crystals – Bean Thinking

biscuits gone wrong, crystals in the oven — Expanding biscuits are a 2D example of a close packed crystal lattice.

Blaise Pascal once wrote of the benefits of contemplating the vast, “infinite sphere”, of Nature before considering the opposite infinity, that of the minute¹. And although the subject of today’s Daily Grind involves neither infinitesimally small nor infinitely large, a consideration of biscuits and coffee can, I think lead to what Pascal described as “wonder” at the science of the very small and the fairly large.

The problem was that my biscuits went wrong. Fiddling about with the recipe had resulted in the biscuit dough expanding along the tray as the biscuits cooked. Each dough ball collapsed into a squashed mass of biscuit, each expanding until it was stopped by the tray-wall or the other biscuits in the tray. When the biscuits came out of the oven they were no longer biscuits in the plural but one big biscuit stretched across the tray. However looking at them more closely, it was clear that each biscuit had retained some of its identity and the super-biscuit was not really just one big biscuit but instead a 2D crystal of biscuits. The biscuits had formed a hexagonal lattice. For roughly circular elements (such as biscuits), this is the most efficient way to fill a space, as you may notice if you try to efficiently cut pie-circles out of pastry.

salt crystals — Salt crystals. Note the shape and the edges seem cuboid.

Of course, what we see in 2D has analogues in 3D (how do oranges stack in a box?) and what happens on the length scale of biscuits and oranges happens on smaller length scales too from coffee beans to atoms. Each atom stacking up like oranges in a box (or indeed coffee beans), to form regular, repeating structures known as crystal structures. To be described as a crystal, there has to be an atomic arrangement that repeats in a regular pattern. For oranges in a box, this could be what is known as “body centred cubic”, where the repeating unit is made up of 8 oranges that occupy the corners of a cube with one in the centre. Other repeating units could be hexagonal or tetragonal. It turns out that, in 3D, there are 14 possible such repeating units. Each of the crystals that you find in nature, from salt to sugar to chocolate and diamond can be described by one of these 14 basic crystal types. The type of crystal then determines the shape of the macroscopic object. Salt flakes that we sprinkle on our lunch for example are often cubic because of the underlying cubic structure on the atomic scale. Snowflakes have 6-fold symmetry because of the underlying hexagonal structure of ice.

It is possible to grow your own salt and sugar crystals. My initial experiments have not yet worked out well, but, if and when they do, expect a video (sped up of course!). In the meantime, perhaps we could take Pascal’s advice and wonder at the very (though not infinitesimally) small and biscuits. And if you’re wondering about where coffee comes into this? How better to contemplate your biscuit crystals than with a steaming mug of freshly brewed coffee?

¹Blaise Pascal, Pensées, XV 199

Coffee ring bacteria

coffee ring, ink jet printing, organic electronics — Why does it form a ring?

We have all seen them: Dried patches of coffee where you have spilled some of your precious brew. The edge of the dried drop is characteristically darker than the middle. It is as if the coffee in the drop has migrated to the edge and deposited into a ‘ring’. It turns out though that these coffee rings are not just an indication that you really ought to be cleaning up a bit more often. Coffee rings have huge consequences for the world we live in, particularly for consumer electronics. Various medical and diagnostic tests too need to account for coffee ring effects in order to be accurate. Indeed, coffee rings turn up everywhere and not just in coffee. Moreover, the physics behind coffee rings provides a surprising connection between coffee and the mathematics of bacteria growth. To find out why, we need to quickly recap how coffee rings form the way they do.

When you spill some coffee on a table it forms into droplets. Small bits of dust or dirt or even microscopic cracks on the table surface then hold the drop in the position. We’d say that the drop is pinned in position.

artemisdraws, evaporating droplet — As the water molecules leave the droplet, they are more likely to escape if they are at the edge than if they are at the top. Illustration by artemisdraws.com

As the drop dries, the water evaporates from the droplet. The shape of the drop means that the water evaporates faster from the edges of the drop than from the top (for the reasons for this click here). But the drop is stuck (pinned) in position and so cannot shrink but instead has to get flatter as it dries. As the drop gets squashed, water flows from the centre of the drop to the edges. The water flow takes the coffee particles with it and so carries them to the edge of the drop where they deposit and form into a ring; the coffee ring. You can see more of how coffee rings form in the sequence of cartoons below and also here.

However in this quick explanation, we implicitly assumed that the coffee particles are more or less spherical, which turns out to be a good assumption for coffee. The link with the bacteria comes with a slightly different type of ‘coffee’ ring. What would happen if we replaced the spherical drops of coffee particles with elliptical or egg shaped particles? Would this make any difference to the shape of the coffee rings?

Artemisdraws — As water evaporates from A, the drop gets flatter. Consequently, the coffee flows from A to B forming a ring. Illustration by artemisdraws.com

In fact the difference is crucial. If the “coffee” particles were not spherical but were more elliptical, the coffee ring does not form. Instead, the elliptical particles produce a fairly uniform stain (you can see a video of drying drops here, yes someone really did video it). The reason this happens is in part due to a pretty cool trick of surface tension. Have you ever noticed how something floating on your coffee deforms the water surface around it? The elliptical particles do the same thing to the droplet as they flow towards the edge. (Indeed, the effect is related to what is known as the Cheerios effect). This deformation means that, rather than form a ring, the elliptical particles get stuck before reaching the edge and so produce a far more uniform ‘coffee’ stain when the water dries.

E Coli on a petri dish — A growing E. Coli culture. Image courtesy of @laurencebu

By videoing many drying droplets (containing either spherical or elliptical particles), a team in the US found that they could describe drying drops containing elliptical particles with a mathematical equation called the Kardar-Parisi-Zhang equation (or KPZ for short). The KPZ equation is used to describe growth process such as how a cigarette paper burns or a liquid crystal grows. It also describes the growth of bacterial colonies. Varying the shape of the elliptical particles in the drying drop allows scientists to test the KPZ equation in a controllable way. Until the team in the US started to ask questions about how the coffee ring formed, it was very difficult to test the KPZ equation by varying parameters in it controllably. Changing the shape of the particles in a drying drop gives us a guide to understanding the mathematics that helps to describe how bacterial colonies grow. And that is a connection between coffee and bacteria that I do not mind.

As ever, please leave any comments in the comments section below. If you have an idea for a connection between coffee and an area of science that you think should be included on the Daily Grind, or if you have a cafe that you think deserves a cafe-physics review, please let me know here.