Very soft, soft matter

With excellent timing, just as I prepared to tell students in my biological physics lectures that we are roughly 20% protein (by mass), and that gelatin is a very abundant protein, which holds our tissues together, The Guardian ran an article on how cool gelatin desserts, aka jellies are. The article included the, slightly odd in my opinion, YouTube video above.

Odd or not, this video is a perfect illustration of something from a living organism, gelatin, being literally soft matter — the other half of the course I am teaching is on soft matter. To see how soft this soft matter is we can use the video to obtain a simple Fermi estimate of the (shear) modulus E, which quantifies how soft a material is.

Roughly speaking a material that is wobbling back and fore as above is a bit like a spring in the sense that when it wobbles an amount x then this costs an energy of about

energy of stretching ~ Elx²

for l the size of the jelly, here around 5 cm I guess. This is just the formula for a spring but with a spring constant k = El, in other words, a material with a modulus E and that is l across, behaves as a spring with spring constant k = El. For l = 5 cm and a wobble of say 1 cm, the energy of stretching is about 5 × 10^-6 E J (NB I had to convert from cm to m).

Just as with a mass on a spring this energy is equal to the

maximum kinetic energy ~ m v²

the jellies are probably about 0.1 kg and speed looks to be a few cm/s, so say v ~ 3 × 10^-2 m/s. If we equate the two energies, E is the only unknown and we get E ~ 20 Pa* (Pa = Pascal the unit of a modulus like E).

This is very soft. For comparison, for a metals E is about a billion times larger. Soft matter like jellies, is often genuinely very soft and wobbly. And the same applies to the tissues like muscle and brain that our bodies are made of. This similarity should not surprise us as both jellies and our tissues are held together by more-or-less the same stuff: the protein collagen, which when treated is sold as gelatin.

* I suspect this rough estimate may be a bit low, E may be close to a few hundred Pa.