The Physics of the Deformation of Densely Packed Granular Materials. M A C Koenders

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      Figure 1.6. Illustration of the potential VT scaled to 2πεaψ²₀ for a value of κa = 20 and A₁₂/(2πεaψ²₀) = 0.3.

      The example in this graph is chosen to highlight some features. Some numbers are relevant. Suppose the particle radius is 0.2 μm, then the double layer thickness is κ^–1 = 10nm. For distances less than a few nanometres the theory is unreliable. In the figure that corresponds to H/a ≈ 0.03. The sum of the two contributory potentials VT is then not accurately represented for very small H/a. Keeping that in mind, two features of the combined potential are clearly visible. Firstly, there are two attractive wells, one very close to the particle (where the theory is not valid) and one around H/a = 0.18. Secondly, moving the particles closer together from the latter minimum, there is a potential to overcome. It must be pointed out that these features are specific to the choice of parameters that has been made.

      For much thicker double layers there are no potential minima in the relevant range and the interactive force is always repulsive. This is illustrated in Fig. 1.7 where κa = 3. Note that the interaction is highly non-linear

      Figure 1.7. Illustration of the potential VT scaled to 2πεaψ²₀ for a value of κa = 3 and A₁₂/(2πεaψ²₀) = 0.3.

      The plethora of behaviours of colloidal substances is largely due to the variety of possible outcomes for the interactive potential curve and whether there are minima or maxima in the ambient mechanical (and thermal) environment.

      One consequence of the existence of an interactive potential is that there is always a force active between neighbouring particles and as a result considerations relating to the isostatic state are not as acute as in the case of an interaction that is solely due to contact.

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