16 Line Defects

Prof. P. N. Kotru

16.1 Line Defects

Line defects are one-dimensional defects. Dislocations are examples of this type of defect. These are of different types:

(i)Edge dislocations,

(ii)   Screw dislocations and

(iii) Mixed type of dislocations

Before going into further details of these defects, it is necessary to have the background of how the existence of dislocations was realized.

We know that on application of stress solids undergo deformation. The deformation is of

two types:

(i)     Elastic deformation and

(ii)   Plastic deformation

In the case of elastic deformation the solid regains its original shape when the external stress F is removed as shown in figure 16.1

Figure 16.1: Illustration of Elastic deformation

Elastic deformation is caused by strains which obey Hooke’s law. According to Hooke’s law, atomic displacements are proportional to the applied force. When the elastic limit of a solid is exceeded, permanent or plastic deformation sets in .In many crystals it occurs by slip. Theoretical estimates of the elastic limit of perfect crystals yields values 103 or 104 higher than the lowest observed values. Hardness is a term which is defined as a measure of resistance to deformation. The atomic interpretation of plastic flow of crystals requires the introduction of a new type of lattice defect viz., dislocation.

16.2 Determination of Shear Strength of Single Crystals (Frenkel Model)

Let us estimate the stress that is necessary to produce deformation in crystals. There exists a critical resolved shear stress τc . For pure metals τc ranges between 10 ⁶ ─ 10⁷ dynes cm^─2. Calculation of theoretical shear stress based on perfect lattice is much larger than the observed values for pure crystals. However, theoretical estimates of the elastic limit of perfect crystals yields values 103 or 104 higher than the lowest observed values. This huge discrepancy between theoretical estimates based on perfect crystals and the actually observed values of real crystals necessitated to bring in the concept of a lattice defect known as dislocation.

Frenkel model is the one which gives us the theoretical estimate of critical resolved shear stress in the case of an ideally perfect crystal.

Figure 16.2: Frenkel model for calculation of theoretical shear strength of a perfect crystal

Frenkel in 1926 proposed a simple model for calculating theoretically shear strength of single perfect crystal. The model as proposed by him is shown in the figure. It shows two adjacent planes with interplanar distance d. On application of shear stress τ in x-direction, all the atoms in the upper plane get displaced. Stress τ is zero when shear stress has moved the planes a/2 with respect to each other. At this point it could go either way when displacement is large such as to bring atom A directly over atom B. Plot of shear stress against the relative displacement of the planes from their equilibrium position is shown at the bottom of the figure 16.2. It is clear that τ will vanish for x=0, a/2, a….etc. where a is the interatomic distance in the direction of x. As first approximation stress─displacement relation is represented by sine function:

τ = (G a/2 π d) sin (2 π x/a)

When x → 0 ,     τ              → G. x/d

Critical shear stress τc at and above which the lattice becomes unstable is given by the maximum value of τ .

Or, τc     = Ga/2 π d, when θ  = (2 π x /a) =π /2

Or, x= a/4    and if a = d

Then τc  = G/2 π  ≈ G/6,

That means the ideal critical resolved shear stress is ≈  one sixth of the shear modulus. Shear modulus G for single crystals of tin (Sn), silver (Ag) and aluminium (Al) is approximately of the order of 10¹¹ dynes cm^─2. The experimental value of the elastic limit is much smaller than suggested by the above equation given as τc  ≈ G/6, since G ≈ 10¹¹ dynes cm─2. The theoretical shear stress τc  is of the 10^─2 order of 10 dynes cm which is several orders of magnitude larger than the observed values

for pure crystals. However, the experimental values for the maximum resolved shear stress as required to start the plastic flow in metals were then reported to be approximately of the order of 10─4 G, which is not in agreement with the theoretical results obtained on the basis of Frenkel model. The recent work has shown that for bulk copper and zinc plastic deformation starts at stresses of the order of 10─9 G.

The observed low values of the shear strength cannot be explained without the presence of imperfections that can act as source of mechanical weakness in real crystals. The Frenkel model considers the crystals to be perfect. In 1934, Orowan, Polanyi and Taylor proposed the existence of a new defect viz., edge dislocation, while in 1939 it was Burgers who proposed existence of another type of dislocation known as screw dislocation to explain the discrepancy between theoretically estimated and experimentally measured values of τc.It is now well established that a type of defect known as dislocation exist in crystals which is responsible for slip on application of small stresses to the crystal.

