Dislocation

Generating dislocations

When metals are subjected to cold working (deformation at temperatures which are relatively low as compared to the material's absolute melting temperature, ${\displaystyle T_{m}}$ i.e., typically less than ${\displaystyle 0.4T_{m}}$) the dislocation density increases due to the formation of new dislocations. The consequent increasing overlap between the strain fields of adjacent dislocations gradually increases the resistance to further dislocation motion. This causes a hardening of the metal as deformation progresses. This effect is known as strain hardening or work hardening.

Dislocation density ${\displaystyle \rho }$ in a material can be increased by plastic deformation by the following relationship:

${\displaystyle \tau \propto {\sqrt {\rho }}}$.

Since the dislocation density increases with plastic deformation, a mechanism for the creation of dislocations must be activated in the material. Three mechanisms for dislocation formation are homogeneous nucleation, grain boundary initiation, and interfaces between the lattice and the surface, precipitates, dispersed phases, or reinforcing fibers.

Interaction and arrangement

Movement

Edge

A crystalline material consists of a regular array of atoms, arranged into lattice planes. An edge dislocation is a defect where an extra half-plane of atoms is introduced midway through the crystal, distorting nearby planes of atoms. When enough force is applied from one side of the crystal structure, this extra plane passes through planes of atoms breaking and joining bonds with them until it reaches the grain boundary. The dislocation has two properties, a line direction, which is the direction running along the bottom of the extra half plane, and the Burgers vector which describes the magnitude and direction of distortion to the lattice. In an edge dislocation, the Burgers vector is perpendicular to the line direction.

The stresses caused by an edge dislocation are complex due to its inherent asymmetry. These stresses are described by three equations:^[15]

${\displaystyle \sigma _{xx}={\frac {-\mu \mathbf {b} }{2\pi (1-\nu )}}{\frac {y(3x^{2}+y^{2})}{(x^{2}+y^{2})^{2}}}}$

${\displaystyle \sigma _{yy}={\frac {\mu \mathbf {b} }{2\pi (1-\nu )}}{\frac {y(x^{2}-y^{2})}{(x^{2}+y^{2})^{2}}}}$

${\displaystyle \tau _{xy}={\frac {\mu \mathbf {b} }{2\pi (1-\nu )}}{\frac {x(x^{2}-y^{2})}{(x^{2}+y^{2})^{2}}}}$

where ${\displaystyle \mu }$ is the shear modulus of the material, ${\displaystyle \mathbf {b} }$ is the Burgers vector, ${\displaystyle \nu }$ is Poisson's ratio and ${\displaystyle x}$ and ${\displaystyle y}$ are coordinates.

These equations suggest a vertically oriented dumbbell of stresses surrounding the dislocation, with compression experienced by the atoms near the "extra" plane, and tension experienced by those atoms near the "missing" plane.^[15]

Screw

A screw dislocation can be visualized by cutting a crystal along a plane and slipping one half across the other by a lattice vector, the halves fitting back together without leaving a defect. If the cut only goes part way through the crystal, and then slipped, the boundary of the cut is a screw dislocation. It comprises a structure in which a helical path is traced around the linear defect (dislocation line) by the atomic planes in the crystal lattice. In pure screw dislocations, the Burgers vector is parallel to the line direction.^[16] An array of screw dislocations can cause what is known as a twist boundary. In a twist boundary, the misalignment between adjacent crystal grains occurs due to the cumulative effect of screw dislocations within the material. These dislocations cause a rotational misorientation between the adjacent grains, leading to a twist-like deformation along the boundary. Twist boundaries can significantly influence the mechanical and electrical properties of materials, affecting phenomena such as grain boundary sliding, creep, and fracture behavior^[16] The stresses caused by a screw dislocation are less complex than those of an edge dislocation and need only one equation, as symmetry allows one radial coordinate to be used:^[15]

${\displaystyle \tau _{r}={\frac {-\mu \mathbf {b} }{2\pi r}}}$

where ${\displaystyle \mu }$ is the shear modulus of the material, ${\displaystyle \mathbf {b} }$ is the Burgers vector, and ${\displaystyle r}$ is a radial coordinate. This equation suggests a long cylinder of stress radiating outward from the cylinder and decreasing with distance. This simple model results in an infinite value for the core of the dislocation at ${\displaystyle r=0}$ and so it is only valid for stresses outside of the core of the dislocation.^[15] If the Burgers vector is very large, the core may actually be empty resulting in a micropipe, as commonly observed in silicon carbide.

