Foam

Espresso crema

A nice espresso in the morning does wonders for a person. The golden crema over a strong caffeine rich liquid is the secret to a good espresso. The crema is a foam. Also cappuccino froth is a milk foam. Have you ever beaten egg whites? After some time a foam appears and when you do it long enough the foam is firm enough to stay in its given form. When baking bread or a cake with yeast or baking soda, the batter becomes a foam when it is heated in the oven as carbon dioxide is produced inside. Today I want to talk about foam.


What actually is a foam? A thermodynamic two-phase system consisting of gas bubbles trapped inside a solid or liquid matrix with the gas cells randomly close packed. In a simple static model a bubble is determined by an equilibrium of the internal gas pressure and the wall tension. The gas pressure depends on the temperature and volume and the surface tension depends on the wall material and the curvature of the bubble. A foam is many bubbles stuck together. Simple? It is not directly clear that with these ingredients we also have a simple solution for a foam model. Since the parameters are interdependent it could be hard to make any calculations about for example the stability of the foam. Especially if the parameters act non-linearly, which they actually do.

Cappuccino with micro foam!

The configuration of one or more bubbles in a steady state is described by the Young-Laplace equation.* One can calculate the bubble shape with this non-linear partial differential equation $\Delta p = - \gamma\nabla \cdot\hat{n}= \gamma({1\over{R_1}}+{1\over{R_2}}) $. Here $\Delta p$ is the pressure difference across the fluid interface and is equal to the surface tension $\gamma$ times the sum of the inverse of two principal radii ($R_1$,$R_2$) of the bubble. Since the pressure is related to the radius of the bubble it makes the equation non-linear, and hard to calculate a general solution.  Does it mean we have to give up hope to understand the physics of foam now? Well there are always the numerical solutions for a given condition. Since this is a static equation it doesn't tell us how the foam will evolve in time. Still we can play a bit with different boundary conditions and see what happens. It should be noted that a two-phase (quasi-static) flow system (a stationary foam) is notoriously difficult to analyse.

Foam from detergent

First of all during the formation the bubbles get perturbed by whisking or aerating. When taking one bubble and stretching it along one axis, we get a sort of a tube. One can imagine taking two ends of a soap bubble with iron rings and stretching it so we will have open ends and a cylindrical surface. What will happen is that at the centre of the tube the films will get closer to one another. This is because of the fact that the pressure difference at both sides of the surface is zero. This is also known as the Plateau-Rayleigh instability more commonly seen when a stream of water breaks into droplets. The surface tension overcomes the straining forces and smaller bubbles will form.


When leaving a foam from a liquid to rest, the strength of the "domain walls" will degrade and fail to maintain structural integrity and the bubble breaks. If this happens the foam will disappear.  How does this work? Dynamically the lamellae between the older bubbles will start to get thinner due to drainage, capillary action, and when the film becomes too thin it collapses, creating a cavity (larger bubble).

The smaller bubbles sustain a larger pressure differential across the interface, hence they have thicker walls. A foam made of smaller bubbles is more rigid, which might be desirable for certain food applications. In coffee parlance it is a so-called microfoam. However smaller bubbles tend to coalesce to larger bubbles over time in a process called coarsening. As small bubbles have a large pressure than the larger ones when there exists a gap between the two the smaller one disappears.

To make a good foam we need the interface between the two bubbles to be able to handle the surface tension without breaking too quickly. This means that the molecules that make up the surface must stick together rather well and/or act as surfactant to reduce the surface tension. Molecules that do that are often polar and hydrophobic and because of the electrical attraction will stay together. Soap is one example of a polar hydrophobic molecule, and we have seen that soap foams rather well. Other substances which have polar and hydrophobic properties are fatty acids with long tails. In fact natural soap is made of fatty acids.

The crema on top of the coffee is made from several sources: Fatty acids, cholesterol, sugars. There are polymer molecules called melanoidines that give the coffee its brown look, and form after roasting the beans due to the Maillard reaction. It is hard to say if they are solely responsible for the crema. Funnily enough most melanoidines are apolar but they are hydrophobic. Here an interesting article about crema.

Beautiful foams can be made from proteins as well. Beating egg white, or albumin, produces a nice stiff foam from the denatured proteins. Denaturing means that the protein folding structure is changed. Proteins are made of amino acids which determine the primary structure, the proteins can fold in the larger structures, named secondary and tertiary structures. Beating an egg changes the tertiary structure and straightens out the molecules so they can act as a surfactant! However, when beaten too much, or left to its own for a while we see the foam disintegrating because of the reasons above. Some proteins in egg white contain sulfide groups which can make disulfide bonds. In the tertiary structures these play a stabilising role, hence they must be deactivated to prevent the tertiary structure from forming.

When one uses a copper bowl to whisk the eggs, the protein complex formed stays strong even after the water has drained from the lamellae. It's not uncommon for metal-ions to bind with proteins, for example haemoglobin, an iron-protein complex. The copper ions bind with the sulfur in the proteins to prevent disulfide bridges from forming in the tertiary structural folding hence making the foam less susceptible to decay. In eggs ovotransferin has an ability to react with cations and hence be more stable (especially at higher temperatures - good for souflés!) Several cations can be used: Copper, aluminium, iron and zinc are known to stabilise the foam.

The milk for a cappuccino is best foamed at low temperature. The casein proteins in milk have the largest surface activeness. Casein has little tertiary structure because it lacks disulfide bonds. As a result, it cannot be denatured and is reasonably stable and well suited for foaming. The other proteins in milk, whey proteins, are less suited. For example, beta-lactoglobulin the most significant whey protein in milk, does have disulfide bonds and tends to form gels at higher temperature. (Incidentally, in the whey proteins the calcium is bonded in milk through the same metal-sulfur complex. Also casein bonds calcium, although differently.) The composition of cows milk is 20% whey proteins and 80% casein, while human milk contains 60% whey 40% casein. This would mean human milk might not be the best choice for a cappuccino ;)

To make the best foam for a cappuccino would hence be: inject air/steam into the milk (usually steam), froth it, making sure you get small bubbles ("micro foam") and only then warm the foam. Do not let the milk become too warm or it will affect the stability of the foam.

Think about the intricate physics that is going on next time you beat your egg whites or wash the dishes. There's a lot more than meets the eye.



* Incidentally, the Young-Laplace equation is similar to the Gullstrand equation which relates the optical power of a thick lens to its geometry and media properties. The Gullstrand equation is provable through Fermat's principle. Hence I'm very much inclined to think that the YL equation is probably the result of a variational technique of a thermodynamic potential.