16.3  Edge Dislocation

Taylor and Orowan were the first who interpreted plastic deformation of solids by introducing the concept of edge dislocation. A dislocation is a line imperfection forming the boundary within the crystal of a slipped area. Ideally, a crystal, on application of stress, deforms when planes of atoms slide over one another just like cards in a deck. We say one unit of slip on a slip plane has occurred if every atom on one side of that plane has moved into a position which was originally occupied by its nearest neighbour in the direction of slip. As such, the unit slip does not disturb the crystal perfection. However, in real crystals slip is not uniform over a slip plane. Suppose unit slip has occurred over only a part of a slip plane as shown in the figure 16.3, while the rest of the plane has remained unslipped.

Figure 16.3: unit slip over only a part of slip plane LMNO.

The figure shows unit slip has occurred over the area LMNO.LO is a line which divides slipped and the unslipped part and so forms the boundary within the crystal. The boundary LO of the slipped area is an edge dislocation. The edge dislocation LO is normal to the slip vector. In fact, both parts of the crystal are displaced relative to each other by an amount whose magnitude and direction is given by the vector b called the Burger’s vector which is perpendicular to the dislocation line .This kind of dislocation is called an edge dislocation. The dislocation does disturb the crystal perfection. If one looks at the atoms as viewed along LO, the situation appears to be as shown in figure 16.4.The dislocation is a region where one notices a severe atomic misfit and the atoms are not properly surrounded by their neighbours; the exact arrangement of atoms at the centre of dislocation not being exactly known. It may, however, be visualized as being an edge of extra half plane of atoms. Edge dislocation is symbolized by the sign┴  as shown in figure16.4. The dislocation can be made experimentally observable by several techniques like chemical etching, thermal etching, decoration, x-ray diffraction topography and electron microscopy.

Figure 16.4: Edge of an extra half plane of atoms

16.4 Screw Dislocation

Burger’s introduced the idea of screw dislocation in 1939. It is also called Burger’s dislocation. In this case the displacement b (Burgers vector) is parallel to the dislocation as shown in figure 16.5. In this type of dislocation the crystal is not composed of a set of parallel planes but consists of a screw ramp which rises along the dislocation line.

Figure 16.5: Illustration of screw dislocation

One way is to visualise formation of a screw dislocation by referring to figure 16.6; the slip that results into creation of a screw-type dislocation. Here also, the figure shows unit slip has occurred over the area LMNO. LO is a boundary within the crystal of the slipped area LMNO. The screw dislocation LO is parallel to the slip vector.

To understand why this dislocation is called screw dislocation. Let a sharp cut be made part way through a crystal and press the crystal so that one side of the cut goes down by one atomic spacing with respect to the other as shown in figure 16.7.The line LO along the edge of the cut is by definition screw dislocation. The line LO is parallel to the line MN which is parallel to the slip vector. As a result of this type of distortion the crystal is not made up of parallel atomic planes one above the other; in fact it is a single atomic plane in the form of a helicoid or a spiral ramp.

Figure 16.6: Unit slip having occurred over LMNO has resulted into screw dislocation.LO is screw dislocation which is parallel to the slip vector

Figure 16.7: Illustration of screw dislocation

16.5 Mixed Dislocations

The two types of dislocations viz., edge and screw dislocations, described so far are straight dislocation lines. The boundary separating slipped from the unslipped part is straight in both the types of dislocations. If the boundary between the deformed and undeformed part of the crystal has a curved form of the type shown in figure 16.8, then at L the dislocation is parallel to the slip vector and as such is in the screw orientation. Near about the point N the dislocation is in the edge orientation. This type of dislocation has both the components i.e. edge and screw dislocations and so is called as mixed dislocations.

Figure 16.8: Illustration of screw dislocation

References.

1. Read,W.T.: “ Dislocations in crystals”, McGraw-Hill, N.Y.,1953.
2. Cottrell, A.H. “ Dislocation and Plastic Flow In Crystals”,OxfordUniv.Press, N.Y.(1953)
3. Shockley, W, Holloman,J; Maurer,R. and Seitz,F.(Eds)” Imperfections in Nearly Perfect
4. Kittel,C.: “ Introduction to Solid State Physics”, Wiley, N.Y.,1971.
5. Dekker,A.J.: “ Solid State Physics”, Macmillan,London,1958.

Interesting & Informative Reading.

1. Gatos,H.C. “ Etching Phenomena and the Study of Dislocations” : in Crystal Growth & Characterization( Eds. Ueda,R. &Mullin,J.B.)Proc. ISSCG2 Spring School,Japan, 1974. North-Holland Publishing Co.,Amsterdam/AmericanElsevierPub.Co. ,N.Y., 1975.