Lecture 15:
Aggregation
What did we cover in the last
lecture?
Hydrogen bonds and hydrophobic
interactions are stronger than simple
dispersion interactions (~4-30 kJmol-1
vs ~1kJmol-1)
They have more specificity than
dispersion
forces
and
have
directionality. They can be used to build
nanoscale structures
The hydrophobic effect arises due to
 D
2 S
changes in the entropy of water F   1 exp   
o
 o 
surrounding nonpolar molecules.
In this lecture…
1) Aggregation
2) Why do aggregates form?
3) What determines the shape of an aggregate?
4) 1D aggregation – rods
5) 2D aggregation – sheets
6) 3D aggregation - spheres
Further reading
Intermolecular and Surface Forces, J.
Israelachvili, chapter 16
Aggregation and phase
separation
We have seen that particles and
surfaces can interact via many different
physical interactions.
The subtle balance between these
interactions determines whether the
particles/surfaces will remain
separated in solution.
When the interactions between
particles favour the attraction of
particles, they will tend to aggregate
If these interactions become really
strong, they will eventually drive phase
separation in the system.
Why do aggregates form?
When the energy per particle/molecule
inside an aggregate is less than the
energy of a free molecule, then
aggregates will start to form
The average free energy per
particle/molecule is often referred to as
the chemical potential and is given
the symbol, m
As we will see the chemical potential
depends upon the shape of the
aggregates and the number of particles
in the aggregate (N)
1 Dimensional aggregates (rods)
Consider a linear aggregate
where all the N
particles/molecules are joined
together in a line.
Number of
particles, N
If each of the (N-1) ‘bonds’ in the
aggregate has an energy of -u
relative to the unbonded state.
The total energy of the aggregate
becomes
Number of
‘bonds’, N-1
U agg  Nm   N  1u

N  1u

m
  1
N


or
1
u
N
2D aggregation
R
The case of 2D aggregation is
slightly more complicated.
As a disk-like aggregate
grows the area of the surface
on which the particles attach
increases with radius of the
disk. The production of this
excess surface area has an
energy penalty associated
with it.
t
Number of particles, N
m   mbulk 
2 t 
1
2
1
2
A consideration of the bulk
N
and surface contributions of
u is molecular/particle volume and  is
the energy per molecule gives excess surface energy of unbonded
(see OHP)
particles around circumference
3D aggregation
Similarly for 3D aggregation we consider volume and bulk
terms and we obtain (See OHP- or leave as exercise for
reader)
2
 3  3 1
m   mbulk  4  
1
4



N3
R
Number of particles, N
What happens as the
aggregation number increases?
Rods
1
u

m  1  u   mbulk 
N
 N
Disks
m   mbulk 
2 t 
N
Spheres/droplets
1
2
1
2
 3 

 4 
m   mbulk  4 
2
3
1
N
1
3
In all cases the average energy per particle/molecule
decreases with increasing aggregation number N (tending to
–mbulk in an infinite aggregate)
Aggregates will tend to grow once formed
Critical aggregation number
There is a critical aggregation number Nc below which
aggregates will not grow
Aggregate growth is only favourable when m < 0
In each case, we have competing surface and bulk terms
2
3
 3  1
  mbulk  m surface

1
 4 
3
N

m   mbulk  4 
surfaceterm
When msurface ≥ |mbulk |, m ≥ 0 and aggregation becomes
unfavourable
Aggregates with N< Nc will therefore tend to ‘dissolve’
Calculation of Nc
When m = 0 the aggregation number is N=Nc

1 
u
0  1 
 Nc 
For 1D aggregates
or Nc =1
i.e. linear aggregates will have
a tendency to grow and form
large structures
For 3D aggregates
Protein
fibrils
300nm
2
3
 3  1
0   mbulk  4  
1
4

  N3
c
3
or
 4   3 
 
N c  

 mbulk   4 
2
How does Nc influence
aggregation?
A critical cluster size has to form in
order for the aggregates to grow
 4 

N c  
 mbulk 
3
 3 


 4 
This means that the local molecule/particle concentration
has to be increased so that an aggregate nucleus can be
formed
This can be caused by the
presence of foreign surfaces
(nucleating agents) or by
local fluctuations in
composition (due to e.g.
thermal fluctuations)
2
Problem
Calculate the critical aggregation number for a
spherical aggregate made of molecules of a
material with a surface energy of 15 mJm-2, if the
bulk cohesion energy per molecule is 15 x 10-20 J
and the molecule has a radius of 1.5 nm
Calculate the radius above which aggregates of
this molecule will start to grow
Summary of key concepts
When the free energy per
molecule/particle inside an aggregate
becomes less than that of a free
molecule/particle, aggregates will start
to form
The form of the free energy per
molecule (chemical potential) depends
2
upon the shape of the aggregate and
 3  3 1
m   mbulk  4  
1
the aggregation number
 4 
N3
A critical cluster size sometimes has to
3
2


4

3


form before growth can occur. This is
 
N c  

needed to overcome the contributions
 mbulk   4 
due to the excess surface energy of the
aggregate.