Thread by @MarkusDeserno

Shout-out to my former visiting student Yiheng Zhang, and his new paper in PRE, in which I had a minor part. A fantastic story about nematics, topology, and effective field theory.

Let me offer you some CLIFF NOTES!

#CMUBiophysics #BeijingNormal

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journals.aps.org/pre/abstract/1…

Nematic liquid crystals are soft matter systems in which elongated molecules align and form long-range oriented fields, in 2 or 3 dimensions. They have a long history, many practical applications, and are increasingly recognized for the many roles they play in biology.

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The local “director field” of liquid crystals tends to be nice and smooth, but it is possible that a little “defect” sneaks in. For instance, these pictures show a +1/2 and a -1/2 defect, where locally the smooth field is disturbed.

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The fun bit is that these defects cannot be “locally removed”: there is no way to turn these sticks a bit at the center of that cowlick and get rid of the defect. The defect is “topological”. Its presence is encoded in the entire field.

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But the reverse is also true: knowing the defects, you also know the field! (At least its minimum energy configuration.)

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That’s A BIT like saying that charges determine the electrostatic field, and conversely, the electrostatic field can tell us where the charges are. In both cases by Gauss’ law. Even though we typically don’t think of this situation as topological.

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Just like charges are conserved, so are defects. But wait, we can create charges! We just must also create charges of opposite sign! Well, the same is true for defects. We can create a +1/2 and a -1/2 defect TOGETHER, locally, and then move them apart.

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All this suggests that a fantastic way to think about liquid crystals is to, well, not think about the actual liquid crystals, but instead to look at the DEFECTS. Just like you can think of the electrostatic field by instead just thinking about the charges that make it.

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And yes, charges interact via the electrostatic field, and topological defects interact via the elastic deformations imposed on the liquid crystal. The analogies go on and on. Soft matter physics is incredibly rich in so much beautiful stuff like that.

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But I’m digressing. Back to the paper. Yiheng, his advisor Zhanchun Tu, (and myself) look at such a liquid crystal on the surface of a sphere, and WHOA, new cool stuff happens, again because of topology.

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There is a famous theorem due to Poincaré that states you cannot have a smooth everywhere non-vanishing vector field on the surface of a sphere. It’s cockily knows as the “Hairy Ball Theorem”, because it can be summarized thusly: “You can’t comb a sphere”.

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But wait: isn’t our nematic liquid crystal a vector field on a sphere? Sorta, almost, and that shows you we’ll run into trouble: There will be cowlicks! Sorry, defects I mean.

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But since the nematic field is not a VECTOR field but a DIRECTOR field (i.e., orientation does not matter), we can have defects of HALF-INTEGRAL nature, like the +1/2 and -1/2 images from above. (That would not be possible if direction mattered; try it!)

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Topology furthermore states that the sum of all defects needs to add up to 2. (Topology says lots of amazing things.) And this means that our nematic liquid crystal on the surface of a sphere must have at least four +1/2 defects.

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Moreover, in the lowest energy configuration (smallest number of defects) it will be EXACTLY 4 defects. You will not be surprised to learn that they’ll sit at the corner of a tetrahedron.

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But wait, there’s more! I’m STILL summarizing what the ancients knew. We haven’t even STARTED… (Sorry, bear with me!)

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You can now make the system ACTIVE. Meaning, you can wave a magic wand and then the individual stick molecules will actively try to move pasts one another.

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Magic wand doesn’t sound much like physics. But biology does that all the time. If your nematic field is made from microtubules, and you sprinkle in kinesis motors, and some ATP, the microtubules will push past one another. The field starts to move on its own!

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What do the defects do? I’m glad you asked: they will of course also move. More precisely: the +1/2 defects will move on their own, but the -1/2 ones will not! (They just get passively shoved around, while then +1/2 actively zoom along.)

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As the whole liquid crystal field gets stirred around, incredibly complex motion happens. It looks like salt water taffy being kneaded. Or turbulent flow lines in a liquid. Even though (for the expert) this all happens at vanishingly small Reynolds number!

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Now imagine you want to describe that. Phew! You have a partial differential equation for the field, and active driving forces. Forget analytics. This goes onto a computer! But EVEN THEN it’s hard: those PDEs take A LOT of effort and computing power to solve!

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But haven’t I told you that the field determines the defects, and the defects determine the field? If I want to understand the motion of a bunch of electrons, I don’t solve Maxwell’s equations. I solve the motion of a few particles subject to Coulomb forces!

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This is what Yiheng and Zhanchun set out to do: find equations of motions for the defects, as they actively zoom about. This should be possible, and it had been looked into for PLANAR liquid crystals. But spheres are more complicated. And MUCH more interesting!

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There’s some technical beauty here, e.g. how to use a variational principle to get equations of motions in dissipative thermodynamic systems (“Minimize the Rayleighian!”), and you should look at it. But I’ll skip that for this summary.

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Long story short: Yiheng found these equations. And he can therefore look at the motion of defects, e.g. as a function of how strong the active driving force is.

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For weak forces, nothing moves. Push the gas pedal a bit more, and you suddenly get rather intricate periodic motions. The four +1/2 defects execute a very beautiful periodic dance around each other. More gas and, BOOM, chaos!

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What I find most fascinating is where this could ultimately lead. Recall that defects behave pretty much like charges, they are conserved, but they can be created and destroyed in pairs. Does that remind you of something? Quantum Electrodynamics maybe?

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Yes, what we have here is really a field theory in which the elementary excitations can be pair created and annihilated, like electrons and positrons in QED. But the underlying field is much more complicated, so a lot more fun stuff could happen.

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Notice in particular that, in QED parlance, the “vacuum” on the sphere is crazy: It is NEVER EMPTY, because you cannot get rid of all defects (topology!). And it is NEVER QUIESCENT, because the +1/2 defects zoom about when active.

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That’s one hell of a field theory to ponder. And Yiheng’s paper did the first step: we have the equations of motion for those defects. But so much more needs to be done. If High Energy Theory is not exotic enough for you, welcome to Soft Matter Physics!

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