September 18th Probability Calculations

A few days ago, I posted a picture of abnormal grain growth (AGG) in the electrodeposited nanocrystalline nickel I've been working with. This picture, is not only extremely pretty, but features two unique features in the field of AGG:

• The abnormal grains are extremely faceted with {100} type planes
• The abnormal grains exhibit a high fraction of Sigma3 boundaries between them

The second point introduced an interesting mathematical calculation for us to work on. For these bi-crystals of abnormal grains that exhibit a Sigma3 boundary between them, do these arise based on random probability? Or do they occur because of some other hidden mechanism behind them...

Optical Image of Nano-Ni

To perform this calculation, we must first discretize the number of possible misorientations. Based on this value, we then determine the number of possible twin-related misorientations (note this includes both coherent and incoherent). Calculations have been done by previous researchers, which show that the probability of finding a twin-related grain is approximately 1/50, or 0.2%

Our next approach then is for a given volume of space (although I've started this calculation with just area first), determine the number of starting grains. We pick one grain, and allow it to grow abnormally. While in most cases, we assume that the grain growth occurs by consuming one grain at a time, I've taken it a "step" further, and assume that the grain grows n^2 to maintain its faceted nature.

For each step that the grain grows, it meets an increased number of neighbors. At the same time, as the grain grows, the number of grains encountered in the starting system decreases. Therefore the probability of meeting a twin-related grain (originally 0.2%) increases overtime with each growth step. When a grain reaches a certain size then, the probability that it should have met another related grain with a twin orientation more or less exceeds one.

We can approach this calculation in two ways, one is to determine the number of steps necessary before this occurs, and the second is to determine the volume necessary for a given abnormal grain size. The second has a value that can actually be determined, whereas the first shows a trend of probability over time. The preliminary calculation does seem to "agree" with some of the current experimental results.

An issue that becomes immediate obvious based on the picture to the left, is that the abnormal grain growth is NOT homogenous. In fact the grains are coarser towards the bottom and nucleate extensively around the crack.

This would seem to imply that the nucleation event and encounter with a twin-related grain is not purely statistical based on the size gradient in the sample. Unfortunately, no answer is offered at the moment, but does raise some interesting questions on how we may manipulate the "degree" of abnormal grain growth in our samples.