Forced Wetting Transition and Bubble Pinch-Off in a Capillary Tube

Immiscible fluid-fluid displacement in partial wetting continues to challenge our microscopic and macroscopic descriptions. Here, we study the displacement of a viscous fluid by a less viscous fluid in a circular capillary tube in the partial wetting regime. In contrast with the classic results for complete wetting, we show that the presence of a moving contact line induces a wetting transition at a critical capillary number that is contact angle dependent. At small displacement rates, the fluid-fluid interface deforms slightly from its equilibrium state and moves downstream at a constant velocity, without changing its shape. As the displacement rate increases, however, a wetting transition occurs: the interface becomes unstable and forms a finger that advances along the axis of the tube, leaving the contact line behind, separated from the meniscus by a macroscopic film of the viscous fluid on the tube wall. We describe the dewetting of the entrained film, and show that it universally leads to bubble pinch-off, therefore demonstrating that the hydrodynamics of contact line motion generate bubbles in microfluidic devices, even in the absence of geometric constraints.

Figure 1

Fluid-fluid interface of air (black) displacing glycerol (white) under increasing capillary numbers (top to bottom) in a wetting capillary tube with ${\theta }_{\mathrm{eq}}=25°\pm 5°$ (left column) and a weakly wetting capillary tube with ${\theta }_{\mathrm{eq}}=68°\pm 5°$ (right column). The orange arrows indicate the direction of interface displacement. At small Ca, the meniscus deforms slightly from its equilibrium shape, but remains as a spherical cap. At large Ca, however, a wetting transition occurs and the invading air forms a single finger that advances along the center of the tube, leaving a macroscopic trailing film of the viscous liquid on the tube walls. The critical capillary number ${\mathrm{Ca}}^{*}$ at which film formation occurs is controlled by the wettability [pane (r) versus pane (e)].

Figure 2

(a) Fluid-fluid interface profiles for $\mathrm{Ca}<{\mathrm{Ca}}^{*}$ in the less wetting capillary tube (${\theta }_{\mathrm{eq}}=68°$). As Ca increases, the fluid-fluid interface deforms further from its equilibrium shape. The symbols and dashed lines correspond to the experimental and numerical results for $\mathrm{Ca}=0.003$, 0.006, 0.012, respectively. (b) Deformation of the fluid-fluid interface can be quantified by the apparent contact angle ${\theta }_{\mathrm{app}}$, which decreases asymptotically towards zero as Ca approaches the critical capillary number ${\mathrm{Ca}}^{*}$. This marks the onset of the wetting transition. The diamond and circle symbols represent the experimental data and theoretical predictions, respectively. The dashed lines above and below the data points represent the experimental uncertainty.

Figure 3

The contact line capillary number ${\mathrm{Ca}}_{\mathrm{cl}}=\mu U_{\mathrm{cl}}/\gamma$ as a function of the macroscopic capillary number $\mathrm{Ca}=\mu U/\gamma$, where $U_{\mathrm{cl}}$ and $U$ are the contact line velocity and the displacement velocity, respectively. The green and blue circles represent the experimental measurements in the more wetting tube (${\theta }_{\mathrm{eq}}=25°$) and in the less wetting tube (${\theta }_{\mathrm{eq}}=68°$), respectively. The triangles show the corresponding theoretical predictions of Eq. (2). The vertical dashed lines represent the critical capillary numbers ${\mathrm{Ca}}^{*}$ as predicted by Eq. (1). For $\mathrm{Ca}<{\mathrm{Ca}}^{*}$, the fluid-fluid interface deforms slightly while remaining a spherical cap, and the contact line and the interface tip travel at the imposed displacement velocity $U=4Q/\pi d^2$ (gray dashed line). For $\mathrm{Ca}>{\mathrm{Ca}}^{*}$, the air forms a finger that advances along the center of the tube, leaving behind a film of the more-viscous defending liquid. The incomplete displacement of the defending liquid results in an interface tip velocity that is larger than the imposed displacement velocity. The contact line that trails behind travels at a velocity that, remarkably, is independent of the imposed flow rate. Instead, the contact line velocity is controlled by the wettability of the capillary tube.

Figure 4

(a) Top: experimental images of the fluid-fluid interface profile just before and after pinch-off. Bottom: the dewetting rim profile in the frame comoving with the receding contact line for $\mathrm{Ca}=0.096$ and ${\theta }_{\mathrm{eq}}=68°$. The dashed and solid lines correspond to the numerical and experimental data, respectively. (b) The maximum height of the rim ${\tilde{h}}_{\max }$ grows linearly at early times and nonlinearly accelerates close to the pinch-off time $\tau$. The color-coded circles represent different experimental realizations of rim growth that led to pinch-off, while the dashed line represents the theoretical prediction of Eq. (2). The inset shows the pinch-off time as a function of the capillary number and the wettability condition, where the green and blue symbols correspond to ${\theta }_{\mathrm{eq}}=25°$ and 68°, respectively. The theoretical predictions (diamonds) match closely to the experimental data (circles) for ${\theta }_{\mathrm{eq}}=68°$. Given the limited experimental window, we did not directly observe pinch-off in the more wetting tube since its $\tau$ is predicted to be over an order of magnitude larger than that in the less wetting tube. (c) The bubble length $L_b$ as a function of the capillary number and the wettability condition, where the green and blue symbols correspond to ${\theta }_{\mathrm{eq}}=25°$ and 68°, respectively. The theoretical predictions (diamonds) agree well with the experimental data (circles) for ${\theta }_{\mathrm{eq}}=68°$. See the video [22].