Multi-principal element alloys (MPEAs) have excellent mechanical properties, such as high strength, good ductility, structural stability, and wear resistance (George et al., 2019;Li et al., 2022a;Miracle and Senkov, 2017;Rao et al., 2022;Senkov et al., 2018;Ye et al., 2016;Zhang et al., 2024Zhang et al., , 2015)). They are composed of multiple elements in equimolar or approximately equimolar proportions (Cantor et al., 2004;Mak et al., 2021;Santodonato et al., 2015;Yeh et al., 2004). The high mixing entropy of MPEAs reduces the Gibbs free energy, and hinders the formation of intermetallic phases (Li et al., 2020;Senkov et al., 2015Senkov et al., , 2014;;Soler et al., 2018). Thus, MPEAs tend to show single-phase solid solutions, including face-centered cubic (FCC) and body-centered cubic (BCC) structures. A major class of MPEAs is the Cr-Mo-Nb-Ta-V-W-Hf-Ti-Zr family of BCC alloys, which have high yield strength and high temperature strength retention (Senkov et al., 2014).
In recent years, the mechanical properties of MPEAs have been extensively studied by combining experiments, theoretical models, with computational simulations (Zhang et al., 2023a). For instance, based on the theoretical guidance and thermodynamic tools, an alloy design process is proposed to discover new single-phase BCC refractory MPEAs that satisfy speci昀椀c mechanical properties (Rao et al., 2022). Based on the understanding of traditional refractory metals and alloys, the in昀氀uence of the interstitial compositions on the mechanical properties of refractory complex concentrated alloys has been thoroughly investigated (Belcher et al., 2023). According to the classical Peierls-Nabarro (P-N) model, a stochastic P-N model has been proposed to explain the origin of high strength in MPEAs (Zhang et al., 2019a). The decomposition of the BCC structure into β+β 7 through the spinodal decomposition produces a modulation of the chemical composition, which achieves an excellent combination of strength 1.1 GPa and ductility 28 % in HfNbTiV MPEAs (An et al., 2021). Combining the 昀椀rst-principles calculations and a set of physical descriptors, the surrogate models are developed to 昀椀nd the candidate alloys that have the enhanced strength-ductility synergies (Hu et al., 2021). The addition of the W element increases the yield stress of TiZrHfNbTa MPEAs from 1064 MPa to 1726 MPa for TiZrHfNbTaW MPEAs (Huang et al., 2022). Based on the theoretical model, the effect of the severe lattice distortion on the mechanical properties reveals that the in昀氀uence of the atomic-radius mismatch on the solid-solution strengthening is the primary factor to govern the yield stress in the BCC Al x HfNbTaTiZr MPEA, surpassing the impact of the shear-modulus mismatch (Li et al., 2020). The orientation-dependent tensile behavior of HfNbTaTiZr MPEAs, the characteristics of deformation twinning, and phase transformations have been investigated at the nanoscale using molecular dynamics (MD) simulations (Jian and Ren, 2024). The above work has made crucial progress in elucidating the close connection between the components, microstructures, and mechanical properties in MPEAs.
There are new theoretical advances to study the fracture behaviors and failure mechanisms (Guo et al., 2024;Li et al., 2022b;Shen et al., 2024;You et al., 2021;Zhao et al., 2023). In terms of the theory of the brittle-ductile transition (BDT) of metals and alloys, the effect of the crack extension on the BDT behavior is emphasized in the previous work (Heslop andPetch, 1956, 1958). It is considered that the BDTT is decided by the P-N stress associated with a free dislocation. The BDT behavior of metals is studied by the mechanism of dislocation source nucleation, which is controlled by dislocation mobility at a constant loading rate (Hirsch and Roberts, 1996;Hirsch et al., 1989). The fracture toughness and the BDTT in the as-cast TiVNbTa MPEA are measured by conducting four-point bending tests at temperatures ranging from -139 ç C to 20 ç C and a strain rate of 1 × 10 -3 s -1 (Scales et al., 2020). Based on the competition between dislocation source operation and crack extension, an analytical method to calculate the BDTT is proposed in the pure metals (Zhang et al., 2019b). The ratio of the screw dislocation velocity to edge dislocation velocity is regarded as a controlling factor to the BDT (Lu et al., 2021). A probabilistic model that incorporates temperature-dependent constitutive relationships has been developed to accurately describe the competition between cleavage and ductile void failure. This model successfully predicts the temperature-dependent fracture toughness and the BDTT in ferritic steels with irradiation effects (Chen et al., 2020). A quantum-mechanical dimensionless metric is proposed to accurately predict the ductility of various MPEAs. This metric aligns well with results from existing tensile experiments (Singh et al., 2023). A room temperature ductility criterion, namely, the ratio of the stress intensity factor for dislocation emission to the stress intensity factor for cleavage, is proposed to analyze the ductility of the existing MPEAs (Mak et al., 2021). Based on a uni昀椀ed thermodynamic framework, a coupled crystal plasticity and phase 昀椀eld model is proposed to study the BDT process and predict the experimental results well in the pure W (Li et al., 2022b). The crystal plasticity models have been used to reveal the temperature dependence of deformation localization in the irradiated W (Li et al., 2021b). A discrete-continuum model, combining 3D discrete dislocation dynamics with the 昀椀nite element method, provides an effective way to understand the behavior of high-speed dislocations under the complex shock loading conditions (Cui et al., 2022). In addition, the role of slow screw dislocations on the fast strain burst events in submicron W is studied using the discrete dislocation dynamics simulations, and the results show the external load mode control can transform the complex collective dynamics of dislocations (Cui et al., 2016(Cui et al., , 2020)). Through different methods, the previous studies have made signi昀椀cant progress in understanding the deformation behavior and BDT of the alloys.
