17 Motion of dislocations & dislocation density

Prof. P. N. Kotru

Learning Objectives

• Here we learn about the role of dislocations in influencing the mechanical property of crystals.

• On application of uniform shear stress along the Burgers vector at a crystal, the dislocations line

experiences force such that the slipped area tends to grow.

• The force per unit length on the dislocation is given by F= τ b where b is the burgers vector. The dislocation

line sweeping across a slip plane leads to displacement of crystal planes.

• Large no. of dislocation sweeping across several slip planes is explained to be responsible for appreciable

plastic deformation in crystals.

• Concept of dislocation density is given

• The slip process resulting from a moving edge dislocation leading to displacement of a part of the crystal

and deform it, is explained through schematic diagrams.

• The slip process resulting from a moving edge dislocation leading to displacement of a part of the crystal

and deform it, is explained through schematic diagrams.

CONTENTS

17.1   Slip, Motion of Dislocations & Dislocation Density

17.2   Strain Energy of a Dislocation.

17.3   Stress Field of an Edge Dislocation

17.1   Slip, Motion of Dislocations & Dislocation Density

Suppose a uniform shear stress τ is applied to the crystal along the direction of the Burgers vector. It is shown that this leads to a force on the dislocation line such that the slipped area tends to grow. The force per unit length on the dislocation is F= τ b, where b is burgers vector. This force lies in the slip plane, is perpendicular to the dislocation line all along its length and is directed towards the unslipped part of the plane. A single dislocation line sweeping across a slip plane results into a displacement of the order of few Angstroms. It means that any appreciable plastic deformation should be due to large number of dislocations sweeping across many slip planes. Clearly, the rate of plastic flow would depend on the rate at which dislocation lines sweep through the slip planes. In this regard an important concept of dislocation density is introduced. It is defined as ρ=S/V, where S stands for total length of the dislocation lines and V is the volume of the crystal. It may be noted that ρ has the dimensions length ─2 .So, the quantity ρ = S/V has the dimensions of inverse area and is called the density of dislocations. Suppose we take simple distribution of dislocations in which the dislocation lines are all straight and parallel, extend from one side of the crystal to the other. In that case, the number of dislocations intersected by a plane of unit area normal to them is the density of dislocations. Techniques for the direct observation of dislocations and determination of dislocation density will be dealt with later in the relevant section.

The slip process resulting from a moving edge dislocation results into displacement of a part of the crystal and deform it. The mechanism behind the mobility of dislocation is shown in a schematic diagram of figure 17.

Figure 17.1: Movement of positive edge dislocation  to the right leading

The motion of an edge dislocation through a crystal may be taken as being analogous to the situation in which a wrinkle passes across a rug. The wrinkle is able to move easily than the whole rug, but the passage of the ruck or wrinkle across the rug results into the same displacement as sliding over the whole rug on the floor. If atoms on one side of the slip plane are moved with respect to those on the other side, atoms at the slip plane will experience repulsive forces from some neighbours and attractive forces from others across the slip plane. These forces cancel to a first approximation. The external stress required to move a dislocation has been calculated and is quite small, probably below 10 5 dynes cm─2 provided that the bonding forces in the crystal are not highly directional. Thus, dislocation may make a crystal plastic. Passage of a dislocation through a crystal is equivalent to a displacement of a part of the crystal. This is how the crystals would deform if the dislocation were to move across the slip plane and hence explains the connection between dislocations and plastic deformation. When an edge dislocation moves from one lattice site to another on the slip plane , the atoms in the core move slowly and the extra half plane of atoms at one lattice position gets connected to a plane of atoms below the slip plane and the nearby plane of atoms becomes the new extra half plane. The process repeats itself till the upper half of the crystal block completes the slip or glide by Burgers vector b. Climb of a dislocation corresponds to its motion up or down from the slip plane.

Edge dislocation for which the extra half plane lies above the slip plane is referred to as positive whereas the one for which the extra half plane is below the slip plane is called as negative edge dislocation. In figure 17.1 we had taken up the slip process resulting from a positive dislocation moving to the right. The result can also be achieved by motion of a negative edge dislocation of the same strength to the left. Motion of dislocation is possible either by a climb or by a slip or by a glide.

