Lecture 15:

Aggregation

What did we cover in the last lecture?

oo

DSF

exp2 1

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 changes in the entropy of water 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,

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.

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

uNNU agg 1 or

u

NN

uN

11

1

Number of particles, N

Number of ‘bonds’, N-1

2D aggregationThe 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.

A consideration of the bulk and surface contributions of the energy per molecule gives(see OHP)

2

1

2

1

2

N

tbulk

is molecular/particle volume and is excess surface energy of unbonded particles around circumference

Number of particles, N

R

t

3D aggregation

Similarly for 3D aggregation we consider volume and bulk terms and we obtain (See OHP- or leave as exercise for reader)

3

1

3

2

1

4

34

Nbulk

Number of particles, NR

What happens as the aggregation number increases?

N

uu

N bulk

11

3

1

3

2

1

4

34

Nbulk

2

1

2

1

2

N

tbulk

Rods

Disks

Spheres/droplets

In all cases the average energy per particle/molecule decreases with increasing aggregation number N (tending to –bulk in an infinite aggregate)

Aggregates will tend to grow once formed

Critical aggregation numberThere is a critical aggregation number Nc below which aggregates will not grow

Aggregate growth is only favourable when < 0

In each case, we have competing surface and bulk terms

surfacebulk

termsurface

bulk

N

3

1

3

2

1

4

34

When surface ≥ |bulk |, ≥ 0 and aggregation becomes unfavourable

Aggregates with N< Nc will therefore tend to ‘dissolve’

Calculation of Nc

When = 0 the aggregation number is N=Nc

For 1D aggregates uNc

1

10 or Nc =1

i.e. linear aggregates will have a tendency to grow and form large structures

For 3D aggregates

3

1

3

2

1

4

340

c

bulk

N

or

23

4

34

bulkcN

Protein fibrils

300nm

How does Nc influence aggregation?

23

4

34

bulkcN

A critical cluster size has to form in order for the aggregates to grow

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)

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 upon the shape of the aggregate and the aggregation number

A critical cluster size sometimes has to form before growth can occur. This is needed to overcome the contributions due to the excess surface energy of the aggregate.

3

1

3

2

1

4

34

Nbulk

23

4

34

bulkcN