While there have been recent signi昀椀cant theoretical advancements that explain the relationship between yield stress and composition in BCC MPEAs, there is a signi昀椀cant lack of research on ductility properties, such as BDTT and fracture toughness (Mak et al., 2021). The alloying elements and interstitial impurities have a signi昀椀cant effect on the BDTT (Scales et al., 2020;Zhang et al., 2020). The chemical short-range ordering (SRO) has been observed in the atomic simulations and experimental studies (Chen et al., 2021a(Chen et al., , 2021b;;Körmann et al., 2017;Li et al., 2019;Singh et al., 2015;Tamm et al., 2015). The previous research has demonstrated that SRO has the potential to enhance the strength of the alloy (Jouiad et al., 1999;Li et al., 2023b;Pettinari-Sturmel et al., 2002). The SRO plays an impact on the dislocation friction stress based on the classic P-N model. The potential strengthening mechanism involves increasing the volume of mismatch, facilitating dislocation cross-slip and multiplication, and phase transformation (Wu et al., 2021). In addition, the interaction between dislocation and SRO has a signi昀椀cant impact on the thermal activation process of dislocation slip (Tanaka et al., 2012). With an increase in the Cu content, the low SRO degree decreases the activation energy required for dislocation slip, ultimately lowering the BDTT in the FeCrNCu MPEA (Tanaka et al., 2014). However, there is a lack on the quantitative impact of SRO on the BDTT and fracture toughness in the MPEA.
In the present work, a dislocation theory-based model is developed to quantify the in昀氀uence of the alloying element and SRO on the dislocation motion, BDTT, and fracture toughness in MPEAs. The developed model gives a quantitative prediction for the compositional contribution to the BDTT and fracture toughness. In addition, the microstructure-based brittle-ductile transition criteria is proposed.
The distinction between ductile and brittle fracture in crystalline metals is primarily determined by whether or not yielding occurs. In essence, the brittle fracture typically occurs prior to yielding, whereas ductile fracture occurs after the metal has yielded. In the case Z. Han et al. of polycrystalline materials, stress concentrations caused by clusters of dislocation pile-ups can trigger dislocation sources in neighboring grains and facilitate crack propagation. If the crack propagation is the primary initiation process, fracture occurs before yielding, resulting in a brittle fracture. On the other hand, if the dislocation source is initiated 昀椀rst, fracture occurs after yielding, leading to a ductile fracture. The dislocation mechanism theory is widely adopted to investigate the BDT behavior (Bonnekoh et al., 2019;Heslop andPetch, 1956, 1958;Lu et al., 2021;Petch, 1958;Zhang et al., 2023b), which depends on grain size, lattice friction stress, yield strength, and applied stress. Hence, the models of dislocation friction stress and yield strength are determined in MPEAs due to their complex element composition and distribution. For example, from the previous experiments (Chen et al., 2024;Zhu et al., 2024), MPEAs have a random distribution or locally chemically ordered distribution of atom types at several nanometers, leading to atomic severe lattice distortion. However, the classic P-N model hardly predicts the dislocation friction stress due to the constantly changing lattice parameters at several nanometers. Thus, the theoretical framework is established in this work for BDT based on the dislocation theory-based model considering the modi昀椀ed lattice friction stress model, the composition-dependent strength model, and the dislocation mechanism-based critical energy model.
Here, the BDT depends on the competition between crack propagation priority and crack tip dislocation source activation priority in MPEAs. The ductile fracture occurs when the dislocation source 昀椀rstly operates, and the brittle fracture takes place when the crack begins to extend. Fig. 1 shows a reasonable model from the previous experiment for crack propagation and dislocation pile-up in the adjacent grains of CrMnFeCoNi MPEAs (Suzuki et al., 2020).
Hence, the effective potential barrier for the initiation of dislocation sources at the crack tip in the MPEA is
where E 1 is the energy for the dislocation line tension, E 2 is the consumed energy for the dislocation movement, E 3 is the energy of the (Suzuki et al., 2020). (e) The schematic diagram of two adjacent grains in MPEAs containing a crack. The length of the crack is c, the distance between the pile-up position and the crack tip is l, and the dislocation source exists at the point, P, in the front of the crack tip. Assuming that a large amount of slipping dislocations pile-up on the boundary between grains A and B.
