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. In a recent paper [Phys. Rev. E 102<\/strong>, 012607 (2020)<\/a>], my former visiting student Yiheng Zhang<\/a>\u00a0(\u5f20\u4e00\u6052), his supervisor Zhanchun Tu<\/a>\u00a0(\u6d82\u5c55\u6625), (both at Beijing Normal University, and myself have investigated a particular question involving nematic liquid crystals, namely: what if we put an active version of such a system on the surface of a sphere\u2014how can we efficiently describe what\u2019s going on? Let me unpack what this question even means.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> The local \u201cdirector field\u201d of liquid crystals tends to be nice and smooth, but it is possible that a little \u201cdefect\u201d sneaks in. For instance, the two pictures below show a +1\/2 and a \u20131\/2 defect, where locally the smooth field is disturbed. The number \u201c+1\/2\u201d means the following: if you walk around the defect in a circle, say counterclockwise, then the local direction of the field also rotates, but it only rotates half way around<\/em>, by 180 degrees, while you do a full turn, 360 degrees. For \u201c\u20131\/2\u201d you also get a half rotation, but the field rotates in the opposite<\/em> direction in which you traverse the circle: if you go around counterclockwise, the field rotates clockwise.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> A +1\/2 defect in a nematic liquid crystal.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div> A \u20131\/2 defect in a nematic liquid crystal.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> The fun bit is that these defects cannot be \u201clocally removed\u201d: there is no way to turn the molecules a bit at the center of that cowlick and get rid of the defect. The defect is \u201ctopological\u201d. Its presence is encoded in the entire field. But the reverse is also true: knowing the defects, you also know the field! (At least its minimum energy configuration.)<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> That\u2019s a bit<\/em> 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\u2019 law. Even though we typically don\u2019t think of this situation as topological. And just like charges are conserved, so are defects. But wait, we can create<\/em> 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 \u20131\/2 defect together<\/em>, locally, and then move them apart to make two individual defects.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> 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<\/em>. Just like you can think of the electrostatic field by instead just thinking about the charges that make it.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> And yes, charges interact via the electrostatic field, and topological defects interact via the elastic deformations imposed on the liquid crystal. The more these curves deviate from simple parallel lines, the higher the elastic deformation energy (called \u201cFrank energy<\/a>\u201d). And the analogies go on and on. Soft matter physics is incredibly rich in so much beautiful stuff like that.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> Back to the paper. We look at such a liquid crystal on the surface of a sphere, and this adds another interesting level of complication, again because of topology. There is a famous theorem due to Poincare\u0301 that states you cannot have a smooth everywhere non-vanishing vector field on the surface of a sphere. It\u2019s cockily knows as the \u201cHairy Ball Theorem<\/a>\u201d, because it can be summarized thusly: \u201cYou can\u2019t comb a sphere without leaving cowlicks\u201d.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> But wait: isn\u2019t our nematic liquid crystal a vector field on a sphere? Sort of, almost<\/em>, and that shows you we\u2019ll run into trouble: There will be cowlicks (sorry, defects<\/em>!). But since the nematic field is not a vector<\/em> field but a director<\/em> field (i.e.<\/em>, orientation does not matter), we can have defects of half integral nature, like the +1\/2 and \u20131\/2 images from above. That would not be possible if orientation mattered: if we rotate a vector by only 180 degrees as we transport it around a defect, it would return to the starting point oriented into the opposite<\/em> direction from the vector it started with. So it\u2019s aligned in direction, but we\u2019re off by a minus sign. If we only care about the direction in the first place, that minus sign is not there.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> Topology furthermore states that the sum of all defects needs to add up to 2. (This is because of the Poincar\u00e9-Hopf Theorem<\/a>. Turns out, 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. Actually, it would also be possible to have two +1 defects, but this is a situation that has a higher energy. It would also be possible to have 6 defects of type +1\/2, and then to cancel two of the superfluous ones by also adding two \u20131\/2 defects. But again, such a situation would have a higher energy. That means, in its lowest energy configuration, the system will have exactly<\/em> 4 defects of type +1\/2. And you will not be surprised to learn that they\u2019ll sit at the corner of a tetrahedron.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> But wait, there\u2019s more! I\u2019m still<\/em>\u00a0just summarizing what the ancients knew.<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div> You can now make the system active<\/em>. Meaning, you can wave a magic wand and then the individual stick molecules will actively try to move pasts one another. Sure, \"magic wand\" doesn\u2019t sound much like physics. But biology does that all the time<\/em>. If your nematic field is made from microtubules<\/a>, and you sprinkle in kinesis<\/a> motors, and some ATP, the microtubules will push past one another. The field starts to move on its own<\/a>!<\/p><\/div>\n\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div>![\"\"](\"http:\/\/research.phys.cmu.edu\/biophysics\/wp-content\/uploads\/sites\/2\/2020\/08\/plus-half-defect.jpeg\" "\\"plus-half-defect\\"")
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