# Single-particle Brownian dynamics for characterizing the rheology of fluid Langmuir monolayers

## Introduction

The authors attempted to probe the rheology of fluid langmuir monolayers by tracking the diffusion of single particles at air-water interface. It is similar in spirit to the tracking of particles to study the rheology of three dimensional materials. The set-up is shown in figures 1 and 2. Polystyrene beads are interpersed in a bead of water in the presence of 3 different monolayers: N-palmitoyl-6-n-penicillanic acid (PPA), pentadecanoic acid (PDA) and L-α-dipalmitoyl- phosphatidylcholine (DPPC). The contact angle that the polystyrene beads make with the water can be measured using the set-up shown in figure 2 using the formula

where β is the water-glass contact angle.

The separation <L^{2}> between pairs of particles trapped between the water-air interface are measured with respect to time. Measuring separation between pairs of particles instead of the absolute displacement of one particle helps to remove any drift, as discussed in problem set 1. If the motion is random and isotropic one expects the squared separation distance to increase linearly with time. The slope gives the surface diffusion coefficient D that is converted to a surface viscosity by using an appropriate hydrodynamic theory.

For brownian motions of particles completely submerged in a newtonian fluid, D is simply related to η by Einstein-Smoluchowski's relation, but in this case, the theoretical analysis is much involved due to two factors:

1. Addition of surfactants results in viscoelastic behaviour of monolayers

2. Polystyrene beads transverse all three phases: air, monolayer, bulk water

## Keywords

brownian motion, brownian dynamics, langmuir monolayers, rheology

## Results

The phase diagram of the monolayers are obtained by measuring the surface pressure as a function of surface area per molecule. There is clear phase transitions from incompressible to compressible monolayers as we increase area per molecule, reminiscent of the phase transition between liquid to gas in ideal gas. The diffusion constant was measured as described in the introduction. D are expressed relative to the diffusion coefficient D_{0} for a clean water interface (i.e. an inviscid and compressible interface). In the experiments, D_{0} = 1.26 ± 0.19 μm^{2}s^{-1}. The experimental value of D (blue crosses with error bars) is then compared to various theoretical work that attempt to capture the physics of the Brownian dynamics.

The different work uses different approaches to model the brownian dynamics. For example, Danov et al studied numerically the motion of spheres with various immersion depths, while Stone appealed to perturbation theory. Some groups have also modelled the spheres as discs of radius R embedded in the monolayer. The results are summarized in figure 4. The Brownian dynamics of the monolayer system requires involved theoretical analysis and there is still no full understanding about this. Nevertheless, the authors' methodology allows reliable measurements of reliable values of surface viscosity, which highlights the potential usefulness of this method of Brownian particle rheology for characterizing fluid membranes.

## Personal Thoughts

There are many reasons why we would be interested in the rheology of langmuir monolayers. The intrinsic proteins found in the phospholipid bilayers in biological cells are analogous to the polystyrene beads in a sea of surfactant. Important phenomena, such as the fusion of vesicles, the breakdown of foams and emulsions, and the opening of transitory pores in membranes, are governed by the viscoelastic properties of films of amphiphilic molecules. Allowing reliable measurements of the rheology of such ampiphilic monolayers is a first step to understanding the dynamics of all these phenomena.

## References

1. Single-particle brownian dynamics for characterizing the rheology of fluid langmuir monolayers, Sickert et al, EPL 2007