# The ‘‘Cheerios effect’’

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## Overview

Authors: Dominic Vella and L. Mahadevan

Keywords: Surface tension

Fig.1 Cheerios clump together in the bowl
Fig.3 A drawing pin ﬂoating upturned on water

## Summary

The Cheerios effect is the tendency for wettable floating objects to attract or repel one another. The attraction or repulsion of particles at interfaces depends solely on the wetting properties, namely, the contact angles, of the particles. An example for the Cheerios effect is the behavior of bubbles on the surface of the water or the tendency of cereals to clump together or stick to the sides of a bowl of milk. It is named for the breakfast cereal Cheerios.

## Explanation

Fig.2 A buoyant bubble near the wall
Fig.4 Typical interfacial proﬁles for the different conﬁgurations of plate wettabilities. (a) Two wetting plates. (b) Two nonwetting plates. (c) A wetting and nonwetting plate at intermediate separation. (d) A wetting and nonwetting plate at short range.

The force between the objects originates from the deformation of the liquid air interface due to the presence of the object. If we consider a gas bubble in the liquid near the wall it has a resulting force that is pulling it up. The wetting properties of the solid surface deform the liquid-air interface as shown in Fig. 2. Because the bubble cannot rise vertically across the interface, it can lower it's energy by moving upward along the meniscus. Thus we se that a buoyant object tends to move towards the wall if the contact angle between liquid and wall is <$\pi/2$ If we consider a case of floating object that is more dense than liquid, it will deform the interface as shown in Fig. 3. From the similar considerations we would expect a bubble to repel and floating object to attract.

To generalise this idea we consider two infinite plates in the liquid as shown in Fig. 4. The pressure inside the liquid between the plates in Fig 4(a) is lower than atmospheric, because of the curvature of the interface, thus two plates attract. The opposite situation is shown in Fig.4(b). The situation is more complicated when one plate is wetting and another nonwetting. If the plates are on large distance (Fig. 4(с)) they repel each other but if they come close (Fig4(d)) they attract.

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.

## Discussion

We can find ‘‘Cheerios effect’’ in many aspects of everyday life like self arrangement of floating objects. Also this phenomena plays a curtail role in the life water walking creatures who use it to prevent themselves from drowning and also to move across the water surface. Here we come up with an easy explanation and also a quantitative description of the dynamics.

## Reference

Dominic Vella and L. Mahadevan, The ‘‘Cheerios effect’’ [Vella; L. Mahdevan; Am. J. Phys.; 73(9), 817, 2005.]