The 'Cheerios Effect'
Entry by Andrew Capulli, AP225 Fall 2011
The ‘Cheerios Effect’ D. Vella and L. Mahadevan, American Journal of Physics, 73, 817-825, 2005.
As the physics and engineering sciences (all sciences really) have gone increasingly 'nano' in the hopes of addressing fundamental questions in the respective fields, there has been debate over what phenomena are most important (ie what should should we be looking into as scientists?) In this paper "The Cheerios Effect," which appears friendly enough with such a title, the authors address some fundamental questions concerning capillarity and buoyancy using some simple experiments to demonstrate their claims. Why do cheerios or bubbles seem to 'attract' and further, are they really 'attracted' to the side of the bowl or glass? Vella and Mahadevan put the science behind these (and more) phenomena demonstrating the importance of buoyancy of small particles in contrast to the traditional mindset of capillarity dominance.
Capillarity vs Buoyancy: Attraction and Repulsion
Soft Matter Discussion The authors begin with a discussion of buoyancy and capillarity as it pertains to bubbles and, living up to the title, cheerios. To generalize my discussion I'll just refer to the bubbles and cheerios as particles. The authors argue that the typical mentality of purely analyzing particle behavior at the liquid-particle-gas interface in terms of capillarity does not suffice (this is simply too 1 dimensional, it neglects the the up/down forces due to buoyancy). So what effects does buoyancy have and how can this explain the 'attraction' of floating particles to each other and to the side of the container? Particles light enough (small and less dense than the liquid they displace) have a positive buoyancy. This positive buoyancy 'wants' to push the particle up and out of the liquid but the particle is kept at the interface by surface tension and capillarity which come to a 'compromise' with the particle somewhat submerged and somewhat protruding from the liquid. This is a very boiled down summary of what is happening to the particle; Vella and Mahadevan more eloquently establish this argument in the first few pages of the article and demonstrate their point via the description of simple-cereal-experiments. Below I've attempted to draw these points out to help with my description. So how does buoyancy explain particle aggregation and 'attraction' to container walls? The positive buoyancy seen in the drawing [A] results in, as previously described, an upward net force on the particle out of the liquid. However, since the particle cannot simply float away into the air (because of the capillarity and surface tension of the liquid) it does, as the authors describe, "the next best thing." The "next best thing" is to rise up the meniscus of the liquid on the container wall [C] or the meniscus (contact angle) on another particle [B] (see my power point sketches below).
Rising up the meniscus of another particle explains particle aggregation and rising up the wall meniscus explains the particle's apparent 'attraction' to the wall. So, interestingly enough, there is a reason cereal, bubbles, and other 'particles' group together and settle at the wall of their container. Similarly, there is an 'attraction' between negatively buoyant particles or substances. As the authors describe, a negatively buoyant tack is kept from sinking by the surface tension of the liquid this time causing an upward force of the tack. Another tack floating nearby has the same negative buoyancy and, of course, the same upward force due to surface tension. As shown in Figure 3 below, the tack has substantial negative pressure and the perturbation it causes in the liquid results in a substantial upward force due to a surface tension that 'wants' to return to its equilibrium state of less energy. This is why the two tacks in the thought experiment will attract; with the tacks closer together, the liquid will be less disturbed and thus less surface tension is needed to keep the tacks afloat. Vella and Mahadevan describe the 'attraction' as one tack "falling down" another tack's contact angle which reduces the energy of the system.
Even more interesting is the authors' discussion of particle 'repulsion.' Vella and Mahadevan describe this repulsion as a difference in buoyancy. Positively buoyant particles have an downward force on them due to surface tension while negatively buoyant particles have an upward force on them due to surface tension. These opposing surface tensions cause the apparent repulsion of particles. The authors give the example of the tack-cap (negative buoyancy) versus the cap (positive buoyancy). In both the tack-cap and the cap, the only surface touching the liquid is the plastic cap. However, because of the weight of the tack, the tack-cap has a negative buoyancy while the cap alone has a positive buoyancy (two different contact angles for one material!). This means contact angle is dependent on buoyancy and since these two complexes have opposing buoyancies, they will 'repel'.
Previous wiki postings on this article describes the author's derivation of the force between two infinite plates (wetting/non-wetting). [] The two main points of this discussion being that as surface tension of the liquid increases, particle 'attraction' or 'repulsion' is reduced and that like contact angles (menisci) 'attract' while unlike contact angles 'repel' (as discussed above, positive/negative buoyancy attracts positive/negative buoyancy while positive/negative buoyancy repels like buoyancy).
While cereal and bubbles become exciting when the phenomena describing their behavior is explained, even more interesting is the physics behind water bugs. The authors describe how a water spider may exit the surface of the water. The spider, as seen below in Figure 12, has the same upward force on its legs as the tacks described above. However, as the spider goes to exit the water, its front legs are lifted and consequently more force is applied to the same area on its back legs. This should cause the spider to sink but it doesn't. The thought is, the spider uses its front legs to pull up on its back legs and consequently the back legs then have a positive buoyancy. This positive buoyancy pushing the legs up allows for the hind legs to ride up the meniscus at the edge of the water just as the cheerios rose up the meniscus at the edge of the bowl.
More Thoughts... Connection to Physiology
I like to try to connect the physics we learn to physiology and bioengineering so here's my attempt: So, how can the cereal relate to human physiology? Vella and Mahadevan have discussed phenomena that have seemingly infinite applications which aren't limited to cereal, tacks, and water spiders. Take the circulation system in the body for example. When there is a rupture in the vasculature, if small enough, the body can seal it off with its natural cascade of clotting. The chemical cascade of coagulation is extremely complicated process of the assembly and destruction of factors and co-factors, proteins, enzymes, cells, and the list goes on and on (see the wiki on coagulation: http://en.wikipedia.org/wiki/Clotting). But maybe, in addition to the chemical assembly of a clot, there is something physical there as well. Could the platelets, some of the first on the scene, be physically 'attracted' to the wound? Perhaps there is a meniscus that forms at the wound edges and like the cheerios around the side of the bowl, platelets 'attract' each other and are further 'attracted' to the meniscus at the wall due to their (possible) positive buoyancy. I've tried to illustrate my thoughts below in (what else?) power point drawings:
But this theory relies on platelets having a positive buoyancy in blood which may or may not be true. When blood is 'spun down' in a centrifuge the red blood cells (the heaviest) are at the bottom and many of factors including platelets are at the top. It may be possible that is the case... that the platelets undergo the same phenomenon that the cheerios undergo. Of course its not as simple as this. The chemical cascade is well studied and further the shearing of the blood on platelets helps to initiate the coagulation cascade. But, how does that first platelet get to the scene? Its possible, I suppose, that it 'rides' up the meniscus at the edge of the wound...