In some circumstances, the surface defect, namely, the grain boundary, can be considered to be made up of line defects, namely, the dislocations. Though such dislocation models of grain boundaries date back to early 1940s, Read and Shockley, in a classic paper in 1950 [1], used such a model to calculate the energy of a grain boundary — which continues to be a landmark study. In this post, I will discuss this paper of Read and Shockley.

As the regular readers of this series of posts might have noticed, it is really difficult to credit ideas to any one person or group. This paper of Read and Shockley is no exception. Though the phrase “dislocation models of grain boundaries” conjures the names of Read and Shockley to the minds of materials scientists, they are not the first ones to propose such a model. In the very first sentence of their paper, Read and Shockley refer to the papers of J M Burgers [2] and W L Bragg [3]. In fact, in Burgers’ paper, there is a very suggestive drawing that I produce below, which clearly shows what he had in mind — quite close but not exactly what Read and Shockley discuss in their paper, resulting in the credit being given to Read and Shockley with Burgers and Bragg being the precursors. By the way, this Burgers is the father and not the uncle of dislocations!

The transition lattice models of Burgers

Getting back to Read and Shockley, their main contribution is to use such a dislocation based model to calculate some quantity which can be measured experimentally: here is the last sentence of the first paragraph of their paper:

Of special interest are grain boundaries between crystallites with a small difference in orientation; we shall show that for these the grain boundary energy can be determined as a function of the angle of misfit and the orientation of the boundary.

With this in mind, they brush aside general results that they could have tried to prove, but move on to calculating the energies pretty quick; here are a couple of sentences from the second paragraph of their paper:

It is always possible, at least for small orientation differences, to join the two grains with a suitable array of dislocations lying on the prescribed plane of the grain boundary. We shall not prove this general result here, however, but shall deal in detail with grain boundary models for certain specially simple cases, pointing out how the results may be extended so as to apply more generally.

So, what are these simple cases? Read and Shockley begin with the following assumptions/givens:

• Simple cubic lattice;
• The grains are rotated about z-axis making the problem essentially one of 2D;
• Isotropic elasticity; and,
• Of course, small misorientations.

The idea behind the calculation is summarised neatly in their first figure; compare and contrast this with that of Burgers shown above!

Read and Shockley's description of grain boundary in terms of dislocations

Once you have such a model, the grain boundary energy is but the energy of arrays of dislocations — which can be calculated in several ways — Read and Shockley list three in the paper, and derive their famous expression.

The now famous Read-Shockley formula

And, immediately, of course, they compare their analytical results with the then available experimental results of C G Dunn:

Read-Shockley comparison of analytical and expeirmental results

Thus, within the first few pages of their 15 page long paper, Read and Shockley changed our way of understanding and looking at grain boundaries for ever. The rest of the paper goes on to treat other interesting aspects of their work; one of them, which I liked a lot, is their explanation of what was then known as “veining” in aluminium in terms of these dislocation arrays!

Lacombe observed that in a single crystal grain there are faint lines or veins which are revealed as rows of separated etch pits under suitable etching conditions. He has also observed that there are small differences in orientation between the regions separated by the veins. We have proposed that each etch pit originates at a dislocation, where the free energy of the stressed material will be somewhat higher than elsewhere; the pit then grows to a large size so that it is observed optically.

Read and Shockley suggest that the distances between these etch pits can be compared with their theory knowing the misorientation.

The rest of the paper looks at large angle grain boundaries and the dynamics of the grain boundary model — with several appendices towards the end where all the mathematical details of the calculations are fleshed out.

The classic

The model of Read and Shockley and their expression for the energy of small angle grain boundaries is of fundamental importance; for example, when one models grain boundaries and their migration, the first step is to check that the model gives the energetics according to the predictions of Read and Shockley. Moreover, at that point of time when it was proposed, their model also lead to some nice experiments by Aust and Chalmers [4] on the measurement of dihedral angles at triple junctions, which, as Cahn notes in his book [5], would not have been carried out but for the theoretical underpinning provided by Read and Shockley. That the results of Aust and Chalmers is not the final word was shown much later in another classic (that is called “one of the most elegant experiments in materials science” by Cahn), the discussion of which we will defer for some other time. In the meanwhile, have fun with Read and Shockley.

