Lecture 13

Steric Interactions and Entropic repulsion

What did we cover in the last lecture?

forces

dispersionattractivepressureosmotic

B

ootot D

AD

Tk

VeznP

3

222

6)exp(

The pressure exerted between charged surfaces in an electrolyte is the sum of osmotic pressure and dispersive terms

In this lecture…1) Steric (Entropic) stabilisation of particles and surfaces

2) Surfaces with grafted molecules

3) Boltzmann form of entropy

4) Free Energy of grafted molecules

5) Entropic forces between sterically stabilised surfaces

Further ReadingGases, Liquids and Solids, 3rd Edition, D. Tabor, Cambridge University Press, p326-321, 1993

Intermolecular and Surface forces, J. Israelachvili, chapter 14

Colloidal stability

It is sometimes not possible to use charges to keep particles and surfaces apart in solution

This is true when using organic solvents as the suspending medium (e.g. in ferrofluids), as ionic solids (electrolytes) are often poorly soluble in these solvents

An alternative route to stabilisation is to use Steric (or entropic) repulsion effects between surfaces

Entropic (Steric) Forces

Molecules are tethered to a surface by one end and can wave around freely in solution. As a result they have a large number of configurations (directions in which they can point)

Surfaces can be decorated with long molecules (e.g. polymers)

When two surfaces come close together, the number of configurations that the molecules can adopt is reduced. This reduces the entropy (or measure of disorder).

The system resists this reduction in entropy by generating a repulsive force…

Boltzmann form of Entropy

Boltzmann derived a statistical form of the entropy of a system, S, such that

WkS B lnwhere W is the number of available (micro) states corresponding to a given macrostate and kB is Boltzmann’s constant (JK-1 ). In this case…

Macrostate → molecule tethered to the surface

Microstate → a single orientation of the molecule

The entropy of a tethered rod Suppose we treat the molecule as a rigid rod with length, l, and area, a, at each of its ends (See OHP)

If we allow the rod to point in any direction it can sweep out half a sphere with a total area, A1

The number of different possible microstates (orientations of the rod), W, is then given by dividing ratio of A1 to a

So the entropy (S1) of the tethered rod becomes

22

1 22

4l

lA

a

l

a

AW

21 2

a

lkS B

2

1

2ln

Confinement of tethered molecules

When a solid surface approaches the tethered rod, its motion becomes restricted and it can sweep out a much reduced area

At a separation D between the surfaces, the total area that can be swept out is A2 where

The entropy of these ‘confined’ molecules is therefore

lDA 22

a

lDkS B

2ln2

How do we extract a force?

We can extract a force if we know how the entropic contribution to the interaction energy , U, depends upon separation D dD

dUF

The change in the Gibbs free energy of the system at constant temperature is given by

STHU Where H and S are the change in enthalpy (energy of interaction) and entropy caused by bringing the surfaces together

Entropic repulsion Force per molecule

If we consider only the entropy contribution (for now) the force per molecule is

dD

SdT

dD

STdF

)()(

where

l

DkS

a

lk

a

lDkSSS

B

BB

ln

2ln

2ln

2

12

So the force F is

D

TkF B

Pressure due to entropic forces

The force between the surfaces is positive and therefore repulsive D

TkF B

If the separation between tethering points is, d, then the effective area occupied by one molecule is d2

The force per unit area (Pressure) is P, where

Dd

TkP B

2

Problem

Two perfectly flat surfaces are brought into close proximity such that dispersion forces attract them towards one another. One of the surfaces is coated with a layer of rod-like molecules of length 3nm with the highest possible surface grafting density that allows the free end of the molecules to rotate freely

1)Calculate the magnitude of the pressure exerted on the surfaces due to repulsive steric (entropic) forces, at room temperature, if the surfaces are separated by a distance of 2.5nm

2)Calculate the equilibrium separation of the surfaces at room temperature if the Hamaker constant is 1x10-20 J

Summary of key concepts

When the motion of molecules near a surface is confined it reduces the entropy of the system

This gives rise to repulsive force which acts to push surfaces apart and increase the entropy again

For rod like molecules on a surface which occupy an area d2, the pressure between surfaces at a separation, D, is given by

Dd

TkP B

2