Wetting dynamics in two-liquid systems: Effect of the surrounding phase viscosity

This paper reports the experimental results of a water droplet spreading on a glass substrate submerged in an oil phase. The radius of the wetted area grows exponentially over time forming two distinct regimes. The early time dynamics of wetting is characterized with the time exponent of 1, referred to as the viscous regime, which is ultimately transitioned to the Tanner's regime with the time exponent of 0.1. It is revealed that an increase in the ambient phase viscosity over three decades considerably slows down the rate of three-phase contact line movement. A scaling law is developed where the three-phase contact line velocity is a function of both spreading radius and mean viscosity, close to the geometric mean of the droplet and ambient fluids’ viscosities. Using the proposed scaling and mean viscosity, all plots of spreading radius for different viscosity ratios collapse to a master curve. Furthermore, several cases with multiple rupture and spreading points, i.e., wetting in a nonideal system, are considered. The growth of an equivalent wetting radius in a multiple point spreading system is predicted by the developed scaling law.

Figure 1

Schematic of the experimental setup. A Petri dish filled with oil is placed on an inverted microscope coupled with a camera. A drop of water with a volume of 2 µl is injected in the middle of the Petri dish, 7 mm above the glass substrate. The drop moves downward by gravity and spreads over the solid surface.

Figure 2

(a) Spreading radius $\left(r\right)$ as a function of time for the spreading of aqueous drops under the oil phase with different viscosity ratios, $0.86,12,178,\mathrm{and}1000$, in blue, red, green, and black, respectively. Increasing the viscosity of the outer fluid decreases the rate of spreading. All curves have a sharp slope at the initial time reaching a plateau. (b) The evolution of $r$ as a function of $t$ in log-log scale. The initial dynamics is identical within the experimental error. Embedded triangles show the slopes. Lines passing through the data points in the first regime are only a visual guide. The black dashed line passing through the late time data has the slope of 0.1. (c) Evolution of measured apparent exponent $\alpha$ as a function of time for the same data as in (b). (d) Evolution of $\alpha$ as a function of $r/R$ for the same data as in (b). The black dashed line is the prediction from Eq. (1)).

Figure 3

(a) $r/R$ as the function of $t/{\tau }_c$ and the resulting master curve. Blue, red, green, and black points are related to the fluid viscosity ratios of $0.86$, 12, 178, and 1000, respectively. The black dashed line is the predicted values from Eq. (6)). (b) $r/R$ as a function of dimensionless time $C_{1}t/{\tau }_c$. The experimental data and related fluid properties are collected from the literature for glycerin ($1120\mathrm{mPa}\mathrm{s}$), glycerin-water ($220\mathrm{mPa}\mathrm{s}$) and water droplet spreading in air [27], and dibutyl phthalate ($16\mathrm{mPa}\mathrm{s}$) and laser oil ($200\mathrm{mPa}\mathrm{s}$) droplet spreading in water [28]. The black dashed line is the predicted values from Eq. (6)).

Figure 4

(a) The transition radius and transition time as a function of the mean viscosity. The transition radii from the viscous regime to the inertial regime for $0.86,12,178,\mathrm{and}1000$ are $307.1\pm 6.24, 364\pm 5, 369\pm 17.11$, and $381\pm 5\mu \mathrm{m}$, respectively. The transition time depends on the viscosity of the ambient phase. For the viscosity ratios of $0.86,12,178,$ and 1000, the viscous regime lasts for $0.04, 0.19, 0.8$, and $6\mathrm{s}$, respectively. (b) The average contact line velocity as a function of the mean viscosity.

Figure 5

The results of $r/r_c$ as a function of $t/t_c$. All data collapse into a single master curve for spreading the water drop on a glass surface submerged in an oil phase.

Figure 6

Spreading of droplets at multiple points. The viscosity of the surrounding oil is 1000 mPa s. The first spreading initiates from a single rupture point. At $t=3\mathrm{s}$, another rupture point touches the solid and spreads throughout the sites. The spreading continues through these points until a portion of the initial liquid (oil) is trapped between them $\left(t=6\mathrm{s}\right)$ and remains as a residual oil $\left(t=15\mathrm{s}\right)$.

Figure 7

Multipoint rupture spreading. The equivalent radius is plotted in black circles. The red line shows the predictive master curves from the dimensional analysis (left and bottom axis). The blue line is the master curve from scaling based on the transition time and length scales (right and top axis).