Particles with an undulated contact line at a fluid interface: Interaction between capillary quadrupoles and rheology of particulate monolayers

We consider solid particles attached to a liquid-fluid interface. If the solid-liquid-fluid contact line (at the particle surface) is undulated, the convex and concave local deviations of the meniscus shape from planarity can be treated as positive and negative capillary charges, which form capillary multipoles. Correspondingly, the meniscus shape can be expressed as a multipole expansion. The leading terms of the capillary multipole expansion turn out to be the charges and quadrupoles. In the present article, we derive theoretical expressions, which enable one to calculate the energy and force of interaction between two capillary quadrupoles for arbitrary interparticle distance. Depending on the amplitudes of the contact-line undulations, and on the particle mutual orientations, the interaction is either monotonic attraction or attraction at long distances but repulsion at short distances. As a rule, the interaction energy is much larger than the thermal energy kT, even for undulations of nanometer amplitudes. This implies the existence of a specific rheological behavior of adsorption monolayers made of such particles. For example, an equilibrium monolayer of identical particles exhibits a plastic behavior upon dilatation; an expression for the respective yield stress is derived. In addition, expressions for the contributions of the capillary interactions to the surface dilatational and shear modules of elasticity are derived. The results of this paper could be helpful for the understanding of some phenomena related to the aggregation and the ordering of particles adsorbed at a fluid interface and for the interpretation of the rheological behavior of monolayers from such particles.