Dislocation

In materials science, a dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allows atoms to slide over each other at low stress levels and is known as glide or slip. The crystalline order is restored on either side of a glide dislocation but the atoms on one side have moved by one position. The crystalline order is not fully restored with a partial dislocation. A dislocation defines the boundary between slipped and unslipped regions of material and as a result, must either form a complete loop, intersect other dislocations or defects, or extend to the edges of the crystal. A dislocation can be characterised by the distance and direction of movement it causes to atoms which is defined by the Burgers vector. Plastic deformation of a material occurs by the creation and movement of many dislocations. The number and arrangement of dislocations influences many of the properties of materials. The two primary types of dislocations are sessile dislocations which are immobile and glissile dislocations which are mobile. Examples of sessile dislocations are the stair-rod dislocation and the Lomer–Cottrell junction. The two main types of mobile dislocations are edge and screw dislocations.…

In materials science, a dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allows atoms to slide over each other at low stress levels and is known as glide or slip. The crystalline order is restored on either side of a glide dislocation but the atoms on one side have moved by one position. The crystalline order is not fully restored with a partial dislocation. A dislocation defines the boundary between slipped and unslipped regions of material and as a result, must either form a complete loop, intersect other dislocations or defects, or extend to the edges of the crystal. A dislocation can be characterised by the distance and direction of movement it causes to atoms which is defined by the Burgers vector. Plastic deformation of a material occurs by the creation and movement of many dislocations. The number and arrangement of dislocations influences many of the properties of materials. The two primary types of dislocations are sessile dislocations which are immobile and glissile dislocations which are mobile. Examples of sessile dislocations are the stair-rod dislocation and the Lomer–Cottrell junction. The two main types of mobile dislocations are edge and screw dislocations. Edge dislocations can be visualized as being caused by the termination of a plane of atoms in the middle of a crystal. In such a case, the surrounding planes are not straight, but instead bend around the edge of the terminating plane so that the crystal structure is perfectly ordered on either side. This phenomenon is analogous to half of a piece of paper inserted into a stack of paper, where the defect in the stack is noticeable only at the edge of the half sheet. Screw dislocations create faults in a crystal that looks similar to that of a spiral staircase. These types of dislocations can be formed by cutting halfway through a crystal and sliding those regions on each side of the cut parallel to the cut to create spiraling atom planes. The dislocation line would be located in the central axis of the spiral. The theory describing the elastic fields of the defects was originally developed by Vito Volterra in 1907. In 1934, Egon Orowan, Michael Polanyi and G. I. Taylor, proposed that the low stresses observed to produce plastic deformation compared to theoretical predictions at the time could be explained in terms of the theory of dislocations. \tau_m = \frac {G} {2 \pi}. The shear modulus in metals is typically within the range 20 000 to 150 000 MPa indicating a predicted shear stress of 3 000 to 24 000 MPa. This was difficult to reconcile with measured shear stresses in the range of 0.5 to 10 MPa. In 1934, Egon Orowan, Michael Polanyi and G. I. Taylor, independently proposed that plastic deformation could be explained in terms of the theory of dislocations. Dislocations can move if the atoms from one of the surrounding planes break their bonds and rebond with the atoms at the terminating edge. In effect, a half plane of atoms is moved in response to shear stress by breaking and reforming a line of bonds, one (or a few) at a time. The energy required to break a row of bonds is far less than that required to break all the bonds on an entire plane of atoms at once. Even this simple model of the force required to move a dislocation shows that plasticity is possible at much lower stresses than in a perfect crystal. In many materials, particularly ductile materials, dislocations are the "carrier" of plastic deformation, and the energy required to move them is less than the energy required to fracture the material. The creation of a dislocation by homogeneous nucleation is a result of the rupture of the atomic bonds along a line in the lattice. A plane in the lattice is sheared, resulting in 2 oppositely faced half planes or dislocations. These dislocations move away from each other through the lattice. Since homogeneous nucleation forms dislocations from perfect crystals and requires the simultaneous breaking of many bonds, the energy required for homogeneous nucleation is high. For instance, the stress required for homogeneous nucleation in copper has been shown to be \frac {\tau_{\text{hom}}}{G}=7.4\times10^{-2} , where G is the shear modulus of copper (46 GPa). Solving for \tau_{\text{hom}} \,\! , we see that the required stress is 3.4 GPa, which is very close to the theoretical strength of the crystal. Therefore, in conventional deformation homogeneous nucleation requires a concentrated stress, and is very unlikely. Grain boundary initiation and interface interaction are more common sources of dislocations. Irregularities at the grain boundaries in materials can produce dislocations which propagate into the grain. The steps and ledges at the grain boundary are an important source of dislocations in the early stages of plastic deformation. v = A\tau^m where A is a material constant, \tau is the applied shear stress, m is a constant that decreases with increasing temperature. Increased shear stress will increase the dislocation velocity, while increased temperature will typically decrease the dislocation velocity. Greater phonon scattering at higher temperatures is hypothesized to be responsible for increased damping forces which slow the dislocation movement. ==Geometry== )]] Two main types of mobile dislocations exist: edge and screw. Dislocations found in real materials are typically mixed, meaning that they have characteristics of both. ===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: \sigma_{xx} = \frac {-\mu \mathbf{b}} {2 \pi (1-\nu)} \frac {y(3x^2 +y^2)} {(x^2 +y^2)^2} \sigma_{yy} = \frac {\mu \mathbf{b}} {2 \pi (1-\nu)} \frac {y(x^2 -y^2)} {(x^2 +y^2)^2} \tau_{xy} = \frac {\mu \mathbf{b}} {2 \pi (1-\nu)} \frac {x(x^2 -y^2)} {(x^2 +y^2)^2} where \mu is the shear modulus of the material, \mathbf{b} is the Burgers vector, \nu is Poisson's ratio and x and 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. ===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. 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 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: \tau_{r} = \frac {-\mu \mathbf{b}} {2 \pi r} where \mu is the shear modulus of the material, \mathbf{b} is the Burgers vector, and 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 r=0 and so it is only valid for stresses outside of the core of the dislocation. 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 \varphi , the angle between the line direction and Burgers vector, where \varphi=\pi/2 for pure edge dislocations and \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. 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. 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. 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. ==Observation== \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, \mathbf{F} , and the sense vector, \mathbf{s} . \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, \sigma , and shear stresses, \tau . \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} 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. \begin{aligned} F_x &= - \frac{\partial U_{\rm int}}{\partial x} \\ F_y &= - \frac{\partial U_{\rm int}}{\partial y} \end{aligned} F_x = b\tau_{zy} = -\frac{Gb^2}{4\pi d} where d is the distance from free surface in the x direction. The force for an edge dislocation with y = 0 can be expressed as: F_x = b\tau_{xy} = -\frac{Gb^2}{4\pi (1 - \nu)d} ==External links== Defects in Crystals/ Prof. Dr. Helmut Föll website Chapter 5 contains a wealth of information on dislocations; DoITPoMS Online tutorial on dislocations, including movies of dislocations in bubble rafts; Difference between Edge dislocation and Screw dislocation Difference between Edge dislocation and Screw dislocation in detail; Scanning Tunneling Microscope – Gallery Image gallery, including a dislocations page, seen at the atomic level of metal surfaces, by the surface physics group at the Faculty of Physics, Vienna University of Technology, Austria. Volterra, V., "On the equilibrium of multiply-connected bodies," trans. by D. H. Delphenich Somigliana, C., "On the theory of elastic distortions," transl. by D. H. Delphenich

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