# Cheerios effect

Five paper clips cling to each other, supported on the water’s surface by surface tension. One red paperclip has sunk to the bottom of the cup, showing that these paper clips don't float.

In fluid mechanics, the Cheerios effect is the tendency for small wettable floating objects to attract one another. An example of the Cheerios effect is the phenomenon whereby breakfast cereal tends to clump together or cling to the sides of a bowl of milk. It is named for the breakfast cereal Cheerios and is due to surface tension and buoyancy. 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. This can include a multitude of things, including hair particles in shaving cream and fizzy beer bubbles. The effect is not noticeable for 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 nanoscopic scale, and boats and other large objects floating in a choppy sea are subject to its classical equivalent; both are caused by waves, not surface tension.)

## Explanation

The quality of surface tension allows the surface of a liquid to act like a flexible membrane. A variety of weak forces act between liquid molecules to cause this effect.

At the interface between water and air, water molecules at the surface are pulled forcefully by water molecules beneath them but experience only a weak outward pull from the air molecules above. Therefore, the surface of the water caves in slightly, forming a curve known as a meniscus.

Water adjacent to the side of a container curves either upward or downward, depending on whether the liquid is attracted to or repulsed by the material of the wall. For example, since water is attracted to glass, the water surface in a glass container will curve upwards near the container walls, as this shape increases the contact area between the water and the glass. A floating object which is less dense than water, seeking the highest point, will thus find its way to the edges of the container. A similar argument explains why bubbles on surfaces attract each other: a single bubble raises the water level locally, causing other bubbles in the area to be attracted to the first. Conversely, dense objects like paper clips can rest on liquid surfaces due to surface tension. These objects deform the liquid surface downward. Other dense objects, seeking to move downward but constrained to the surface by surface tension, will be attracted to the first.[2] Objects denser than water will repel objects less dense than water: dense objects deform the water surface downward, and less dense objects tend to move upward, away from the dense object. Objects with an irregular meniscus also deform the water surface, forming "capillary multipoles". When such objects come close to each other, they first 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 self assembly of small structures.[4] They calculate the force between two spheres of density $\rho_s$ and radius $R$ floating distance $\ell$ apart in liquid of density $\rho$ as

$2\pi\gamma RB^{5/2}\Sigma^2K_1\left(\frac{\ell}{L_c}\right)$

where $\gamma$ is the surface tension, $K_1$ is a modified Bessel function of the first kind, $B=\rho gR^2/\gamma$ is the Bond number, and

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

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