References

[1] W T Read and W Shockley, Dislocation models of crystal grain boundaries, Physical Review, Vol. 78, No. 3, pp. 275-289, 1950.

[2] J M Burgers, Geometrical considerations concerning the structural irregularities to be assumed in a crystal, Proceedings of Physical Society, Vol. 52, pp. 23-33, 1940.

[3] W L Bragg, The structure of cold-worked metal, Proceedings of Physical Society, Vol. 52, pp. 105-109, 1940.

[4] K T Aust and B Chalmers, Surface energy and structure of crystal boundaries in metals, Proceedings of the Royal Society of London A. Mathematical and Physical Sciences, Vol. 204, No. 1078, pp. 359-366, 1950.

[5] R W Cahn, The coming of materials science, Pergamon, 2001.

It’s hosted on Ning.com, which offers a fairly complete suite of social networking features: groups, forums, events, and yes, blogs!

Over there, Materialia Indica — perhaps we should call it MI 2.0 — has a chance to become a true community, a network of folks with a common interest: materials engineering research and education in India.

So, come on over there and join us!

Membership in MI 2.0 is open to everyone — students, faculty, researchers, engineers — with an interest in materials engineering research and education in India. And everyone is equal when it comes to contributing to the site; more important than the lack of hierarchy in the forums is the fact that everyone has a blog! All the blog posts get aggregated on the main page.

* * *

When we launched Materialia Indica almost four months ago, we were certain that it should be an open forum, with participation by as many people as possible. However, due to my ignorance and/or inexperience (and against very good advice from Phanikumar), I ended up choosing the blog format which, it is now clear, is totally inappropriate for the purpose of networking and community-building. There’s no way the blog format could have allowed Materialia Indica to become something like iMechanica, which has grown to over 9000 users, each with posting privileges on his/her own blog in that Drupal-based site.

So, what about Materialia Indica, the blog? Now that the networking and India-centric aspects have moved to the second home, this blog can now become a theme-based blog with a focus on materials science and engineering.

And I think that’s a fine goal for this blog. And I hope you’ll all agree …

The conventional analytical treatment of actual systems can in fact only be successfully performed in the simplest cases. It fails when a more or less realistic model of multiphase alloys in considered. As a matter of fact, computer simulation is a new approach to the probelm and is applicable to real alloys. The computer makes it possible to solve problems that have been considered unsolvable. The use of high-speed computers for modeling the processes occurring in alloys cannot be obviated, and I believe that computer-simulated research will form a new field in the material sciences.

Even the first attempts to simulate, with the help of computers, the martensitic transformation and the strain-induced coarsening in decomposed alloys using but idealized crude models gave an encouragingly good description of those processes, and the results obtained are sometimes in excellent agreement with electron microscopic observations. We may hope that the use of more realistic (and therefore more elaborate) models will make it possible to come close to theoretically predicting the structures of two-phasealloys applied in industry.

There is another problem that should be mentioned. The physical processes occurring in alloys are affected by varied factors that sometimes conceal the important characteristics of the phenomenon. Eliminating them may be a difficult problem which can hardly be solved in all cases. From this standpoint computer simulation serves as an “experiment” carried out under ideal conditions in the absence of any interfering factors.

– A G Khachaturyan, Theory of structural transformations in solids, 1983.

I will be back to share specific materials stories with you soon, but, here’s some food for thought before I leave-
Did you know: The widely known beneficial effect of Boron in enhancing ductility in Ni-base alloys was actually a serendipitous discovery through contamination from crucibles containing B during melting of the alloy. Here’s a snapshot of the original 1958 article:

In the realm of science fiction, the idea of tiny nanorobots that can enter the human body and seek out and destroy unwanted elements has been prevalent. This idea has been made a reality by a research team led by Bin Kang and Yaodong Dai at Nanjing University and the Georgia Institute of Technology. They have used what is called the photoacoustic effect of carbon nanotubes (CNTs) to make what are essentially nanogrenades that can target and destroy cancer cells.

