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, le• 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 = ao

(optimal head group area)

N(min)=2ao ao=(K/)

2)(2 ooN aaa

a

aKaN /

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, lc.

• This maximum length is somewhat empirical, and it corresponds to the length beyond which the chains can no longer be regarded as fluid.

• Given ao, v and lc (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/ao.

Physically, it must have

r lc

hence

• Different molecules, with different geometrical ratios, will therefore favour different-shaped micelles.

3

1

cola

v

Slide 5: Spherical micelle.

v

R

a

RM

o 3

44 32

Slide 6: Amphiphiles assembling into a bilayer. Reprinted with permission from “Intermolecular and Surface Forces”, Jacob N. Israelachvili, Academic Press 1991. 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, εα, at the size α*, which has the optimal curvature for packing, then we can write the approximation

Using this in the general expression for xα 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.