Consider a sphere.
A sphere is a surface. A surface consisting of a set of points at a particular radius from a center, like so:
Figure 1. The 2-sphere is embedded in 3D space, but is itself only a two-dimensional surface. From Geek3 via Wikipedia.
A sphere doesn’t have to be in three dimensions, though. We can apply the same rule, equidistance from a center-point, just as easily to one or two-dimensional space.
For one-dimensional space we simply get two points on a line, which can be considered a 0-sphere because a single point has no dimensions.
In two dimensional space we get a circle, or more precisely a 1-sphere, since the boundary (the circumference of the circle) is a one dimensional line.
Figure 2. Left: a 0-sphere. Right: a 1-sphere.
In fact, spheres can exist in literally any number of dimensions at all. In 4-dimensional space (or perhaps spacetime) we get the 3-sphere – a remarkably interesting character in and of itself that looks like a dying spaghetti when you try to draw it and is totally tangential to the point of this article.
Figure 3. A stereographic projection showing the parallels, meridians, and hypermeridians of a 3-sphere. From Claudio Rocchini via Wikipedia.
For now, let’s content ourselves with the rather less exotic 1-sphere, or, as some would call it: A circle!
We could have used a 2-sphere, but really it’s just another level of complexity and unlikely to show us anything a 1-sphere wouldn’t. That’s the nice thing about spheres: They’re friendly beasts. What you can do with one kind of sphere usually works more or less the same way with any other.
Right. Here’s our sphere:
What can we do with a singular 1-sphere? Short answer: Not a lot. We could cough up some sine waves, we could give it a velocity and a mass and model some projectile motion, but basically we’ve just got a circle.
Let’s make things a bit more fun, and give it a friend:
Better. Now let’s slap in some vertical gravity and a spot of chunky maths to make our two circles bounce off each other, in such a way that every collision conserves both momentum and energy. If you’d like to read all about that collision algorithm, here's a lovely article about it [1], so go ham. For everyone else, I’ll spare you the math and get to the cool part:
Let’s go ahead and drop a whole bunch of spheres into a box, slap each of them with a random velocity, sprinkle in some simple elastic collision mechanics, and see what we get:
Short answer: We get a gas.
Remember the ideal gas law, PV=nRT? Turns out that really all it does is conjure up some juicy macroscopic values to explain what happens when you stick a whole bunch of spheres in a box.
We can sum up the forces of every collision with a wall that happens in some unit of time, and simple as that we’ve got a measure of pressure. Take the average kinetic energy of all the spheres in some region of the box, and now we’re measuring temperature as well! If I bump an edge with the side of the box, I can send a sound-wave through it. Now, if we were to drop in some more spheres and shrink the box a little bit we could have tons of fun calculating Avogadro’s constant, all from a page or so of Java code that calculates the collisions of spheres!
But this isn’t a first year SSP report. This is Jeremy! If you want to read about people doing things ‘a little bit’, you’re in the wrong place.
Let’s shrink the box a lot:
That, there, is a solid. A hexagonally close-packed crystal lattice, to be precise. Forced into beautiful, rigid structure by the sorts of pressure that forge diamonds from graphite, deep within the earth!
Actually, that’s almost exactly what we’ve done. Behold my two dimensional diamonds!
Figure 4. Replace each little ‘c’ there with a comfy yellow dot, and you might see something familiar. From M. Zulqarnain, Science Solve.
But we’re not done yet. My two dimensional diamonds were just squished into existence out of a gas. They’re unimaginably hot, and under almost as much pressure as your average physics student on that first week back after mid-sem!
Let’s be nice and give them a little elbow room:
Increase the volume, and ol’ pal Avogadro comes to the party. Him and his PV=nRT. (I always knew he was a rock’n’roll kind of fellow.)
With the volume up and the pressure lowered, some of spheres in the top center break out of their rigid lattice structure and start rolling over each other.
Houston: We have a phase change!
Some of our spheres are behaving like a liquid. Not all of them, yet; I haven’t made the box big enough for that. No, our Sphereium is right on the turning point. Some spheres melt into the liquid phase, others snap back to their rigid, space-saving hexagonal lattice. It’s basically the thermal equivalent of a saturated solution.
Now, what else can liquids do? Diffuse! Did you notice those green fellows I dropped in on the left up there? That’s what they’re doing: diffusing through the liquid components of our material, constantly increasing entropy until the sheer, unfathomable pressure of our box forces them to lock into place when a part of the lattice recrystallizes.
Right. I’ve had enough of being nice. Semester’s over. It’s exam week. The pressure is back on. Fast.
And that, right in the middle, is a lattice defect. The whole thing is a solid, but it was forced there from a liquid so rapidly that two misaligned crystal structures have formed: grains of the crystal. This is actually one of the foundational principles of metallurgy: If you cool a lattice slowly, you can get a nicely aligned structure, strong and with minimal imperfections. Cool it fast and your grains (your crystals) become smaller, and your material becomes harder. More brittle. More imperfections.
And here we have it: the grain structures of a metallic lattice. The foundations of metallurgy, from nothing but a couple dozen lines of Java code calculating elastic collisions between spheres.
But I’m not done yet! Ho ho no I am not!
Exams are over, let’s finally give our Sphereium some room to stretch its metaphorical legs.
Now we’ve got a proper liquid, which takes the shape of its container and everything! Liquid Sphereium. But what’s even more interesting than just this mundane liquid behaviour are the two spheres bouncing about up the top. Those are well and truly out of the liquid, behaving like a gas, but the Sphereium is nowhere near its boiling point. No, those two just happened to be jostled out of the menagerie by pure chance.
Ever noticed how water can evaporate slowly from a cup on a warm day, despite never reaching 100 degrees? That happens because the water has a vapor pressure. Particles of H2O can go into or out of the main body of liquid, just like what has happened in our Sphereium, to maintain some concentration in the gaseous phase. It’s basically diffusion, but with a phase transitions slapped in the middle.
And there we have it: Vapor pressure, phase transitions, the foundations of the entire field of metallurgy, and the whole life’s work of the renowned Sir Amendo Avogadro (or, at least, all the interesting bits), simulated with nothing but some equations for sphere collisions that I could fit on the back of my hand! Hardly more than simple energy and momentum conservation.
Now, no doubt most of you will be wondering why this article has another paragraph after so dramatic and satisfying a conclusion. Fair point. However, the few of you who aren’t wondering that are, I’d hazard a guess, wondering this: Where can I get one of those boxes!? Throughout the whole article, I have had a very convenient, indestructible box, nicely realizable and with the handy property of linearly accelerating everything within it downward at 9.8 m/s/s. More astounding still, my box nicely isolated every single sphere from every other – none of this ‘wave-particle duality’ business, and not so much as a whiff of an electromagnetic interaction to be seen!
In part two of this article, we are going to fix that. And an old friend of ours, the dying-spaghetti 3-sphere, might just help us do it.
The end of:
Sphere Collision.
References
[1] Circle-Circle Collision Tutorial, Presented at the 2010 Junior Science and Humanities Symposium. https://ericleong.me/research/circle-circle/
By Murray Jones.
(The guy with the hat.)