Calculation of the Peierls barrier of screw dislocations in bcc metals and its dependence on stress


Source of funding: Czech Science Foundation
Number: P204/10/0255
Duration of funding: 3 years (01/2010-12/2012)

Main results

Calculation of the Peierls barriers of screw dislocations in BCC metals

We demonstrate that the straightforward application of the Nudged Elastic Band (NEB) method does not determine the correct Peierls barrier of 1/2<111> screw dislocations in BCC metals. Although this method guarantees that the states (images) of the system are distributed uniformly along the minimum energy path, it does not imply that the dislocation positions are distributed uniformly along this path. In fact, clustering of dislocation positions near potential minima occurs which leads to an overestimate of both the slope of the Peierls barrier and the Peierls stress. We propose a modification in which the NEB method is applied only to a small number of degrees of freedom that determine the position of the dislocation, while all other coordinates of atoms are relaxed by molecular statics as in any atomistic study. This modified NEB method with relaxations gives the Peierls barrier that increases smoothly with the dislocation position and the corresponding Peierls stress agrees well with that evaluated by the direct application of stress in the atomistic modeling of the dislocation glide.

Fig. 1: The initial image of a block with a 1/2[111] screw dislocation used for the calculation of the Peierls barrier using the NEB method. The atoms are depicted as circles with the colors that distinguish the three successive (111) atomic planes.
Fig. 2: Variation of the energy of a straight dislocation along the minimum energy path between the fixed states at ξ/a0 = 0 and 1, where a0 is the distance of the two neighboring minimum energy lattice sites in the (101) plane in the [121] direction. The black data (curve) has been obtained by the standard NEB method that operates in the configurational space of the dimension 3N . The red data (curve) have been obtained using the NEB+r method in which only the z coordinates of the five atoms marked 1-5 in Fig. 1 were adjusted by the NEB force (1), while the remaining DOFs were obtained by atomic relaxations using the standard molecular statics. In both cases, the barrier is shown to be independent of the number of images used to discretize the path.
We define the transition coordinate ξ as the distance of the dislocation from its position in the image I=0 to M+1, measured along the path of the dislocation. Unfortunately, the position of the dislocation cannot be determined unambiguously from the coordinates of all atoms in the block and thus also the path of the intersection of the dislocation line with the (111) plane is not known unequivocally. We thus approximate the dislocation path as a straigh line connecting the two neighboring minimum energy lattice sites, and a the lattice parameter. This assumption of a straight path of the dislocation was adopted also by Rodney and Proville (2009), and Ventelon and Willaime (2007). The variation of the energy of the dislocation, Eq. 2, along this transition coordinate is plotted in Fig. 2 in black. The same barrier has been obtained when the path of the dislocation was discretized using M = 5, 15 and 32 images. The barrier increases slowly at first but then rises to its maximum rapidly. Clearly, this abrupt change of the slope calls for a deeper understanding.

Studies of anisotropic Eshelby twist in thin films of BCC metals

We address the question as to whether the core structure of screw dislocations in Mo in the bulk can be obtained from high-resolution electron microscopy (HREM) images of such dislocations viewed end-on in a thin foil. Atomistic simulations of the core structure of screw dislocations in elastically anisotropic Mo were carried out using bond order potentials. These simulations take account automatically of the effects of the surface relaxation displacements (anisotropic Eshelby twist). They show that the differential displacements of the atoms at the surface are different with components perpendicular to the Burgers vector about five times larger than those in the middle of the foil, the latter being characteristic of the bulk. Nye tensor plots show that the surface relaxation stresses strongly affect the incompatible distortions. HREM simulations of the computed structure reflect the displacements at the exit surface, modified by interband scattering and the microscope transfer function. Nye tensor plots obtained from the HREM images show that interband scattering also affects the incompatible distortions.

Since the HREM image reflects the displacements at the exit surface, the fundamental questions are:
  1. Whether the HREM observations relate to the core displacements in the bulk.
  2. Whether the stresses associated with the Eshelby twist produce such significant changes in the dislocation core structure that the relation to the core structure in the bulk becomes obscured.
The objectives of this work have been to elucidate these points by carrying out an atomistic simulation of the 1/2[111] screw dislocation in a thin foil of Mo of thickness typical for HREM observations, and to simulate the HREM images of the calculated structure. The atomic displacements of the relaxed structure automatically contain the surfaceinduced relaxation displacements, and the effects of the surface relaxation stresses on the core structure. These displacements can then be identified as the difference between the relaxed positions of atoms in the foil and their relaxed positions in a block that is periodic in the direction parallel to the dislocation line, i.e. without free surfaces. Finally, the calculated relaxed atomic positions in the foil can be used as input for the image analysis. This provides the positions of image peaks at the exit surface, which reflect the Eshelby twist and core displacements, plus the displacements due to the interband scattering and the microscope transfer function.

