W. Richard Bowen and Nidal Hilal 4
W. Richard Bowen and Nidal Hilal 4
W. Richard Bowen and Nidal Hilal 4
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68 2. MEASUREMENT OF PARTICLE ANd SURFACE INTERACTIONS<br />
where E s, E f <strong>and</strong> v s, v f are the Young’s modulus <strong>and</strong> Poisson ratios for the<br />
sphere <strong>and</strong> flat surface respectively. The JKR model applies well for large<br />
probes with soft samples <strong>and</strong> large adhesions. For the case of relatively<br />
small tips with surfaces with high Young’s moduli <strong>and</strong> low adhesion, the<br />
DMT model may apply better. For the DMT theory, a slightly different<br />
relationship is found:<br />
a F R W<br />
FAd R W<br />
R R E<br />
DMT<br />
c<br />
p a<br />
� p a �<br />
p<br />
�<br />
2 ( 2π<br />
)<br />
2π<br />
3 2<br />
*<br />
p<br />
2<br />
3<br />
(2.64)<br />
The JKR <strong>and</strong> DMT models are really descriptions of the two ends of a<br />
continuum. Tabor suggested a way of deciding which of these two models<br />
would be the best to apply to a certain situation. From the following<br />
relationship, if the factor � R is greater than unity, then the JKR theory<br />
would be best applied, with DMT being appropriate for � R values less<br />
than unity [153, 154]:<br />
� R<br />
2<br />
a p<br />
0 3<br />
1<br />
⎞ 3<br />
* ⎟<br />
⎛<br />
⎜W<br />
R<br />
� 2. 92 ⎜<br />
⎝⎜<br />
E z ⎠<br />
(2.65)<br />
where � R is effectively the ratio of the elastic deformation due to the applied<br />
load <strong>and</strong> adhesion to the effective range of the surface forces (z 0) [153, 155].<br />
For intermediate systems where values of � R are close to unity, then a treatment<br />
using the Maugis–Dugdale theory is more appropriate [155].<br />
The AFM has been used to measure adhesive forces between particles<br />
<strong>and</strong> process surfaces. One important example is adhesion between particles,<br />
including both inorganic colloidal particles <strong>and</strong> bacterial cells, <strong>and</strong><br />
filtration membranes. This interaction is of great importance when considering<br />
the fouling <strong>and</strong> biofouling of such surfaces. Particles adhere to<br />
the process membranes <strong>and</strong> reduce flow through the membrane, greatly<br />
reducing filtration, the efficiency <strong>and</strong> working lifetime of the membranes.<br />
The process testing of new membranes is potentially expensive <strong>and</strong> time<br />
consuming. The quantification of adhesion forces between colloids <strong>and</strong><br />
membranes can provide an important contribution to developing the<br />
theoretical prediction <strong>and</strong> optimisation <strong>and</strong> control of many engineering<br />
separation processes. As a result, the development of AFM methods to<br />
quantify the adhesion of different materials to membranes of different<br />
compositions can potentially be very useful for membrane manufacturers<br />
<strong>and</strong> engineers [156]. When particle sizes are greater than the pore size<br />
in the absence of repulsive double layer interactions, such particles may<br />
plug the pores very effectively, leading to a catastrophic loss in filtration<br />
flux. Of the many established membrane characterisation techniques,