Layered cake and floating crystals

Mille crepe. By Laitr Keiows - Own work, CC BY-SA 3.0,

The soil we stand on is like a mille crepe, a layered cake made by the slow deposition of solid matter on an ocean bottom, each era adding a layer of a different nature. The process that makes particles even slightly denser than water settle down is called sedimentation.

A particle is pulled down by gravity, slowed down by the viscosity of the solvent. It also gets kicked randomly by the atoms around. For a large and heavy particle like a canon ball this random motion is negligible and the particle sediment to the bottom. For a small and almost buoyant particle like a protein, this random motion dominates and the particle diffuses in any direction. In between we have the so called sedimentation-diffusion equilibrium. Particles settle down, but also diffuse up, and we observe that the concentration of particles changes depending on height. At the bottom we count more particles than at the top. This is what we call a density profile.

Equilibrium density profiles are a great tool for physicist. By measuring them, you can learn how your particles behave as a system. For example, if you observe a density that decreases exponentially with altitude, you known that the suspension behaves like an "ideal gas", which means that the particles almost do not interact. That's more or less the density profile of the gases in the atmosphere.

If you observe a sudden jump in a density profile, it means that you have an interface between two phases. For example between a gas of particles and a liquid of particles.

A colloidal gas-liquid interface. Picture by Paddy Royall.

If your particles are all the same size, you can even observe two consecutive jumps, from gas to liquid and then from liquid to crystal, where the particles are neatly aligned. Particles with different sizes would jumble the alignment. In general, it is quite difficult to make particles of different size crystallize.

There are several ways to get to this triple coexistence situation.  One possibility is that you first have the gas and the liquid that separate, and then the crystal forms from the liquid. A second possibility is crystals condensing from the liquid, settling down in sufficient quantities and only then does the liquid evaporates to form a gas layer on top. A third possibility is the crystals forming at the same time as gas bubbles, racing to the bottom or the top respectively. Only when gas and crystal layers sit on top of each other does some of the crystal melts to form a liquid layer in between.

My contribution was to add some more complexity to the first scenario. What if I add a few large particles (green) in the suspension of small particles (red) ?

At first, nothing changes: on top a gas that has almost no particle and on the bottom a mixture of many small and a few large particles.  If there was only small particles crystals would form at the bottom. But the large particles get in the way and no crystallisation occurs at the bottom.

Meanwhile the large particles settle faster than the small ones. So at the top of the liquid we soon have a layer devoid of large particles. Only small particles? Easy to make crystals then (big red blobs on the video below). Crystals are large, compact, and fall even faster than large particles. They outpace them and dive in the dense mixture of large and small particles. Splash!

And here we have something unexpected: the crystals float! I mean, yes, ice floats over water, we are accustomed to this. But water is an exception. Solid metal sink down into molten metal.

Actually we demonstrates that the mixture of small and large particles can get so dense without crystallizing that crystals made only of small particles can float in it.

The crystals are reasonably happy in there, not melting but not growing either. Since crystals are dropped continuously from the top, they end up filling pretty much the whole pool (where the large particles are) and even piling up over the level of the large particles.

Now the crystals that are over the level of the pool have no large particles to prevent their growth, so they grow and make a dense "ice pack" on top of the pool.

Final state of the limit between floating crystals (below) and the ice pack (above). This is the same place as the video above.

At the end, you get a pretty layered cake: gas on top, then a layer of liquid, then the ice pack, then the crystals made of small particles floating in the pool of large and small particles.

Details of the full layered sediment. Top: gas-liquid interface. Middle: ice pack. Bottom: crystals made of small particles in a small+large amorphous matrix.

