Line-length-dependent dislocation glide in refractory multi-principal element alloys

Dislocation gliding in body-centered cubic (BCC) pure metals has been studied extensively. At low temperatures, a long screw dislocation (> 1 nm) is known to glide via the kink-pair mechanism.¹ Kink-pair migration was also found during long edge dislocation gliding in BCC Fe^2–5 and Mo,⁶ yet with less pronounced kinks. For example, on the {110} plane in Fe, the kink width on an edge dislocation, 0.7 or 1.5 nm,^2,3 is smaller than that on a screw dislocation, 4, 4.8, 5, or 6 nm.^3,7–10 For edge dislocations in Fe, the kinks on {112} planes are more pronounced than those on {110} planes.^4,5 Since a longer dislocation (tens of nm) can contain more kinks, its critical resolved shear stress (CRSS) is usually lower than that of a short one (⁠$≈1$ nm) of the same character.¹¹ For the same length, the CRSS of a screw dislocation is at least one order of magnitude higher than that of its edge counterpart.¹² A practical result of all these dislocation behaviors is that plasticity in BCC pure metals at low temperatures is mainly controlled by the gliding of screw dislocations.

The motion of long dislocations in BCC refractory multi-principal element alloys (RMPEAs)¹³ is more complicated. First, subject to zero applied stress, the static screw dislocation core in RMPEAs can be compact or non-compact, depending on the local chemical environment.^14–16 This contrasts the compact static core structure of the screw dislocation in multiple BCC pure metals.^17,18 As a result of the chemistry-dependent core structure, partial kinks are spread on varying {110} planes along the screw dislocation line. Consequently, screw dislocation glide in RMPEA is controlled by cross kinks, i.e., collision of kinks on different glide planes,^19,20 instead of kink-pair formation as in BCC pure metals. Wavy screw dislocations in RMPEAs have been confirmed by transmission electron microscopy (TEM) experiments.²¹ Edge dislocation glide in RMPEAs is also significantly affected by kinks,²² which are much more pronounced than those in BCC pure metals. Therefore, the screw-to-edge ratio in CRSS is significantly lowered in RMPEAs and BCC concentrated binary alloys: In MoNb₃, NbTi₂Zr, and NbTiZr, respectively, atomistic simulations or experiments found that the ratios in the CRSS between the long screw and long edge dislocations are 2, 6.3, and 2.^12,23,24 These results suggest that both edge and screw dislocations play an important role in the plasticity of BCC concentrated alloys.^25,26

The kinks on long dislocations in both BCC pure metals and RMPEAs indicate that the dislocation lines in simulations need to be at least several times longer than the kink width for the dislocation dynamics to be length-independent, i.e., free of any length-related artifact.²² In what follows, we term this minimum length as the characteristic length of the dislocation, $Lc$⁠. Physically, $Lc$ is the minimum length that can represent key dynamic features of a long dislocation, and so simulations of dislocations shorter than $Lc$ are not very meaningful. Analytical models predicted that, for screw dislocations in Nb_xTi_1−2xZr_x RMPEAs, $Lc=22$–36 nm;²⁷ for edge dislocations, in MoNbTaVW family and NbTaV RMPEAs, respectively, $Lc=16$–25 and 12 nm,^22,23 with the former validated by atomistic simulations. The fact that $Lc$ is generally smaller for edge dislocations than screw ones is consistent with the former's narrower kinks. However, it remains unknown whether a dislocation's CRSS would converge to a near constant when it is longer than $Lc$⁠. The main goal of this Letter is to answer this question through systematic atomistic simulations.

3D simulation cells, illustrated in Fig. 1, are constructed with three chemical elements randomly assigned to BCC lattice sites. In Fig. S1 of the supplementary material, selected atomistic structures in NbTiZr are presented and the Warren–Cowley parameters are found to be small, confirming the very low degree of chemical short-range order (CSRO) in our models. A single dislocation is built into each cell by applying the corresponding isotropic elastic displacement fields to all atoms.^28,29 Edge and screw dislocations on {110} and {112} planes are studied, except the {110} screw dislocation because it may be unstable in either BCC pure metals¹⁸ or RMPEAs.^30,31 We remark that most prior atomistic simulations of long dislocation gliding in RMPEAs only considered slips on {110} planes, with a few exceptions that studied slips on {112}^27,32 and {123}³² planes. On the other hand, a TEM study in MoNbTi³² found that slips on high-order planes, such as {112}, {123}, and {134}, predominate. As will be shown later, the characteristic length of the edge dislocation is the same for {110} and {112} planes. Hence, we expect that the characteristic length would be similar for the higher-order planes, i.e., {123} and {134}.

