Toward a 3D coupled atomistic and discrete dislocation dynamics simulation: dislocation core structures and Peierls stresses with several character angles in FCC aluminum
 Jaehyun Cho^{1},
 Till Junge^{1},
 JeanFrançois Molinari^{1} and
 Guillaume Anciaux^{1}Email author
DOI: 10.1186/s4032301500286
© Cho et al. 2015
Received: 29 December 2014
Accepted: 5 May 2015
Published: 18 June 2015
Abstract
Background
We present a robust method to obtain the displacement field of a dislocation core, which is one of the building blocks for the development of a direct multiscale method coupling an atomistic domain to a discrete dislocation dynamics engine in 3D (e.g. CADD3D).
Methods
The core structure is achieved by modeling of a straight dislocation with an arbitrary mixed angle using atomistic simulation. In order to validate the obtained atomistic core structures, a variational PeierlsNabarro method is extended to include arbitrary characters.
Results
Both methods show comparable dislocation core structures for all studied angles. We provide also the Peierls stress for a wide range of character angles.
Conclusions
The obtained displacement fields for the dislocation cores were fully validated. These can consequently be employed to construct the described CADD3D coupling scheme.
Keywords
CADD3D Mixed dislocation Character angle Shockley partial Variational PeierlsNabarro method Peierls stressBackground
The understanding of the collective motion of line defects called dislocations is important in order to predict plastic deformations of crystalline materials. This can be studied using several computational tools, including discrete dislocation dynamics (DDD) [1–4]. In DDD, a dislocation is simply represented as a set of nodes connected by straight segments [5]. Due to this simple representation, this method can handle much larger simulation sizes (order of microns) than atomistic simulations (MD), and yields results on sample sizes which are comparable to insitu TEM experiments [6]. DDD requires several parameters, which can be obtained from atomistic models, such as a dislocation core energy and nodal mobility law [7]. With these parameters, dislocation dynamics are reasonably well approximated. However, DDD has important limitations which require adhoc treatments. For instance, dislocation nucleation is only possible by inserting nucleation seeds such as frankread sources [8–10]. On the other hand, MD describes naturally dislocation nucleation as it represents dislocations explicitly as crystalline line defects, although the much larger computational cost is an important drawback. Therefore, it is attractive to combine the advantages of both approaches within a multiscale framework.
Currently, several multiscale methods exist including quasicontinuum [11], bridging domain [12] and finite element (FE) combined with atomistic modeling [13]. All these methods couple energies or displacements between two different domains: MD and FE. The authors are developing a 3D directcoupling method between MD and DDD, which is named CADD3D^{1} and is an extension of the CADD2D [14] approach. In this approach, MD is used where dislocation nucleation are expected (and where DDD would require an adhoc treatment), whereas in the remaining zones standard DDD is employed. The key point of the CADD3D method is to deal with dislocations passing seamlessly between these two different domains. One notable difficulty is that a dislocation line can cross the interface between the coupled models. Then, the distinct representation of a dislocation, being either an atomistic structure or a set of DDD segments should match perfectly at the coupling interface. This can be achieved effectively by imposing respective boundary conditions to both MD and DDD regions. In this paper, we focus on the MD boundary conditions imposed by DDD: an adequate position and displacement field matching the MD core structures has to be imposed to all boundary atoms within the influence region of dislocations. The details of this approach, commonly called the core template technique will be described in Section “Coupled atomistic and discrete dislocations in 3D: core template”. Because the dislocation structure is varying with its character angle^{2}, the core template of an arbitrary mixed angle has to be built.
We explain in Section “Coupled atomistic and discrete dislocations in 3D: core template” the importance of the core template boundary condition for CADD3D to successfully build the multiscale framework. In order to obtain this core template, we provide comprehensive modeling techniques to create a dislocation with arbitrary character angles (Section “Method 1: Atomistic dislocation modeling”). We also provide an extended variational PeierlsNabarro (PN) method [15] to validate the obtained MD core structures (Section “Method 2: Variational PeierlsNabarro method”). Interestingly, the extended PN method can be used to predict core structures of mixed dislocations which cannot be obtained with MD simulations due to limitation of computational cost. With the successful extension of PN method, we show favorable core structures when compared to MD, and analyze the details of dislocation core structures of all the studied character angles (Section “Results 1: Comparison between the MD and PN models core structures”). We also measure Peierls stresses for various mixed dislocations (Section “Results 2: Peierls stresses”), and show that Peierls stresses are largely influenced by the atomistic structures of the dislocation core. Finally, the paper concludes with a discussion on future work towards a fully working implementation of CADD3D.
