# Dynamics of surfactant-driven fracture of particle rafts

Original entry by Joerg Fritz, AP225 Fall 2009

## Contents

## Source

"Dynamics of Surfactant-Driven Fracture of Particle Rafts"

Dominic Vella, Ho-Young Kim, Pascale Aussillous, and L. Mahadevan: *Physical Review Letters*, 2006, 96, pp 178301-1 to 178301-4.

## Keywords

Particle rafts, Surfactant, Fracture, Cracking, Particulate interface, Surface tension

## Summary

A densely packed monolayer of particles (a particle raft) at the interface between two fluids with a large difference of surface tension (such as air and water) generally shows a behavior that is similar to an ideally elastic two-dimensional solid. While this simple idea has been surprisingly successful in a large regime of parameter space there are obviously limits where this is no longer a good approximation. If a particle layer comes into contact with surfactants it exhibits a highly nonlinear and complex cracking behavior that is very much unlike that of an elastic solid. The formation of cracks and their propagation are determined by the interaction between the surfactant and the layer of particles and are not a simple function of material properties as in the classic mechanics analog. This behavior can for example be observed when drinking black tea, where tea particles hanging together at the surface are being broken apart by the addition of milk.

The paper investigates this nonlinear regime experimentally and explains the observations with simple scaling arguments. The most surprising result is that the propagation speed of the crack is set by the transport of surfactant in the layer. This means that the propagation speed is not constant and a detailed analysis shows that the length of the crack <math>L_c</math> scales like

<math>L_c \approx t^{4/3}</math>

## Particle rafts in the elastic regime

The fact that particle rafts behave very similar to a 2D elastic solid has first been discovered and repeatedly tested in experiments. For this discussion the most relevant case is out of plane buckling for compression in the plane of the raft. The particle raft essentially acts like a solid with an apparent shear modulus of

<math>G \approx \frac{\gamma}{d}</math>

where *d* is the particle diameter and <math>\gamma</math> the surface tension. This behavior implies that the particle raft can sustain finite shear stress distributions and makes it plausible that this would also be the case outside of the linear regime, where cracks usually form to relive high stresses in one direction.

## Experimental results for surfactant-induced cracking

The experiments where performed with a densely packed layer of non-Brownian particles at the interface between air and a mixture containing water and glycerol (to allow large changes in fluid viscosity). After addition of a drop of surfactant, the propagation of cracks was observed for different initial conditions.

Some observations that are interesting in the context of soft matter and the following discussion are:

- the size of the drop of surfactant does not have an influence on the crack propagation
- the cracks show surprising kinks and frustrated branches that are usually associated with inhomogeneities of the material in solid mechanics
- the speed of crack propagation is much slower than the speed of sound waves in the medium
- if the raft is compressed until close to its buckling threshold, no cracks form after the addition of surfactants.
- once a crack has reached its equilibrium value a further compression leads immediately to buckling
- cracks maintain their final length and form over several hours

## Different scales of speeds in the problem

The combination of all these observations leads to the conclusion that the cracking behavior is not due to inertial effects, which has formerly been proposed. The different time scales in the problem more over indicate that parts of this behavior should be explainable if the transport of the surfactant at the tip of the crack is investigated in more detail.

Information about the crack is propagated by sound waves that scale like the shear wave speed of the raft which is

<math>v_s \approx \sqrt{G/\rho}</math>

which is on the order of meters per second. The observed speeds of propagation are in every case at least one order of magnitude lower. This means that the particles away from the tip of the crack learn of its existence long before the crack actually arrives and reorder themselves into a dynamical equilibrium.

The experiments also show that ahead of the crack, the velocity of the raft relative to the medium is negligible, even compared to the propagation speed, we thus have three distinct velocity scales in this problem. Combining these results we can follow that the displacements at the crack tip are essentially determined by the geometry of the crack itself, since the particles had time to find an equilibrium due to the shear sound wave and the speed of the raft relative to the medium is negligible (at least for the behavior at the tip, we will later learn that it is important in another context).

## The length of a single crack

All his suggests that the speed of advection of the surfactant is what limits the propagation.

The surfactant lowers the surface tension of the area where the crack will appear and thus increases the free energy of the system. If <math>\Delta \gamma</math> is the change in surface tension due to the addition of surfactant the free energy change scales like

<math>\dot{E}_c = \Delta \gamma A_c</math>

where <math>A_c</math> is the area of the crack. This energy has to be dissipated. Two good candidates for the mechanism by which this dissipation occurs are possible. The first, more obvious one, is the flow of fluid between particles in the raft, which are displaced during crack propagation. If we assume that this is essentially a 2D lubrication layer and the shear rate between particles is on the order of the rate of compression for the raft <math>\dot{\gamma}</math>, the dissipation scales like

<math>D_{lub} = \mu L_c^2 \dot{\gamma}^2</math>

The second, less obvious option can again be understood by observation of a real world system (think again of tea particles after the addition of milk). The raft start to move on the surface while the cracks form. Since the liquid far away from the interface is essentially static, there is a boundary layer (of size <math>\delta</math>) with a velocity gradient close to the boundary. The dissipation in this layer scales like

<math>D_{lub} = \mu \frac{L_c^2}{\delta} \dot{L}_c</math>

according to the classic Blasius boundary layer solution.

Surprisingly, an order of magnitude comparison shows that this second dissipation mechanism is clearly dominant. We can balance the two dominant terms and receive a scaling for the change of crack length over time

<math>L_c = \left (\frac{\Delta \gamma^2}{\mu \rho} \right)^{1/4} t^{3/4}</math>

so the length of the crack changes over time in a non-linear fashion. A comparison with measurements, as shown in figure two, confirms this theory.

Understanding the dynamics of this process is important, small particles on interfaces are common in many applications and their behavior in the presence of surfactants influences the stability of the drops or bubbles that are defined by this interface. A particluar promissing example of this, which is also mentioned in the paper, is the delivery of drugs by inhalation as described in the paper "Trojan particles: Large porous carriers of nanoparticles for drug delivery" by the Weitz group. The drug particles are contained in drops which are coated by a layer of particles, basically a particle raft. Once these particles are inhaled they come into contact with the inner lining of the lungs which acts as a surfactant and might induce cracks very similar to the ones described in this paper, thus releasing the drug in a very efficient manner.