On the equation of the maximum capillary pressure induced by solid particles to stabilize emulsions and foams and on the emulsion stability diagrams

The knowledge of the adequate equation for the maximum capillary pressure ( P c max ), stabilizing relatively large bubbles and drops by relatively small, solid particles, is essential to control the stability of foams and emulsions. The idea to introduce this quantity to explain emulsion stability was published in the break-through paper by professor I.B. Ivanov and his co-workers [N.D. Denkov, I.B. Ivanov, P.A. Kralchevsky, D.T. Wasan, A possible mechanism of stabilization of emulsions by solid particles, J. Colloid Interface Sci. 150 (1992), 589–593]. A higher positive value of P c max ensures that a thin liquid film between the droplets (of an emulsion), or between the bubbles (of a foam) can withstand a higher pressing force. In the present paper, different equations for P c max are reviewed, published since the above cited paper. The exact form of this equation depends on the arrangement of particles in the liquid film between the drops (in the case of emulsions) or between the bubbles (in the case of foams). In the present paper the following general equation is derived for the maximum capillary pressure: P c max = ± 2 p σ ( cos ⁡ θ ± z ) / R , with a ‘+’ sign, referring to o/w emulsions and foams, and with a ‘−’ sign, referring to w/o emulsions; R, the radius of the spherical solid particle; σ, the interfacial energy between the two liquids (in case of emulsions), or between the liquid and gas (in case of foams); θ, the contact angle of the water droplet in the environment of the oil phase on the solid particle (in case of emulsions) or of the liquid in gas phase on the solid particle (in case of foams), parameters p and z are functions of particle arrangement. Particularly, z = 0 for the single layer of particles, and z = 0.633 (at θ > 90°) for the closely packed double layer of particles. The above equation was jointly analyzed with the well-known equation for the energy of removal of the particles from the liquid/liquid interface. As a result, emulsion stability diagrams (ESD) have been created with the contact angle and volume fraction of the water phase on its axes, indicating the stability intervals for o/w and w/o emulsions. The emulsion stability diagram was used to explain the phenomenon of ‘catastrophic phase inversion’ (i) due to solely changing the volume fraction of water, and (ii) due to solely changing the particle concentration, in the system of the same composition (water, oil, solid). The emulsion stability diagram was also used to rationalize the transitional phase inversion due to changing the ratio of hydrophobic to hydrophilic particles as function of water content. Stabilization of emulsions and foams by a 3D network of solid particles is also discussed.