Mathematical modelling of multilayer structures in surfactant systems
Multilayer structures at interfaces are increasingly being recognized as important structural features for technological and biological applications. Such systems, which exhibit self-assembly at surfaces, include surfactants, polyelectrolytes, polymer-surfactant mixtures, lipids, and phospholipids. Specular reflectivity of neutrons and X-rays are currently being used to probe multilayer structures.
I co-supervised D.Phil. student Cara Morgan alongside Chris Breward and Peter Howell in this research project. This research is in collaboration with the Thomas/Penfold Group.
Surfactants, meaning 'surface active', are chemical molecules that contain both hydrophobic, 'heads' and hydrophilic 'tails'. A single surfactant molecule is called a monomer. Surfactants in aqueous solution arrange themselves in such a way so that their hydrophobic heads are shielded by water molecules. They do this by forming a monolayer at an interface, so that their tails remain in the aqueous solution but their heads do not. This process of arrangement at an interface is known as adsorption and has an effect on the surface tension of the fluid they are in. It is this property that has many beneficial applications in biological and industrial systems. If the surfactant concentration of a solution is locally high enough it can be energetically more favourable for monomers to combine in such a way so that their tails are shielded from the water molecules. They do this by forming spherical structures consisting of about 70~90 monomers called micelles. The minimum concentration at which this occurs is called the Critical Micelle Concentration (CMC).
Polymer-surfactant systems can be characterized into two main types: weakly interacting and strongly interacting systems. Weakly interacting systems contain neutral polymers. In aqueous solution surfactants will locally form complexes with the polymer molecules if the surfactant concentration is high enough. These complexes consist of a polymer backbone with a necklace of micelles attached along them and are called polymer-micelle aggregates. A single polymer molecule can form an aggregate with 20-30 micelles attached. The minimum concentration at which this occurs is called the Critical Aggregation Concentration. Strongly interacting systems consist of charged polymers which allow the formation of polymer-micelle aggregates but also polymer-monomer aggregates where the polymer acts as a backbone to a necklace of surfactant monomers. In strongly interacting polymer-surfactant systems the polymer-surfactant aggregates may also adsorb at the interface.
We are interested in the behaviour of systems containing complex structures such as micelles, vesicles, bilayers and polymer-surfactant aggregates. In particular the dynamic formation of such complexes is of interest in determining the structure and stability of multilayers. Cara's research uses mathematical models to describe the static behaviour of a range of representative systems and thus show how surface tension is related to the multilayer structures. The dynamics of surface adsorption, rearrangement and reorganization can be described by simple time-dependent models for polymer-surfactant systems in the bulk phase and can be linked to the behaviour of molecules at the surface. We use a combination of asymptotic, analytic and numerical methods to examine the models describing the different polymer-surfactant systems.
The theoretical models from Cara's research are complemented by ongoing neutron scattering experiments at (ISIS). The reflection of neutron beams on a flat surface is widely used to investigate thin films. The reflected neutrons from a thin film may undergo interference depending on the wavelength of the light and the structure of the surface of the film. The intensity of the reflected neutrons is measured and used to calculate the surface tension of the thin film. Experimental measurements of the surface tension characterize the behaviour and structure of a range of different multilayer systems at air-liquid and solid-liquid interfaces.
You can learn more about our work in our papers:
- Straining flow of a weakly interacting polymer–surfactant solution
C.J.W. Breward, I.M. Griffiths, P.D. Howell & C.E. Morgan
Eur. J. Appl. Math., 26, 743
- Mathematical modelling of multilayer surfactant self-assembly at interfaces
C.E. Morgan, C.J.W. Breward, I.M. Griffiths & P.D. Howell
SIAM J. Appl. Math., 75, 836
- The Marangoni flow of soluble amphiphiles
M. Roché, Z. Li, I.M. Griffiths, S. Le Roux, I. Cantat, A. Saint-Jalmes & H.A. Stone
Phys. Rev. Lett., 112, 208302
- The spreading of hydrosoluble surfactants on water
M. Roché, Z. Li, I.M. Griffiths, A. Saint-Jalmes & H.A. Stone
Phys. Fluids, 25, 091108
- A new pathway for the re-equilibration of a micellar surfactant solution
I.M. Griffiths, C.J.W. Breward, D.M. Colegate, P.D Howell & C.D. Bain
Soft Matter, 9, 853
- Kinetics of surfactant desorption at the air–solution interface
C.E. Morgan, C.J.W. Breward, I.M. Griffiths, P.D. Howell, J. Penfold, R.K. Thomas, I. Tucker, J.T. Petkov & J.R.P. Webster
Langmuir, 28, 17339
- An asymptotic theory for the re-equilibration of a micellar surfactant solution
I.M. Griffiths, C.D. Bain, C.J.W. Breward, S.J. Chapman, P.D. Howell & S.L. Waters
SIAM J. Appl. Math., 72, 201
- On the predictions and limitations of the Becker-Döring model for reaction kinetics in micellar surfactant solutions
I.M. Griffiths, C.D. Bain, C.J.W. Breward, D.M. Colegate, P.D. Howell & S.L. Waters
J. Coll. Interf. Sci. 360, 662