the boundary into a compensating configuration. The fluid molecules, however have a far greater mobility than those in the solid. Moreover, their equilibrium state — far from the solid boundary — is determined by the type of molecule in the fluid and its temperature.
The mobilisation of the ions in the fluid is achieved by either turning the dipoles of the fluid molecules in the direction of the solid boundary, or by attracting or repelling ionic charges. This can only be partially successful, as the thermal motion tends to make the alignment less effective. Also, if in a fluid a layer of molecules has a more or less aligned dipole moment, the next layer of fluid will respond by turning its dipoles in the opposite direction in order to achieve charge neutrality. Thus, a double layer is created. The electrical potential in the fluid as a function of the distance from the boundary will be a declining function.
Now, if two particles are brought together there are two declining potentials and the charges inside the fluid will act on that, effectively causing a repulsive interaction. This is called the double layer interaction and it is part of a multi-aspected interaction, the so-called DLVO theory — named after its main contributors Debije, Landau, Verwey and Overbeek. The analysis of the complete theory involves a large number of approximations, basically taking account of the repulsive double-layer interaction and an attractive van der Waals interaction.
The literature on this subject is vast. The classic is [Kruyt and Overbeek, 1969]. Good textbooks that treat the basics and a plethora of applications are [Hunter, 1987, 2001].
The theory of the double layer interaction is extremely well-researched in the colloid literature and all that needs to be done here is to communicate the results.
A measure for the thickness of the double layer is some chosen multiple of κ–1 and κ is approximately
where e is the electron charge, n(0) the bulk concentration of ions, Z the valency of the ions, ε the electrical permittivity of the fluid, kB Boltzmann’s constant and T the absolute temperature. If there are more than one type of ions in the fluid the concentration and valences are simply summed. Now, the interaction between two particles depends on the separation of the particles H and the parameter κ; the simplest non-dimensional combination is Hκ. Thus, the double layer interaction is a function of Hκ. The actual form of the interaction is exposed in two approximations involving the particle radius a. The first approximation pertains to the case in which κa is large (say, larger than 10). Defining the surface potential as ψ0, the interactive potential is
The second approximation is relevant for interactive potential takes the form κa < 5, in which case the interactive potential takes the form
In these formulas the surface potential as ψ0 depends on the type of surface and the ionic content of the fluid. The interactive force is obtained from –∂V/∂H.
The van der Waals contribution has also been evaluated. Here the interactive potential for two equal particles is given with A12 a constant called the Hamaker constant (the analysis is due to [Hamaker, 1937])
If the two particles are very close together (H/a
The contributions from the double layer interaction and the van der Waals interaction can be added to give the main contributors to the DLVO theory. The total effect depends on the coefficients, which reflect the exact type of system that is relevant. An example of the sum of the two contributions is given in Fig. 1.6.
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.
References
Amontons, G. (1699) De la resistance causée dans les Machines, tant par les frottemens des parties qui les composent, que par roideur des cordes qu'on y employe, & la maniere de calculer l'un & l'autre (On the resistance caused in machines, both by the rubbing of the parts that compose them and by the stiffness of the cords that one uses in them, and the way of calculating both), Mémoires de l'Académie royale des sciences, in: Histoire de l'Académie royale des sciences, pp. 206–222.
Bi, D., Zhang, J., Bulbul Chakraborty, B. and Behringer, R. P. (2011) Jamming by shear. Nature 480 355–358.
Bowden, F.P. and Tabor, D. (1956) Friction and Lubrication. London: Methuen & Co Ltd.
Blumenfeld, R. (2007) Stresses in two-dimensional isostatic granular systems: exact solutions. New Journal of Physics 9 (2007) 160–181.
Coulomb,
