Viscoplastic sessile drop coalescence

The evolution of the liquid bridge formed between two coalescing sessile yield-stress drops is studied experimentally. Surprisingly, we find that the height of the bridge evolves similar to a Newtonian fluid as $h_0\left(t\right)\sim t$, before arresting at long time due the fluid's yield stress. From viscoplastic lubrication theory we find a model for the arrested interface shape based on the balance between capillary pressure and yield stress. We then solve numerically for this final arrested profile shape and find it to depend on the fluid's yield stress ${\tau }_y$, the surface tension coefficient $\sigma$ and the coalescence angle $\alpha$, represented by a modified Bingham number. We also present a scaling argument for the bridge's temporal evolution using the length scale found from this arrested shape analysis and present a similarity solution for the spatial evolution of the liquid bridge.

Figure 1

Side (a) and bottom (b) view viscoplastic sessile drop coalescence with yield stress ${\tau }_y=30\mathrm{Pa}$. High speed imagery captures the coalescence event. Postcoalescence (b) the contact line spreads characterized by $r_0\left(t\right)$, and (a) the bridge grows characterized by its minima $h_0\left(t\right)$, before halting due to the drop's yield stress. The initial height of the drop $h_{\mathrm{drop}}$ and its footprint $a$ are related through the coalescence angle $\alpha$. Bottom and side view time sequences are presented from two different experiments with similar initial conditions.

Figure 2

(a), (b) Rheological data for Carbopol 940. A fit of the Herschel-Buckley model (2) recovers $n=[0.44,0.4,0.37,0.37]$ and $K=[1.5,4.0,10.2,14.9]\text{Pa}·{\text{s}}^n$ for yield stress ${\tau }_y=[2,14,30,50]\text{Pa}$ samples respectively. (a) Measurement of viscosity against shear rate. (b) Measurement of yield stress using the constant oscillatory shear rate test. (c) Postcoalescence ($t=\infty$) final arrested shapes for increasing yield stress ${\tau }_y$, each scale bar is $500\mu \mathrm{m}$. For similar coalescence angles $\alpha$, increasing ${\tau }_y$ decreases the arrested bridge height $h_0\left(\infty \right)$. Increases in $\alpha$ serve to increase the arrested bridge height, as shown for the 50 Pa case. (d) Comparison of the experimentally measured liquid/gas interfacial profiles $h(x,\infty )/h_{\mathrm{drop}}$ (circles) against $x/a$ with their corresponding numerical solutions (solid lines) obtained from a scaled (4), $\widehat{h}\left(\widehat{x}\right)d^3\widehat{h}\left(\widehat{x}\right)/d{\widehat{x}}^3=J/{(h_{\mathrm{drop}}/a)}^3$, with boundary conditions $d\widehat{h}\left(\widehat{x}\right)/d\widehat{x}=0\text{at}(\widehat{x}=-1,0)\text{and}\widehat{h}(\widehat{x}=-1)=1$ for the experiments shown in (c).

Figure 3

Bridge evolution and self-similarity. (a) Unscaled bridge $h_0\left(t\right)$ evolution data. (b) Scaled bridge $h_0\left(t\right)$ evolution data. At early times the bridge growth is suggestive of $h_0\left(t\right)\sim t^n$ with $n\approx 0.4$ before evolving linearly as $h_0\left(t\right)\sim t$. At late times the growth ceases due to the yield stress ${\tau }_y$. (c) Scaled drop interface $\mathcal{H}\left(\xi \right)=h(x,t)/h_0\left(t\right)$ against the similarity variable $\xi$ from (5) with $\beta =1$ in the linear regime of (d). The solid line is the numerically determined similarity solution obtained when $Y(x,t)\approx h(x,t)$ and $n=1$ from (1), $\mathcal{H}\left(\xi \right)-\xi {\mathcal{H}}^{\prime }\left(\xi \right)+1/V{\left({\mathcal{H}}^3\left(\xi \right){\mathcal{H}}^{″\prime }\left(\xi \right)\right)}^{\prime }=0$ with one unknown parameter $V$ and boundary conditions $\mathcal{H}\left(0\right)=1$, ${\mathcal{H}}^{\prime }\left(0\right)={\mathcal{H}}^{″\prime }\left(0\right)=0$, ${\mathcal{H}}^{″}\left(\infty \right)=0,$ and ${\mathcal{H}}^{\prime }\left(\infty \right)=1$ [32], ${\left(\right)}^{\prime }=d/d\xi$. Excellent agreement can be seen despite increasing ${\tau }_y$.