Collapsing a Cylinder-Sphere
In today’s post I collapse my lignin cylinders. In my previous post I had presented the uncollapsed model of cylinder stacking into a dodecagon (quasi-hexagon) pyramid. In this model I took an array of close-packed spheres, with a cylinder inside each sphere and showed their alignment once crystallized, but before the actual crystallization.
In this model the radius of the cylinder is exactly √0.5, or ca. 0.707 the radius of its rotational sphere. And equally, the length of the cylinder is exactly √2, or ca. 1.414 the radius of its rotational sphere. Because I didn’t take this into consideration in my first stacking model, the (yellow) cylinders were overlapping, as you can see in figure a) below. However, the dodecagon pyramid stack is only a partial solution, as it depicts a very transient state. In figure b) below I draw the edges of the rotational sphere around the cylinder, but intentionally not along the y-z plane, even though the picture is a y-z projection. Instead, the edges are tilted by sin¯¹ (1/√3) or ~35.26°, or the tilt between the rotational spheres. This tilt converts the edge into an ellipse, with a width of 2 r and a height of √(8/3) r ≈ 1.63 r. And the fun thing about using these ellipses is that they depict an actual close-packed hexagonal plane, despite the cylinders themselves not being aligned along this plane.
The next step is to figure out how the cylinders collapse. For a while I lived under the illusion that the cylinders would still touch their horizontal neighbor as in figure a), but once I started playing with the model in Blender, it became obvious that isotropic collapse where the spherical array contracts equally along each dimensions of the y-z plane, making the cylinders collide diagonally before reaching their horizontal neighbors, as seen in figure c). The diagonal distance between cylinders is twice the radius of the cylinder, but √2 r, or ca. 1.414 r (r = still the radius of its rotational sphere). The horizontal distance between cylinders is collapsed from 2r to 2 √2 rcos(tan¯¹(√2)) = √(8/3) r ≈ 1.63. Whether this comes directly from the size of the ellipse, or indirectly, I’m not sure. This means that the distance between cylinders reduces by ~18.4 % in the collapse.
While I thought that the cylinder sphere collapses isotropically, there is a catch. While the outer dimensions of the cylinder sphere indeed reduce isotropically, the thickness of individual zig-zag crystallites reduces more than its diameter. As the cylinder is has a length of ~0.707 of the diameter of its rotational sphere, the crystallite can shrink by ~29.3 %. If the cylinders collapse without rearrangement, this leaves significant grooves between the zig-zag crystallites. However, these gaps could be at least partially (or fully) filled, if the cylinders continued to collapse into rounded rhombuses, as depicted in figure d).
And just for clarity, this is what figures a) to d) look like from the side:
And next I illustrate the collapsed crystallite with cylinders following figure c):
The biggest reason for not incorporating figure d) in the above model was that I only realized this feature upon writing this post. And the model is so processor-heavy, that I can’t easily change it for this post. It’ll take hours to change, even if I didn’t have to figure out anything new. But I think I still have to eventually make this model as well.
After these steps, it's quite clear that the collapsed cylinder model offers a very credible theory of how hollow nanotubules can crystallize into spheres. Whether the theory is correct or not will require indepemdent scrutiny from the scientific community. However, there is nothing mathematically wrong with it.
Next, I need to start writing this into a manuscript, where I also introduce the theory of quantum pressure. While I could probably tweak the theory ad infinitum, I think it's good enough to send for peer review. The only thing I have to consider is which paper the send the manuscript to. Before, I always sent my manuscripts to Nature just to be desk rejected. Then I would possibly try something else, but eventually end up submittting to Scientific Reports. However, I'm thinking of the comments of the only reviewer who actually read the paper. To paraphrase from memory, the comment was roughly "Scientific Reports should publish experimental papers and this paper is too theoretical for it". Perhaps I shoul consider submitting directly to a more theoretical journal. Perhaps to Journal of the American Chemical Society. The journal is still quite prestigious, but at least it has a history of publishing theoretical papers, which apparently isn't a the case with Scientific Reports.
But, I'll start writing the manuscript and as the writing progresses, I'll probably have a better idea about where I want to submit it to.
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