Association of Percolation Theory with Princen’s Approach To Model the Storage Modulus of Highly Concentrated Emulsions

Abstract

An approach based on percolation theory is associated to the rigorousness of the two-dimensional (2D) model of close-packed, monodisperse, cylindrical emulsions established by Princen, to obtain an equation to model the dispersed-phase volume fraction (ϕ) dependence of the storage modulus (G′) of highly concentrated emulsions. A first-order Taylor expansion of this general percolation model leads to an expression similar to the three-dimensional (3D) model proposed by Princen and Kiss for real polydisperse emulsions, with a more satisfactory derivation of the function E(ϕ). Finally, the robustness of this model is tested against different experimental data collected from the literature.

Evaluation of the repartition of the particles in Pickering emulsions in relation with their rheological properties

“…The model is not able to account for the abrupt growth of elastic modulus in the vicinity of φ = φc(P). Similar conclusions were already reported by other authors [30,31,43]. It appears that the power law model ("G' Power law") is better suited to describe the experimental data.…”

“…Therefore, a structural approach mathematically close to the previous one but based on physical principles, even if they are only phenomenological, is preferable. On the basis of these remarks, the percolation approach has recently been used to successfully describe the rheology of highly concentrated emulsions [31]. This approach makes it possible to deal with the viscosity and elasticity of suspensions in the same formal framework and appears to be a good candidate to model the rheology of Pickering emulsions.…”

“…The percolation model considers that the elastic modulus of the emulsion becomes nonzero above a critical volume fraction of dispersed phase called the percolation threshold (φc) [31]. At this volume fraction, the number of droplets in contact is large enough to form a continuous network of interconnected droplets allowing the transport of elastic interactions quantified by G' which is related to the number of links between the droplets (Fig.…”

“…In other words, the percolation threshold (φc (η)) is the critical volume fraction of dispersed phase for which the zero-shear viscosity (Newtonian plateau) starts to diverge. The viscosity and the elastic modulus can be expressed as [31]:…”

“…However, it does not work efficiently for all the systems, mainly when a very steep increase of G' with φ takes place [30]. Recently, we have applied a percolation approach which represents satisfactorily the evolution of both the viscosity and elasticity with φ [31].…”