(149g) 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.