Stability of toroidal droplets inside yield stress materials

We study the stability of toroidal droplets inside a yield stress material. Similar to toroidal droplets in a viscous liquid, the slenderness of the torus controls whether it breaks into spherical droplets or grows thicker towards its center to coalesce onto itself and form a single spherical droplet. However, unlike tori generated in a viscous liquid, the elasticity of the outer medium can prevent either or both of these instabilities; this depends on the slenderness of the torus. Interestingly, we find that the value of the tube radius needed to prevent breakup is always larger than the value of the radius of the handle to prevent growth. This reflects the different deformations experienced by the yield stress material in either process. A simple model balancing the surface tension stress, which drives the evolution of the torus, and the yield stress, which favors its stability, accounts for all of our observations.

Figure 1

(a) Tube radius, $a$, and inner radius, $R$, of a toroidal droplet. (b) State diagram for the transformation of toroidal droplets in viscous liquids. The line $a=R$ separates regions where the torus either shrinks (closed symbols) or breaks (open symbols). Injected liquid: 10 cSt PDMS oil. Continuous phase: water with 60 mM of the surfactant sodium dedecyl sulphate. (c) State diagram for the transformation of toroidal droplets inside yield stress materials. In addition to the $a=R$ dashed line, there is a horizontal and a vertical line separating regions where the torus is stable ($•$) from regions where the torus either breaks ($\circ$) or shrinks (△). Injected liquid: 10 cSt PDMS oil. Continuous phase: yield stress material with $c=0.3$ wt %, corresponding to ${\tau }_{y,s}=9$ Pa.

Figure 2

Evolution of a torus: (a)–(c) Toroidal droplet made of water undergoing both breaking and shrinking simultaneously inside a 30 000 cSt PDMS oil. (d)–(f) Toroidal droplet made of 10 cSt PDMS oil undergoing only breaking inside a yield stress material with $c=0.1$ wt %. Note that the handle does not shrink while it moves to the side to induce breakup. (g)–(i) Toroidal droplet made of 10 cSt PDMS oil undergoing only shrinking inside a yield stress material with $c=0.1$ wt %. Note that the handle shrinks while remaining stationary. Scale bar: 5 mm.

Figure 3

Stability diagram for thin toroidal droplets made with 10 cSt PDMS oil inside yield stress materials with different ${\tau }_{y,s}$. The line separates stable (open symbols) and breaking (closed symbols) tori.

Figure 4

(a) Side view of a shrinking torus. The box highlights the region where the yield stress material is elongated. (b) Top view of the breaking torus. The box highlights the region where the yield stress material is sheared. $T_{ij}$ represents the stress along the $j$ direction acting on a surface element with normal along the $i$ direction. (c) Critical tube radius $a_c$ versus critical inner radius $R_c$. The line is the linear fit of the experimental points.

Figure 5

Snapshots of a toroidal droplet (a) right after breaking and (b) after equilibrated. We note that we waited for 12 hours in order to guarantee that the droplet had reached its final equilibrium shape. Beyond this time, we did not detect any further droplet evolution. The insets in (b) highlight the osculating circles used to determine the smallest tube radius of the equilibrium droplet. Scale bar: 5 mm. (c) Average radius of the osculating circles fitted at the ends of the droplet in the final crescentlike shape, $a_t$, as a function of the ratio $R/a$ of the toroidal droplet before breakup. Continuous phase: yield stress material with $c=0.3$ wt %.