[Original work: Kang et al., Small (2009), doi: 10.1002/smll.200801820]

A soap bubble is an extraordinarily beautiful thing and yet it requires virtually no skill to produce. This is because surface tension does all the work for you, making sure that a perfect spherical membrane is produced every time. In fact it is impossible to blow imperfect bubbles. Even if you try to blow the bubble through a noncircular orifice you may at first achieve a temporary nonequilibrium shape [...] but the end result is always a perfect sphere.

In this chapter we will show how such beautiful but tyrannical surface tension effects can be investigated using the computational equivalent of soap bubble solution: the Potts model. Like soap bubble solution is is easy to use and provides fundamental insights into surface tension phenomena; but also like soap bubble solution, it can lead to a sticky mess.

– Mark Miodownik, Chapter 3. Monte Carlo Potts Model, in Computational Materials Engineering: An introduction to microstructure evolution, Edited by K G F Janssens, D Raabe, E Kozeschnik, M A Miodownik, and B Nestler, Academic Press, 2007.

Potts model is widely used in the materials science literature to simulate a wide variety of microstructural features and phenomena: be it grain growth — normal and abnormal, recrystallization or Zener pinning (and, the article by Miodownik, from which I quote above, discusses all these and much more). In fact, to quote Miodownik again,

At the moment for large 3D systems, with complicated textures and pinning phases, the model has no equal.

In this Classics in Materials Science post, I would like to talk about Potts model and the paper in which Potts outlined it [1].

Potts model, as wikipedia notes, is a model describing spins that are interacting on a crystalline lattice:

The model is named after Renfrey B. Potts who described the model near the end of his 1952 Ph.D. thesis. The model was related to the “planar Potts” or “clock model”, which was suggested to him by his advisor Cyril Domb. The Potts model is sometimes known as the Ashkin-Teller model (after Julius Ashkin and Edward Teller), as they considered a four component version in 1943.

Not surprisingly, the paper of Potts is communicated to Cambridge Philosophical society by Cyril Domb who is also thanked in the acknowledgement for suggesting the problem and helpful suggestions.

The Antecedents

The paper of Potts begins with a reference to a result obtained by Kramers and Wannier [2,3] for 2D Ising models:

In considering the statistics of the ‘no-field’ square Ising lattice in which each unit is capable of two configurations and only nearest neighbours interact, Kramers and Wannier [...] were able to deduce an inversion transformation under which the partition function of the lattice is invariant when the temperature is transformed from a low to a high (‘inverted’) value. The important property of this inversion transformation is that its fixed point gives the transition point of the lattice.

The result that is being alluded to, in the words of Kramers and Wannier is this:

… which permits location of the Curie temperature if it exists and is unique. It lies at

if we denote by J the coupling energy between neighbouring spins.

That is, Kramers and Wannier, calculated the Curie temperature for a 2D Ising lattice in terms of the coupling constant for spins, and showed that it is unique. (The reference to ‘no-field’ by Potts means that there was no externally applied magnetic fields).

One restriction that Kramers and Wannier put on the spins (which is what makes their model a Ising model) is the following:

Let each of the spins be capable of two orientations which we characterize by and .

Potts relaxed that condition and solved for the Curie temperatures:

It is the purpose of the present paper to consider the inversion transformations for a square lattice in which each unit is capable of a general number r of configurations.

Contribution of Potts

Potts was, of course, not the first one to introduce more than two possible spins per lattice site; that was done, apparently, by Domb. Further, much earlier, Ashkin and Teller [4], introduced a system in which each lattice can have any one of the four different types of spins, and calculated the Curie temperature for such a lattice (as part of Julius Ashkin’s Ph D thesis, I understand — which is a reason why Potts model is sometimes also known as Ashkin-Teller model); more interestingly, Ashkin and Teller even talked of four different types of atoms when they considered the four possible spins:

We have considered a two-dimensional square net consisting of four kinds of atoms supposing that only nearest neighbours interact and that there are only two distinct potential energies of interaction, one between like and one between unlike atoms.

