George Tucker

Harvey Mudd College Mathematics 2008

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Thesis Proposal: Interphase Line Tension in Langmuir Films
Thesis Report: Domain Relaxation of Langmuir Films
Thesis Advisor: Prof. Andrew J. Bernoff
Second Reader: Prof. Elizabeth Mann

Domain Relaxation of Langmuir Films with Dipole-Dipole Repulsion

A Langmuir film is a molecularly thin fluid layer on the surface of a subfluid; a classic example is the lipid bi-layer that forms the external membrane of a biological cell. Langmuir layers can have multiple phases, which can evolve a variety of morphologies. If the molecules in a phase have a strong vertical dipole moment (often a result of hydrophilic/hydrophobic chains orienting themselves on the subfluid), exotic morphologies such as dog-bone shapes and labyrinth patterns manifest themselves. Similar morphologies are observed in numerous other physical systems, including type-1 superconductors, chemical reaction-diffusion systems, and films of ferrofluids. The current hypothesis, which has been confirmed qualitatively, is that these patterns are caused by the competition between line tension forces and dipolar repulsion. The goal of my research is to develop an accurate qualitative and quantitative understanding of the dipolar repulsion forces in a Langmuir film. This entails extending a model for Langmuir films [1] to incorporate dipolar repulsion and observing changes in equilibrium states as dipole forces increase.

Literature Review

Heinig et al. (2004) have investigated systems driven by competition between repulsive long-range and attractive short-range interactions in models developed by others (see [4]). In these systems, they observe the formation of dog-bone and labyrinth patterns. Their numerical simulations with dipolar repulsion achieve qualitative agreement with experiments. Alexander et al. (2006) note, "Heinig et al. (2004) were able to qualitatively reproduce many experimental results, although the scheme exhibited moderate area loss," and develop a stable, accurate and efficient numerical model for Langmuir films dominated by line tension using a boundary integral method [3]. By comparing their model to experimental data, they were able to estimate the line tension force to within 10%. However, long-range dipolar repulsion forces were absent from their model.

Research Plan

Alexander et al. (2006) find that solving the boundary integral is numerically stiff and explicit methods (such as the work by Heinig et al. (2004)) are susceptible to instabilities. They resolve this problem by operator splitting (cf. Hou et al. (1994)). I plan to extend this formulation to include dipolar repulsion forces, while preserving the numerical benefits from this formulation. To validate the results, I plan to compare the results from the model to experimental data collected on Langmuir films (by Prof. Elizabeth Mann's group at Kent State). I can experimentally determine how changes in the dipole moment affect the equilibrium state. In addition, from experimental data, I can determine when the domain changes from a dog-bone shape to a labyrinth pattern.

References

  1. Alexander, J.C., Bernoff, A. J., Mann, E. K., Mann Jr, J. A., Wintersmith, J.R. & Zou, L. 2007 Domain relaxation in Langmuir films. J. Fluid Mech. 571, 191-219.
  2. Heinig, P., Helseth, L. E. & Fischer, T. M. 2004 Relaxation of patterns in 2D modulated phases. New J. Phys. 6, 189.
  3. Hou, T., Lowengrub, J. & Shelley, M. 1994 Removing the stiffness from interfacial flow with surface tension. J. Comput. Phys. 114, 312-338.
  4. Lubensky, D. K. & Goldstein, R. E. 1996 Hydrodynamics of monolayer domains at the air- water interface. Phys. Fluids 8, 843-854.