Statistical Mechanics and Soft Condensed Matter

Amphiphile aggregation: critical micelle concentration by Pietro Cicuta

Slide 1:

The formation of micelles and bilayers can be described in terms of statistical thermodynamic functions, such as chemical potential and free energy. (Copyright E Perez, Laboratoire IMRCP, Toulouse.)

• • • • An amphiphile has a hydrophilic head group and a hydrophobic tail. Lipids are one class of amphiphiles.

There are many synthetic amphiphiless that have the same basic properties.

The head groups can be polar or charged.

SDS aerosol OT • • • Surfactants are amphiphiles.

They sit at interfaces (e.g. air/water or water/ oil) and stabilise them and/or reduce their surface tension.

In terms of self-assembly, synthetic amphiphiles and lipids behave in the same way.

Slide 2:

Amphiphiles.

hydrophilic head hydrophobic tail

Slide 3:

Schematic, chemical and physical structures of a phospholipid molecule.

Copyright © 1994 From Molecular Biology of the Cell by Bruce Alberts, et al. Reproduced by permission of Garland Science/Taylor & Francis Books, Inc.

• • • • Self-assembly occurs because the hydrophilic heads all point towards the aqueous phase, while the hydrophobic tails try to avoid it.

In a bilayer, the tails sit in the centre of the layers away from the aqueous phase.

Lipid membranes form in this way.

Many molecules behave similarly.

Slide 4:

Lipid bilayers.

• • • • • Forming a bilayer implies a significant loss of entropy .

So other changes in the system must lead to an overall decrease in free energy .

Enthalpic interactions are not dominant.

The driving force is the entropy of the solvent molecules.

Grouping solute molecules can increase the entropy of the solvent molecules.

• • • This entropic driving force due to a gain in water molecules is known as the

hydrophobic force

.

This is also important for protein structure.

It is a key component of self assembly in biological systems.

Slide 5:

Hydrophobic force.

• • • • Consider a solution of

N

molecules.

N

a is the number of molecules belonging to aggregates of size a . The number of micelles of size

α: n

a =

N

a / a.

Assuming “ideal” non-interacting micelles, the partition function of the system is: L 3 a

z

int a : volume of an a micelle. : partition function corresponding to internal degrees of freedom.

Slide 6:

Partition function for surfactants and micelles.

• The resulting

free energy

is where the internal free energy of each a micelle is • The

chemical potential

is where the internal free energy per surfactant molecule in an a micelle is

Slide 7:

Free energy for surfactants and micelles (1).

• • • Change variable to the molar fraction. Now

x

a is the molar fraction of the molecules in any a micelle.

In terms of the molar fractions, the chemical potential is where the free energy change of putting a molecule from the bulk into an a micelle is and the mean volume of molecules in solution is

v = V/N

.

Slide 8:

Free energy for surfactants and micelles (2).

• • In equilibrium, the

chemical potential of all different aggregation numbers must be the same.

We can call this value μ.

Therefore we have a key, useful, result Note: we have assumed ideal mixing, i.e. that inter-aggregate interactions can be ignored. In practice, this means that the system is dilute.

Slide 9:

Free energy for surfactants and micelles (3).

k

1

k

N • • • Let size

X

1 molecules in monomer form and micelles of

N

and

X

N be the mole fractions of , respectively.

The rate of association =

k 1 x 1 N

The rate of dissociation =

k

N (

x

N /

N

) • •

X N



N

  

X

1 exp    1 -

k B T



N

    

N

In equilibrium, the backward and forward rates are equal.

The equilibrium constant

K

is

K



k

1

k N

 exp   -

N

( 

N

 1

k B T

)  

Slide 10:

Aggregation as a reaction.

• No

x N

can exceed unity, so

X

1  exp       1 -

k B T



N

     • • • • There comes a point when the number of monomers cannot increase, and molecules

must

be involved in aggregates.

At low concentrations, almost all of the molecules exist as monomers, but beyond a certain concentration, aggregates form. This concentration is known as the critical micelle concentration or

CMC

.

The concentration of monomers at all higher concentrations is given by the equality above.

Slide 11:

Aggregate size and

CMC

.

Slide 12:

There is an optimal packing number for phospholipids to form a spherical micelle.

• • In practice, it is found that above the

CMC

, the spherical micelles are reasonably monodisperse, i.e. we do not simply have a random collection of micelles of any size.

If the peak of aggregates is very sharp at a size

x

a* =

x

1 = CMC/2.

• Using this condition together with and assuming large α* and small

CMC

, a *, then at

CMC

we have

CMC

 2 exp   (  1  a * )  

k B T

Slide 13:

CMC

for spherical micelles.