Z. Han et al. crack propagation, and E 4 is the crack surface energy (Zhang et al., 2019b). When the thermal activation energy overcomes the maximum effective potential barrier, the BDT behavior occurs.
The energy for the dislocation line tension is
where F = μb 2 /2 is the line tension. μ is the shear modulus. θb is the elongation of the arc "CD". Here, the dislocation segment "CD" is expanded from the radius r 1 to radius r 2 , as shown in Fig. 2. The consumed energy for the dislocation segment "CD" movement is expressed as
where σ is the ampli昀椀ed stress of the pile-up group at the point of P, ΔS is the area of the pink area swept by the dislocation source. σ y is the yield stress, which is given by Eq. ( 24). σ f = M(σ PN +τ 1 +τ 2 +τ 3 ) is the stress acted on the dislocation motion, where σ PN is determined by Eqs. (22,23). τ 1 is the stress of the elastic interaction with other dislocations, τ 2 is the stress of the jogs formation, and τ 3 denotes the stress of the elastic stress 昀椀eld of the dislocation. Here, τ i = α i μb /x, α i is approximately between 0.2 and 0.3. The value of x is approximately 10 -7 m. d is the average grain size. The radius r 2 is expressed by
The elastic energy released by crack expansion for plane stress conditions is approximated as U = πσ 2 cb 2 /(2μ) (Lawn, 1993). When the dislocation source opens 昀椀rst at point P, the unreleased elastic energy will become the energy barrier for the dislocation source to open. Thus, the energy of the crack propagation is
The surface energy of the crack is
where 2b 2 is the incremental crack surface area, and γ s is the speci昀椀c surface energy. Here, the crack extends from c to c + b.
The relationship between strain rate and BDTT has been determined by (Tanaka et al., 2008)
where Û ε is the strain rate, A is a pre-factor, E a is the activation energy for the BDT behavior, k is the Boltzmann's constant, and T c is the BDTT.
According to the thermal activation theory, the strain rate can be given by (Argon, 2007) Z. Han et al. where ρ is the mobile dislocation density, b is the Burgers vector, s is the dislocation moved distance, and v d is the natural vibration frequency of the dislocation. Thus, the BDTT is
where
The fracture toughness of metallic materials varies with the temperature. When the temperature is below the BDTT, the fracture toughness decreases rapidly. In the temperature-transition region, the relationship between the fracture toughness and temperature is (Sailors and Corten, 1972)
where K IC is the fracture toughness, the constant k 1 is 14.6, χ is the Charpy V-notch value, χ 0 and B are obtained from the experimental data, and T is the temperature.
The dislocation core plays an important role in the dislocation slip and dislocation interaction (Alkan et al., 2018;Fan et al., 2021;Krasnikov and Mayer, 2018;Li et al., 2021a;Yang et al., 2020). In the classical P-N model (Nabarro, 1947;Peierls, 1940), it integrates an atomic-level description of dislocation core and a long-range description of the dislocation strain 昀椀eld (Hirth et al., 1983). Fig. 3 shows a single edge dislocation in MPEAs, and this model considers the random atom occupancy and SRO (Zhang et al., 2019a).
The total energy, stress 昀椀eld, mis昀椀t energy, and generalized stacking-fault energy (GSFE) in a single edge dislocation are given by (Anderson et al., 2017)
where σ yx (x) is the shear stress on the slip surface, ν is the Poisson ratio, and γ(φ) is the periodic interplanar potential. Assuming that the dislocation line in a grain of MPEAs is along the z-axis and the Burgers vector is along the x-axis, and the slip plane is the x-z plane. Two continuous and linear elastic half spaces (y > 0 and y < 0) are separated by the slip plane. The disregistry function in the x direction along the slip plane is expressed by a function, φ(x); φ(x→ -∞) = 0 and φ(x→
. a is the spacing of the atomic plane perpendicular to the glide plane. Z. Han et al.
The equilibrium dislocation disregistry function and core width are derived by δE total /δ[φ(x)] = 0. The disregistry function and Peierls stress are expressed as follows
where w 0 = a /[2(1 -ν)] is the dislocation core width.
In MPEAs, the random lattice occupancy varies from the site to site, and thus the mis昀椀t energy related to the GSFE is different from that of traditional alloys. To understand the effect of the random site occupancy and SRO on the intrinsic strength, a random variable, ω(x), with the short-range spatial correlation is introduced. Thus, the mis昀椀t energy is expressed as
The random variable is described by a normal distribution, and its probability density function is
where Δ is the standard deviation, and it is written as
Here, the standard deviation explicitly depends on the short-range spatial correlation λ and standard deviation Δ (Zhang et al., 2019a). The reasons for choosing a normal distribution are explained in Appendix A.
