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Dislocations 2

This piece is a continuation of my previous article on the same subject. The aim of this article is to overcome some of the shortcomings of the previous articles in terms of the lack of illustrations (which we all know are more effective, since “A picture is worth a thousand words.”) and also, I will try to go beyond the level at which the other writing stopped.

First, I would like to explain, with the help of pictures, the structures of the edge and the screw dislocation. The pictures below show what they look like.




Fig 1 An Edge (top) and a Screw Dislocation

The image on top is an edge dislocation, while the one on the right is a screw dislocation. These are two extreme kinds of dislocations, other dislocations, and most dislocations we observe are of a mixed character, that is they have both an edge and a screw component.
 
As you can easily observe in the pictures, the edge dislocation is formed from the insertion of an extra half plane into the crystal lattice, while the screw dislocation is made by the relative shearing of the atoms within the crystal.
 
As  is  evident  from  the  pictures,  the  disturbance  in  the  atomic  positions  is  localized,  and  the crystal  adopts  the  normal  atomic  position  at  a  large  distance  from  the  dislocation.  How  does  a dislocation then arise within a crystal? Well, in our universe, where the laws of physics as we know them are  valid,  any  system  reaches  equilibrium  when  it  attains  a  state  of  minimum  potential  energy.  The presence  of  potential  energy  in  a  system  indicates  an  ability  to  do  work,  and  whenever  that  ability  is present, the system does work. However, there are present, almost always, certain physical, and thus energetic barriers to the doing of this work. If the system has adequate potential energy to overcome this  barrier,  it  is  able  to  do  work.  Otherwise,  it  remains  in  its  state  of  equilibrium.  The  formation  of  a dislocation  is  also  a  method  of  minimizing  the  energy  of  the  crystal  system.  When  the  crystal  formed from a melt, or some other amorphous or crystalline form, there may have been some stresses. If the formation of a dislocation would relieve this stress, then a dislocation would form, and thus reduce the total potential energy of the system.

In my previous article, I also mentioned that the presence of dislocations causes a reduction in the strength of metals, and thus renders them ductile enough that we may form them easily. I will now explain, partly though analogy, how this happens. Even though the dislocation was formed in order to reduce  the  strain  energy  of  the  crystal,  the  dislocation  has,  associated  with  itself,  a  certain  amount  if strain energy. A perfect crystal lattice, on the other hand has none. Therefore, when an external stress is applied, a perfect crystal lattice requires a higher stress to reach the energy level at which deformation can start occurring. However, the presence of the dislocation increases the internal strain energy of the crystal,  and  a  relatively  small  stress  is  enough  to  start  the  movement  of  atoms  within  the  crystal  to cause deformation.
 
The previous paragraph may be somewhat difficult to follow for those readers who are not that comfortable with this concept of potential and strain energy. If you are such a reader, then I hope that this  analogy  will  serve  to  form  a  good  picture  of  the  situation.  In  crystals,  the  mechanism  by  which deformation occurs is known as slip. The simple reason for this name is that we consider the dislocations to  slip  over  a  certain  plane  and  this  results  in  an  overall  deformation.  To  explain  with  the  help  of  an analogy,  take  the  example  of  a  heavy  carpet  lying  on  the  floor.  In  order  to  move  this  carpet  by  two inches, you try to pull it one of its ends. From common experience we know that this is quite difficult because of the friction that the carpet has with the floor. Moving it by even a small distance requires a large amount of effort. Instead of pulling it so, you could also just pinch the carpet a little bit and create a small hump which extends throughout the length of the carpet, parallel to one of its sides. Now you may push this hump easily from one side to the other, and if the span of the hump was two inches, then viola!  Your  carpet  has  moved  by  the  required  two  inches,  and  you’re  not  too  tired  at  the  end  of  the exercise too. The crystal is like the carpet and the dislocation is exactly like this hump. The plane of the crystal over which the slip occurs is called the slip plane.
 
The  above  analogy  is  also  very  useful  in  forming  a  very  handy  definition  of  a  dislocation.  A dislocation is also defined as the boundary between a slipped and unslipped region in a crystal. By using this definition, we arrive at several useful properties of a dislocation. One of the very useful results of this definition is that a dislocation cannot end abruptly within a crystal. It must end either at a foreign particle within the crystal, or at the grain boundary, or even at another dislocation, but never abruptly within  the  crystal.  The  dislocation  may  even  form  a  loop,  and  these  dislocations  loops  are  observed quite  often.  These  loops  often  act  as  sources  of  more  dislocations.  The  most  commonly  quoted dislocation  loop  is  the  Frank‐Read  source,  which  keeps  generating  more  dislocations,  while  it  remains intact.
Most of the other results are out of the scope of this article, but the interested reader may refer to the book by Johannes and Julia Weertman, “Elementary Dislocation Theory” to learn more. For the more mathematically inclined reader, I would suggest “Theory of Dislocations” by John Price Hirth.


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Adhish Majumdar,
Mar 11, 2011, 1:18 AM
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