Most of the mechanical properties of crystals, including elastic modulii, slip and plastic deformation, hardening by alloying and heat treatment, annealing properties and work hardening can be explained in terms of the motion of dislocations and the interaction of dislocations with one another and with impurity atoms. Any detailed discussion on this topic is beyond the scope of this e-book. Anybody interested to have further details is advised to refer to books “D islocations in Crystals” (McGraw-Hill, New York 1953) by W.T. Read and “Dislocations and Plastic flow in Crystals” (New York, Oxford 1953).

Coming back to figure 17.1 regarding deformation of a crystal due to motion of dislocations on application of stress. When a dislocation of strength b sweeps over an entire slip plane, the two half- crystals which meet on the plane become displaced relative to each other by the amount b in the direction of slip.

17.2   Strain Ene rgy of a Dislocation.

The energy of a dislocation, be it of edge or screw dislocation, can be estimated by assuming that the crystal behaves as an elastic solid during the process of creation of the dislocation. We start considering a perfect crystal, then let a cut be made in an appropriate way ( depending on whether a screw or edge dislocation is to be dealt with ), and let the two sides of the cut be made in an appropriate way ( depending on whether a screw or edge dislocation is to be dealt with ), and let the two sides of the cut be displaced with respect to each other by the distance b in the manner as required. In order to create displacement, a distribution of forces is required to be exerted over the surface of the cut, and the work done by the forces in making the displacement b is equal to the energy of the dislocation, Ed.

Ed = ∫ F. b dA …………………..…………17.1

The integral is evaluated over the area of the surface of the cut. The force F is the average force per unit area at a point on the surface during the displacement. Use of the average value is justified because the force at a point builds up linearly from zero to a maximum value as the displacement is carried out.

Hooke’s law suggests that the density of strain energy U in an elastic body is one-half of the product of stress and strain. It is usually easier to find the strain energy by considering the fact that this strain energy originates from the work done on the body by the applied forces that cause the strained condition.

We proceed by considering the process of forming dislocation. This is done by taking it step wise as follows:

(i)    We cut into the body to the line of the intended dislocation.

(ii)  Next we apply forces to the faces of the cut, starting from zero and increasing it gradually until the

dislocation is formed.

As the dislocation forms, the faces slide past each other and the forces on them do work. The energy enters the body and becomes the strain energy of the dislocation. Let the surface forces acting when the deformation is complete be F per unit area. The force is a function of position on the surface A of the body. The work done, and hence the strain energy is:

½ ∑ force x displacement = ½ ∫F.b.dA ……………………………17.2

Where, b is the displacement at the    surface A.

The factor ½ is taken because the surface forces build up from zero to their final values as the displacement takes place. Therefore, the average force is taken and used here.

Next, we define the core of dislocations. It is a region in the range of few lattice constants of the centre of dislocation and it is actually this region where the regular atomic arrangement of the crystal is severely affected. It is this region where maximum breakdown in the orderly arrangement of atoms is noticed.

Assuming an elastic isotropic medium, it becomes easier to calculate the strain field of a screw dislocation. For this let us consider a thin cylindrical shell of some material with radius r and length l around a screw dislocation in the Z-direction as shown in figure 17. 2

Figure 17.2: displacement  of material  in a cylindrical shell around a screw dislocation

For a screw dislocation the burger vector (or the slip vector) b is parallel to the line of the dislocation. Here, the screw dislocation is in the Z-direction. The elastic strain in the material at r corresponding to a displacement b in the direction of Z is assumed to be uniformly distributed over entire circumference 2πr. No strain occurs in the r or Φ directions. The strain in the Z-direction is of pure shear type and is given by:

Є  Φ z =b/2π r………………………………17.3

The corresponding shear stress on the Φ face in the z-direction is:

τ  Φ Z  =G.  Є Φ z = G. b/2 π r……………………….17. 4

Where G represents shear modulus or modulus of rigidity of the concerned material. The elastic strain energy is evaluated by taking sum of contributions from cylinders beginning at the core, r0 =10─7
cm, and extending out to r1 of the order of the dimensions of the crystal, say 1 cm. Here, r1and r0 are taken as appropriate upper and lower limits for the variable r .A reasonable value of r0 is comparable to the magnitude b of the Burgers vector or equal to one or two lattice constants, the value of r1 cannot exceed the dimensions of the crystal. Thus, the average force is half the final value when the displacement is b, i.e,