The dislocation friction stress and its probability distribution in the MPEA are expressed as
where β = b /(2πw 0 ). Here, σ p is de昀椀ned as the normalized dislocation friction stress, σ PN /σ 0 p . Therefore, the dislocation friction stress of a single edge dislocation is a function of the standard deviation of the GSFE perturbation and the correlation length of the spatial component distribution. Hence, the existence of SRO signi昀椀cantly in昀氀uences the standard deviation of the GSFE perturbation and the correlation length of the spatial component distribution, thus ultimately impacting the dislocation friction stress.
For a single-phase BCC MPEA, there are three strengthening mechanisms that contribute to the yield strength, including the solidsolution strengthening, σ ss , dislocation strengthening, σ dis , and grain-boundary strengthening, σ gb . Hence, the yield stress in Eq. ( 4) for the MPEA is expressed as
Solid-solution strengthening originates from various interactions between dislocations and solute atoms (Li et al., 2023c), including Z. Han et al. elastic interaction, modulus interaction, electrical interaction, chemical interaction, and localized ordering interaction. Among these, the elastic interaction and modulus interaction dominate solid-solution strengthening (Courtney, 2005). Here, the contribution of SRO to the yield strength of MPEAs is very limited (Zhang et al., 2020). Previous work has suggested that the contribution of SRO only accounts for about 4 % of the yield strength of BCC MoNbTaW MPEA (Li et al., 2023a), and even SRO results in a decrease in the yield strength of solid solution alloys (Abu-Odeh and Asta, 2022). Moreover, the SRO causes the limited antiphase boundary strengthening (Schön, 2021), while it leads to a decrease in solute concentration which greatly reduces solid solution strengthening (Fang et al., 2022). On other hand, the antiphase boundary energy is very dif昀椀cult to measure and calculate due to the formation of the complex SRO and the obvious changeable solute composition around the SRO structure, thus hardly evaluating the SRO contribution to the yield strength in MPEAs. As a result, this work primarily investigates the dominant solid-solution strengthening mechanism, while disregarding the weaker SRO strengthening effect.
Here, the solute atom in the BCC matrix induces the lattice distortion to impede dislocation motion (Fig. 4). Based on Vegard's law (Li et al., 2020;Toda-Caraballo and Rivera-Díaz-del-Castillo, 2015;Vegard, 1916), for MPEA composed with n elements, the contribution of solid solution strengthening to the yield stress is summarized as:
where c i is the atomic percentage of the element, i. The strength contribution, σ i ss , of the element, i, is expressed as
where A = 0.04 is the material constant, and μ = 3 n i=1 c i μ i is the shear modulus following the average rule (Senkov et al., 2010). The mismatch parameter, δ i , is given by (Toda-Caraballo and Rivera-Díaz-del-Castillo, 2015)
where ξ = 2.5 in BCC metals (Li et al., 2020), δμ i is the shear modulus mismatch, and δr i is the atomic-size mismatch. The coef昀椀cient β is dependent on the kind of dislocations (Labusch, 1970). For screw dislocations, 3 < β < 16 is used, while for edge dislocations, β > 16 is employed. MPEA, ijkl, is assumed to be composed of a multi-principal matrix, jkl, and an additional element, i. The expressions of the atomic-size mismatch, δr i , as well as the modulus mismatch, δμ i , are given:
and δμ i =
, δr ave ijkl and δμ ave ijkl are the average atomic size mismatch and average modulus mismatch of the ijkl MPEA, respectively. The expressions δr ave and δμ ave are calculated by
where
) and δr ij = 2 ( r i -r j ) / ( r i +r j ) are the shear-modulus mismatch and the atomic-size mismatch between two elements, i and j, respectively. μ i and r i are the shear moduli and atomic radii of the element, i. There are no atomic-size mismatch and shear-modulus mismatch between atoms i. Therefore, δr ii and δμ ii are zero. The probability of mismatch between atom i and atom j is c i c j .
According to the Taylor relationship (Kim et al., 2019), the dislocation strengthening is written as
where M=3.06 is the Taylor constant, and ς=0.33 is an empirical constant.
The grain boundary strengthening is written as (Liu et al., 2013)
where k y is the Hall-Petch constant.
Based on Eqs. (2-7), the relationship between the effective potential barrier and microstructure is obtained as
Z. Han et al. Hence, from Eqs. (10, 32), the microstructure-based BDT criteria can be expressed by
where f(d, λ, Δ) = σ y -σ f is de昀椀ned as the microstructure parameters. The microstructural parameters are in昀氀uenced by various factors, including the average grain size, the standard deviation of the interplaner potential perturbation, and the short-range correlation length. In addition, Eq. ( 33) is simpli昀椀ed as
Here, for a given temperature T, when Eq. ( 34) is satis昀椀ed, the material is ductile; otherwise, the material is brittle. Therefore, the BDT behavior is predicted in MPEAs based on the above developed physical model. Considering the critical energy model, the modi昀椀ed lattice friction stress model, and the composition-dependent yield strength model, Fig. 5 shows the 昀氀owchart of the calculation process for the dislocation theory-based model. It is important to note that the critical energy model based on dislocation theory is not dependent on the materials used. The modi昀椀ed lattice friction stress model takes into account the component randomness and SRO of MPEAs. Additionally, the composition-dependent strength model considers the effects of composition, lattice distortion, and component concentration in MPEAs. Therefore, the developed physical model provides accurate and reasonable predictions for the BDT behavior. The speci昀椀c calculation details are described as follows: i) To begin with, we would provide de昀椀nitions for the following parameters: atomic radii, shear modulus, atomic fractions, dislocation density, Hall-Petch constant, and grain size, as outlined in Eqs. (24-31). Subsequently, we will make predictions regarding the yield strength, taking into account solidsolution strengthening stress, grain-boundary strengthening stress, and dislocation strengthening stress. ii) Secondly, determine the dislocation friction stress using Eqs. (18-23). The average dislocation friction stress is determined by the phenomenological parameters Δ and λ. iii) Finally, predict the BDTT of MPEA of MPEA using Eqs. (1-10). As the composition of MPEA varies, the yield strength, shear modulus, and dislocation friction stress also vary, leading to changes in the effective potential barrier and ultimately in昀氀uencing the BDTT. Z. Han et al.
