Line imperfections are called dislocations. There are two primary types of dislocations. These are:
- Edge dislocation, and
- Screw dislocation.
To understand an edge dislocation, consider a perfect crystal shown in Figure 1(a). Atoms (not shown in diagram) are located on each cross-point in it. A perfect crystal is free from dislocations.
Now consider the above crystal with an extra or incomplete plane of atoms ABCD as shown in Figure 1(b). The spacing of the atoms is slightly different in the portions above and below the plane S1S2 containing atom at B. This is because of the presence of dislocations. The line BC is called dislocation edge.
Stress field: In a perfect crystal, the atoms are in equilibrium positions. Just-above the edge of the incomplete plane, the atoms are squeezed together and are in a state of compression. The bond length is smaller than the equilibrium value (r < ro). Contrary to this, the atoms are pulled apart just below the edge and are in tensile state. The bond length is more than the normal value (r > ro). This distorted configuration extends all along the dislocation edge BC in the crystal.
Reason of extra strain energy: The potential energy increases for an increase as well as a decrease in bond length. Thus there is extra strain energy due to the distortion in the region surrounding the edge of an incomplete plane. As the region of maximum distortion is centered around the edge of the incomplete plane, this distortion represents a line imperfection and is called an edge dislocation.
Positive and negative type edge dislocations: Depending on whether the incomplete plane starts from the top or from the bottom of the crystal, these two configurations are referred as `positive’ and ‘negative’ edge dislocations respectively. These are generally shown by inverted T and T symbols. Representation of dislocation in two-dimensional and three-dimensional cases is also done as shown in Figure 1(c).
Slip plane: The plane S1BS2S3S4 is called slip plane. It is shown in Figure 1(b). The region over this plane is called the slipped part, and the region below the slip plane is called un-slipped part. The dislocation line S1S2 is then defined as the boundary between slipped and un-slipped parts of the slip plane.
Burgers Circuit and Burgers Vector
The direction and magnitude of an edge dislocation is determined by a vector called the Burgers vector, named after of its originator. It is normally denoted by B.V. or b. Burgers vector of a dislocation is found by drawing a Burgers Circuit described as follows.
Burgers circuit in a perfect crystal: Let us take a perfect crystal, Figure 2(a). Starting from the point M, we go vertically up by steps to reach point N. Then we move horizontally right in q steps to reach point 0. Each step is equal to one atomic distance in the crystal model under consideration. Steps are p = 5 and q = 6 in Figure 2(a).
Now we come down vertically from 0 to point P in p steps and move horizontally left in q steps to reach point Q. The MNOPQ is the Burgers circuit. Point Q coincides with point M. As the starting and end points M and Q respectively are the same, we conclude that for an ideal crystal
|B.V.| = |b| = 0
Burgers circuit in an imperfect crystal: Now consider an imperfect crystal as shown in Figure 2(b). Take a loop around the dislocation edge BC by taking p x q steps in the same manner as in case of perfect crystal. These steps p and q are the minimum essential lattice translation to enclose dislocation edge BC. The Burgers circuit has been drawn clockwise viewing in the direction of arrow marked along line BC.
It should be clearly understood that we should not move rightward from M1 or come down from N3; otherwise the dislocation edge cannot be encircled. In this case MM1 = M1M2 …, and NN1= N1N2 = …., are the steps. The loop MNOPQ is the Burgers circuit. The magnitude and direction of Burgers vector is given by:
B.V. = vector QM = b
In the model under consideration, |b| = 1 atomic distance. It is seen that the Bergers vector QM for an edge dislocation is perpendicular to the dislocation edge BC.
Contrary to edge dislocation, there is no extra plane in the case of screw dislocation in a crystal. Screw dislocation forms when a part of the crystal displaces angularly over the remaining part. The plane of atoms converts into a helical surface, or a screw. The angular displacement is similar to movement of a screw when turned. These are of two types – ‘clockwise’ and ‘anticlockwise’ or ‘positive’ and ‘negative’ screw dislocations.
Stress-strain field: Model of a crystal having screw dislocation is shown in Figure 3. Part of it MNB’C has displaced over part FGHJ from right to left direction. Displacement has occurred along the slip plane CDOH. Vertical edge ADE is unaffected by this shifting.
Originally the right hand side face ADEFGB has angularly deformed. Line DG has angularly displaced to DC, and edge AB has shifted to AB’. The screw dislocation line is marked in the direction of arrow l. The antenna bonds in the vicinity of dislocation line undergo through shear deformation. This gives rise to shear stress-shear strain field.