Leocmach, M., Royall, C. P., & Tanaka, H. (2010). Novel zone formation due to interplay between sedimentation and phase ordering. EPL (Europhysics Letters), 89(3), 38006. doi:10.1209/0295-5075/89/38006
http://arxiv.org/abs/1402.0315

Count on your neighbour

Counting how many stuff you have is important

Scrooge counting his money

... but boring

counting sheeps

During last week, I saw a few times one of my fellow lab member printing a picture like this

Phase contrast microscopy picture of nucleation

and putting a cross on each white object. He was counting them. The first time I saw this, I thought he had to do it for one or two pictures. But at the end of the week, I asked him what he was doing and if I could help.


The above picture is taken when a phase A nucleates into a phase B. This appends for example if you cool a liquid below it's crystallization temperature. A crystal nucleus will appear from time to time and grow. The probability to form a nucleus (nucleation rate) is a very important physical parameter: if nucleation is extremely rare, you will have a single nucleus in you bottle that will grow to form a single crystal before the birth of the next nucleus. This is exactly what you want for example when you make a silicon wafer for microelectronics. If nucleation rate is high, then you will have many nuclei growing at the same time and at the end a material that is made of many different crystals. You may want this in ice creams, because small crystals have a more pleasant texture than big ones.




The only method to measure the nucleation rate in a given system is to count the number of nuclei function of time. So my colleague was counting ... for the whole week. He had done two dozens of experiments at different temperatures and compositions, and took a series of picture for each (like every couple of second for a few minutes). This makes hundreds if not thousands of pictures to analyze. And his plan was to do it by hand.

Try to count how many nuclei are in the above picture. This is a task that need careful attention: large nuclei have a good contrast, but there are many smaller ones very difficult to tell from the background. That's why my colleague was printing and crossing the counted nuclei.

As I told you in a previous post, this kind of procedure can be fully automatized. The programming takes time, so if you have only a few pictures to analyze, this may not be a good idea. In addition, this counting is tricky because the objects can have very different sizes and contrasts. However I, sitting 3 steps away, had already developed and tested such a program. The physical signification is different (I am tracking polydisperse colloidal particles) but the technology is the same. So yes, I could help.

An hour later my colleague had in his computer a script counting the nuclei for him, a picture per second or less, automated to treat a whole time series automatically without human intervention. Setting-up Python and dependencies on his computer took half of the time. We should have communicated earlier, before he had spent a week doing what the script could do in an hour.

Result of the localization. Original image (red) superimposed with localized positions (cyan squares)

As you can see on the picture above, the result is not 100% perfect, but quite close. For example there are problems when nuclei are fusing and there are also (very few) centers counted multiple times. I think I know how to adapt better my program to this situation, but my colleague told me it was enough precision for him.

This gives an other motivation to explain (in a future post) how this counting/localizing method is working.

Mikan stacking fault

Never leave a crystallographer with a pile of fruits.

HCP(left)-FCC(right) stacking fault

Even if I do not consider myself a crystallographer, I spent most of my PhD identifying local symmetries in materials. And the pioneers of the land of symmetries in materials are the crystallographers, so I owe them most of my analysis tools ... and the tetris-o-philia.

The FCC side, note how the 3rd layer's fruits sit on top of voids in the first layer

During new year's vacation in my family in law, my daughter (in the background of the upper picture) started emptying the reserve of mandarin to bring them on the living room's table. This got me started at piling the fruits.

The HCP side. 1st layer and 3rd layers sit on top of each other

The pile begins by a layer where the fruits form hexagons. This is the most compact way of packing disks of the same size in 2D, and thus spheres of the same size on the same plane. Real fruits have different sizes, but anyway.

For the second plane, you have two possibilities that are mirror image of each other, a translation, a rotation of 30 degree, etc. In short, this is not a real choice because you have no reference point.

The same alternative has richer consequences in the third plane. Depending on your choice, you end whether with

• 3rd layer's fruits sitting on top of 1st layer's fruits
• 3rd layer's fruits sitting on top of 1st layer's voids

Because you have the first layer as a reference, the choice is no more silent. You end with two different crystals: Hexagonal compact (HCP) and Face-centered cubic (FCC).