To verify the reasonableness of the BDT model, we calculate the BDTT of the TiVNbTa MPEAs. The experimental data is obtained from the existing TiVNbTa MPEAs (Scales et al., 2020). Table 1 shows the nominal chemical compositions of the TiVNbTa MPEAs. The fracture testing of the TiVNbTa MPEA is essential to measure the fracture toughness and BDTT. This testing is conducted over a range of temperatures to ensure its performance and prevent any potential catastrophic failures during service. By analyzing the fracture test and examining the fractography, the fracture toughness and the BDTT are obtained in Fig. 6, where the temperature of the four-point bending test is in the range of -139 ç C to 20 ç C and the strain rate is 1 × 10 -3 s -1 (Scales et al., 2020). The fracture toughness versus temperature shows a soft BDT behavior (Fig. 6). The variation in fracture toughness with the temperature is observed in four distinct stages: brittle, semi-brittle, brittle-ductile transition, and ductile. When the temperature is low, the fracture toughness remains relatively stable at 43 MPa⋅m 0.5 , but it increases to 65 MPa⋅m 0.5 within the semi-brittle range. When the temperature is higher than -40 ç C, the as-cast TiVNbTa MPEA exhibits high ductility and does not easily fracture. The soft transition phenomenon is a characteristic feature of semi-brittle BCC alloys, where the BDTT ranges from -47 ç C to -27 ç C in TiVNbTa MPEAs.
For the TiVNbTa MPEA, the related parameters are listed in Table 2. The full dislocation in the BCC crystal is b =
Poisson ratio is ν = 0.3, and the half core width of the dislocation is w 0 = 0.75a 0 . Thus, based on Table 2 and Eq. ( 10), the calculated BDTT of the TiVNbTa is T c = -39 ∘ C (Fig. 6), which is well in line with the experimental results (Scales et al., 2020). Then, the effects of component 昀氀uctuation, component randomness, and SRO on BDTT would be analyzed.
Based on the modi昀椀ed P-N model, the normalized dislocation friction stress is a statistical quantity with its own probability density distribution. Fig. 7 illustrates the normalized dislocation friction stress probability density function corresponding to different component randomness standard deviations and spatial correlation lengths. The average dislocation friction stress is calculated for different parameters, and the average dislocation friction stress is used as a description of the actual dislocation friction stress in MPEAs. Fig. 7a shows that the increasing standard deviation leads to a wide distribution of dislocation friction stress at a consistent spatial correlation length. This, in turn, leads to high average dislocation friction stress. Fig. 7b demonstrates that the increasing spatial correlation length at a 昀椀xed standard deviation leads to a wide distribution of dislocation friction stress and an increase in average dislocation friction stress.
Here, the variation of average dislocation friction stress with standard deviation and correlation length is presented in Fig. 8a. Here, when the standard deviation and correlation length is 0, it represents the pure metal; when the standard deviation and correlation length is small value, it represents the random alloys; when the standard deviation and correlation length is large value, it represents the MPEA with SRO. Fig. 8a shows the high dislocation friction stress in the MPEA compared to the pure metals (Pei et al., 2021;Xu et al., 2021), for the given standard deviation and correlation length. The increasing standard deviation at a 昀椀xed correlation length leads to the high dislocation friction stress. Furthermore, the increasing spatial correlation length and standard deviation both contribute to an increase in the dislocation friction stress, where these 昀椀ndings are con昀椀rmed in the previous work (Zhang et al., 2019a). Fig. 8b shows the distribution of BDTT for different component randomnesses and SRO degrees. Compared to the pure metals and random alloys, MPEA has a high BDTT. As the SRO degree increases, the BDTT rises (Tanaka et al., 2014). This trend would provide a way to control the BDTT for meeting the special service environment. Fig. 8c reveals the variation of fracture toughness at the BDTT in the MPEAs. As the spatial-correlation length and standard deviation increase, the fracture toughness of MPEAs is enhanced (Zhang et al., 2020). Fig. 8d reveals the variation of fracture toughness at room temperature in the MPEAs. As the spatial-correlation length and standard deviation increase, the fracture toughness of MPEAs is reduced at room temperature. In other words, the regulation of the chemical element distribution not only changes the BDTT, but also alters the fracture toughness in MPEAs.
