Department of Applied Mathematics, **Machine** CM

Research School of Physical Sciences and Engineering

**Co-Investigator** Peter Pieruschka

Department of Applied Mathematics,

Research School of Physical Sciences and Engineering

**Monte Carlo Simulation of Curvature-Elastic Interfaces**

Surfactant molecules in solution spontaneously form two-dimensional elastic membranes, which then serve as building blocks for different phases ranging from perfectly ordered minimal surfaces to partially ordered dispersions with characteristic length scales. These structures appear in biological context (cell membranes, organelles) and in surface chemistry (detergents, foams) and have many practical applications. In this project we studied their statistical, mechanical and topological properties.

At the beginning of the project a surface Monte Carlo algorithm was developed in order to study topology and stability of different phases. Interfacial systems were equilibrated using the energy expression based on the bending elasticity of the films. During 1994 we analysed the scattering properties of many such systems. The analysis confirms the existence of various kinds of local order in isotropic amphiphilic sponge structures. For example, the structures may favour tubular, lamellar or saddle-shaped local arrangement. Our findings are in agreement with neutron scattering and freeze-fracture electron microscopy experiments, thus confirming that such systems are controlled by the elastic bending energy of the constituent membranes.

**What are the basic questions addressed?**

What will be the characteristics of surface structures obtained by thermal equilibration of elastic surfactant membranes?

**What are the results to date and the future of the work?**

The algorithm in itself is new, based on the Fourier representation of surface structures and calculation of their elastic energies. It enabled us to equilibrate the system and obtain the energy-entropy relationships which drive membranes (specified by their elastic moduli) into specific shapes (layers, sponges, bubbles etc.). During 1994 we proceeded with the analysis of equilibrated systems in order to compare the results with data from neutron scattering experiments.

The first stage of the project has now been completed and summarised in the PhD thesis of Peter Pieruschka. A report on scattering properties of equilibrated phases has been submitted for publication. Future work may include membranes with electrostatic interactions and improvement of the Monte Carlo algorithm using wavelet representations.