FCC is left, HCP right, and the grain boundary in the middle (hole in the 3rd layer)

Now let's be messy. I made two different choices of 3rd layer in two different places. The line where the two stacking meet is a stacking fault. It is not possible to pack same-size spheres efficiently on this line. Evidence is the hole you can see on the pictures.

FCC is right, HCP left, and the grain boundary in the middle (hole in the 3rd layer)

Finally, I added a fourth layer, with fruits sitting on top of the ones of the 2nd layer. No fault in this fourth layer, I can continue my stack if I want. However the fruits immediately over the fault line have a little more space to rattle. The same holds for the fruits immediately below (2nd layer).

If you think about the stacking of a crystal of hard spheres, the price paid to have such a stacking fault is the spheres you could not fit in because of the line. In my very small crystal, I could have fit 2 more spheres without the fault. This is a global penalty.

On the other hand, the space gained to rattle by the spheres neighbouring the fault line increases their (vibrational) entropy and thus decreases their free energy. This is a local gain.

When you balance the global penalty with the local gain, you end up with quite a lot of stacking lines.

Seminar and meetings in France

I'm giving a seminar in the Ecole Normale Supérieure (ENS) in Lyon, France the 6th of December. Just after that I'll be in Paris for 2 consecutive meetings:

• The French-Japanese meeting on Jamming, Glasses and Phase transitions
• Unifying Concepts in Glass Physics V,  where I will be presenting a poster.

Both seminar and poster are about the same stuff I talked about in Kanto-softmatter workshop and in a previous post. Here is the more formal abstract.

A link between local structural ordering and slow dynamics has recently attracted much attention from the context of the origin of glassy slow dynamics [1, 2]. There have been a few candidates for such structural order [3, 4], icosahedral order, exotic amorphous order, and crystal-like order. Each type of order is linked to a different scenario of glass transition. Thus, revealing the order responsible for slow dynamics is crucial for our understanding of the glass transition. Here we experimentally access local structural order in polydisperse hard spheres by its particle-level observation with confocal microscopy. We identify the key structures as icosahedral and face-centred-cubic(fcc)-like order, excluding any other simple local symmetry. We find that both types of order are statistically associated with slow particles. However, when approaching the glass transition, the icosahedral order does not grow in size whereas crystal-like structures grow. It is the latter that governs the dynamics and is linked to dynamic heterogeneity. This questions the direct roles of the icosahedral ordering in glassy slow dynamics and stresses the importance of the structural order compatible with the avoided first order transition, crystallization. Our finding also suggests that the growing lengthscale of structural order is essential for the slowing down of dynamics and the nonlocal cooperativity in particle motion.

References

1. Cavagna, A. Supercooled liquids for pedestrians. Physics Reports 476, 51124, 2009.
2. Berthier, L. & Biroli, G. Theoretical perspective on the glass transition and amorphous materials. Rev. Mod. Phys. 83, 587, 2011.
3. Steinhardt, P., Nelson, D. & Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784805, 1983.
4. Tarjus, G., Kivelson, S. A., Nussinov, Z. & Viot, P. The frustration-based approach of super-cooled liquids and the glass transition: a review and critical assessment. J. Phys.: Condens. Matter 17, R114R1182, 2005.
5. Lubchenko, V. & Wolynes, P. Theory of structural glasses and supercooled liquids. Annu. Rev. Phys. Chem. 58, 235266, 2007.
6. Tanaka, H., Kawasaki, T., Shintani, H. & Watanabe, K. Critical-like behaviour of glass-forming liquids. Nature materials 9, 324Ð31, 2010.

Reconstruction from confocal microscopy coordinates. Only structured particles are shown for clarity.

Kanto softmatter talk

A busy week is ending ... almost. I give a talk tomorrow at a workshop (yes, a Saturday !), on Monday I submit a research proposal to be paid from April. And after that I will have to work again on a paper that has been rejected.