To investigate the element distribution, the single crystal MPEA MoNbTaW sample is built using the large-scale molecular dynamics massively parallel simulator (LAMMPS) for Monte Carlo and molecular dynamics calculations. The MPEA sample with the size of 33.3 × 3.1 × 27.2 nm 3 has about 171,000 atoms. To construct the desired MPEA MoNbTaW sample containing the random elements, the atoms within the single crystal Mo structure are randomly substituted with atoms of Nb, Ta, and W, as presented in Fig. 9. The periodic boundary conditions are applied to all dimensions. The annealed MPEA samples are equilibrated using a hybrid Monte Carlo/ molecular dynamics (MC/MD) approach (Fig. 9). In every MC step, a random atom is swapped with another random atom, following the Metropolis algorithm in the canonical ensemble. Each MC/MD step consists of 100 MC swaps followed by up to 10 MD relaxations. The system is maintained at a temperature of 300 K during these processes.
The Warren-Cowley (WC) parameter is widely utilized to describe the SRO degree in the MPEAs (Cowley, 1950), which is expressed
. Here, N represents the Nth nearest-neighbor shell of the central atom i, α N ij is the WC parameter of i-j type
The elemental content of the as-casted TiVNbTa MPEA.
Element Ti Ta Nb V O C N at.% 23.592 26.797 25.041 24.439 0.0963 0.0276 0.0073 Z. Han et al.
atomic pair in the Nth nearest-neighbor shell, C j is the average concentration of element j in MPEAs, p N ij is the probability of locating a j-type atom within the Nth nearest-neighbor shell of an i-type atom, and δ ij is the Kronecker delta. The 昀椀rst-principles simulations show that the GSFE in the MPEAs with different SRO degrees follows a normal distribution (Ding et al., 2018). Here, the SRO parameters are Table 2 Parameters used in the BDTT calculations of TiVNbTa. Parameters Symbol Magnitude Yield strength (MPa) σy 1095 Lattice constant (nm) a0 3.237 (Scales et al., 2020) Burgers vector (nm) b 3 : a0/2 Shear modulus (GPa) μ 49.5 Grain size (μm) d 34.5 (Raman et al., 2021) Distance to crack tip (μm) l 0.1d(Zhang et al., 2019b) Dislocation density (/cm 2 ) ρ 10 11 Strain rate (/s) Û ε 10 -3 (Scales et al., 2020) Speci昀椀c surface energy (J/m 2 ) γ s 1 (Zhang et al., 2019b) Vibration frequency of dislocation (/s) ν d 10 13 Boltzmann constant (J/K) k 1.3806505 × 10 -23 Grif昀椀th crack size (cm) c 2 × 10 -4 (Zhang et al., 2019b) Coef昀椀cient β 3 Fig. 7. (a) The normalized dislocation friction stress distribution versus the correlated random coef昀椀cient for several values of standard deviation at 昀椀xed spatial correlation length λ /w 0 = 1. (b) The normalized dislocation friction stress distribution versus the correlated random coef昀椀cient for some values of spatial correlation length at 昀椀xed standard deviation Δ = 0.1 (Zhang et al., 2019a).
computed using the MD simulation in the BCC MoNbTaW MPEA (Fig. 10). The positive value of α 1 w-w increases signi昀椀cantly in the MPEAs, indicating an increase in the degree of the local W segregation (Fig. 9).
The stacking fault energy curves of the random MPEA and SRO MPEA are presented in Fig. 11a-c, and the distributions of the corresponding two other elements are described in Fig. 11d. For different ID samples, the stable stacking fault energy shows signi昀椀cant 昀氀uctuations, which is 63.1 ± 0.8 mJ/m 2 in the random MPEA and 78.4 ± 1.1 mJ/m 2 in the SRO MPEA. By computing the SFE deviation under a given SRO MPEA, the corresponding correlation length can be obtained. For example, the short-range correlation length increases from 0.53 to 3.68 as the WC parameter increases from 0.18 to 0.24. Thus, when the correlation length is four times the dislocation core width and the standard deviation is 0.1, the BDTT is -22 ç C in the MPEA TiVNbTa (Fig. 8). This comparison indicates that the atomic simulation results are consistent with the predicted results of the theoretical model, namely, the randomness to increase BDTT to a certain extent.
To assess the rationality and accuracy of the composition-dependent yield strength model, the predicted yield stress is compared to the result of experiment obtained from the BCC TiVNbTa MPEA. In the TiVNbTa MPEA, the Hall-Petch coef昀椀cient is taken as 592 MPa μm :
according to previous studies (Cordero et al., 2016;Raman et al., 2021). Table 3 presents the atomic radius and shear modulus of each element for the solid solution strengthening.
