Crystallography, Defects

Dislocations – Why even care?

To start off on an exploration of dislocations, it’s important that we first understand why is it that we need to know about dislocations.

It all started with a rather simple question : What is the minimum shear force that’s required to start plastic deformation1 of a single crystal2?

Ideal Shear Strength

Figure 1: The shearing of a material.

When a material is subjected to a shearing force, it can deform like in Figure 1. It’s important to quantify the ability of the material to withstand shearing force and this quantity is the shear strength of the material. The shear strength is defined as the The applied shear stress needed to plastically deform a single crystal. There were attempts made by scientists to estimate the shear strength of materials theoretically. This value (obtained theoretically) is the ideal shear strength.

Figure 2: The sinusoidal function that can be approximated to represent stress reaches maximum at x = \frac{a}{2} when each row is assumed to be infinite,  due to symmetry.

Obtaining this theoretical value is simple enough. Suppose there is an atom isolated in space. When another atom is brought into its vicinity, the new atom is initially attracted by the first atom (gravitation), however, when the new atom comes very close the first atom, it gets repelled (electrostatic interactions between both the atoms) and there is a spot in between where the atoms are ’at equilibrium’ (the same thing also known as a ’chemical bond’). If there are a lot of atoms lined up as a bunch of rows (assume, for now, infinitely), they arrange themselves such that they are all at equilibrium. If one of these lines of atoms are perturbed, they are initially repelled, but soon, they sit in the next available equilibrium spot (previously occupied by their neighbour). If we look at the energy of the atoms, it’ll first increase and then decrease. This can be approximated to a sinusoidal function like in Figure 2.

Figure 3: The notation used in the problem.

We can now model the ideal shear strength using this input. The notation for the problem is defined in Figure 3. The assumption made is that: All the bonds break simultaneously. This can be visualised like in Figure 3. Thus, the energy (and therefore stress) of an atom can be represented by S = S' sin(\frac{2 \pi x} {a}). For small values of $sin(\theta)$, $sin(\theta) \approx \theta$. Therefore, S = S' ( \frac{2 \pi x} {a}). We know from Hooke’s Law that for small strains, the stress is directly proportional to the strain and the constant of proportionality is the Shear Modulus (G) (if the stress and strain are shear stress and shear strain respectively). The strain is \frac{x}{d}. Thus, stress is given as S = G(\frac{x}{d}). Equating both stresses, we get S' = \frac{G}{2 \pi}, if a \approx d. This is, by definition, the ideal shear strength because the material “deforms” beyond the maximum of the stress sinusoidal curve.

Figure 4: (a) The initial position.
(b) The top half-planes “moving”.
(c) A new configration showing plastic deformation.

Though the derivation doesn’t have flaws, the resultant shear strength differs by orders of magnitude compared to the observed experimental shear stress. This difference says that the assumption that all bonds break at once is incorrect. Thus the search was on to find a better explanation for plastic deformation.

A Novel Plan – Introducing Missing Half-planes

In 1934, three scientists – Orowan, Polanyi and Taylor, independently came up with a new solution. They envisioned missing half-planes of atoms. Imagine that in the line up of rows of atoms, there is a missing half plane of atoms or an additional half plane of atoms. This is depicted in Figure 5. The assumption that all the bonds must break simultaneously is not necessary once half planes are introduced.

In fact, these half-planes can ‘move’ and the end result is the same deformation of the material, as can be seen in Figure 6. The reason for the ideal shear strength to be much larger than experimental values was the assumption that all the bonds break at once, and breaking bonds is not easily achieved. By introducing half planes, with only one bond breaking while another being formed immediately, the values of shear strengths calculated began to match the experimental values. These half-planes were therefore a very successful explanation for the problem.

Connection with Dislocations

The missing (or extra) half-planes have an edge (a line of atoms that don’t have neighbouring atoms below (or above) them). This edge came to be known as the Edge Dislocation3. Soon, all the mathematics of dislocations was established. The mathematics was so convincing that despite the fact that no one had ever ‘seen’ a dislocation before (microscopy techniques needed more development), the scientist community was next to certain on the existence of dislocations. When dislocations were finally ‘seen’ in the 1950’s, it only provided the final formality of a confirmation. From then on, there have been more additions to the theory and now, to learn about dislocations in detail, you might have to read thick books running for more than 1000 pages!

So, we care.

And hence, there’ll be more on dislocations to follow.

1Plastic deformation is the permanent deformation beyond the elastic limit of a material.
2More on single crystals later.
3There’s another kind of dislocations called Screw Dislocations. Dislocations which exhibit traits of both are called Mixed Dislocations. More on these types and the definition of dislocations later.