. Ozisik, Intermetallics, 2014, 50, 1–7.
7. H. Niu, X.-Q. Chen, P. Liu, W. Xing, X. Cheng, D. Li and Y. Li, Sci. Rep., 2012, 2, 718.
8. D. Clark, S. Zinkle, M. Patel and C. Parish, Acta Mater., 2016, 105, 130–146.
9. K. Trachenko, J. Phys.: Condens. Matter, 2004, 16, R1491.
10. K. Trachenko, J. Pruneda, E. Artacho and M. T. Dove, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 71, 184104.
11. K. Trachenko, M. T. Dove, E. Artacho, I. T. Todorov and W. Smith, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 174207.
12. J. Xiao, T. Yang, C. Wang, J. Xue and Y. Wang, J. Am. Ceram. Soc., 2015, 98, 1323–1331.
13. S. Zhao, J. Xue, Y. Wang and Q. Huang, J. Appl. Phys., 2014, 115, 023503 and references therein.
14. K. Whittle, M. Blackford, R. Aughterson, S. Moricca, G. Lumpkin, D. Riley and N. Zaluzec, Acta Mater., 2010, 58, 4362–4368.
15. G. Kresse and J. Hafner, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 47, 558.
16. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50.
17. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50.
18. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953.
19. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758.
20. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, B864.
21. W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133.
22. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
23. O. Jepson and O. Anderson, Solid State Commun., 1971, 9, 1763–1767.
24. P. E. Blöchl, O. Jepsen and O. K. Andersen, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49, 16223.
25. Y. Page and P. Saxe, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 65, 104104.
26. X. Wu, D. Vanderbilt and D. Hamann, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 035105.
27. X. Wang, X. Cheng, Y. Zhang, R. Li, W. Xing, P. Zhang and X. Chen, Phys. Chem. Chem. Phys., 2014, 16, 26974 and references therein.
28. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188.
29. A. Otero-de-la Roza, M. Blanco, A. M. Pendás and V. Luaña, Comput. Phys. Commun., 2009, 180, 157–166.
30. A. Otero-De-La-Roza, E. R. Johnson and V. Luaña, Comput. Phys. Commun., 2014, 185, 1007–1018.
31. X. Wang, R.-Z. Qiu, Y.-J. Xian, Y.-T. Zhang, P.-C. Liu and P.-C. Zhang, RSC Adv., 2016, 6, 19150–19154.
32. I. Papadimitriou, C. Utton and P. Tsakiropoulos, Acta Mater., 2015, 86, 23–33.
33. T. Ledbetter, Mater. Sci. Eng., 1977, 43, 225–230.
34. S. Kamran, K. Chen, L. Chen and L. Zhao, J. Phys.: Condens. Matter, 2008, 20, 085221.
35. B. Dick, Phys. Rev., 1963, 129, 1583.
36. G. Leibfried and H. Hahn, Z. Phys., 1958, 150, 497–525.
37. A. Herpin, J. Phys. Radium, 1953, 14, 611–620.
38. S. O. Lundqvist, On the elastic Constants of cubic ionic crystale, Almqvist u. Wiksell, 1952.
39. M. Eberhart, D. Clougherty and J. MacLaren, J. Mater. Res., 1993, 8, 438–448.
40. M. Eberhart, Acta Mater., 1996, 44, 2495–2504.
41. M. Eberhart, D. Clougherty and J. MacLaren, J. Am. Chem. Soc., 1993, 115, 5762–5767.
42. A. Verma, Mater. Chem. Phys., 2012, 135, 106–111.
43. D. L. Henann and L. Anand, Acta Mater., 2009, 57, 6057–6074.
44. X.-Q. Chen, H. Niu, D. Li and Y. Li, Intermetallics, 2011, 19, 1275–1281.
45. P. Dederichs, C. Lehmann and A. Scholz, Z. Phys. B: Condens. Matter, 1975, 20, 155–163 and references therein.
46. P. Dederichs, C. Lehmann, H. Schober, A. Scholz and R. Zeller, J. Nucl. Mater., 1978, 69, 176–199.
47. P. Dederichs and R. Zeller, Phys. Rev. B: Solid State, 1976, 14, 2314.
48. D. Men, M. K. Patel and I. O. Usov, J. Nucl. Mater., 2013, 443, 120–127.
49. M. Jiang, H. Xiao, H. Zhang, S. Peng, C. Xu, Z. Liu and X. Zu, Acta Mater., 2016, 110, 192–199 and references therein.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra05194d

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