Thus, the main contribution of Potts seems to be that (a) he considered a general model in which the spins can take ‘r’ possible values (r = 2, 3, 4, …) and (b) obtained the Curie temperature for the lattice for the cases r=2, 3, 4 (See the figure below):

The current relevance

A tutorial review written in early eighties notes [5]:

The problem attracted little attention in its early years. But in the last ten years or so there has been a strong surge of interest, largely because the model has proven to be very rich in its contents. It is now known that the Potts model is related to a number of outstanding problems in lattice statistics; the critical behaviour has also been shown to be richer and more general than that of the Ising model. In the ensuing efforts to explore its properties, the Potts model has become an important testing ground for the different methods and approaches in the study of the critical point theory.

In addition to such inherent interest, as we noted earlier, Potts model has also been found to be of great use to those who try to simulate microstructures. In the case of microstructural evolution, our interest in the model is not so much towards obtaining analytical solutions of the type noted above, but to get an idea of the microstructure as a function of time — go to this page, for example, for an Open Source Potts model code for grain growth simulations in 3D.

I can do no better than quote Miodownik [6] on the beauty of Potts models (with liberal sprinkling of modelling philosophy) to end this post:

The beauty of the Potts model is that it is a simple way to model complex systems by modeling local physics (in particular, the effect of surface tension phenomena on the development and evolution of microstructure). … At all times we have been concerned with comparing the model with theory and experimental data; this instinct is essential to any modeler. It is easy to make a model yield pretty pictures that appear to have a correspondence with a “real” phenomenon, but quantification of the simulation is the only way to use the model as a method to gain physical insgihts and understanding. Finally it is important to note that a model is just that, a model, and the benefit of it is as much to guide experiments and to hone the intuition about physical phenomena as it is to make predictions. The major role of computer models is to reduce the number of experiments that need to be carried out and to highlight what variables are key to understanding the results. The Potts model should be seen in this light; it is a guide to the intuition, and above all it is a medium of communication between experimentalists and theoreticians.

References

[1] R B Potts, Some generalized order-disorder transformation, Proceedings of the Cambridge Philosophical Society, Vol. 48, pp. 106−109, 1952.

[2] H A Kramers and G H Wannier, Statistics of the two-dimensional ferromagnet. Part I, Physical Review, Vol. 60, pp. 252-262, 1941.

[3] H A Kramers and G H Wannier, Statistics of the two-dimensional ferromagnet. Part II, Physical Review, Vol. 60, pp. 253-276, 1941.

[4] J Ashkin and E Teller, Statistics of two-dimensional lattices with four components, Physical Review, Vol. 64, pp. 178-184, 1943.

[5] F Y Wu, The Potts model, Reviews of Modern Physics, Vol. 54, pp. 235-268, 1982.

[6] M Miodownik, Chapter 3. Monte Carlo Potts Model, in Computational Materials Engineering: An introduction to microstructure evolution, Edited by K G F Janssens, D Raabe, E Kozeschnik, M A Miodownik, and B Nestler, Academic Press, 2007.

Overcoming the difficulties of nanotechnology water treatments means developing innovative solutions to engineer a usable product. One answer could be to develop methods where the nanoparticles need not be suspended in the water. For example, one can stick — or ‘immobilise’ — the nanoparticles onto suitable materials such as steel or polymer sheets,which in turn can be easily dipped in and out ofwater tanks to work as catalysts or adsorbents.

This needs care to retain nanoparticles’ size and integrity so as to keep the benefits of operating at the nanoscale. But if it can be done, immobilisation is a good technique since it would keep water free of nanoparticles during and after water treatment. We, here at the Indian Institute of Science, Bangalore, are developing such a technique for degrading organic molecules using nano titanium dioxide. The results have been very promising and we are working closely with the University of Johannesburg to scale up the process in the near future.

That’s from this article in SciDev.Net by my friend and colleague Prof. Ashok Raichur.

