Capillary Breakup of Discontinuously Rate Thickening Suspensions

Using discontinuously rate thickening suspensions (DRTS) as a model system, we show that beads-on-a-string morphologies can arise as a result of external viscous drag acting during capillary-driven breakup of a non-Newtonian fluid. To minimize the perturbative effect of gravity, we developed a new experimental test platform in which the filament is supported in a horizontal position at the surface of an immiscible oil bath. We show that the evolution of thin DRTS filaments during the capillary thinning process is well described by a set of one-dimensional slender filament equations. The strongly rate-dependent rheology of the test fluid and the aspect ratio of the filament couple to control the thinning dynamics and lead to a simple criterion describing the localized arrest of the capillary thinning process and the subsequent formation of complex, high aspect ratio beads-on-a-string structures.

Figure 1

Typical beads-on-a-string structures. (a) Solution of poly(ethylene oxide) (PEO) in water, surrounded by air. (b) Solution of PEO in water, in corn oil bath. (c) Solution of polystyrene in styrene oil, in glycerol bath. (d) Suspension of cornstarch in glycerol, in corn oil bath. (e) Suspension of silica particles in water, in corn oil bath. Scale bars: 1 mm.

Figure 2

(a) Experimental setup showing a DRTS catenary supported by two rods and a floating filament attached to glass slides at both ends. The filament is held at the surface of the oil bath by surface tension. (b) Black dots: shear rheology of 55 wt % suspension of cornstarch in glycerol, measured in a parallel plate geometry. Dashed red line: constitutive relation ${\mu }_1$, with $\alpha =1$, $k=5$, and ${\dot{\gamma }}_{\mathrm{crit}}=0.1$. Solid red line: constitutive relation ${\mu }_2$, with $\beta =12$, $\xi =1/6$, ${\dot{\gamma }}_{\mathrm{crit}}=0.29$. (c) Dimensionless radius of a typical filament as a function of time, with ${\dot{\gamma }}_{\exp }=0.1{\mathrm{s}}^{-1}$.

Figure 3

(a) Experimental observations of capillary breakup of a DRTS filament and (b) numerical solutions of Eq. (1) using ${\mu }_1$ shown in Fig. 2b, and $\lambda =8.45$, $\mathrm{Wi}=10.99$, and $C_d=0.078$ (right). (c) Dimensionless minimum radius of the filament at bead formation ($h_b/R$) as a function of dimensionless length of the filament ($l_b/R$) for external oil viscosities ${\mu }_{\mathrm{high}}=3.0\mathrm{Pa}\mathrm{s}$ ($△$) and ${\mu }_{\mathrm{low}}=0.147\mathrm{Pa}\mathrm{s}$ ($▽$).

Figure 4

Initial (thin dashed line) and late-time (solid line) filament shapes and velocity fields from a numerical solution of (1) with $C_d=0$ and $\lambda /\mathrm{Wi}<2/3$. Thick dashed red line represents theoretical shape in domain $⟨\lambda /2,\lambda ⟩$. The solution domain is enclosed in the dashed rectangle in the upper inset. Relative size of the initial disturbance $ϵ$ is exaggerated for clarity. The velocity field reaches a steady state, where the velocity gradient $\partial v/\partial z$ is limited to $\pm {\mathrm{Wi}}^{-1}$ in the region $⟨0,\lambda ⟩$. Lower inset: Experimental photograph.

Figure 5

Comparison of simulation and experimental results. Blue squares: simulations with ${\mu }_1$, $C_d=0$. Black crosses: simulations with ${\mu }_1$ and $C_d\ne 0$. Green triangles: simulations with ${\mu }_2$ and $C_d\ne 0$. Red circles: experimental results. Each mark represents formation of the first, central bead. The value of the ordinate is $3{\tau }_{\max }\lambda$ for simulations with $C_d=0$ and $3{\tau }_{\mathrm{bead}}\lambda$ for simulations with $C_d\ne 0$ and for experiments.