Mixed

In many materials, dislocations are found where the line direction and Burgers vector are neither perpendicular nor parallel and these dislocations are called mixed dislocations, consisting of both screw and edge character. They are characterized by ${\displaystyle \varphi }$, the angle between the line direction and Burgers vector, where ${\displaystyle \varphi =\pi /2}$ for pure edge dislocations and ${\displaystyle \varphi =0}$ for screw dislocations.

Partial

Partial dislocations leave behind a stacking fault. Two types of partial dislocation are the Frank partial dislocation which is sessile and the Shockley partial dislocation which is glissile.^[3]

A Frank partial dislocation is formed by inserting or removing a layer of atoms on the {111} plane which is then bounded by the Frank partial. Removal of a close packed layer is known as an intrinsic stacking fault and inserting a layer is known as an extrinsic stacking fault. The Burgers vector is normal to the {111} glide plane so the dislocation cannot glide and can only move through climb.^[1]

In order to lower the overall energy of the lattice, edge and screw dislocations typically disassociate into a stacking fault bounded by two Shockley partial dislocations.^[17] The width of this stacking-fault region is proportional to the stacking-fault energy of the material. The combined effect is known as an extended dislocation and is able to glide as a unit. However, dissociated screw dislocations must recombine before they can cross slip, making it difficult for these dislocations to move around barriers. Materials with low stacking-fault energies have the greatest dislocation dissociation and are therefore more readily cold worked.

Stair-rod and the Lomer–Cottrell junction

If two glide dislocations that lie on different {111} planes split into Shockley partials and intersect, they will produce a stair-rod dislocation with a Lomer-Cottrell dislocation at its apex.^[18] It is called a stair-rod because it is analogous to the rod that keeps carpet in-place on a stair.

Jog

A Jog describes the steps of a dislocation line that are not in the glide plane of a crystal structure.^[17] A dislocation line is rarely uniformly straight, often containing many curves and steps that can impede or facilitate dislocation movement by acting as pinpoints or nucleation points respectively. Because jogs are out of the glide plane, under shear they cannot move by glide (movement along the glide plane). They instead must rely on vacancy diffusion facilitated climb to move through the lattice.^[19] Away from the melting point of a material, vacancy diffusion is a slow process, so jogs act as immobile barriers at room temperature for most metals.^[20]

Jogs typically form when two non-parallel dislocations cross during slip. The presence of jogs in a material increases its yield strength by preventing easy glide of dislocations. A pair of immobile jogs in a dislocation will act as a Frank–Read source under shear, increasing the overall dislocation density of a material.^[20] When a material's yield strength is increased via dislocation density increase, particularly when done by mechanical work, it is called work hardening. At high temperatures, vacancy facilitated movement of jogs becomes a much faster process, diminishing their overall effectiveness in impeding dislocation movement.

Kink

Kinks are steps in a dislocation line parallel to glide planes. Unlike jogs, they facilitate glide by acting as a nucleation point for dislocation movement. The lateral spreading of a kink from the nucleation point allows for forward propagation of the dislocation while only moving a few atoms at a time, reducing the overall energy barrier to slip.

Transmission electron microscopy (TEM)

Transmission electron microscopy can be used to observe dislocations within the microstructure of the material.^[22] The first observation thereof with a robust explanation of the contrast was in 1956 by Hirsch, Horne and Whelan at Cambridge in 1956.^[23] Thin foils of material are prepared to render them transparent to the electron beam of the microscope. The electron beam undergoes diffraction by the regular crystal lattice planes into a diffraction pattern and contrast is generated in the image by this diffraction (as well as by thickness variations, varying strain, and other mechanisms). Dislocations have different local atomic structure and produce a strain field, and therefore will cause the electrons in the microscope to scatter in different ways. Note the characteristic 'wiggly' contrast of the dislocation lines as they pass through the thickness of the material in the figure (dislocations cannot end in a crystal, and these dislocations are terminating at the surfaces since the image is a 2D projection).