Periodic boundary conditions are applied on the x and z directions, while the y boundaries are non-periodic. This configuration, called “periodic array of dislocations,” was found to provide a better estimate of the Peierls stress in Nb than other configurations.³³ Let L_x, L_y, and L_z be the edge lengths of the cell along the three directions, respectively. For all cells, $Lx≈40$ nm and $Ly≈50$ nm, following Ref. 34. The dislocation line length $L0=Lz$⁠. To study the effects of L₀, we choose seven L_z: 1, 5, 10, 20, 30, 40, and 50 nm. Accordingly, the smallest and largest cells, respectively, contain about 0.09 × 10⁶ and 6.5 × 10⁶ atoms.

The embedded-atom method potentials used in our recent work^30,32,37 are employed here to describe the interatomic interactions in the two alloys, two artificial pure metals, and one natural pure metal, Nb. For each dislocation on each slip plane in each alloy, five random atomic distributions of the three elements are studied and their mean and standard deviation in the critical stresses are reported. This number is sufficiently large, compared to the 3–5^38,39 and 3¹² random structures, respectively, used to study dislocation gliding in face-centered cubic (FCC) FeNi and BCC NbTiZr. We will not study the effects of the CSRO on dislocation behaviors, which were recently investigated in several BCC MPEAs.^40–43 For each pure metal, only one atomic structure is used for each dislocation. The two artificial pure metals—homogeneous MoNbTi and homogeneous NbTiZr—are simulated via A-atom potentials,³⁷ and the results are denoted by MoNbTi_A and NbTiZr_A, respectively.

Atomistic simulations are carried out using LAMMPS.⁴⁴ First, a dynamic relaxation step was conducted for 10 ps using an NPT ensemble at 5 K and zero strain with a time step size of 1 fs. Then, a shear strain tensor $Δϵ$ with one component being non-zero is applied to each cell and with all other eight strain components set to zero. For a screw and an edge dislocation, respectively, the non-zero strain components are $Δϵyz$ and $Δϵxy$⁠. In all cases, the strain rate is 10⁷ s⁻¹. The constant applied strain rate loading mode was shown to be a better choice in critical stress calculations than the constant applied stress loading mode.³⁴ On the {112} plane, the dislocation, either edge or screw, moves along the twinning direction.³³ Along the y axis, the simulation cell is divided into three types of regions: two boundary regions, two buffer regions, and a central region which contains the dislocation. During the deformation, atoms within the boundary regions are continuously displaced to create shearing. An NVT ensemble is adopted in the buffer regions to keep a constant temperature of 5 K while an NVE ensemble is applied to the remaining three regions.³⁴ The low temperature is selected because we want to focus on the line-length-dependence without the additional influence of temperature. To account for the thermal fluctuation, three different temperature damping parameters are used for each case in Nb. Simulations stop when the non-zero strain component reaches 0.035 and 0.05, respectively, for the edge and screw dislocations, which are sufficient for the dislocation to move through the entire simulation cell at least once. Effects of some model settings are analyzed in Sec. II of the supplementary material.

Figure 2(a) presents stress–strain curves measured based on glide of dislocations (⁠$L0=50$ nm) in NbTiZr. Figures 2(b)–2(d) show that, subject to zero strain, both edge and screw dislocations possess a wavy configuration as a result of the complex energy landscapes,⁴⁵ in contrast to the straight dislocations in pure metals.^32,46 On each stress–strain curve, multiple local maxima and minima exist. In pure metals, the first local stress maximum is a result of the entire dislocation line moving.^18,33 However, in NbTiZr, the first local peak stress, i.e., points B1, C1, and D1 in Fig. 2(a), correspond to the bow-out of a short segment from its original position [gray atoms in Figs. 2(b)–2(d)]. We refer to this first local peak stress as the “initial CRSS (ICRSS).” As the strain increases, the remaining segments of the dislocation move as well. However, in most cases, the dislocation arrests after gliding over a short distance, e.g., at points B2, C2, and D2 in Fig. 2(a). Higher stresses, e.g., the stresses at points B3, C3, and D3, are then required to drive the dislocation to move further. The morphology of the gliding dislocation is wavy and constantly changing due to the local chemical composition fluctuations, as shown in Figs. S2–S4 in the supplementary material and as confirmed by our recent phase-field dislocation dynamics simulations in MoNbTi.^47,48 Eventually, at a sufficiently high stress, the dislocation glides through the entire simulation cell. This critical stress, e.g., at points B4, C4, and D4 in Fig. 2(a), is the CRSS.

