In Lecture 3, we explored the formation of micelles by looking at the energetics of their formation. In this lecture, we consider the factors that govern the size and shape of a micelle (Slide 1).

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.

**4.1 Aggregate shape**

What determines the shape of an aggregate? Slide 2 shows common states of aggregation and indicates how these are related to the geometry of the constituent amphiphiles. Here we characterise the geometry of an amphiphile using just two parameters: the area, a, of its head group and its effective tail length, *l*_{e}.

First we need to understand, in a more quantitative way relating to the shape of the molecules involved, why aggregation should be thermodynamically favoured at all.

An amphiphile can be characterised by three parameters:

1. The optimum head area, *a*_{0}. This results from a balance of repulsions between the surfactant molecules (electrostatic, excluded volume, etc) and an effective attraction to minimise the contact area between hydrocarbon tails and water (see below).

2. The critical chain length, *l*_{c}. This is the maximum length of the hydrocarbon chain if stretched.

3. The hydrocarbon volume, *v*. This is the volume occupied by the chain, irrespective of the conformation.

*4.1.1 Forces *

There are two opposing effects that control the head group area, a (the interfacial area):

1. Attractive forces at the interface arise from hydrophobic and surface tension effects, both tending to pack molecules closely. The interfacial hydrophobic free energy contribution to the chemical potential, ε_{N}, is proportional to the head group area so can be written as γ*a* , where γ is the energy per unit area, i.e. the interface tension.

2. Repulsive forces arise from a range of sources, including charge repulsion (for charged head groups) and steric effects. This effect is inversely proportional to the head group area, so that its contribution to ε_{N} is *K/a*, where *K* is a constant.

We therefore have

Equation 1

The optimum head area is that which minimises the energy (Slide 3), that is

Equation 2

and in this case the energy is

η_{N}(min) = 2γ*a*_{0}

Equation 3

Near the minimum, the energy will be harmonic in small changes of the area per molecule, and we can expand Equation 1 as

Equation 4

*4.1.2 Geometry*

We should also consider the packing constraints on the hydrophobic tails. These will occupy a volume, *v*, are assumed to be fluid and incompressible, and have a maximum effective length, *l*_{c}. This maximum length is somewhat empirical and corresponds to the length beyond which the chains can no longer be regarded as fluid.

Given *a*_{o}, *v* and *l*_{c} (all measurable or estimable quantities), the shape into which the lipids pack can be determined. Given different possible arrangements with comparable free energy, η_{N}, entropy will always favour the smallest aggregate (Slide 4). We will illustrate this by looking at some possible micelle shapes.

Consider a spherical micelle of radius *r* containing *M* molecules (Slide 5). Its volume is

Equation 5

and also

Equation 6

The surface area is

Equation 7

This sets the condition

Equation 8

In order for the sphere to be physically possible, the radius has to be less than the critical length,

*r* < *l*c.

Equation 9

These relations for a sphere (Equations 5–9) are summarized as

Equation 10

For cylindrical micelles, a similar argument to the one for spheres gives the upper limit ½ in this relation,

Equation 11

A bilayer can be considered as an "unwrapped" cylinder. If

Equation 12

then the geometry of the amphiphile favours the formation of bilayers (a planar or gently curved membrane). This condition means that for a given optimal a0, the volume is large and the critical length is small. This is achieved in practice by molecules with a double hydrocarbon chain, like biological phospholipids. A bilayer can join its edges together to form a vesicle, gaining some edge energy but at a cost that comes from bending the bilayer (see Slide 6). The thickness of a biological membrane is around 3 nm for phospholipids with 18 carbon atoms in the hydrophobic chain.

Slide 6: Amphiphiles assembling into a bilayer. Reprinted with permission from “Intermolecular and Surface Forces”, Jacob N. Israelachvili, Academic Press 1991. Copyright 2011, Elsevier.

*4.2 Geometry, size and CMC*

The geometry of aggregates affects the α dependence of the term εα, which was introduced in Lecture 3 as the free energy change when a molecule is put into an aggregate from solution. This is important, and will determine if the aggregates are of finite or infinite size.

Equation 13

Equation 14

(Equations 13 and 14 are Equations 9 and 10 from Lecture 3.)

(If ε_{α} were a monotonically decreasing function of α, then the system would aggregate (i.e. phase separate) into infinitely large domains. This is the well-known case, for example, for a pair of fluids phase separating, discussed in these terms in Jones (2002).)

For amphiphiles, the drive to optimise the arrangement of hydrophilic and hydrophobic components, shielded and in contact with water, respectively, leads to ε_{α} being a non-monotonic function of α.

In the case of spherical micelles, there will be a sharp minimum in the energy ε_{α} at the size α*, which has the optimal curvature for packing. If this minimum is sharp, then at the CMC the aggregates will be predominantly of size α and we can write

Equation 15a

Equation 15b

(Equation 15a is Equation 13 from Lecture 3.) Using this in the general expression for *x*_{α} (Equation 11 from Lecture 3) enables us to evaluate the critical concentration (Slide 7),

Equation 16

Equation 17

under the approximations that α* is a large number and that x_{α} is small. These are justified because a micelle typically has hundreds of molecules, and because the molar concentrations for the CMC are typically 10^{–5} to 10^{–3}.

The case of cylindrical micelles is a little different from the spherical micelles, because the end-caps contribute an important energetic cost, and there is not a single well defined preferred length for the micelles. Jones (2002) treats this case explicitly. The case of bilayers is also treated in the references, where it is shown that in the case of a bilayer, there is no preferred size, and the membrane extends indefinitely. Slides 8 and 9 summarise how phospholipid geometry determines the shape of a self-assembled structure.

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

*4.2.1 Micelles, nematic, hexagonal and lamellar phases*

The balance of translational entropy versus curvature energies gives rise to very complex phase diagrams when the amphiphile concentration is increased (see Slide 10). At sufficiently high concentrations, the interactions between the assembled structures will begin to play a role as well. With the further complication of extending these ideas to more than a single component, this approaches the complexity of real-world biological structures, and also forms the basis on which detergent and personal-care products are formulated.

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.

These are typical examples of complex fluids, of practical (soaps) and technological (nanostructure materials) importance.

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