Why isn’t it hard to rip a piece of paper in half?

Tearing Paper

Interesting scientific questions come in at least two categories. The first category comprises those big questions that everyone knows. Things like the origin of the universe, how consciousness works, ways to ensure that our brother and sister species – and we – survive and prosper, how to build better computers with better algorithms, build better roads and buildings, conquer disease, and so forth.

The second category also comprises those things that everyone knows, but they are things we do not see as questions. I’ve written about some of these before. For example, while everyone agrees consciousness is a big question, we rarely stop to appreciate how truly strange the way we think is. While everyone agrees the origin of the universe is a big question, we rarely stop to appreciate how truly strange the universe’s complexity and richness is.

Here is another example of one of these strange things. The energy binding atoms together is, ballpark, 20 to 50 electron Volts, regardless of the kind of atom. So, if the binding energy determines a material’s strength, such strengths would hardly differ. Consider a concrete example. Imagine we are hanging a hunk of concrete from a piece of paper, let’s calculate how big the rock has to be to rip the paper. As seems appropriate, we’ll do a back of the envelope calculation. To unbind atoms, we have to separate them by about 1 nanometer. The work the weight must do to unbind all the atoms is the product of that distance and the applied force, which is the mass of the rock multiplied by gravitational acceleration, 9.8 meters per second squared. And, of course, we have to separate a large number of atoms, proportional to Avogadro’s number, 6.02 x 1023. To rip the paper, the total binding energy must be matched by the work done. I won’t go through the multiplications and divisions and unit conversions, but it is not hard. The answer the calculation gives is that the size of the rock you must hang from a piece of paper to cause it to rip corresponds to a rather large building, or somewhat small asteroid. And since, as I mentioned, binding energies are much the same for steel, paper, wool, and butterscotch candies, we would get pretty much the same answer for everything. This is one of the paradoxes at the heart of materials science.

Let us look at the same thing experimentally. Rip a piece of paper (note you do not need a small asteroid), and you will see the pattern of the rip follows a zig-zaggy path as the apex of the tear propagates forward as it presumably hits different random weak spots. Now, listen. Depending on how fast you rip the paper, the rip’s pitch frequency is higher (fast rip) or lower (slow rip), and the pitch is proportional to the speed. What you are hearing is the sproinging of the atomic bonds as the atoms let go of their neighbours. You cannot hear individual atomic bonds of course. What you are hearing is a coherent structure of atomic bonds sproinging together with enough force to move the air and get to your ear. It is easy to estimate how big these coherent regions are. Rip at a speed of 1 centimeter per second, the pitch of the ripping sound is around 1000 Hertz. This gives the size of the coherent regions as about 0.01 millimeters.

So, by fiddling with a few equations, and sacrificing a few sheets of paper, we find that the strength of a piece of paper, and indeed the strength of almost any material is not determined by an atomic binding energy, it is determined by the local weaknesses of the material on length scales which are significantly larger than the atomic size, but significantly smaller than macroscopic scales. The challenge of materials science then is to understand the processes giving rise to such structures, and then to use that knowledge to design materials by tuning the structure on those scales.

The latter – tuning the structure – is harder than it sounds. One cannot position Avogadro’s number of atoms one by one like Lego blocks. Instead we let Mother Nature help, as many systems far from their normal operating conditions assemble themselves. Even our simple example of ripped paper shows emergent behaviour. The pitch frequency of the rip was around 1000 Hertz. But in fact, that was an oversimplification; it is more like an angry buzz than a well-defined pitch, as many frequencies of comparable power contribute to the sound. This is more readily understood by looking at the pattern of the ripped paper. I called it zig-zaggy above, but it is much more zig-zaggy than one might expect. Take a look. The rip goes back and forth, and within the large back and forth parts, there are smaller back and forth parts, and within them, even smaller structures; there is a spectrum of length scales involved. This self-similarity reflects the underlying self-organized structure, and is a common feature of interacting systems far from equilibrium, which often tune themselves into emergent structures spontaneously. By emergent, I mean the properties of the interacting system of many atoms can be quite different from the simple and relatively uninteresting properties of the individual atoms themselves.

It is not only our example of ripped paper which does this, I draw your attention – again – to other emergent behaviour, where Mother Nature takes many simple individual constituents and gives you something from nothing, sometimes. Indeed, as I write this, in the Canadian sky far above me, super-cooled randomly fluctuating water-vapour molecules stand ready to crystallize into beautifully symmetric, and almost infinitely variably-shaped snowflakes.

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