# Elasticity of interfacial particle rafts

Original Entry by Xu Zhang for AP225, Fall 2009

## Contents

## Reference

Elasticity of interfacial particle rafts, Vella, D., P. Aussillous and L. Mahadevan, Europhysics Letters, 68 (2), 212-18, 2004

## Key Words

monolayer, elasticity, Young's modulus, Poisson ratio

## Summary

### Solid-like properties of particle rafts

In this paper, the collective behavior of a closed-packed monolayer of non-Brownian particles at a fluid-liquid interface is studied. Such particle layers behave like 2D elastic solids:

1.A monolayer buckles under sufficient static compressive loading demonstrating its ability to support and anistropic stress. This stress can only be supported by a material with a non-zero shear modulus, which is the signature of solid.

2.A monolayer fractures under relatively small tensions, which is also a properties of solid.

Capillary forces are responsible for the formation of the solid monolayer via the aggregation of particles trapped at the interface and give the monolayer cohesion under deformation. The particle monolayer studied here are distinct from other two-dimensional interfacial systems such as Langmuir monolayers because the large size of the particles(diameters of up to 6mm) makes them non-Brownian objects which interact solely via capillary attraction.In this respect they are similar to conventional bubble rafts, so they are called particle rafts here.

In this system, it is reasonable to assume that the particle raft behaves approximately as an isotropic solid, in which case its elastic behavior can be completely characterised in terms of Young's modulus, E, and Poisson ratio, <math>\nu</math>. These parameters are theoretically determined here.

### A simple model

**Young Modulus**

For a conventional 2D elastic solid, the mean stress <math><\sigma>=\frac{\sigma _1+\sigma _2}{2}</math>, and strain <math><\varepsilon >=\frac{\varepsilon _1+\varepsilon _2}{2}</math> ( Where the subscripts 1,2 denote the two principal directions), are related by

<math>\ \ \ \ \ \ \ \ \ \ \ <\varepsilon>=\frac{1-\nu}{E}<\sigma></math>

If we replace the stress by a thickness-averaged isotropic tension <math>\tau \equiv <\sigma>d</math>, then this relation reads

<math>\ \ \ \ \ \ \ \ \ \ \ <\varepsilon >=\frac{1-\nu}{Ed}\tau</math>

<math>\ \ \ \ \ \ \ \ \ \ \Longrightarrow \frac{1-\nu}{Ed}=\frac{d<\varepsilon>}{d\tau}=\frac{1}{A}\frac{dA}{d\tau}=\frac{1}{A_1+A_s}\frac{d(A_1+A_s)}{d\tau}</math>

<math>A_1</math>: the area of the system covered by liquid.

<math>A_s</math>: the area covered by solid particles.

<math>A=A_1+A_s</math>: the total area of the system.

<math>\Longrightarrow E\propto \frac{1-\nu}{1-\phi}\frac{\gamma}{d}</math>

(Here <math>A_s \ \ is\ \ constant,\ \ A_1\frac{d\tau}{dA_1}\propto \gamma,\ \ \phi=\frac{A_s}){A}</math>

**Poisson Ratio**

Figure 2 demonstrates the origin of the Poisson effect for a single rhombic cell in the lattice in which the two central particles are displaced by a distance of <math>2\varepsilon</math>. From elementary geometry, the rhombus joining the centres of the four particles in the undeformed state has width <math>2\sqrt{3} R</math> and height 2R, where R is the particle radius. During the deformation, the width decreases to <math>2\sqrt{3} R\sqrt {\frac{1-2\varepsilon}{3R}}\approx 2 \sqrt{3} R(1-\varepsilon/3R)</math> and the height increases to <math>2(\varepsilon +R)</math> so that the Poisson ratio is simply <math>\nu =1/\sqrt{3}</math>.