# Cheerios effect

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Five paper clips cling to each other, supported on the water’s surface by surface tension. One of the clips has sunk to the bottom of the cup, showing that these paper clips normally do not float.

In fluid mechanics, the Cheerios effect is the phenomenon that occurs when floating objects that don't normally float attract one another. Wetting, an example of the "Cheerios effect," is when breakfast cereal clumps together or clings to the sides of a bowl of milk. It is named after the common breakfast cereal Cheerios and is due to surface tension. The same effect governs the behavior of bubbles on the surface of soft drinks.[1]

## Description

This clumping behavior applies to any small macroscopic object that floats or clings to the surface of a liquid. Examples of such objects are hair particles in shaving cream and fizzy beer bubbles. The effect is not noticeable in boats and other large floating objects because the force of surface tension is relatively small at that scale. (The Casimir effect, with a similar result, occurs at a nanoscopic scale, boats and other large floating objects in a choppy sea are subject to its classical equivalent. Both are caused by waves, not surface tension.)

## Explanation

At the interface between a liquid and air, molecules of the liquid are subject to greater attractive forces from those below than from air molecules. Opposing these forces is the attraction of the liquid molecules to the surface of the container. The result is that the liquid's surface forms a meniscus which exhibits surface tension and acts as a flexible membrane. This membrane may be curved with the center either higher or lower than the edges.

A floating object will seek the highest point of the membrane and thus will find its way to either the center or the edge. A similar argument explains why bubbles on surfaces attract each other: a single bubble raises the liquid level locally causing other bubbles in the area to be attracted to it. Dense objects, like the paper clips in the photograph, can rest on liquid surfaces due to surface tension. These objects deform the liquid surface downward. Other floating objects that are seeking to sink but are constrained by surface tension will be attracted to the first.[2] Objects with an irregular meniscus also deform the water surface forming "capillary multipoles". When such objects come close to each other they rotate in the plane of the water surface until they find an optimum relative orientation. Subsequently they are attracted to each other by surface tension.[3]

Writing in the American Journal of Physics, Dominic Vella and L. Mahadevan of Harvard University discuss the Cheerios effect and suggest that it may be useful in the study of the self-assembly of small structures.[4] They calculate the force between two spheres of density ${\displaystyle \rho _{s}}$ and radius ${\displaystyle R}$ floating distance ${\displaystyle \ell }$ apart in liquid of density ${\displaystyle \rho }$ as

${\displaystyle 2\pi \gamma RB^{5/2}\Sigma ^{2}K_{1}\left({\frac {\ell }{L_{c}}}\right)}$

where ${\displaystyle \gamma }$ is the surface tension, ${\displaystyle K_{1}}$ is a modified Bessel function of the first kind, ${\displaystyle B=\rho gR^{2}/\gamma }$ is the Bond number, and

${\displaystyle \Sigma ={\frac {2\rho _{s}/\rho -1}{3}}-{\frac {\cos \theta }{2}}+{\frac {\cos ^{3}\theta }{6}}}$

is a nondimensional factor in terms of the contact angle ${\displaystyle \theta }$. Here ${\displaystyle L_{C}=R/{\sqrt {B}}}$ is a convenient meniscus length scale.