Fig. 12 shows the comparison of the yield strength from the theoretical model and experiment in TiVNbTa MPEAs. The yield strength predicted from the composition-dependent strength model is consistent with the experimental result (Lee et al., 2018). The yield strength of the TiVNbTa MPEA is composed of three components, namely solid solution strengthening caused by the severe lattice distortion, grain boundary strengthening, and dislocation strengthening. The yield strength experimentally obtained is 1273 MPa (Lee et al., 2018), and the result predicted from the composition-dependent yield strength model is 1095 MPa. The experimental and theoretical results deviate from each other by 14 %. Among these components, the solid solution strengthening contributes signi昀椀cantly, dominating the overall yield strength; the grain boundary strengthening is 100 MPa, which contributes the least to the yield strength; the dislocation strengthening is 443 MPa. The composition-dependent strength model is used to explore the effect of element concentration on the yield stress, atomic size mismatch, shear modulus mismatch, and solid solution strengthening in the BCC Ti x V 50-x NbTa MPEAs. Thus, the current model is utilized to predict the yield stress of the Ti x V 50-x NbTa MPEA when Ti element varies in the range of 15 % to 35 %. Fig. 13a demonstrates the calculated yield stress decreases with the increase Ti content in the Ti x V 50-x NbTa MPEAs. Figs. 13b, c depict the variations of the atomic radius mismatch and shear modulus mismatch with the increasing Ti content. At the given Ti concentration, the atomic radius mismatch of V element is larger than that of other elements, including Ti, Nb, and Ta. The atomic radii of other elements are very close to each other, while the atomic radius of V element is signi昀椀cantly smaller compared to the other elements (Li et al., 2020). The shear modulus mismatch of Ta element is higher than those of other elements, as exhibited in Fig. 13c. The shear moduli of the other elements are signi昀椀cantly smaller than that of the pure Ta. Fig. 13d shows the variation of solid solution strengthening due to the large atomic radius mismatch and shear modulus mismatch. From Figs. 13b-d, the V element to reduce the solid solution strengthening is the main reason for the decreasing yield strength of Ti x V 50-x NbTa MPEA. The atomic radius mismatch induces strong strengthening rather than the modulus mismatch, which agrees with the previous 昀椀ndings (Dou et al., 2024;Lee et al., 2018).
The effect of the solid solution strengthening induced by the lattice distortion on BDTT is investigated, as shown in Fig. 14. The composition effect on the BDTT and fracture toughness is presented in Figs. 14b-d. Here, the yield stress, the BDTT and the fracture toughness are studied in Ti x V 50-x NbTa, Ti x VNb 50-x Ta and TiV x Nb 50-x Ta MPEAs. From Fig. 14a, with the increasing Ti content, the yield strength of Ti x V 50-x NbTa decreases due to the reducing solid-solution strengthening contributed by the V element, and the yield strength of TiV x Nb 50-x Ta increases. In addition, the high V content in Ti x V 50-x NbTa leads to the large yield stress and fracture toughness, in agreement with the experimental results (Wang et al., 2021). Fig. 14b displays the variation of BDTT versus the element 昀氀uctuation. The BDTT of Ti x V 50-x NbTa 昀椀rstly decreases, and then increases with the increasing element content. The BDTT trend of TiV x Nb 50-x Ta is similar to that of Ti x V 50-x NbTa, while the BDTT of Ti x VNb 50-x Ta shows a decreasing trend. The BDTT reaches the minimum value in the Ti x VNb 50-x Ta MPEAs. It indicates that the increase Ti and the decrease Nb contribute to the reduction of BDTT, while the decrease Ti and the increase V signi昀椀cantly increase the BDTT. Fig. 14c demonstrates the variation of fracture toughness at the occurrence of BDT behaviour. When the Ti content is between 20 % and 25 %, the fracture toughness approaches the maximum value in the Ti x V 50-x NbTa MPEAs. It suggests that the decrease Ti and the increase V lead to the improvement of fracture toughness. Fig. 14d demonstrates the variation of fracture toughness at room temperature. When the Ti content is between 15 % and 25 %, the Ti x VNb 50-x Ta MPEA shows the high fracture toughness compared to the Ti x V 50-x NbTa and TiV x Nb 50-x Ta MPEAs. It reveals that the increasing Ti and the reducing V lead to enhance the fracture toughness at room temperature.
Here, the impact of atomic radius and shear modulus from the other elements on the mechanical properties of MPEAs is evaluated Fig. 10. Average SRO parameter obtained from MD simulation in the BCC MoNbTaW MPEA. Z. Han et al.
