Anandh Subramaniam & Kantesh Balani MATERIALS SCIENCE & ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide

Core of the dislocation and the Peierls Stress Near the dislocation line the stress fields and associated strains are so large that linear elasticity theory breaks down → this region is known as the core of the dislocation. The minimum shear stress required to move a dislocation is the Peierls stress  can be visualized as a lattice friction. For all practical purposes it is equivalent to the Critical Resolved Shear Stress (CRSS) (except for the T aspect). The lattice friction stress* or the Peierls Stress is a sensitive function of the structure of the core. The structure of the core is determined by the bonding in the crystal and the crystal structure. When the core is planar (lies on the slip plane) the Peierls stress can be described by an exponential function. When the core is non-planar then atomistic calculations are required to calculate the Peierls stress (e.g. screw dislocations in BCC materials). The original model is due to Peierls and Nabarro (PN) wherein they derived the lattice friction stress as an exponential function of the ‘width of the dislocation’ and the Burgers vector (as below). ◘ In their model the width (w) is the ‘effective extent; of the dislocation. ◘ ‘Wider dislocations’ have a lower PN stress ◘ Different slip systems have different values of Peierls stress Understanding the origin of the Peierls stress: ◘ The dislocation is in a local metastable equilibrium → sits in a Peierls valley ◘ Stress has to be applied to ‘pull’ the dislocation out of the valley (→ into the next valley) * Stress required to move a dislocation on its slip plane

What is the connection between Peierls Stress and Critical Resolved Shear Stress? There are two very similar quantities which we have seen:  Peierls stress (or PN stress or Lattice friction stress)  Critical Resolved Shear Stress (CRSS) Both of them are stress to cause plasticity at the microscopic level. How are these quantities related? (Answer in diagram below) Peierls stress may corrected for an increase in temperature and hence the concept may be extended to finite temperatures. Hence, often these two terms are interchangeably used  this is in some sense justified as they are a measure of the same physical effect  inherent lattice resistance to the motion of dislocations. Peierls stress At zero K, theoretically/computationally derived Stress to cause microscopic plasticity CRSS At finite temperatures, experimentally determined

Stress to move a dislocation: Peierls – Nabarro (PN) stress We consider the original Peierls-Nabarro model (though this has been superseded by better models and computations). Width of the dislocation is considered as a basis for the ease of motion of a dislocation. Two extreme ‘widths’ are shown below for illustration. Extreme situations Unrealistic Unrelaxed condition- stiff Smaller width of displacement fields  atomic adjustments required (for any one atom) for dislocation motion are large ‘Relaxed’ condition Large width of displacement fields  atomic adjustments required for dislocation motion are small

Peierls – Nabarro stress (PN) → P-N stress → Lattice Friction The PN stress required to move a dislocation depends exponentially on the width of the dislocation (w) & the modulus of the Burgers vector (b). Being an exponential function of both ‘w’ and ‘b’; PN stress is a sensitive function of these factors:  ‘w’ is determined by the bonding characteristics (metallic, ionic…)  ‘b’ is determined by the crystal structure (superlattices have a large ‘b’; ordered structures have a larger ‘b’ as compared to their disordered counterparts). G → shear modulus of the crystal w → width of the dislocation !!! b → |b| Effect of w on PN w b 5b 10b PN G G / 400 G / 1014 G / 1027 Hence, ► narrow dislocations are more difficult to move than wide ones ► dislocations with larger b are more difficult to move Peierls – Nabarro stress (PN) → P-N stress → Lattice Friction

Though the Peierls original formula has been superseded by more sophisticated theoretical models and computational calculations; it worthwhile noting that if the core of the dislocation is planar then the Peierls stress can be described by an exponential function similar to the one originally conceived by Peierls. Additionally, a better feel can obtained for the PN stress by connecting the width of the dislocation to the bonding characteristics of the material. Core splitting in BCC crystal is well studied by atomistic computational methods.

Dependence of width of the dislocation on bonding of crystals Nature of chemical bonding in the crystal determines the → extent of relaxation & the width of the dislocation Covalent crystals Strong and directional bonds → small relaxation (low w) → high PN Usually fail by brittle fracture before PN is reached Metallic crystals Weaker and non-directional bonds → large relaxation (high w) → low PN E.g. Cu can be cold worked to large strains Transition metals (e.g. Fe) have some covalent character due to ‘d’ orbital bonding → harder than Cu Ionic crystals Moderate and non- directional bonds Surface cracks usually lead to brittle fracture Large b (NaCl: b = 3.95Å) → more difficult to move Intermetallic compounds / complex crystal structures Intermetallic compounds and complex crystal structures (Fe3C, CuAl2) do not have good slip systems → favorable planes & directions → usually brittle Ordered compounds may have very large b In CuZn (an ordered compound) dislocations move in pairs to preserve the order during slip Quasicrystals have 4, 5 or 6 dimensional b and the 3D component is not sufficient to cause slip in the usual sense