Coupled atomistic and discrete dislocations in 3D: core template
The coupled atomistic and discrete dislocation (CADD) method [14] is a partitioneddomain, concurrent multiscale coupling method that couples a molecular dynamics or statics subdomain to a discrete dislocation dynamics subdomain and allows dislocations to transition between them. This possibility of passing dislocations between the subdomains makes CADD the only coupling method that, in principle, can handle the development of large dislocation networks at scalable computational cost.^{3}
Even though only planestrain twodimensional implementations of CADD exist to date, the method has been successfully used for the study of such problems as dislocation emission from crack tips, void growth or nanoindentation in quasistatic (zero temperature) problems [14], fully dynamic problems at finite temperature [16] and intermediate problems where finite temperature atomistic is coupled to quasistatic discrete dislocations [17].
Far from any dislocation core, nonlinear effects become negligible, and the smallstrain linear elasticity solution \(\tilde {\vec {u}}\) converges with the atomic solution \(\vec {u}_{\mathrm {A}}\).

Dislocation spacing: Implicit in the template definition (1) is the assumption that dislocations lines in the pad are spaced far from one another, i.e. no two dislocation cores overlap. If this assumption is violated and two dislocation lines approach one another, there will be spurious forces acting on them in the pad.

Accuracy and simplicity: The template is only useful if it can be precomputed and looked up during the simulation. This means that the displacement field for dislocation cores with arbitrary character angle and curvature needs to be accurately interpolatable from a finite number of precomputed dislocation cores.
In this paper, we focus on the details of dislocation core structures with respect to the character angle in order to build valid core templates. Consequently, this study starts with a rigorous atomistic modeling of arbitrary mixed dislocations as explained in the following section.
Method 1: Atomistic dislocation modeling
Selected eight angles, lattice coordinates, number of replicas and atoms of each simulation box
θ (°)  x  y  z  Replicas (x,y,z)  Atoms 

90 (edge)  \([11\bar {2}]\)  [111]  \([1\bar {1}0]\)  (6, 10, 60)  86040 
109.107  \([14\bar {5}]\)  [111]  \([3\bar {2}\bar {1}]\)  (2, 10, 20)  66900 
120  \([01\bar {1}]\)  [111]  \([2\bar {1}\bar {1}]\)  (10,10, 30)  71700 
130.893  \([\bar {1}5\bar {4}]\)  [111]  \([3\bar {1}\bar {2}]\)  (2,10, 20)  66960 
139.107  \([\bar {1}3\bar {2}]\)  [111]  \([5\bar {1}\bar {4}]\)  (4,10, 12)  80400 
150  \([\bar {1}2\bar {1}]\)  [111]  \([10\bar {1}]\)  (6, 10, 60)  86220 
160.893  \([\bar {2}3\bar {1}]\)  [111]  \([41\bar {5}]\)  (4, 10, 12)  80520 
180 (screw)  \([\bar {1}10]\)  [111]  \([11\bar {2}]\)  (10,10, 30)  72000 
First, the step created by the edge displacement fields (u _{ y } and u _{ z }) can be avoided by removing the extra plane (a slab of length \(\vec {b}\sin \theta \)). Second, as seen in Fig. 5(b), the opposite boundaries in direction z mismatch each other by half of the screw Burgers vector \(\vec {b}/2\cos \theta \). The dashed black lines are the boundaries of the bottom surface. This mismatch created by the screw displacement field (u _{ z }) is fixed by tilting the simulation box with respect to the y axis in the x direction by half of the corresponding screw component \(\vec {b}/2\cos \theta \). Consequently, the PBC in the z direction is satisfied without any change of the original elastic displacement field. Finally, the surfaces in the y direction remain free boundaries.
In order to find the equilibrium state of dislocation core structures, the simulation box has to be relaxed. We choose the latest Aluminum EAM potential (Mendelev et al. [20]) to evaluate the interatomic forces. The comparison of this potential with other EAM ones will be described in Section “Method 2: Variational PeierlsNabarro method”. We relax by using a quenching process which is stopped when a norm of forces of all the atoms is below 10^{−10} e V/Å.