While Fig. 2 is for NbTiZr only, we find that stress–strain responses (Fig. S8 in the supplementary material) and dislocation behaviors in MoNbTi are similar. Also, in both RMPEAs, when $L0>5$ nm (a similar critical length was found in Fe),⁴⁹ the gliding screw dislocation leaves behind a large amount of vacancies and interstitials on the slip plane, while the gliding edge dislocation on either slip plane does not for any L₀. Since the screw dislocation does not cross slip on the {112} plane, the former phenomenon is a result of the collision of cross kinks and has been observed in prior atomistic simulations and TEM studies of gliding screw dislocation or dislocation loop expansion in Co_16.67Fe_36.67Ni_16.67Ti₃₀,^14,40,46 HfNbTiZr,²¹ NbTaV,²⁷ and NbTiZr,^12,40 but much less pronounced in MoNbTaW.^42,46 Note that the formation of debris in the wake of a gliding screw dislocation may be unique to BCC MPEAs, since it has not been observed in FCC MPEAs.^50,51

Values of all CRSS and ICRSS in both RMPEAs are presented in Figs. 3(a) and 3(b). We first compare some of them with those in the literature. For an edge dislocation on the {110} plane with $L0=50$ nm in NbTiZr, the CRSS is 250 MPa, which agrees well with 280 MPa measured by prior atomistic simulations¹² of the same dislocation with the same length at the same temperature (5 K). On the {112} plane in MoNbTi, when $L0=50$ nm, the calculated CRSS for an edge and a screw dislocation at 5 K are 498 and 1034 MPa, respectively, both of which are, as expected, higher than the CRSS measured by a room temperature TEM study, 455 MPa.³² Generally, for the same type of dislocations, the critical stresses in MoNbTi are about twice those in NbTiZr.

Regarding the dislocation line length L₀-dependence, both CRSS and ICRSS decrease sharply when L₀ increases from 1 to 5 nm. The decrease in the critical stress is associated with the change from no kink formed (when $L0=1$ nm) to a single kink-pair formed on the dislocation (when $L0=5$ nm). Then, when $L0=10$–20 nm, two or three narrowly spaced kink-pairs were formed, and both critical stresses decrease further. As L₀ increases further, however, both CRSS and ICRSS become length insensitive within the statistical dispersion, and the stress oscillation is reduced (Fig. S8 in the supplementary material). At these L₀, widely spaced kink-pairs were formed, as indicated by arrows in Figs. 2(b)–2(d). Taken together, our atomistic simulations determine that, in MoNbTi, the characteristic length $Lc=25$ nm for both edge and screw dislocations and in NbTiZr, $Lc=15$ and 25 nm, respectively, for edge and screw dislocations.

Recently, Maresca and Curtin proposed analytical models to estimate the characteristic length of the edge (superscript “E”)²³ and screw (superscript “S”)²⁷ dislocations in BCC MPEAs, i.e.,

where $ζc$ is the characteristic length for kink formation, b is the magnitude of the Burgers vector, μ is the isotropic shear modulus, $Ek$ is the kink formation energy, and $ΔẼp$ is the net dislocation/solute interaction energy over adjacent Peierls valleys. Values of the last four parameters in two RMPEAs are taken from Refs. 27 and 37 and are summarized in Table I. Values of $Ek$ and $ΔẼp$ for MoNbTi are taken from those for Mo_0.69Nb_0.31,²⁷ which has the same atomic size mismatch⁵² as MoNbTi. Substituting these parameters into Eqs. (1) and (2) yields $ζc$ and $Lc$⁠, whose values are presented in Table I. While analytically derived $Lc$ is for CRSS only, its values, 7.67–22.7 nm, agree well with that from present atomistic simulations for both CRSS and ICRSS, 15 or 25 nm.

The differences between CRSS and ICRSS, $Δσc$⁠, are summarized in Fig. 3(c). In the same RMPEA, the screw dislocation generally possesses a higher $Δσc$ than the edge one. When $L0=50$ nm, $Δσc=68$–75 MPa in MoNbTi and 34–105 MPa in NbTiZr, suggesting that CRSS is 7%–68% higher than ICRSS.