The atomic radius, shear modulus, and atomic fraction of Ti, V, Nb, and Ta. for a constant composition content. In fact, based on the lattice distortion related strengthening model, the effects of atomic radius and shear modulus on the yield strength, BDTT, and fracture toughness are quantitatively evaluated in the TiVNbTa MPEA. For example, the other elements (denoted as X) can form VNbTaX, TiVTaX, and TiVNbX MPEAs. Then, the in昀氀uence of the atomic radius and shear modulus on the mechanical properties is quantitatively assessed in VNbTaX, TiVTaX, and TiVNbX MPEAs. In Fig. 15a, when the element radius is less than 148pm, the yield strength improves with the increase atomic radius in the VNbTaX, TiVTaX, and TiVNbX MPEAs. It is evident that the VNbTaX MPEA has the high strength. As shown in Fig. 15b, the BDTT decreases in both TiVTaX and TiVNbX MPEA, as the atomic radius increases. However, the BDTT for VNbTaX is intricate, and it shows a complex relationship with the increasing atomic radius. This trend shows a pattern of initially decreasing, then increasing, and ultimately decreasing. Fig. 15c shows that the fracture toughness at the BDTT initially decreases, and then increases with the increasing atomic radius in VNbTaX.
Similarly, the fracture toughness of TiVTaX and TiVNbX follows a similar trend as the atomic radius increases. Here, the other elements lead to the improved strength and fracture toughness in VNbTaX. Fig. 15d shows that the trend of fracture toughness in relation to atomic radius is almost the same at room temperature for TiVTaX and TiVNbX. It is worth noting that the MPEAs exhibit a high fracture toughness for the large atomic radius at room temperature. Fig. 16 presents the effect of shear modulus of the other elements on the yield stress, BDTT, and fracture toughness in the MPEAs. From Fig. 16a, as the shear modulus increases, the yield strength of VNbTaX exhibits a parabolic trend, while TiNbTaX and TiVTaX show similar trends. By substituting Zr for Nb, experimental results reveal that the hardness of TiVZrTa surpasses that of TiVNbTa (Kareer et al., 2019). This enhancement in hardness also signi昀椀es a corresponding increase in the strength. TiNbTaX MPEA exhibits superior yield strength under the condition that the atomic radius of the other element remains constant and the shear modulus is below 45 GPa. In Fig. 16b, the BDTT of TiNbTaX exhibits a more pronounced variation in response to the shear modulus change. When the shear modulus is below 45 GPa, VNbTaX exhibits a low BDTT. Fig. 16c shows that the trends of fracture toughness at the BDTT in relation to shear modulus for VNbTaX and TiVTaX are quite similar. However, it is worth noting that TiNbTaX exhibits a notably high level of fracture toughness when the shear modulus is below 45 GPa. Fig. 16d shows that the trends of fracture toughness at room temperature in relation to shear modulus for VNbTaX and TiVTaX are quite similar. However, it is worth noting that VNbTaX exhibits a notably high level of toughness when the shear modulus is below 45 GPa.
Fig. 17 illustrates the brittle/ductile characteristic distribution of TiVNbTa MPEA at different temperatures. When the given temperature is T 1 = -39 ç C, the microstructural parameters including standard deviation and spatial correlation length fall within the blue shaded region, indicating that the material is ductile; when the given temperature is T 2 = -15 ç C, the microstructural parameters fall within the yellow shaded region, indicating that the material exhibits brittle properties. Based on the microstructure-based BDT criterion, it is possible to analyze the conditions satis昀椀ed by the microstructural parameters of the MPEAs in order to maintain ductility at a given temperature. Thus, such results provide the opportunity to adjust the mechanical properties through alterations in the microstructure. It is anticipated that this work would aid in the prediction of the BDTT and fracture toughness of MPEAs, and developing materials for extreme working conditions.
In the current work, a dislocation theory-based model coupling the critical energy model, the modi昀椀ed lattice friction stress model, and the composition-dependent strength model is developed to predict the BDTT and fracture toughness in MPEAs. The yield stress of TiVNbTa predicted by the composition-dependent strength model shows good agreement with the experimental results. Furthermore, the analytical model accurately predicts the BDTT in the TiVNbTa MPEA, consistent with previous experimental observations. The impact of standard deviation, short-range spatial correlation length, elemental concentration 昀氀uctuation, and element substitution on the toughness is investigated in the TiVNbTa MPEA. A high degree of short-range spatial correlation length increases dislocation friction stress, but it does not necessarily reduce the BDTT in TiVNbTa. The solid-solution strengthening is identi昀椀ed as the main mechanism for strengthening in TiVNbTa. In the Ti-V-Nb-Ta MPEAs, the ductility of Ti x VNb 50-x Ta is most affected by changes in concentration. A microstructure-based BDT criterion for predicting the necessary microstructural parameters for MPEA is proposed to exhibit ductility at a speci昀椀c temperature. These 昀椀ndings support the use of the developed methods for computationally guided design of advanced BCC MPEAs with superior strength and toughness.
Zebin Han: Writingoriginal draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Data curation. Bin Liu: Writingoriginal draft, Supervision, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation. Qihong Fang: Writingoriginal draft, Validation, Supervision, Methodology, Investigation, Funding acquisition. Peter K Liaw: Writingoriginal draft, Supervision, Methodology, Investigation, Funding acquisition. Jia Li: Writingoriginal draft, Software, Methodology, Investigation, Funding acquisition, Conceptualization. Z. Han et al.