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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