In the three pure metals, the first local peak stresses are taken as the CRSS, which are much less dependent on L₀, as shown in Fig. 3(d), suggesting that $Lc$ is near zero. Note that Eqs. (1) and (2) are only applicable to alloys where the dislocation/solute interactions are strong as they are in the two RMEPAs of interest, not pure metals. Once the CRSS is reached, the dislocation glides continuously. In most cases, the CRSS in Nb is lower than that in MoNbTi_A but higher than that in NbTiZr_A. In addition, compared with their alloy counterparts, artificial pure metals have much lower CRSS due to their lack of lattice distortion. For example, for a screw dislocation on the {112} plane with $L0=50$ nm, the CRSS in MoNbTi and MoNbTi_A is 1034 and 554 MPa, respectively. For an edge dislocation of the same length on the same slip plane, the two CRSS are 498 and 21 MPa, respectively.

Our recent work on local slip resistances (LSR) of short dislocations (⁠$L0≈1$ nm) found that differences in LSR among various glide planes and character angles were much lower in five RMPEAs than in their A-atom counterparts and six constituent BCC metals: Cr, Mo, Nb, Ta, V, and W.^30,31 Here, we explore whether this is also the case for the CRSS of long dislocations. To this purpose, two quantities are calculated to exhibit the slip resistance anisotropy. The first one is the ratio of the CRSS on the {112} plane to that on the {110} plane, both for the edge dislocation. The second quantity is the ratio of the CRSS of a screw dislocation to that of an edge dislocation, both on the {112} plane. Figure 4 shows that both ratios are lower in the two RMPEAs than in the three pure metals and that these ratios are largely insensitive to L₀. In particular, when $L0=50$ nm, the second ratios are 2.08 and 2.42, respectively, in MoNbTi and NbTiZr. Prior atomistic simulations of long dislocation gliding calculated the screw-to-edge ratio in CRSS on {110} planes, finding it to be 2 in both MoNb₃²³ and NbTiZr.¹² 

In this Letter, we conduct atomistic simulations to calculate the CRSS and ICRSS for dislocation glide in two RMPEAs: MoNbTi and NbTiZr. Edge dislocations on both {110} and {112} slip planes, as well as screw dislocations on the {112} plane, are considered. As references, the CRSS of the same dislocations in homogeneous MoNbTi, homogeneous NbTiZr, and Nb are also calculated. In each case, seven dislocation line lengths, varying from 1 to 50 nm, are chosen. Main findings are summarized as follows:

1. In RMPEAs, small dislocation segments bow out when the ICRSS is reached. As the strain increases further, the entire dislocation line moves, but may be later arrested, until a higher stress, i.e., CRSS, is applied to drive the dislocation to move continuously in the lattice.

2. In RMPEAs, the gliding screw dislocation leaves behind a large number of vacancies and interstitials, as a result of the collision of cross kinks. This phenomenon does not occur in the wake of a gliding edge dislocation.

3. In RMPEAs, the dependence of CRSS and ICRSS on length becomes undetectable within the statistical dispersion once the dislocation line is longer than 25 nm, as a result of the change in dislocation behavior.

4. The CRSS anisotropy, quantifying the differences in CRSS among slip modes, is greatly reduced in RMPEAs than pure metals, providing direct evidence of the homogeneous plasticity in RMPEAs observed in prior TEM experiments.³² 

Despite the large differences in the critical stresses between the two RMPEAs, their stress–strain responses, CRSS anisotropy, characteristic dislocation lengths, and dislocation behaviors are similar. These findings, thus, could likely apply to other three-element RMPEAs. Future work may consider line-length-dependent dislocation mobility, which has been studied in an FCC concentrated alloy⁵³ but not in any RMPEAs.

See the supplementary material for information as referred to in the main text.

The authors thank Professor Fulin Wang, Professor Liming Xiong, Professor Suzhi Li, Dr. Satish Rao, and Dr. Edwin Antillon for helpful discussions. S. X. and I. J. B. gratefully acknowledge support from the Office of Naval Research under contract ONR BRC Grant N0. N00014-21-1-2536. W.-R. J. would like to acknowledge funding from the Office of Naval Research under Grant No. N000141712810. Use was made of computational facilities purchased with funds from the National Science Foundation (No. CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; No. NSF DMR 1720256) at UC Santa Barbara.