Method 2: Variational PeierlsNabarro method
Continuum theory of linear elasticity provides useful analytic solutions for dislocation problems because the theory offers quite accurate dislocation structures when far from the core. However, this theory cannot be employed anymore near the dislocation center, where one finds highly distorted atomistic structures. These nonlinear atomic interactions can be naturally captured by an atomistic model. For these reasons, simulations combining the continuum and atomistic models can be useful not only for the far field but also for the dislocation center. One of the available tools combining the continuum and atomistic models is the variational PeierlsNabarro (PN) method [21, 22], which generally has been used to calculate dislocation core structures by minimizing the total energy of the system. More precisely, the energy in the far field is captured by the energy formulation of continuum linear elasticity theory, while the energy in the core region is obtained by using the crystalline misfit energy and a spread displacement field on the slip plane. The total energy is the sum of both these energy formulations, and the optimal dislocation core structures are found when the total energy is minimum. The PN model is known as a tool providing good descriptions of dislocation core structures in comparison with MD results [5, 15, 23–25].
where K _{edge}=μ/(4π(1−ν)), K _{screw}=μ/4π.
For a dislocation in FCC materials, the generalized stacking fault (GSF) energy curve in \(x=\left <11\bar {2}\right >\) directions is important to understand dislocation slip behaviors. The GSF energy curve was first suggested by Vitek [30] to be composed of an intrinsic stacking fault energy γ _{ I } (ISF) and an unstable stacking fault energy γ _{ U } (USF) [24, 31]. The ISF energy is the local energy minimum of partial dislocations, and the USF energy is the minimum energy required in order to translate partials. The obtained GSF energy curve in \(x=\left <11\bar {2}\right >\) is given in Fig. 7(b) with γ _{ U }=243.3m J/m ^{2} and γ _{ I }=128.6m J/m ^{2}. When compared DFT computations and Mishin & Farkas, our selected Mendelev et al. potential provides an accurate description of γ _{ U } and γ _{ I }.
After constructing the total energy formulation, one can see that the only difference between the arbitrary character angles is the amount of elastic energy E _{el} given by the terms K _{edge} and K _{screw} (see Equation 9), which differ by 1/(1−ν): the dislocations close to edge have larger elastic energies than the dislocations around screw. The total energy E _{tot} is minimized numerically to get the optimal shape of the displacement fields \(\vec {u}\) over the slip plane.
Results 1: Comparison between the MD and PN models core structures
We see that the extended PN method produces dislocation core structures, at much cheaper computational costs, that compare reasonably well with the atomistic core structures. Therefore, we can employ the PN method to predict core structures of mixed dislocations which cannot be obtained by atomistic simulations when the sizes of unit cells become large enough to exclude an atomistic calculation.
Results 2: Peierls stresses
The Peierls stress is defined as the minimum stress required to translate a dislocation at zero temperature [18], and it is known to be influenced by core structures [35–39]. Because we have obtained the details of the dislocation core structures e.g., lattice orientation and stacking fault widths, the influences of the core structures on the Peierls stress can be studied in this section. Several articles [35, 37, 38] have been published regarding the Peierls stresses of FCC Aluminum, measured using direct atomistic simulations. A wide range of results (from 1MPa to 13MPa for an edge and from 1MPa to 82MPa for a screw) is observed. Shin and Carter [38] found that a dislocation dissociated into partials has a smaller Peierls stress than a compact dislocation. Olmsted et al. [37] studied Peierls stresses for mixed dislocations (120° and 150°) including the edge and screw character angles. They found that the screw and 120° dislocations are required to overcome a higher Peierls energy barrier than the edge and 150° dislocations. They argued that such results are due to the density of atoms in the motion direction (Simmons et al. [39] initially suggested this idea). To the best of our knowledge, Peierls stresses have been studied only for the four character angles (90°, 120°, 150° and 180°), whereas we selected other new character angles.
with L _{ Z } the length of the simulation box in direction z. On the top surface, the negative Burgers (−b _{edge} and −b _{screw}) vectors are used, and the positive Burgers (+b _{edge} and +b _{screw}) are employed for the bottom surface. In order to consider the mixed character angles, the linear displacements are rotated by the given character angle θ (Equation 6). The remaining atoms are subjected to the Volterra fields, and are also rotated by θ as we described in Section “Method 1: Atomistic dislocation modeling”. After imposing the various displacement fields on the atoms in each region, we relax the simulation box, and measure the Peierls stress through quasistatic loading. Specifically, we translate the atoms on the top surface along the Burgers direction in small steps, while we fix the atoms on the bottom surface. After each translation, we minimize the bulk of atoms and monitor the final energy value when the minimization finishes. Therefore, we can observe the energy variation of the system as the dislocation moves to different positions along the gliding direction.
