## Some notes on GB migration — I

### March 27, 2007

The following are the notes of Chapter 3 of Grain boundary migration in metals: Thermodynamics, kinetics, applications by Guenter Gottstein and Lasar S Shvindlerman, CRC Press, New York 1999.

Grain boundary displacement vs. non-zero diffusive flux across a boundary

Grain boundaries are structures that separate regions of the same phase and crystal structure which differ in their orientation. Thus, a displacement of grain boundary corresponds to the growth of one at the expense of another grain. This displacement also distinguishes the grain boundary migration from the shrinkage/growth of grains due to a diffusive motion.

Consider a case in which a non-zero flux across the grain boundary leads to a growth of a grain at the cost of the other. In such a case, with respect to an external frame of reference, the grain boundary remains stationary, while the opposite faces of the grains move. On the other hand, during the grain boundary displacement, when atoms from one grain make a jump to another, they change their orientation to that of the grain to which they are jumping to. Thus, there is a non-zero net exchange of lattice sites across the boundary, which results in the grain boundary displacement.

Do grain boundaries have specific properties?

Grain boundaries are usually defined to be layers of defined thickness separating phases of different orientations and the grain boundary phase is assumed to have an associated energy, entropy or mobility. However, in reality grain boundaries are only regions of discontinuities in crystal orientation; thus, for example, depending on the constraints of the adjacent crystalline surfaces, the grain boundary mobility can be different for motions in opposite directions.

Is there a theory of grain boundary migration?

No; all theories are attempts to describe grain boundary motion in terms of the rate of atoms crossing the grain boundary with net energy gain.

Consider a very narrow–single layer wide– grain boundary; thus, each atomic jump displaces the boundary by the diameter of an atom $b$. The grain boundary velocity $v$ in such a system is given by $v = b (\Gamma_{l}-\Gamma_{r})$ where $\Gamma_{l/r}$ are the jump frequencies in the respective directions.

If there is no difference in the Gibbs free energy of the two crystallites, $\Gamma_l - \Gamma_r = 0$, and hence there is no boundary migration. This leads us to the question as to what leads to such differences in Gibbs free energy; in other words, what is the driving force for GB migration? That question has to wait for the next post.