You may also be interested in other articles in the SciDev.Net series on how nanotechnology may be harnessed for making clean water available for all.

From Books and Arts
Nature 457, 31-32 (1 January 2009) | doi:10.1038/457031b; Published online 31 December 2008

Year of astronomy: Voyaging to discovery, alone
David Bodanis1

BOOK REVIEWED
-The Age of Wonder: How the Romantic Generation Discovered the Beauty and Terror of Science
by Richard Holmes

HarperPress/Pantheon: 2008/2009. 380 pp/576 pp. £25/$40

Everyone knows how to be a great scientist. First, you have to be really smart. Having awed your school science teacher is good. Humbling all the others at university is even better. Then you need to find a place where you can let your solitary genius come out. You might escape to a coffee shop to think; or if your parents are really rich, to a trendy loft apartment or, better still, a remote and windswept cottage. There you will engage in the next and most crucial of stages: the creative torture.

There is an art to this. It is important your torment doesn’t end too quickly — otherwise you’d show you had been working on a problem that was too simple. Yet if the torment never ends, you’ll have nothing to tell anyone about and will remain unknown. After some time, ideally a few months, you get to have a ‘eureka moment’. At that point all you have to do is write up your discovery, accept the plaudits, be it from department colleagues or the Swedish Academy, and then if you can bear it, repeat from the start.

We smile at this recap but the basic vision — science as an endeavour of individual creative genius that explodes in an instant of discovery — is one we take for granted. Yet, as Richard Holmes describes in his new book The Age of Wonder, it was not always so. Most founders of modern science in the 1600s, such as Isaac Newton, rarely saw their work this way. For them the process was clinical, building on a slow accumulation of insight.

The big shift took place in the decades around 1800, a period called the Romantic era. The pursuit of progress by thinkers and artists became more a wait for a divine spark of inspiration than the steady toil of uncovering that had been accepted by their predecessors. And any means by which that spark could be nurtured was embraced.

To guide readers through the science and culture of this period, Holmes masterfully dips in and out of the life of Joseph Banks. He is an inspired choice. As a curly haired 26-year-old, Banks was aboard HM Bark Endeavour on the momentous day in April 1769 when it first glided into sight of Tahiti. Ostensibly Banks was just the expedition’s plant collector, but he soon became more fascinated by the island’s human inhabitants.

For most of the British crew — young men who had been away from female company for long months — the explorations in Tahiti were of one sort only, with the initial going rate being one ship’s nail for one sexual encounter. That rate soon changed — as Holmes describes with gentle skill, the Tahitians were quick to grasp the workings of the market economy, and had a keen eye for the other useful metal objects aboard the ship. Hyperinflation set in, and at one point “there was a crisis when one of the Endeavour’s crew stole a hundredweight bag of nails, and refused to reveal its whereabouts even after a flogging”.

Banks looked further. He took up Tahitian mistresses too, but systematically recorded the local language, studied their religious systems, and even hinted at the true functional significance of native actions that, at first, seemed to be merely bizarre. Within a few months he had helped set the stage for the modern science of anthropology.

Back in London, Banks’s insights, energy and inherited wealth eventually led to him becoming president of the Royal Society. From his headquarters, as Holmes gracefully phrases it, “his gaze swept steadily round the globe like some vast, enquiring lighthouse beam”. His own days of direct discovery were over, but couldn’t other like-minded individuals be encouraged to carry on such wondrous, intense investigations?

One of the young men Banks chose to support was Humphry Davy: friend of the Romantic poets, and — in his quick, intense creation of a safe coal-miner’s lamp in response to underground disasters — a man who made himself into a perfect exemplar of the new, Romantic style of discovery. Davy hurried to the mines, spent intense weeks with the miners and then took himself off to an isolated lab where, using his unique genius, he cracked the problem.

Another of Banks’s protégés was William Herschel, the immigrant Hanoverian astronomer. Herschel is most famous now for having measured the orbit of Uranus and establishing this body as a planet, which almost became known as planet George to honour King George III. But Herschel also worked out the shape of the Milky Way and the Sun’s off-centre position in it, and he discovered infrared radiation.