Burgers Circuit in Screw Dislocation
Burgers circuit is drawn using right hand screw convention. For that a clockwise sense has been taken on right side face looking in the arbitrary direction l of the crystal. The Burgers circuit is marked by 1 2 3 4 5 6 1 in the Figure 3. Direction 4-5 shows the direction of Burgers vector b, which is parallel to direction l. Thus we see that the Burgers vector is parallel to the screw dislocation line.
Dislocations having mixed character combining edge and screw dislocations are termed as mixed dislocations. When extra plane of atoms accompanies with angular shift of a part of real crystals, the consequence is a mixed dislocation. Mixed dislocation generally emerges at the curved boundary on which the directional continuity changes. Such curved boundaries are holes, notches, cuts etc. in the materials.
Burgers Circuit for Mixed Dislocation
A crystal model of mixed dislocation is shown in Figure 4. The Burgers circuit on the front face of the crystal is shown for screw dislocation while on the right side face for edge dislocation. Its character will be mixed in between these faces. The figure is self-explaining.
Characteristics of Dislocations
The properties and behavior of dislocations are characterized by certain geometries. These are:
- A crystal normally incorporates large number of dislocations. Hence there exist numerous Burgers vectors. The sum of these Burgers vectors meeting at a point, called nodal point, inside the crystal remains zero.
- A dislocation does not end abruptly within the crystal. It vanishes either at a nodal point or on the surface of the crystal.
- A dislocation under the influence of stress-field may close on as a loop. The profile of the loop may be a circle or otherwise.
- The distortional energy associated with dislocations may be the source of crystals instability. Distortional energy is produced due to tensile and compressive stress-strain field in edge dislocations, and due to shear stress-strain field in screw dislocations.
The elastic strain energy U per unit length of a dislocation is directly proportional to the square of Burgers vector b. It is given by
U = (π/8)Gb2 = Gb2/2 (approximately).
Where G is shear modulus or modulus of rigidity of the crystal.
- Dislocations may have Burgers vector of full lattice translation or partial lattice translation i.e., the Burgers vector may be of b, 2b, 3b, …… and b/2, b/3, …. magnitudes.
- Dislocations have inherent tendency to keep smallest possible Burgers vector. By doing so enhanced stability is developed in the crystals. It can be understood by the following equations. A dislocation of Burgers vector of 3b magnitude tries to break into three dislocations each of b magnitude.
3b –> b + b + b
By doing so it possesses elastic strain energy as:
G(3b)2/2 ≠ G(b)2/2 + G(b)2/2 + G(b)2/2
The amount of energy given by right hand side of above expression is much lower than that of left hand side term. Hence stability is increased.
- Two edge dislocations of opposite sign inverted T and T, of equal Burgers vector, and on the same slip plane cancel-out. This is because the distortional strain energy field super-impose and annihilate each other.
- Edge dislocations travel much faster (approximately 50 times) than screw dislocations.
Burgers Vectors of Dislocations in Cubic Crystals
Probable Burgers vectors of full dislocations in some cubic crystals are as under.
- Monoatomic SC <100>
- Monoatomic BCC 1/2 <111>
- Monoatomic FCC 1/2 <110>
- Diamond Cubic DC 1/2 <110>
- Ionic solid
: NaCl structure 1/2 <110>
: CsCl structure <110>
In NaCl crystal, the Burgers vector cannot be from the centre of a chlorine ion to the centre of a sodium ion. Similarly in CsCl crystal, the Burgers vector cannot be from a chlorine ion at body corner to cesium ion at body centre.
It is because the Burgers vector in above cases will not give a full lattice translation, Burgers vector in ionic crystals are larger than those in metallic crystals.
Sources of Dislocations
There are two main sources of dislocations in the solids. These are:
- Mishandling during grain growth, and
- Mechanical deformation.
Crystals are obtained by process of crystallization. In doing so, the molten metal is solidified. Recovery, re-crystallization and grain growth are the essential features of material’s manufacturing and processes. It is almost impossible to achieve perfection in the process of grain growth, and control on the size and orientation of grains. Process mishandling cannot be ruled out. That is why the dislocations invariably crop-in the solids.
Mechanical deformation is another source of dislocations. Crystals are subjected to various kinds of situations during manufacturing and fabrication in which they get deformed. Deformation may be geometrically linear, angular or complex. Due to these, the dislocations enter into the crystals.
Effects of Dislocations
Effects of dislocations are detrimental to the properties of crystalline materials.
- Their mechanical strengths lower down. Due to this, the structural and machine components become uneconomical.
- Dislocations are responsible for reduced electrical conduction.
