Lecture 6: Interactions between macroscopic bodies II

What did we cover in the last lecture?

1) Dispersion interactions between atoms/molecules and macroscopic bodies

2) Dispersion interactions between two macroscopic bodies

In this lecture…

1) Interactions between macroscopic bodies continued

2) The effects of geometry on dispersion interactions

3) Determining the energy scale of dispersion interactions (Hamaker constant)

4) Range of interactions between macroscopic bodies

5) Magnitude of forces between macroscopic bodies

Interactions between a sphere and a solid

Problem 2: Derive an expression for the dispersion forces which act between a sphere of radius R (for separations

D>>R) and a semi-infinite solid if the number density of atoms in the slab and sphere are n1 and n2 respectively.

Summary of results

36)(

D

nCDU

Dispersion Interaction

energy between an atom/molecule and a semi-infinite solid

Dispersion Interaction energy between a flat slab of area, S, and a semi-infinite solid

Dispersion Interaction energy between a sphere of radius, R, and a semi-infinite solid

221

12)(

D

CSnnDU

D

CRnnDU

3)(

221

All of these are longer range than dispersion interaction between isolated molecules

The Hamaker Constant

The strength of the dispersion interaction between materials is often quantified in terms of the Hamaker constant, A

CnnA 22112

n1 and n2 are the number densities of atoms/molecules in the two interacting materials (m-3)

C is the ‘strength’ of the interaction between an atom/molecule from one material and an atom/molecule from the other (Jm6)

A12 is the Hamaker constant for materials 1 and 2

Measured in Joules

Dispersion interactions and Hamaker constants

Dispersion Interaction energy between a flat slab of area, S, and a semi-infinite solid

Dispersion Interaction energy between a sphere of radius, R, and a semi-infinite solid

212

221

1212)(

D

SA

D

CSnnDU

D

RA

D

CRnnDU

33)( 12

221

We can rewrite our expressions for the dispersion interaction energy between two macroscopic bodies in terms of the Hamaker constant

Dispersion Forces and Hamaker constants

Dispersion force between a flat slab of area, S, and a semi-infinite solid

Dispersion force between a sphere of radius, R, and a semi-infinite solid

312

6)(

D

SA

dD

dUDF

212

3)(

D

RA

dD

dUDF

We can also write the forces in terms of the Hamaker constant

So for a given geometry we can determine the strength of the dispersion interaction A12between two materials by measuring forces

Some other important geometries

2

1

21

21

2

312

212

)(

RR

RR

D

LADU

21

2112

3)(

RR

RR

D

ADU

The geometry of the macroscopic bodies is important in determining the separation dependence of the dispersion interaction

2112

6)( RR

D

ADU

L

Sphere-sphere

Parallel cylinders

Perpendicular cylinders

How big are Hamaker constant values?

P190 Surface and Intermolecular forces by J. Israelachvili

Values of A typically lie in the range 10-21 to 10-19 Joules

Note: kTroom = 4.41x 10-21J

What is the range of dispersion forces between macroscopic bodies?

kTD

RADU

range

3

)( 12

We can calculate the range of the interaction by comparing dispersion and thermal energies i.e. |U(Drange)| ~ kT

For a sphere of radius R and a solid surface we have

kT

RADrange 3

12Hence

For values of T=300K, R=10 nm and A12=10-19J

Drange ~ 10R = 100 nm

Note: Drange(atoms) ~ 0.3 nm

How big are the dispersion forces between macroscopic bodies?

We can also calculate the magnitude of the forces exerted on macroscopic bodies due dispersion forces

Sticking with the sphere and surface we have that

If R=10 nm, A12=10-19J

When D=10nm, F= 3 x 10-12 N ~ 3 pN

When D=1nm, F= 3 x10 -10 N ~ 0.3 nN

But can we measure forces this small?

212

3)(

D

RA

dD

dUDF

Force between two surfaces

312

6)(

D

SADF

In the last lecture we showed that the force between a perfectly flat slab and a semi-infinite solid surface is

If the area of the slab is S=10-4m2 and A12 = 10-19J. What is the force between the slab and the surface at a separation of 0.3 nm?

Ans: ~20,000 Newtons This is a very large force!

So why don’t all surfaces stick together irreversibly?

Summary of key concepts

1) Dispersion interactions between macroscopic bodies occur over significantly longer ranges than for isolated atoms/molecules

2) We can derive expressions for the dispersion interaction energy and forces between macroscopic objects

3) The strength of the dispersion interaction can be quantified in terms of the Hamaker constant A

4) Dispersion forces on nanoscale objects can be on the pN to nN scales