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.)
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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
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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.
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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.
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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.
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Slide 5: Hydrophobic force.
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 selfassembly in biological systems.
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Consider a solution of N molecules.
Na is the number of molecules belonging to aggregates of size a.
The number of micelles of size α: na = Na / a.
Assuming “ideal” non-interacting micelles, the partition function of the system
is:
L3a : volume of an a-micelle.
zinta : partition function corresponding to internal degrees of freedom.
Slide 6: Partition function for surfactants and micelles.
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The resulting free energy is
where the internal free energy of each a-micelle is
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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).
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Change variable to the molar fraction.
Now xa 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).
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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).
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k1
kN
XN
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Let X1 and XN be the mole fractions of
molecules in monomer form and micelles of
size N, respectively.
The rate of association = k1x1N
The rate of dissociation = kN(xN/N)

 N  X 1 exp

 1 -  N

 k T
B





N
In equilibrium, the backward and forward rates are
equal.
The equilibrium constant K is
K 
k1
kN
Slide 10: Aggregation as a reaction.
 N ( N -  1 ) 
 exp  
k BT


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No xN can exceed unity, so
  1 -  N
X 1  exp  - 
  k BT
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



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.
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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 a*, then at CMC we have
xa* = x1 = CMC/2.
• Using this condition together with
and assuming large α* and small CMC,
 ( -  a * ) 
CMC  2 exp  - 1

k
T
B


Slide 13: CMC for spherical micelles.