The ‘Cheerios Effect’

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Authors: Dominic Vella & L. Mahadevan

Source: arXiv:cond-mat/0411688v3 [cond-mat.soft], 2008

Soft matter keywords: bubbles, interfacial tension, wetting


Fig.1 : D.Vella & L.Mahadevan, arxiv 2008

The authors of this publication investigate the physics underlying the 'Cheerios effect', i.e. the aggregation of breakfast cereal floating in a bowl of milk! The phenomenon is not limited to breakfast cereal, but also applies to bubbles floating in all sorts of liquids. Its understanding paves the way for many interesting micro-electromechanical applications. The authors start out by citing the well-known forces at play between infinite plates of different or same wettabilities, immersed in liquid. Next they provide a suggestion on how these laws might differ for spherical objects floating in liquid. Finally they propose a dynamic model for particles floating in a liquid.

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Fig.2 : D.Vella & L.Mahadevan, arxiv 2008

Figure 2 demonstrates four different scenarios of plates immersed in water: (a) features two wetting plates, (b) two non-wetting plates, (c) a wetting and a non-wetting plate and (d) a wetting and a non-wetting plate at short range. On the vertical plane, the plates experience a force balance between their weight and the interfacial tension:

<math>\gamma \frac{d^2h}{dx^2} = \rho g h</math>

Where <math>\gamma</math> is the surface tension coefficient for the liquid-gas interface and <math>\rho</math> the density of the liquid. This differential equation is solved, having as boundary conditions the menisci contact angles <math>\theta_1, theta_2</math>. After some manipulation, the authors derive the horizontal force per unit length experienced by the plates. The case of floating spherical particles however is somewhat different: in addition to interfacial tension and weight, the particles also have to balance the buoyancy force:

<math>\frac{4}{3} \pi \rho_s g R^3 = 2 \pi \gamma R sin\phi_c \frac{z_c'}{\sqrt{1+z_c'^2}} + \rho \pi g R^3 (\frac{z_c}{R}sin^2\phi_c + \frac{2}{3} -cos\phi_c + \frac{1}{3}cos^3\phi_c)</math>

Again, after some manipulation the authors derive the horizontal force between two interacting, floating particles to be:

<math>F(l) = -2 \pi \gamma R \Beta^{5/2} \Sigma^2 K_1 \frac{l}{L_c}</math>

Fig.3 : D.Vella & L.Mahadevan, arxiv 2008

In the above equation D is the relative density <math>D = \frac{\rho_s}{\rho}</math>, <math>L_c</math> is the characteristic length scale <math>L_c = \frac{\gamma}{\rho g}</math>, while <math>\Beta</math> and <math>\Sigma</math> are scaling factors :

<math>\Beta = \frac{R^2}{L_c^2}</math>

<math>\Sigma = (\frac{2D-1}{3} - \frac{1}{2}cos\theta + \frac{1}{6}cos^3\theta)</math>

The authors make a couple of significant observations regarding this result:

  • The sign of the force is positive if the particles have same menisci (attractive force) and negative when the menisci are unlike (repulsive force). Much like the simpler case of infinite plates (fig.2).
  • The strength of the inter-particle interaction decreases as the surface tension <math>\gamma</math> increases. The reader can grasp this intuitively by considering that at high surface tensions