The expression can be differentiated with respect to D⊥ and then ∂F⊥/∂D⊥ as a function of F⊥ may be obtained,
A comparison of the contact stiffnesses for the cases of two spheres and two cylinders is illustrated in Fig. 1.5. It is seen that there is a very substantial variation in the result, especially lightly loaded contacts have a vanishingly small incremental stiffness.
The assumption made here is that the contact areas are immaculately clean and that the contact is perfectly smooth. For two particles made of a natural material — sand particles, for example — that assumption is obviously severely contestable. Further research on two fractal surfaces pressed together is reported in [Hanaor et al., 2015]. As the only purpose of the contact stiffness is to hinder two particles from overlapping (and do so in a controlled manner), it could be argued that any stiffness is fine, as long as the indentation is such that only a very small overlap (compared to the typical radius of the particles) is effected at the typical contact force regime in the assembly. It is also noted that for particles composed of natural materials a number of plastic effects can be expected (including breakage). Therefore, frequently, researchers just take a constant value for the contact stiffness and add friction, for example [Kuhn, 1999]. This is computationally simple and achieves the purpose of rectifying the problems of determining the contact forces in the case that the assembly is not in an isostatic state, at the expense of some physical realism. This is a perfectly reasonable thing to do.
In some sense the details of the normal interaction are not that critical. The tangential stiffness, including frictional effects can be added to the interaction. There are various approaches. A well-known one takes account of slip in an annulus inside the contact area. The extent of the annulus depends on the applied force ratio. A fair amount of ink has been spilt over this problem; quoting [Johnson, 1985]: ‘In a paper of considerable complexity, [Mindlin, 1949], [Mindlin and Deresiewicz, 1953], have investigated the changes in surface traction and compliance between spherical bodies in contact arising from the various possible combinations of incremental change in loads: normal force increasing, tangential force increasing; normal force decreasing, tangential force increasing; normal force increasing, tangential force decreasing; etc.’ The parameters needed are the material stiffness, Poisson’s ratio and a friction coefficient. Despite the complexity of the calculations the result in terms of incremental contact law is not dissimilar from the one obtained from the phenomenological approach as pursued above. All the notes regarding the idealisation of the problem, and therefore the question marks that accompany an application to the ‘dirty’ materials of which the real world is composed, are relevant again. It could also be argued that an analysis meant to explain friction based on the assumption of a friction coefficient is tautological, at best adding details to the mechanism.
1.7Interaction for small particles in a fluid environment
This section deals with small particles, micron- and sub-micron-sized, in a fluid environment. The question is how such particles interact when they come close together. Applications in chemical and environmental engineering (especially filtration), cosmetics, the mechanics of clay, etc. are envisaged. In these applications dense cakes of small particles are created and subsequently manipulated by either sedimentation or filtration methods.
The particles are solids, implying that the constituent molecules are in some sort of crystal structure. On the boundary of the particle solid the crystal structure meets the fluid; the crystal arrangement suddenly ends. There is then a discontinuity in the electric charge distribution, which is accommodated by the recruitment of the ions in the fluid near 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