W. Richard Bowen and Nidal Hilal 4

68 2. MEASUREMENT OF PARTICLE ANd SURFACE INTERACTIONS
where E s, E f and v s, v f are the Young’s modulus and Poisson ratios for the
sphere and flat surface respectively. The JKR model applies well for large
probes with soft samples and large adhesions. For the case of relatively
small tips with surfaces with high Young’s moduli and low adhesion, the
DMT model may apply better. For the DMT theory, a slightly different
relationship is found:
a F R W
FAd R W
R R E
DMT
c
p a
� p a �
p
�
2 ( 2π
)
2π
3 2
*
p
2
3
(2.64)
The JKR and DMT models are really descriptions of the two ends of a
continuum. Tabor suggested a way of deciding which of these two models
would be the best to apply to a certain situation. From the following
relationship, if the factor � R is greater than unity, then the JKR theory
would be best applied, with DMT being appropriate for � R values less
than unity [153, 154]:
� R
2
a p
0 3
1
⎞ 3
* ⎟
⎛
⎜W
R
� 2. 92 ⎜
⎝⎜
E z ⎠
(2.65)
where � R is effectively the ratio of the elastic deformation due to the applied
load and adhesion to the effective range of the surface forces (z 0) [153, 155].
For intermediate systems where values of � R are close to unity, then a treatment
using the Maugis–Dugdale theory is more appropriate [155].
The AFM has been used to measure adhesive forces between particles
and process surfaces. One important example is adhesion between particles,
including both inorganic colloidal particles and bacterial cells, and
filtration membranes. This interaction is of great importance when considering
the fouling and biofouling of such surfaces. Particles adhere to
the process membranes and reduce flow through the membrane, greatly
reducing filtration, the efficiency and working lifetime of the membranes.
The process testing of new membranes is potentially expensive and time
consuming. The quantification of adhesion forces between colloids and
membranes can provide an important contribution to developing the
theoretical prediction and optimisation and control of many engineering
separation processes. As a result, the development of AFM methods to
quantify the adhesion of different materials to membranes of different
compositions can potentially be very useful for membrane manufacturers
and engineers [156]. When particle sizes are greater than the pore size
in the absence of repulsive double layer interactions, such particles may
plug the pores very effectively, leading to a catastrophic loss in filtration
flux. Of the many established membrane characterisation techniques,