- They also influence the surface-sensitive properties negatively.
- Infact, almost all the properties listed in deteriorate mildly to drastically.
However, the density of the solid is negligibly affected.
Remedies to Minimize the Dislocations
Following remedies can be suggested to avoid or minimize occurrence of dislocations
- Use of thermal energy.
- More careful control on crystallization.
- Prevention of undesired mechanical deformation.
- Use of whisker form of material.
Dislocations can be minimized by use of thermal energy. On heating the crystals to a high temperature, many dislocations either annihilate among themselves or drive out of the crystal surfaces.
Other way to remove dislocations will be to change the form of crystalline material. Normally the materials are in bulk form. If they could be produced as whiskers, dislocation density will definitely decrease immensely.
Phenomena Related to Behavior of Dislocations
We shall now study the responses of dislocations under externally applied stress field. Various phenomena pertaining to behavior of dislocations are the following.
- Dislocation glide motion
- Dislocation climb motion (a) climb-up motion (b) climb-down motion
- Dislocation cross-slip
- Dislocation jogs or jumps
- Dislocation interaction with point imperfections.
- Frank-Read source
1. Glide motion of dislocation: A dislocation in a crystal can be moved by externally applied stress. By doing so, the dislocation will disappear on reaching the surface. Only edge dislocation and mixed dislocation can have glide motion.
To understand this phenomenon let us consider a crystal with edge dislocation of positive type, Figure 5(a). It is subjected to a shear stress ‘t’ applied parallel to the slip plane.
Dislocation moves in the direction of shear stress. The incomplete plane of atoms A1B1 (Figure 5a) moves to situation A2B2 (Figure 5b) and then to A3B3 (Figure 5c) successively in response to the applied shear stress. Dislocation, finally, reaches to A4B4 (Figure 5d) and disappears.
Motion of dislocation occurs on slip plane that contains the Burgers vector and the direction vector. This plane is called glide plane and the motion of edge dislocation is known as glide motion. A screw dislocation cannot have glide motion.
A plane perpendicular to the glide plane may be called climb plane. Motion of an edge dislocation on this plane is known as climb motion. When edge dislocation moves above the slip plane in perpendicular direction Y, Figure 6, the motion is called climb-up. If it moves in (—y) direction, the motion is known as climb-down. Movement of atoms and vacancies at high temperatures is the cause of such motions.
Mechanism of climb-up and climb-down motion: The incomplete (or extra) plane of atoms AB shrinks due to subtraction of rows of atoms in climb-up motion. It is just the opposite in climb-down phenomenon.
- There is an increase in the incomplete planes of atoms due to addition of rows of atoms in this case. Due to subtraction and addition of atoms, the climb motion is called non-conservative.
- Subtraction or addition of atoms to the incomplete plane of atoms is from the remaining part of crystal. Thereby, vacancies are formed or filled as the case may be.
- Screw dislocations cannot climb-up or climb-down.
Dislocation climb is a diffusion controlled process. It is a slower process than the glide motion.
Phenomenon of cross-slip occurs in screw dislocation. A screw dislocation can change its slip plane while in motion. Change in its slip plane occurs if obstacles are present on the plane of its motion. The cross-slip of a screw dislocation is shown in Figure 7. Screw dislocation 11 cross-slips to location 44 enrouting location 22 and 33.
Jogs in Dislocation
A dislocation of Burgers vector b lying in a plane other than the slip or glide plane will be called a jog if its Burgers vector is equal to the Burgers vector of a dislocation lying in slip plane. As the dislocation can move from one plane to another, similarly it can jump from one plane to another. Jog may be treated as a shorter dislocation.
Figures 8(a) show a jog and its relation with the edge dislocation. The dislocation edge BC on slip plane SS, Figure 8(a), has jumped up as B1C1 on S1S1 plane in Figure 8(b) CB1 in Figure 8(c) is a jog.
Interaction of Dislocations with Point Imperfections
Point imperfections are thermodynamically stable but dislocations are not. Dislocations interact with point imperfections provided the energies of two interacting agents lower down. Important occurrences of this interaction are given as follows:
- Vacancy diffusion helps in dislocation climb.
- Substitutional impurity interacts with edge dislocation, and changes the tensile and compressive stress-strain field. Smaller substitutional atom reduces the energy of interacting system. Contrary to this, larger substitutional atom destabilizes this system.
- Interstitial atom fits into large spaced region of edge dislocation. Carbon atom occupies locations around BCC crystal of iron. Phenomenon of discontinuous yielding in mild steel is the effect of this interaction.