W. Richard Bowen and Nidal Hilal 4

2.3 INTERACTION FORCES 41
Horn and Israelachvili [17] presented an expression for the Hamaker
constant, which is as follows:
AH � A � Aξ � A
o ξ�
131 1
(2.12)
where the two terms in this equation may be found using the Cauchy
Plot data from:
where
A
ξ
�1
3
⎛εr
( 0) �ε
r ( 0)
⎞
⎜ 1 2
Aξ�kT⎜ o 4
⎜
⎝⎜
εr
( 0) �εr
( 0)
⎠⎟
n
o
2 2
o o
n n
�
�
2
1 3
2 2
o o o
∆n �n �n
1 3
ω1
2 ω3
2
X � ( n o �1) � ( n o �1)
1 3
ω
ω
3
1 2
2 2
o o o o
3ℏ
ωω 1 3 Xn�2XΔnn�Δn ( 3�2Y) �
7/ 4
64no
⎡
1/ 2
/ ⎤
⎢ 2
2
( Y � Y �1)
� ( Y � Y � 1)
⎥
⎣
⎢
⎦
⎥
1 ⎡ ω1
2 ω ⎤
Y � ⎢
3 2
( n o �1) � ( n o �1)
⎥
1 3
4 n ⎢
o ⎣ω
3 ω ⎥
1 ⎦
1
(2.15)
(2.16)
(2.17)
(2.18)
and ε ( 0) ri is the dielectric constant of component i, � is Planck’s constant
divided by 2�, k is Boltzmann’s constant, n is the refractive index of com-
oi
ponent i (calculated from C UV �n o �
2
1), T is the absolute temperature,
�i is the ultraviolet characteristic adsorption frequency of component
i (i.e. � �UVi), subscript 1 refers to the material and subscript 3 to the dispersion
medium.
Therefore, if the dielectric constant and refractive index versus wavelength
data are known for each component, i, the Hamaker constant may
be approximated by use of a Cauchy Plot and the above equations.
2
1 2 3
(2.13)
(2.14)