Dislocations do not have random structures, the local atomic structure of a dislocation is determined by the Burgers vector. One very useful application of the TEM in dislocation imaging is the ability to experimentally determine the Burgers vector. Determination of the Burgers vector is achieved by what is known as ${\displaystyle {\vec {g}}\cdot {\vec {b}}}$ ("g dot b") analysis.^[24] When performing dark field microscopy with the TEM, a diffracted spot is selected to form the image (as mentioned before, lattice planes diffract the beam into spots), and the image is formed using only electrons that were diffracted by the plane responsible for that diffraction spot. The vector in the diffraction pattern from the transmitted spot to the diffracted spot is the ${\displaystyle {\vec {g}}}$ vector. The contrast of a dislocation is scaled by a factor of the dot product of this vector and the Burgers vector (${\displaystyle {\vec {g}}\cdot {\vec {b}}}$). As a result, if the Burgers vector and ${\displaystyle {\vec {g}}}$ vector are perpendicular, there will be no signal from the dislocation and the dislocation will not appear at all in the image. Therefore, by examining different dark field images formed from spots with different g vectors, the Burgers vector can be determined.

Other methods

Field ion microscopy and atom probe techniques offer methods of producing much higher magnifications (typically 3 million times and above) and permit the observation of dislocations at an atomic level. Where surface relief can be resolved to the level of an atomic step, screw dislocations appear as distinctive spiral features – thus revealing an important mechanism of crystal growth: where there is a surface step, atoms can more easily add to the crystal, and the surface step associated with a screw dislocation is never destroyed no matter how many atoms are added to it.

Forces on dislocations

Dislocation motion as a result of external stress on a crystal lattice can be described using virtual internal forces which act perpendicular to the dislocation line. The Peach-Koehler equation^[25][26][27] can be used to calculate the force per unit length on a dislocation as a function of the Burgers vector, ${\displaystyle \mathbf {b} }$, stress, ${\displaystyle \sigma }$, and the sense vector, ${\displaystyle \mathbf {s} }$.

${\displaystyle \mathbf {f} =(\mathbf {b} \cdot \sigma )\times \mathbf {s} }$

The force per unit length of dislocation is a function of the general state of stress, ${\displaystyle \mathbf {F} }$, and the sense vector, ${\displaystyle \mathbf {s} }$.

${\displaystyle \mathbf {f} =\mathbf {F} \times \mathbf {s} ={\begin{vmatrix}{\hat {\imath }}&{\hat {\jmath }}&{\hat {k}}\\F_{x}&F_{y}&F_{z}\\s_{x}&s_{y}&s_{z}\end{vmatrix}}}$

The components of the stress field can be obtained from the Burgers vector, normal stresses, ${\displaystyle \sigma }$, and shear stresses, ${\displaystyle \tau }$.

${\displaystyle {\begin{aligned}F_{x}&=b_{x}\sigma _{xx}+b_{y}\tau _{xy}+b_{z}\tau _{xz}\\F_{y}&=b_{x}\tau _{yx}+b_{y}\sigma _{yy}+b_{z}\tau _{yz}\\F_{z}&=b_{x}\tau _{zx}+b_{y}\tau _{zy}+b_{z}\sigma _{zz}\end{aligned}}}$

Forces between dislocations

The force between dislocations can be derived from the energy of interactions of the dislocations, ${\displaystyle U_{\rm {int}}}$. The work done by displacing cut faces parallel to a chosen axis that creates one dislocation in the stress field of another displacement. For the ${\displaystyle x}$ and ${\displaystyle y}$ directions:

${\displaystyle {\begin{aligned}U_{\rm {int}}&=\int _{x}^{\infty }(b_{x}\tau _{xy}+b_{y}\sigma _{yy}+b_{z}\sigma _{zy})\,dx\\U_{\rm {int}}&=\int _{y}^{\infty }(b_{x}\sigma _{xx}+b_{y}\tau _{yx}+b_{z}\sigma _{zx})\,dy\end{aligned}}}$

The forces are then found by taking the derivatives.

${\displaystyle {\begin{aligned}F_{x}&=-{\frac {\partial U_{\rm {int}}}{\partial x}}\\F_{y}&=-{\frac {\partial U_{\rm {int}}}{\partial y}}\end{aligned}}}$

Free surface forces

Dislocations will also tend to move towards free surfaces due to the lower strain energy. This fictitious force can be expressed for a screw dislocation with the ${\displaystyle y}$ component equal to zero as:

${\displaystyle F_{x}=b\tau _{zy}=-{\frac {Gb^{2}}{4\pi d}}}$

where ${\displaystyle d}$ is the distance from free surface in the ${\displaystyle x}$ direction. The force for an edge dislocation with ${\displaystyle y=0}$ can be expressed as:

${\displaystyle F_{x}=b\tau _{xy}=-{\frac {Gb^{2}}{4\pi (1-\nu )d}}}$