Towards a constitutive relation for emulsions exhibiting a yield stress

Constitutive relations are needed to predict the behavior of complex fluids in nonviscometric flows. This is an area that is largely unexplored for yield stress materials because of the difficulty describing the elastoviscoplastic behavior for arbitrary flows. Here, we measure the shear and extensional rheology of a simple tunable yield stress system: emulsions with different oil volume fractions that allow one to vary the flow properties over a large range. We propose universal concentration scaling laws that produce master curves for the shear and extensional rheology with a minimal number of known emulsion parameters. The extensional viscosity is obtained experimentally using a theory for inelastic shear-thinning materials, demonstrating that elastic stresses are unimportant in the pinching dynamics, and the elastic normal stress differences contribute minimally to the von Mises yield surface. Hence, this shows that material elasticity is unimportant, and an explicit constitutive equation of, for example, Criminale-Ericksen-Filby type, with a Herschel-Bulkley viscosity and a modulus equal to the Laplace pressure is adequate to describe the behavior of such concentrated soft-sphere systems in general steady and low Deborah number unsteady Eulerian and Lagrangian flows.

Figure 1

Universal constitutive laws for shear stress and first normal stress differences in concentrated emulsions. (a) A cone-plate geometry is used to investigate the shear stress and normal stress difference ($N_1$) of an oil in water emulsion. The close up shows a typical confocal microscopy picture of the investigated system and the droplet size distribution. (b) Shear stress as a function of shear rate for different volume fractions $\varphi$ between 0.72 and 0.80. Continuous lines are Herschel-Bulkley fits of the experimental data [Eq. (1)]. (c) Collapse of shear flow curves onto a master curve when plotted according to Eqs. (2) and (3). The black line corresponds to a Herschel-Bulkley fit with $\mathrm{\Delta }/\mathrm{\Gamma }=0.55$. (d) First normal stress difference as a function of shear rate for the same set of emulsions (same color code). Continuous lines are values predicted by Eq. (4). (e) Collapse of the normal stress difference curves when plotted according to Eq. (6). Solid line is the fit using the Herschel-Bulkley parameters.

Figure 2

Universal constitutive laws for shear rheology and elongational rheology in concentrated emulsions. (a) Sequential photographs of liquid bridges between two plates moving apart, for volume fractions $\varphi =0.72$ and 0.80 (top to bottom), at times to breakup $t_c-t=0.15$, 0.05, 0.012, and 0.001 s (left to right). Scale bar is $150\mu \mathrm{m}.$ (b) Viscosity as a function of deformation rate for different values of $\varphi$. Data points are obtained from elongational measurements. Continuous lines are Herschel-Bulkley predictions based on shear rheology measurements [Eq. (1)]. (c) Elongational flow curves collapse onto a master curve when plotted according to Eqs. (2) and (3) with $\mathrm{\Delta }=2.1$ and $\mathrm{\Gamma }=4.1$. The black line corresponds to a Herschel-Bulkley fit with $\mathrm{\Delta }/\mathrm{\Gamma }=0.55$.