Although his work relied on the dull accumulation of observational facts, it was the role of sudden insight and genius that Herschel and others emphasized in their written accounts. In Herschel’s case, it was true to his character: he had risen in society by transforming his own life and moving to England. Wouldn’t he imagine there were fresh realms — new planets, stars beyond our Solar System, light beyond the visible spectrum — to uncover in nature as well?

Our notion of earlier scientists, including Newton, was rewritten to fit this Romantic view. The young poet William Wordsworth, for example, had famously devalued science with his harsh line that when we probe the mechanism of a natural process, “we murder to dissect”. But decades later, as the work of Banks and his protégés became better known, Wordsworth shifted to admire science, seeing Newton as the archetypal Romantic hero: “Voyaging through strange seas of Thought, alone.”

Over time, the simplest Romantic imagery slid away. Few of the hundreds of physicists working today at CERN, the European particle-physics lab near Geneva, Switzerland, must view themselves as voyaging anywhere alone. But the Romantic era left great legacies. Newton’s static Universe was gone, and Herschel’s dynamic one — in which stars evolved and the heavens were ever-changing — was in its place. This mutable view was useful for the next notable young man, one Charles Darwin, when he was ready to embark on his own voyage of discovery on HMS Beagle in 1831. And even among today’s doctoral candidates at CERN, who doesn’t hope that maybe, with enough concentration, something very special and very unique could still burst out?

1.David Bodanis is a writer based in London, and author of Electric Universe. His forthcoming book is on the Ten Commandments.

In her NYTimes article, Natalie Angier has several quotes from Rob Ritchie of UC-Berkeley on the mechanical behaviour of bones (check out his recent publications):

… [H]ealthy bone is disciplined bone, with a structure enviably organized at every scale yet probed, from the caliper calibrations of femurs and phalanges down to the nano dimensions of bone’s constituent atoms. “It’s all in the architecture,” said Robert O. Ritchie, a professor of materials science at the University of California, Berkeley, who studies bone.

Bone is built of two basic components: flexible fibers of collagen and brittle chains of the calcium-rich mineral hydroxyapatite. But those relatively simple ingredients, the springy and the salty, are woven together into such a complex cat’s cradle of interdigitating layers that the result is an engineering masterpiece of tensile, compressive and elastic strength. “We only wish we could mimic it,” Dr. Ritchie said.

[...]

Behind the dissolution of bone with age is a system designed for the itinerant years of youth. The skeleton is a multipurpose organ, offering a ready source of calcium for an array of biochemical tasks, and housing the marrow where blood cells are born. Yet above all the skeleton allows us to locomote, which means it gets banged up and kicked around. Paradoxically, it copes with the abuse and resists breaking apart in a major way by microcracking constantly. “Bone microcracks, that’s what it does,” Dr. Ritchie said. “That’s how stresses are relieved.”

* * *

Image courtesy: Prof. Ritchie’s home page.

……Next year, researchers at the Lawrence Livermore National Laboratory (LLNL) in California hope to tick that box off Gould’s list. Despite his foresight, Gould could not have imagined the lengths to which scientists and engineers would have to go to bring his prediction to reality. LLNL’s National Ignition Facility (NIF), which was officially completed last month, is a laser on a truly epic scale. The building housing it is 10 stories high and covers an area the size of three football fields; for a very brief instant, its beams deliver a power of 500 terawatts, more than the power-generating capacity of the entire United States.

If all goes according to plan, some time in 2010 the power of those beams will be directed at a small beryllium sphere filled with hydrogen isotopes. The resulting implosion will crush the hydrogen to a temperature and pressure higher than in the core of the sun. If NIF’s scientists get everything right, the hydrogen isotopes will do what they do in the sun: fuse together into helium nuclei and release a huge store of energy. NIF’s principal aim is to reach “ignition”: a self-sustaining fusion burn that gives off more energy than was put in to make it happen–something that so far has occurred only in nuclear explosions and stars……