We can first compare Peierls stresses of the dislocations constructed with an equivalent lattice orientation (but with a different character angle). As shown in Table 1, the edge and 150°, 109.109° and 130.893°, 139.109° and 160.893°, 120° and screw dislocations are modeled based on the same lattice orientation. For each lattice orientation, the Peierls stress decreases low as the character angle decreases. These observations were confirmed by Simmons et al. [39], and it can be understood with the magnitudes of dislocation dissociation [38]. As seen in the variation of stacking fault widths in Fig. 11, the dislocations are dissociated into larger stacking fault widths when their character angles become small. Consequently, the Peierls stress decreases as the character angle reduces. In the case of 139.109° and 160.893° dislocations, the above argument is not valid, and we have not found any reasons for the discrepancy.
Second, we confirm qualitatively the Peierls stresses of the dislocations between the different lattice orientations as follows. The two dislocations (120° and 180°) based on the lattice orientation \(x=<\bar {1}10>\), y=<111> and \(z=<11\bar {2}>\) have higher Peierls stresses than the other dislocations. It can be understood with the density of atoms in the gliding direction. We see in Fig. 13 that the densities vary with the lattice orientations of the slip plane. Therefore, the dislocations built within the same family of lattice orientation having small density have high Peierls stresses. Consequently, the 109.107° and 130.893° dislocations have small Peierls stresses compared to the other dislocations.
Conclusion
In this article, we presented one part of the coupled atomistic discrete dislocation formulation for the three dimensional case. This important part, the socalled core template, deals with dislocations traversing the coupling interface in order to impose a matching boundary condition to the MD. This template is defined by the core atomistic positions which depend on the character angle. These positions can be obtained by modeling a straight mixed dislocation via an atomistic simulation. The detailed setup of such simulations has been extensively described in this paper. Furthermore, based on the extensive modeling results, Peierls stresses have been measured.
As expected, the core size of the modeled straight dislocations varied with the character angle. More specifically, the fault widths between the two partials are decreasing when the character angles increase. This can be understood by considering the variation of the elastic energy. Also, the MD core structures were confirmed by using the variational PN model, which was extended to consider arbitrary angles. The comparison of the core structures obtained from both methods showed good agreements. The extended PN method can predict displacement fields for any angle, including these that are difficult to obtain with periodic and finite size MD simulations. Furthermore, the measured Peierls stresses were qualitatively confirmed, and we found that the magnitude of dissociation and the density of atoms in the gliding direction decide the Peierls stress of the dislocation.
As we already mentioned, these results are directly applicable to the CADD3D implementation to provide matching boundary conditions. However, another important component of CADD3D deals with the dynamic motion of dislocations. Moving dislocations  partially MD and partially DDD  need to have comparable speeds in order to avoid nonphysical spurious forces at the interface. The presented core structures can straightforwardly be employed as a starting point for simulating MD dislocation motion under shear stress and thus to carry out mobility laws for arbitrary mixed angle.
Endnotes
^{1} Swiss national science foundation project conducted by three principal investigators, William Curtin, JeanFrançois Molinari and Guillaume Anciaux.
^{2} Due to planestrain restrictions, twodimensional CADD can only handle edge dislocations.
^{3} Another method, AtoDis [41] has similar properties, but it has not been shown to be operational in the presence of more than a single dislocation line.
^{4} The stacking falut sequence of the dislocation core region (A →B →A) allows us to see the second layer of A atoms. These atoms are not seen in the stacking sequence of perfect FCC crystal systems (A →B →C →A).
Declarations
Acknowledgments
This work is supported by the Swiss National Science Foundation (grant no. 200021140506/1).
Authors’ Affiliations
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