Tomorrow is the kanto softmatter workshop, a very local meeting for the soft matter labs around Tokyo. Talks are only given by young researchers, not by big names. That is why I have an opportunity to talk. In larger conferences until now I only got poster presentations. Well, there is no bed of roses.

I will talk about my thesis work, in particular the content of the paper that was rejected: what are the local structures playing a role in a model of glass transition and which one is more important than the other. The answer is rather surprising. A glass is amorphous, so most people think that a glass is the opposite of a crystal. Therefore if glass has a structure this structure must be very different and incompatible with the crystal symmetry. That's why icosahedral order is often exhibited as a typical glass order.

A icosahedron is a solid with 20 identical faces. Like this dice used in Dungeons&Dragons.

via Wikimedia

13 particles forming a perfect icosahedron, from my thesis

As you can see, there are pentagons everywhere in that structure: icosahedron has 5-fold symmetries. The problem with five-fold symmetry is that it cannot pave space (at least in 2D and 3D). Try to pack them together and you will always have gaps.

By JF Sadoc via Wikimedia

However, the icosahedron is very dense and often maximises locally the interaction energy between the particles. Icosahedral order is locally the best structure, so it forms easily in a dense liquid, but cannot spread. That is what is called frustration.

What one can image in a supercooled liquid is icosahedral bits, probably forming a sort of network or fractal, and total disorder in the gaps. The icosahedral structure is stable, so is moves very slowly and slows the overall dynamics. If we are still in the liquid a given icosahedral bit will eventually disappears while order is formed elsewhere, but in the glass even that rearrangement is forbidden, too costly in energy to append in a reasonable time, so everything is stuck. Here is an explanation of the glass transition.

Another explanation (advertise by my boss, so my judgement may be biased) is that a supercooled liquid is by definition metastable to the crystal, so the liquids "wants" to become a crystal. Things are getting in the way (like icosahedron for example) so the crystal is not formed. However, there are stuffs in the supercooled liquid that look like a little bit like crystals. Not very healthy crystal if you pass me the expression; hunchbacks, twisted legs, broken faces, no arms ... still if you look close enough the local structure is closer to the crystal than anything else.

a) displacements b) crystalline order and c) number of neighbours in a 2D shaken granular supercooled fluid. From Keiji Watanabe and Hajime Tanaka,
Physical Review Letters (2008).

Once you have a method to detect these crystal-like stuffs, which has been done in an handful of models, you discover that they are slower than the rest of the liquid and that their size is growing when you get closer to the glass transition. Paradoxically isn't it the crystal that is responsible for the slowing down to the glass ?

Who is slowing down the system ? The locally favoured structure of the fluid or the influence of the crystal ? To answer this question I used a system that has independently icosahedra and crystal-like structures. In a few systems, people have found very slow icosahedral structures and some of them exhibited it as the proof that liquid order was the culprit. However others remarked that the "crystal" in these systems actually contains some icosahedral motifs. For example if the "crystal" is in fact a quasicrystal with five-fold symmetry, you cannot tell if the icosahedra that you see in the supercooled liquid come from the locally favoured structure of the liquid or as crystal-like stuff.

A Frank-Kasper phase, which is a crystal containing icosahedra (large blue spheres). From the Trebin lab in the university of Stuttgart.

A quasicrystal with icosahedral symmetry, via Wikimedia

To avoid that confusion, my system has a well known crystal of face centered cubic structure, without a glimpse of icosahedron in it. In addition, icosahedral order is locally favoured. In that situation, no mistake possible, the slower structure wins.

And at the end, I found that the icosahedral bits play very little role in the slowing down, the crystal-like bits are doing all the slowing work. Of course Icosahedral order plays a role : it is frustrating the crystallisation, and that is thanks to that frustration that we are able to supercool the liquid in the first place. However, that is the influence of the crystal that governs the slowing down and thus the glass transition.

Details in the paper to come ... when accepted.