Classics in Materials Science: Shockley and Read’s dislocation models of crystal grain boundaries

Any crystalline material contains many defects. Depending on the topology of the defects, they can be classified as point, line, surface and volume defects. Of these, only point defects are equilibrium defects — that is, at any temperature above absolute zero, it is energetically favourable for the system to create these defects — a specific number of them, which number, varies from system to system and within a system with temperature. An example of a line defect is the dislocations; an example of a surface defect is the grain boundary.

In some circumstances, the surface defect, namely, the grain boundary, can be considered to be made up of line defects, namely, the dislocations. Though such dislocation models of grain boundaries date back to early 1940s, Read and Shockley, in a classic paper in 1950 [1], used such a model to calculate the energy of a grain boundary — which continues to be a landmark study. In this post, I will discuss this paper of Read and Shockley.

As the regular readers of this series of posts might have noticed, it is really difficult to credit ideas to any one person or group. This paper of Read and Shockley is no exception. Though the phrase “dislocation models of grain boundaries” conjures the names of Read and Shockley to the minds of materials scientists, they are not the first ones to propose such a model. In the very first sentence of their paper, Read and Shockley refer to the papers of J M Burgers [2] and W L Bragg [3]. In fact, in Burgers’ paper, there is a very suggestive drawing that I produce below, which clearly shows what he had in mind — quite close but not exactly what Read and Shockley discuss in their paper, resulting in the credit being given to Read and Shockley with Burgers and Bragg being the precursors. By the way, this Burgers is the father and not the uncle of dislocations!

The transition lattice models of Burgers

The transition lattice models of Burgers

Getting back to Read and Shockley, their main contribution is to use such a dislocation based model to calculate some quantity which can be measured experimentally: here is the last sentence of the first paragraph of their paper:

Of special interest are grain boundaries between crystallites with a small difference in orientation; we shall show that for these the grain boundary energy can be determined as a function of the angle of misfit and the orientation of the boundary.

With this in mind, they brush aside general results that they could have tried to prove, but move on to calculating the energies pretty quick; here are a couple of sentences from the second paragraph of their paper:

It is always possible, at least for small orientation differences, to join the two grains with a suitable array of dislocations lying on the prescribed plane of the grain boundary. We shall not prove this general result here, however, but shall deal in detail with grain boundary models for certain specially simple cases, pointing out how the results may be extended so as to apply more generally.

So, what are these simple cases? Read and Shockley begin with the following assumptions/givens:

  • Simple cubic lattice;
  • The grains are rotated about z-axis making the problem essentially one of 2D;
  • Isotropic elasticity; and,
  • Of course, small misorientations.

The idea behind the calculation is summarised neatly in their first figure; compare and contrast this with that of Burgers shown above!

Read and Shockley's description of grain boundary in terms of dislocations

Read and Shockley's description of grain boundary in terms of dislocations

Once you have such a model, the grain boundary energy is but the energy of arrays of dislocations — which can be calculated in several ways — Read and Shockley list three in the paper, and derive their famous expression.

The now famous Read-Shockley formula

The now famous Read-Shockley formula

And, immediately, of course, they compare their analytical results with the then available experimental results of C G Dunn:

Read-Shockley comparison of analytical and expeirmental results

Read-Shockley comparison of analytical and expeirmental results

Thus, within the first few pages of their 15 page long paper, Read and Shockley changed our way of understanding and looking at grain boundaries for ever. The rest of the paper goes on to treat other interesting aspects of their work; one of them, which I liked a lot, is their explanation of what was then known as “veining” in aluminium in terms of these dislocation arrays!

Lacombe observed that in a single crystal grain there are faint lines or veins which are revealed as rows of separated etch pits under suitable etching conditions. He has also observed that there are small differences in orientation between the regions separated by the veins. We have proposed that each etch pit originates at a dislocation, where the free energy of the stressed material will be somewhat higher than elsewhere; the pit then grows to a large size so that it is observed optically.

Read and Shockley suggest that the distances between these etch pits can be compared with their theory knowing the misorientation.

The rest of the paper looks at large angle grain boundaries and the dynamics of the grain boundary model — with several appendices towards the end where all the mathematical details of the calculations are fleshed out.

The classic

The model of Read and Shockley and their expression for the energy of small angle grain boundaries is of fundamental importance; for example, when one models grain boundaries and their migration, the first step is to check that the model gives the energetics according to the predictions of Read and Shockley. Moreover, at that point of time when it was proposed, their model also lead to some nice experiments by Aust and Chalmers [4] on the measurement of dihedral angles at triple junctions, which, as Cahn notes in his book [5], would not have been carried out but for the theoretical underpinning provided by Read and Shockley. That the results of Aust and Chalmers is not the final word was shown much later in another classic (that is called “one of the most elegant experiments in materials science” by Cahn), the discussion of which we will defer for some other time. In the meanwhile, have fun with Read and Shockley.

References

[1] W T Read and W Shockley, Dislocation models of crystal grain boundaries, Physical Review, Vol. 78, No. 3, pp. 275-289, 1950.

[2] J M Burgers, Geometrical considerations concerning the structural irregularities to be assumed in a crystal, Proceedings of Physical Society, Vol. 52, pp. 23-33, 1940.

[3] W L Bragg, The structure of cold-worked metal, Proceedings of Physical Society, Vol. 52, pp. 105-109, 1940.

[4] K T Aust and B Chalmers, Surface energy and structure of crystal boundaries in metals, Proceedings of the Royal Society of London A. Mathematical and Physical Sciences, Vol. 204, No. 1078, pp. 359-366, 1950.

[5] R W Cahn, The coming of materials science, Pergamon, 2001.

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About Guru

I am an Assistant Professor in a Metallurgical Engineering and Materials Science Department; I also pursue research in the broad area of computational materials science.
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2 Responses to Classics in Materials Science: Shockley and Read’s dislocation models of crystal grain boundaries

  1. Pingback: Revisiting the classics: Read-Shockley model of grain boundaires « Entertaining Research

  2. Surendra Kumar M says:

    sir, This is in regard to the dislocations forming grain boundaries in a real crystal and dislocations originated within the grains. What is the difference between them? I mean in terms of there size, type, effect on crystal etc.

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