17.3 Stress Field of an Edge Dislocation

Let us consider dislocation of figure 17.3 in an elastically isotropic body. Dislocation is made by an operation in which a cut is made along part of slip plane, then displacing the sides of this cut rigidly past each other. The discontinuity of the diagram shown in figure 17.3 represents a positive edge dislocation along the z-axis with a burgers vector b along the x-axis

Figure 17.3: The discontinuity representing a positive edge dislocation along z-axis with Burgers vector along the x-axis

(i) σrr and σΦΦ , the normal stress along the radial ( radial tensile stress i.e., compression or tension along the radius r ) and circumferential (circumferential tensile stress i.e., compression or tension acting in a plane perpendicular to r ) directions.and,

(ii) τrΦ ( = τΦr) the shear stress acting in a radial direction. For an isotropic elastic continuum the general equation for τrΦ =τΦr = Gb  .    cosΦ 2 π r (1─ ν) For an edge dislocation cosΦ =1 for a cut along the slip plane In an isotropic elastic continuum the radial tensile stress i.e., compression or tension along the radius r. (σrr) and the circumferential tensile stress i.e., compression or tension acting in a plane perpendicular to r (σΦΦ) are found to be proportional to sinΦ /r, because of the requirement of a function which varies as 1/r and which changes sign when Y changes sign. However, τrΦ is found to be proportional to cosΦ/r, considering the plane Y =0.

Without going into the rigorous and finer details of calculations which fall outside the scope of the subject here, the stress field of the edge dislocation in terms of r and Φ are given by the following expressions:

Here, the positive values of σ refer to tension and negative values of σ refer to compression .σΦΦ is a compression or tension acting in a plane perpendicular to r. The shear stress τrΦ acts in a radial direction. G stands for shear modulus or modulus of rigidity and ν stands for poisson’s ratio. Above the slip plane σrr is negative and hence compression whereas below the slip plane it corresponds to tension.

It is observed that the stresses fall off as 1/r from the dislocation and becomes zero at infinity. The stress function is thus appropriate for a dislocation in a cylinder of infinite outer radius with an outer boundary free from external forces. It may be emphasized that the stresses become infinite for r=0 and hence a small cylindrical region of r0 around the dislocation is excluded. In an actual crystal this difficulty does not arise, since the material consists of atoms and is not a continuum. On the other hand, the stresses in the immediate vicinity of an actual dislocation will also be large and Hooke’s law is not valid in that region. For example, if we put r=b, the strains are of the order of D/Gb ≈ 1/ 2 π ( 1─ ν ) ≈ 1/4 ≈25% and so are too large to be dealt with accurately by the theory of elasticity.

The energy of unit length of edge dislocation is calculated by taking τ = τΦr

This is an expression for strain energy of a unit length of edge dislocation.

This shows that strain energy per unit length [Ed (edge)] of a dislocation in a cylinder of infinite outer radius (i.e., for a crystal of infinite size) is infinite. For a single slip dislocation in a crystal we may take the values r1= 1 cm. (i.e., crystals of ordinary size say 1 cm on an edge), r0= 10─7 cm.

Loge (r1/r0) ≈16

Taking G = 4×1011 dynes/ cm2, b=2.5×10 ─8 cm. ν =0.34 which are applicable to copper, the strain energy of an edge dislocation is about 5×10─4 erg cm─1 or approximately 8eV for each atom plane threaded by the dislocation line. For screw dislocations the strain energy is about two-thirds of this value.

SUMMARY

References.

1. Read,W.T. “ Dislocations in Crystals”, McGraw-Hill, N.Y. ,1953.
2. Cottrell,A.H. “ Dislocations and Plastic Flow in Crystals”, Oxford University Press,N.Y.1953.
3. Dekker, A.J.” Solid State Physics”, Macmillan, London, 1958.
4. Kittel,C. “ Introduction to Solid State Physics”,Wiley,N.Y.1971.
5. Shockley,W., Holloman,J.,Maurer,R, Seitz,F.(Eds) “ Imperfections in Nearly Perfect Crystals,John Wiley & Sons,N.Y.1952.
6. Brown.F.C. “ The Physics Of Solids”,W.A.Benjamin,Inc.,N.Y.,1967.

Suggested Reading For More Information .

The references given in module XVII are also suggested for the topic dealt with in this module.