## Grain boundaries and Nye dislocation density tensor

### July 2, 2007

Title: Stress-free states of continuum dislocation fields: Rotations, grain boundaries, and the Nye dislocation density tensor

Authors: Surachate Limkumnerd and James P Sethna

Abstract:

We derive general relations between grain boundaries, rotational deformations, and stress-free states for the mesoscale continuum Nye dislocation density tensor. Dislocations generally are associated with long-range stress fields. We provide the general form for dislocation density fields whose stress fields vanish. We explain that a grain boundary (a dislocation wall satisfying Frank’s formula) has vanishing stress in the continuum limit. We show that the general stress-free state can be written explicitly as a (perhaps continuous) superposition of flat Frank walls. We show that the stress-free states are also naturally interpreted as configurations generated by a general spatially dependent rotational deformation. Finally, we propose a least-squares definition for the spatially dependent rotation field of a general (stressful) dislocation density field.

### 4 Responses to “Grain boundaries and Nye dislocation density tensor”

1. Biswajit Says:

Guru,

I was unable to figure out what was exciting about this paper. Could you add some explanatory notes and your take on what’s great and what’s not in the paper?

Thanks.

2. Guru Says:

Dear Biswajit,

My interest in the paper is from the point of view of developing a phase field model for modelling grain boundary migration using Nye dislocation density tensor as the order parameter. From that point of view, the last example that this paper discusses in Appendix B, namely, to show that the boundary of a circular grain embedded in another one can be represented by a superposition of flat cell walls sounded interesting to me.

However, as you might have noticed, making a phase field model not only requires a description of the boundary, but also a functional for energy in terms of the order parameter. My idea is that, if one can come up with the configurational force on such a curved boundary, then an evolution equation can be written down, which can then be used to study grain growth/shrinkage and grain boundary migration.

I am aware of several studies which talk about the description of grain boundaries using dislocation based models; Cermelli and Gurtin (Int. J Sol. Struct., 39, 2002, pp. 6281-6309), several papers in Volume 87, Issues 8&9, 2007 of Philosophical Magazine (including one by Amit Acharya), and this paper of Cahn, Mishin and Suzuki (pdf), for example. However, I do not understand these papers in their entirety; nor do I see a discussion on the energetics associated with each dislocation description in a fashion that would help me come up with a free energy functional.

I would love to hear any comments/suggestions/ideas you might have!

3. Biswajit Says:

Guru,

I can’t claim to understand much, if any, of the details. If the stress at the cell walls is zero in the continuum limit I don’t see how you can have a free energy that drives the motion of the walls. So, what is driving the motion in your case? Am I getting the whole idea wrong?

You could ask Amit for some pointers. He’s thought about similar issues for a while now and will definitely have something nontrivial to say.

Biswajit

4. Guru Says:

Dear Biswajit,

Thanks a lot for the comments and pointers. As far as I understand, the stress fields vanish far away from the boundary deep into the grain interior when the dislocations form a grain boundary. However, closer to the boundary, there should be a finite contribution–I think Sutton and Balluffi show that the stress fields decay exponetially away from the grain boundary.

In this paper, the argument seems to be that for small $\theta$, the misorientation across the boundary, the grain boundary energy goes as $b \theta \ln{\theta}$, which, in the limit $b \rightarrow 0$, vanishes. And, the Nye dislocation density tensor gives the relative rotation of the grains. However, since different planes of the grains are in contact with each other due to such a rotation, there would be some excess energy associated with the grain boundary, I think. Having said that, I do have difficulty is imagining a circular grain boundary made up of dislocations, whose movement is driven purely by curvature; and, I do not have any idea how to model that either.

As you suggest, probably Amit Acharya will be able to give some comments!