Today I heard Prof. Chris Macosko on the Effects of processing flows in polymer-polymer adhesion; if you are interested in such problems, here is some information and some publications from Prof. Macosko’s group page.
Archive for October, 2006
- using Euler’s polyhedron formula,
- assuming that triple junctions are points at which the grain boundaries make 120 degrees with each other (in other words, achieve equilibrium), and,
- in systems with isotropic interfacial energy,
von Neumann showed that if the curvature of any given domain wall is a constant (as it is in soap froths), then six sided grains are stable; the grains with sides less than 6 will shrink while ones with sides greater than 6 will grow. The contribution of Mullins is to show that this result holds even if the curvature is not a constant (mean curvature is what matters) on any given domain wall (as in the case of grain boundaries). This result can also be derived using Gauss-Bonnet theorem. This is a purely topological result.
Apparently, there had been many attempts to generalise von Neumann-Mullins to 3D; however, none of these attempts have been successful in obtaining an exact result. Prof. Srolovitz and R D MacPherson have not only managed to generalise the results to 3D, but to any higher dimension. Unfortunately, I am not able to get any reference to their work on the net. So, here is my attempt at summarising the idea:
- Generalise 2D von Neumann-Mullins to multiply connected domains;
- Volume integrate the result; this is equivalent to considering all possible sections of a 3D structure, and doing a von Neumann-Mullins analysis on each, and putting the results together.
Apparently doing all this introduces a natural measure of length called mean width (which is a Hadwiger measure); and, using such a measure, von Neumann-Mullins can be generalised to N-dimensional domain wall networks and the 2D and 3D networks are just the special cases of such a general result. And, as it turns out, in 3D (and in higher dimensions), curvature driven growth is not purely topological.
Finally, ‘curvature’ in this post means mean curvature (and not Gaussian); by the way, while you are at it, take a look at this wiki page on curvature; and, this history of curvature page which it links to. Have fun!
Software — that is the question; here is a review of a book which tells why “Not”.
Here is an article from LA Times about the generation with falling testerone levels:
People in their 30s and 40s can and do get pregnant—my mom was 39 and my dad 49 when I, swimming past various barrier methods of birth control, arrived to completely ruin their retirement. Generally, though, the older you get the harder it is to run off a Xerox. And yet, I don’t feel alone or unfairly singled out. Even younger couples are struggling to conceive, as fertility rates in Western industrialized societies plummet for reasons that are not well understood.
Link via The Frontal Cortex. The article is hilarious. Don’t miss it.
The relative misorientation between two grains across a grain boundary is the rotation of one of the crystals such that it is brought into the same orientation as the other; such a rotation is completely described by the rotation angle and the axis about which the rotations are carried out.
The rotations can be decomposed into a tilt and a twist rotation. Tilt is a rotation about an axis that lies in the plane of the boundary. Twist is a rotation about an axis that is perpendicular to the boundary plane. This is the tilt-twist description of the grain boundaries.
Consider a tilt boundary; that is a boundary which is completely described by a tilt angle (say, theta). Further, let the tilt angle be small (i.e., less than 15 deg). The energy (E_tilt_boundary) of such a tilt grain boundary is given by by the Read-Shockley equation:
E_tilt_boundary = C theta (A – ln(theta))
where, C and A are constants.
All this is well known and is described in many text books. However, what is not described is the following:
- How far is this description of tilt boundaries is correct? Are there experimental evidence to show that Read-Shockley equations hold?
- What is the dependence of the grain boundary energy on the boundary plane? In other words, will a tilt a boundary about the <111> axes and <100> axes in a cubic material, for example, have the same energy?
These two questions, and one more (namely, the dependence of the relative mobility of a low-angle grain boundary on the boundary plane) are answered in this paper by Yang, Rollet and Mullins:
- The variation of low-angle grain boundary energy does indeed follow the Read-Shockley expression;
- There is a small dependence of energy on the misorientation angle, with axes closer to <100> having the highest energy and those close to <111> the lowest; and,
- The boundaries with misoreintation closer to <111> are relatively more mobile than those closer to <100>.
These, and other information on grain boundary structure and energetics are also described in Recrystallisation and related annealing phenomena by Humphreys and Hatherly.
The listing reminded me of my own reading when I was kid; though I was born and brought up in the later part of the twentieth century our home library (as well as the local library) stacked many of these authors. That is the funny thing about books; it is more like The Shadow of the Wind of Carlos Ruiz Zafon; even if one person enjoys reading a book, it lives forever. Scott (my father’s favourite) and Dickens (my grandfather’s favourite) figure in the list; their discussing these two authors during dinner time made me read Ivanhoe in the original when I was in high school; and, when I was in College my English professor marked my essay to be “a bit Victorian” in style. I discovered Arthur Conan Doyle on my own, and read Emma and Pride and Prejudice once every summer (for four or five consecutive years). The Scott-like novels of Kalki (and later Sandilyan) were also my favourites (as well as my grandmother; she remembered all the novels of Kalki, and the people who read those novels to her, and even what she was doing when it was being read to her, vivdly). I read the popular twentieth century authors much, much later; P G Wodehouse, for example, I read when I was in college, since I read an article in Current Science which mentioned him to be one of S Chandrashekhar‘s favourite authors.
At this point, I should also mention that reading (apart from listening to the radio) was the only pastime you could have in those parts of the country in those days. So, most of us kids read to kill time. But, as they say, the past is a different country; they do things differently there.