View source for Dynamics of surfactant-driven fracture of particle rafts

Original entry by Joerg Fritz, AP225 Fall 2009 

== Source ==
[http://scitation.aip.org.ezp-prod1.hul.harvard.edu/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000096000017178301000001&idtype=cvips&gifs=yes "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 phases|Surfactant]], [[Fracture]], [[Cracking]], [[Particulate interface]], [[Surface tensions|Surface tension]]

== Summary ==

A densely packed monolayer of particles (a particle raft) at the interface between two fluids with a large difference of surface pressure (such as air and water) generally shows a behavior that is similar to an ideally elastic two-dimensional solid. In contrast to this, if this 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. This 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.

[[Image:Fritz_raft_1.jpg|thumb|400px|Fig.1 The addition of a drop of surfactant to a monolayer of particles leads to the propagation of cracks, seen in black.]]

This paper studies how particles rafts behave in 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 not set by the properties of the layer itself, but instead 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 ==

== Experimental results for surfactant-induced cracking ==

some observations:

-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 length of a single crack ==

[[Image:Fritz_raft_2.jpg|thumb|400px|Fig.1 The addition of a drop of surfactant to a monolayer of particles leads to the propagation of cracks, seen in black.]]

The experiments show that crack propagation is much slower than what would be expected based on the speed of sound waves in the medium. This 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. This energy has to be dissipated. Two good candidates for the mechanism by which this dissipation occurs are immediately obvious. The first is the flow of fluid between particles in the raft, which are displaced during the 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>