by Pietro Cicuta Statistical mechanics and soft condensed matter Micelle geometry

Slide 1:The size and shape of a micelle are determined by the geometry of its constituent molecules and by the forces that act between them.

The shape of an amphiphile building block is characterised by – the area of the head group, a –the effective tail length, l e The planar bilayer, which makes up cell membranes, is only one example. Slide 2:States of aggregation for amphiphiles.

Two opposing effects control the head group area, a (interfacial area). Attractive forces from hydrophobic and surface tension effects. Energy directly proportional to the head group area. The repulsive forces arise from a range of sources, including charge repulsion (for charged head groups) and steric effects. The energy is inversely proportional to the head group area. Total interfacial energy per molecule: Minimum energy at a = a o (optimal head group area)  N (min)=2  a o a o =  (K/  ) Slide 3:Optimal head group area.

We have also to consider the packing constraints on the hydrophobic tails. They will occupy a volume, v, they are are assumed to be fluid and incompressible, and they have a maximum effective length, l c. This maximum length is somewhat empirical, and it corresponds to the length beyond which the chains can no longer be regarded as fluid. Given a o, v and l c (all measurable/ estimable quantities), the shape into which the lipids pack can be determined. Given different possible arrangements with comparable  N, the entropy will always favour the smallest aggregate. Slide 4:Packing and shape of aggregates.

For a sphere of radius r, the mean aggregation number, M, is hence r= 3v/a o. Physically, it must have r  l c hence Different molecules, with different geometrical ratios, will therefore favour different-shaped micelles. Slide 5:Spherical micelle.

Slide 6:Amphiphiles assembling into a bilayer. Reprinted with permission from “Intermolecular and Surface Forces”, Jacob N. Israelachvili, Academic Press Copyright 2011, Elsevier. The thickness of a biological membrane is around 3 nm for phospholipids with 18 carbon atoms in the hydrophobic chain.

Slide 7:The CMC in terms of aggregation number and microscopic energies. In the case of spherical micelles, if there is a sharp minimum in the energy, ε α, at the size α*, which has the optimal curvature for packing, then we can write the approximation Using this in the general expression for x α enables us to evaluate the critical concentration, Previously we had

Slide 8:Examples of phospholipids with varying geometry, leading to different self-assembled structures.

Slide 9:Further examples of phospholipids with varying geometry (continued from Slide 8).

Slide 10: Amphiphiles at high concentration assemble to form different phases, depending on their geometry and also on the mechanical bending properties of their basic constituents.