Dislocation Models of Grain Boundaries

We begin our geometric discussion with the case of pure tilt boundaries on the grounds that the connection between dislocations and the boundary is most evident visually in this case. As was discussed in some detail in chap. 9, a tilt grain boundary is characterized by a simple misorientation between the two grains as was discussed in chap. 9. The intent of this section is to illustrate that a simple superposition of the elastic displacements implied by the Volterra solution for straight dislocations discussed in chap. 8 can lead to exactly the same type of misorientation. 

For simplicity, consider the isotropic linear elastic description of the problem. The total stress is obtained by summing up the contributions of the constituent dislocations. If we consider the shear stress components for concreteness, then the total stress for the boundary with dislocation spacing D is 

All this formula instructs us to do is to take the contribution to the total stress of each dislocation using the results of eqn (8.38) and to add them up, dislocation by dislocation. We follow Landau and Lifshitz (1959) in exploiting the Poisson summation formula to evaluate the sums. Note that while it would be simple enough to merely quote the result, it is fun to see how such sums work out explicitly. If we use dimensionless variables a = x/D and yS = y/D, then the sum may be rewritten as 

The next step in the thrust to beat the sum into submission is to recognize that if 

With these clever algebraic machinations behind us, the original problem has been reduced to that of evaluating the sum J a, ). Recall that the Poisson summation formula tells us 

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W.T. Read and W. Shockley. Dislocations models of grain boundaries. In Imperfections in Nearly Perfect Crystals. John Wiley Sons, New York, 1952. 

The structural unit model has been used successfully to predict the structures of grain boundaries in perovskite structured SrTiOs bicrystals [11.31-11.34]. The structural units observed for symmetric SrTiOs [001] tilt boundaries are shown in Fig. 11.8. In a similar manner to the isolated dislocation cores in YBCO, the structural units also appear to contain atomic positions where the cations are too close together. Again, depending on the structural unit, the close separation of the atomic columns can occur for either of the sub-lattice sites,... 

Read, W.T. and Shockley, W. (1950) Dislocation models of crystal grain boundaries, Phys. Rev. 78(3), 275. A classic readable paper with great diagrams. 

Figure 8.10. Bubble model showing grain boundary running diagonally across center of field and dislocation just above center of field.

The operation of ceria under catalytic conditions can place the material under severe mechanical duress and therefore it is important to understand the behaviour of the material under operational conditions, such as vibration, friction, thermal cycling, etc. The mechanical properties of the material may prove pivotal. In particular, it is well known that microstructural features, such as dislocations, defects and grain boundaries, govern the mechanical properties and result in the measured mechanical strength being considerably lower than that predicted based upon the pristine, defect-free material. If one is to simulate the mechanical properties directly then atomistic models, which include all such microstructural features including their synergy of interaction, are needed. And while there are considerable efforts focused in this direction. 

Fig. 16. Dislocation models of small-angle grain boundaries (a) simple tilt boundary, (b) simple twist boundary. The former is an array of edge dislocations, the latter is an array of screw dislocations. From Read. ...

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