The Ball-and-Stick Model of Lignin Crystallization

In my last post I presented the tricolor model of Waterman polyhedra that allowed me to visualize individual planes in this shape much easier than before. Of course, after making the model with spheres, I set about applying it to a Waterman-Lintinen polyhedron made of cylinders, very much like what I had done decently well with two colors.  But once I had converted most of the spheres into cylinders, I began looking at the shape with more skeptical eyes than before. The reason was that I knew I had cheated a bit in all of my previous models. You see, I had converted the spheres into cylinders with the same radius as in the spheres and its depth the same as the diameter of the sphere. But this resulted in cylinder-spheres that were clearly a bit ‘off’. The reason was that this conversion made the neighboring cylinders in the same plane to overlap and this was an issue I needed to solve.

It wasn’t that I had made a mistake and only now realized it. Rather I had made a conscious choice to retain the compact structure of the original Waterman polyhderon, rather than convert each sphere to a cylinder with its dimensions so small that the crystallized structure would no longer look like a uniform crystal, but a loose array of smallish cylinders with a considerable gap between each. However, this is actually my hypothesis on how lignin crystallizes into LignoSpheres. The starting point is a spherical cluster of somewhat freely rotating spheres of solvent with a wet lignin nanotubule trapped inside. Exactly what this means is outside the scope of this post. However, if we consider the sphere of the Waterman polyhedron to be solvent, the tubular lignin solute cannot occupy the whole volume. Again, skipping a bit of the mathematics and still assuming a freely rotating tubule, both the radius and length of the lignin tubule can be only √0.5 ≈ 0.71 times the radius of the solvent sphere. This means that while the crystallite cylinders overlap in the old model, there is actually gap between the crystallites in the new model. Because of hexagonal packing of the spheres in the hexagonal faces (as revealed by the name), the distance between cylinder crystals is not 2r, but actually √3 r  ≈ 1.73 r, whereas the diameter of a cylinder is √2 r ≈ 1.41 r, meaning that there is a ~18 % gap between the cylinders. And in the side with square crystals the distance between cylinders is indeed 2r, which increases the gap between cylinders to ~29 %. But the smallest gap isn’t between cylinders within a plane but between neighboring planes. The distance between neighboring hexagonal planes is

To explain why is again beyond the scope of this post, but this reduces the gap between planes to ~13 %. All in all, this means that once the rotating stub cylinders have stopped rotating and crystallized into long cylinders, there are multiple gaps within the system and there is no way for the shape to contract evenly into a compact sphere of cylindes without rearrangement of the stub cylinders. Assuming that there is no rearrangement of the stubs, the final crystal will have huge gaps everywhere.

And to illustrate this gap state, here is the cylinder crystal with  the correct radius of cylinder stubs, but the rotating cylinders at each corner kept as a sphere, as there is no mathematical way of determining into which orientation the corner cylinders should settle:

https://video.wixstatic.com/video/ed0781_8e03a9d3c95f42a0955c3539799475fe/1080p/mp4/file.mp4

And before you ask, even the above shape isn’t fully accurate. I’ve intentionally kept the length of individual cylinder stub equal to the diameter of the rotating sphere, allowing each cylinder crystal to be uniform, without contracting the shape. This ball-and-stick model resembles those used to describe molecules. Except this is a supramolecular ball-and-stick model.

So, how does this correlate with LignoSpheres? Well, the formation of the loose crystallite requires quite a bit of space to allow for the rotation of the stubs. If I recall correctly, the diameter of this loose crystal sphere is about 650 nanometers. But this loose state is extremely transient. Optimally one wants to change the environment of the loose crystal in a way that the external pressure is sufficiently high that the shape compacts evenly from all sides, pushing the stub cylinders as close to each other as possible. Exactly how this is done is actually both patented and a lot of the details are kept intentionally secret. Suffice it to say that the compact shape is about 450 nanometers in diameter. However, if there’s just pressure, this is not very conducive to the rearrangement of the stub cylinders, which is required for the compressed shape to be maximally packed. For this, you also need thermal motion, or heat as the laymen say. But only just the right amount, because too much heat will break the shape apart and not induce compact rearrangement. And finally, if the rearrangement is conducted perfectly, the shape collapses into a diameter of 395 nanometer. So, if you consider that the contraction takes place along three axes, the reduction in size from 650 to 395 nanometers means a reduction in volume by ~78 %, despite the reduction in radius being just ~39 %. This is even more than the 47 % difference between the volumes of a sphere and a cylinder fitting within the sphere. And all of this could be calculated here, but I’ll spare you the details. Actually there’s a clear mathematical reason for this close-packing of equal spheres gives the density of sphere π/√18 ≈ 0.74048, whereas the close packing of equal cylinders gives a density of π/√12 ≈ 0.9069. Which is exactly the same ratio. The curious thing is that I calculated this before I had electron microscope evidence of the hollow nanotubule nature of lignin, or any measured data on the intermediate stages of the spheres. But once I had these calculations, I tested this by intentionally preparing the spheres inside a dynamic light scattering (DLS) device, so that I could see what their size immediately upon formation would be. And the device gave quite a repeatable size of 650 nm. Usually, I always observed sizes of 450 nm and smaller, because I had observed that I need to collapse the spheres immediately to get the best quality. But because my calculations said that 395 nm would be possible, this led me to explore the collapsing more thoroughly and I indeed did find that this was indeed the lower limit for the size. But getting to this low a value was a hassle, so I would be happy to get sizes between 420 and 430 nm.

While all of this should just sound wonderful, it leaves me just a bit worried. You see, I did talk about this in my Sneaky Quantum Gravity Manuscript, but because I’d left it in the supplementary information, the reviewer 100 % ignored it. And then (probably) he had the audacity to say that I have no data to support my theory. I can just imagine this claim being thrown at me once more. I should have experiments that the reviewer recognizes and not some random microscope images that can’t possibly show nanotubules and definitely not some convoluted calculations that are based on unproven hypotheses.

Oh, and finally, I’m definitely going to create the same shape also with the stub cylinders, with a gap between each stub. And I’m also going to create the collapsed shape without rearrangement as an illustration for the upcoming manuscript, as well as the collapsed shape with rearrangements. Creating the shape shouldn’t require any new ideas, but from previous experience, it is possible that despite the apparent ease, it will require more work than expected. Sometimes this is work is just Blender-drudgery, but there is always the possibility that my computer begins to freeze when the Blender file becomes large, so I might end up waiting for tens of seconds, or even minutes, after simple steps, even though without the computer freezing each of these steps would take a blink of an eye. And when a model contains thousands of objects, this freezing begins to drain the soul. But once the model is ready, the bigger the joy.