The Spreading of Sessile Drops in Miscible Environments

The spreading of liquids is a classical problem in interfacial fluid mechanics. The spreading parameter, S, is defined as the difference in energy of a substrate when it is dry to when it is completely wetted with a liquid. Depending on the sign of the spreading parameter, the liquid either completely wets the substrate and takes on an equilibrium contact angle of zero (S > 0) or it partially wets the substrate and forms a sessile drop with an equilibrium contact angle greater than zero (S < 0). Often, the spreading of liquids is studied in the context of sessile drops. Under the action of gravity or capillary forces, sessile drops in air have been measured to spread with well-known power law dependencies in time as they adjust their shapes and contact angles. Tanner's law, which describes the spreading of a sessile drop under the action of capillary forces, is a prominent example.

Similarly, a sessile drop that finds itself immersed beneath a second fluid in which it is miscible can spread spontaneously as well. This problem, which does not appear to have been previously addressed, is the subject of this study.

As time evolves, a sessile droplet surrounded by a miscible environment will undergo spreading, due to gravitational, capillary, and Marangoni forces, and diffusion due to the chemical potential difference between the two initially distinct, homogeneous phases. The liquid-liquid interface translates with drop spreading and becomes less distinct as the two liquids diffuse into one another. Density differences between the two miscible liquids create a gravitational force that can influence the spreading phenomena. At the contact line, the two miscible liquids compete to wet the solid interface relative to their surface energies and surface tensions. Gradients in interfacial tension over the liquid-liquid interface may arise from drop movement and dissolution at different rates and in different directions, which lead to Marangoni stresses that can act on the spreading miscible sessile drop.

Six different liquid pairs (Drop Liquid-Ambient Liquid: Corn Syrup-Water, Glycerol-Water, Glycerol-Ethanol, Glycerol-Isopropanol, Tricresyl Phosphate-Ethanol, Tricresyl Phosphate-Isopropanol) were studied.  Various droplet sizes were also studied.  Using data from the literature, the ratios of gravitational to capillary (the Bond number, Bo) and gravitational to Marangoni forces (the inverse Marangoni number, Ma^-1) and diffusion to convection time scales (t­_D/t_C) were estimated. Based on these estimated values (Bo ~ 5-1500, Ma^-1 ~ 5-1500, t_D/t_C ~ 10⁵), it appears that gravitational forces dominate the drop dynamics and that convection occurs on a much shorter time scale than diffusion.

The observed shape evolution and dynamics of sessile drops spreading into miscible environments is qualitatively different than those observed for liquid-immiscible environment systems. In addition to a spreading contact line, there exists a portion of the drop that is elevated above the liquid-substrate interface and, in some cases, extends beyond the contact line, as depicted in Figure 1.

Figure 1 – Side view of a miscible sessile drop of corn syrup immersed in water.

We have found in the miscible liquid pairs studied to date that miscible sessile drops also spread with power law dependencies on time, R ~ tⁿ. For corn syrup and glycerol in water, the leading edge radii progresses with n Å 0.40. The leading edge radii for the other liquid pairs have n Å 0.20. The contact line radii do not fall into two groups like the leading edge radii, however, within the same ambient liquid, the contact line power laws decrease with increasing viscosity.

Confocal microscopy experiments were performed to characterize the drop dynamics at the initial contact line and the thicknesses of the drop leading edge and the ambient liquid layer that exists between the drop leading edge and the solid substrate.