(a) (b) (c)
Fig. 1: Map of edge differential displacements for the 1/2[111] screw dislocation calculated for three slices taken from the block representing the foil at (a) z=+h/2 (upper surface), (b) z=0 (centre of the foil), and (c) z=-h/2 (lower surface). The displacements in (b) are scaled by a factor of 10 whereas those in (a) and (c) by a factor of 2.

Fig. 1 shows the maps of the differential displacements perpendicular to the dislocation line (and to the Burgers vector) for three slices taken from the upper and lower surfaces and from the center of the relaxed block that represents the foil. The atomic arrangement is shown in the projection perpendicular to the direction of the dislocation line. The circles stand for the atoms in the three consecutive (111) atomic planes (differentiated by shading) and the lengths of the arrows are proportional to the magnitudes of the relative displacements of neighboring atoms in the direction perpendicular to the dislocation line, i.e. corresponding to the relative displacements in the z direction and in the (x,y) plane, respectively. The colors of atoms are chosen such that they get darker with increasing position along the z direction in the ideal lattice. The edge displacements near the surfaces are about five times larger than those in the centre of the foil and, therefore, the displacements in Fig. 1b are scaled by a factor of 10, whereas those in Figs. 1a and 1c only by a factor of 2.
Fig. 2: Plots of Nye tensor components for three slices taken from the upper (z=+h/2) and lower surfaces (z=-h/2) of the foil and from the middle of the foil (z=0). The colors correspond to the density of the Burgers vectors of continuously distributed infinitesimal dislocations. The component σ33 corresponds to the screw component; σ31 and σ32 to the two edge components parallel to the x and y axes, respectively. The circles represent the relaxed positions of atoms in the three consecutive (111) planes.
Any compatible distortion of the lattice for which the displacement field remains single-valued and the strain field integrable does not affect the Nye tensor. Consequently, if the Eshelby twist did not affect the dislocation core, the Nye tensor plots calculated for the two surface slices would have to be identical to that taken from the middle of the foil. In Fig. 2 we show the three relevant components of the Nye tensor for the upper and lower surfaces and the middle of the foil. Since the dislocation line is parallel to the z direction, α33 corresponds to the screw component of the dislocation density. Similarly, α31 and α32 relate to the two edge components, for the Burgers vectors parallel to x and y, respectively. We represent the dislocation core by a region around the dislocation in which the displacements of atoms relative to their positions in the ideal lattice, and thus also αij, are larger than some threshold. Since the Eshelby twist strongly affects the edge component of this field (as seen in Fig. 2), i.e. the relative displacements of atoms perpendicular to the slip direction, the dislocation core close to the two free surfaces is very different from that in the centre of the foil. We thus conclude that the Eshelby twist strongly affects the dislocation core near the foil surfaces but not in the bulk of the foil. The Eshelby twist thus induces both compatible deformation away from the dislocation core and the incompatible deformation in the core region.

Dissemination of results

Impact papers

  • Gröger R., Vitek V.: Stress dependence of the Peierls barrier of 1/2<111> screw dislocations in BCC metals. Acta Mater. 61:6362-6371 (2013).

  • Srivastava K., Gröger R., Weygand D., Gumbsch P.: Dislocation motion in tungsten: Atomistic input to discrete dislocation simulations. Int. J. Plast. 47:126-142 (2013).

  • Gröger R., Vitek V.: Constrained Nudged Elastic Band calculation of the Peierls barrier with atomic relaxations. Model. Simul. Mater. Sci. Eng. 20:035019 (2012).

  • Gröger R., Dudeck K.J., Nellist P.D., Vitek V., Hirsch P.B., Cockayne, D.J.H.: Effect of Eshelby twist on core structure of screw dislocations in molybdenum: Atomic structure and electron microscope image simulations. Philos. Mag. 91:2364 (2011).

Papers in conference proceedings

  • A new approach to calculate the Peierls barrier of screw dislocations in BCC metals. Int. Symp. on Plasticity and Its Current Applications, San Juan, Puerto Rico (January 3-8, 2012).

Invited talks

  • A new approach to calculate the Peierls barrier of screw dislocations in BCC metals. Int. Symp. on Plasticity and Its Current Applications, San Juan, Puerto Rico (January 3-8, 2012).

Contributed talks

  • Atomistic calculation of the Eshelby twist in BCC metals. University of Oxford, Oxford, UK (April 29-30, 2010).

Conferences and workshops

  • Member of the International Advisory Committee, International Conference on Plasticity (2012).
  • Co-organizer of a minisymposium "Plasticity of BCC metals" at the Int. Symp. on Plasticity and Its Current Applications, San Juan, Puerto Rico (January 3-8, 2012).

Prizes and awards

  • Otto Wichterle prize "for outstanding young researchers of the Academy of Sciences of the Czech Republic", Academy of Sciences of the Czech Republic (2010)