Spreading of miscible liquids

Miscible liquids commonly contact one another in natural and technological situations, often in the proximity of a solid substrate. In the scenario where a drop of one liquid finds itself on a solid surface and immersed within a second, miscible liquid, it will spread spontaneously across the surface. We show experimental findings of the spreading of sessile drops in miscible environments that have distinctly different shape evolution and power-law dynamics from sessile drops that spread in immiscible environments, which have been reported previously. We develop a characteristic time to scale radial data of the spreading sessile drops based on a drainage flow due to gravity. This time scale is effective for a homologous subset of the liquids studied. However, it has limitations when applied to significantly chemically different, yet miscible, liquid pairings; we postulate that the surface energies between each liquid and the solid surface becomes important for this other subset of the liquids studied. Initial experiments performed with pendant drops in miscible environments support the drainage flow observed in the sessile drop systems.

Figure 1

Solid-liquid-fluid system and the force balance of the three surface energies—${\sigma }_{\text{LF}}$, ${\sigma }_{\text{SF}}$, and ${\sigma }_{\text{SL}}$—arising at the three-phase contact line. The three subscripts indicate the surface energies at the solid-fluid (SF), solid-liquid (SL), and liquid-fluid (LF) interfaces. $\theta$ is the contact angle that the spreading liquid forms with the solid substrate. $R$ is the radius of the spreading liquid. Both $\theta$ and $R$ can be time-dependent parameters if the spreading liquid is not at its equilibrium shape.

Figure 2

Experimental setup for performing sessile drop experiments. First, a sessile drop is formed in air on a circular disk, which serves as the sessile drop platform. The circular disk is affixed to a thin metal rod and suspended above a bath of the ambient liquid. The bath of ambient liquid is set upon a platform mounted to a vertical stage that can translate in order to immerse the circular disk and sessile drop to initiate the miscible sessile drop spreading experiment. Cameras are positioned to image the spreading process from the side and below. A laser and cylindrical lens are mounted to produce a sheet of light orthogonal to the side camera to allow for PTV. The setup can accommodate pendant drop experiments by replacing the sessile drop platform with a fixture to hold a syringe needle. Inset: The pattern formed on a substrate for use in sessile drop experiments. There are two regions on each substrate, demarcated by the dashed circle. The first region is solid white, contained within the dashed circle, and the second exists between the dashed circle and the outer edge, striped gray and white. The diameter of the substrate is 30 mm and the width of the outer region is 2 mm. In the case of a hydrophilic surface, the interior, solid white region was protected to remain hydrophilic while the exterior, gray striped region was exposed to a hydrophobic treatment. In the alternate case of a hydrophobic surface, the interior region was exposed to a hydrophobic treatment while the exterior region was protected to remain hydrophilic. Patterning in such a way facilitated a symmetric immersion process.

Figure 3

Snapshot of a corn syrup sessile drop spreading while immersed in water as visualized from the (a) side and (b) bottom. The black arrows indicate the most radially advanced portion of the drop, which is elevated above the solid surface, and we refer to it as the leading edge. The white arrows indicate the most radially advanced portion of the drop that remains in contact with the solid surface, and we refer to it as the contact line. The shape evolution for this sessile drop immersed in a miscible environment is distinctly different from that of one present in an immiscible environment. Refer to the Supplemental Material [24] for the full movie from which these frames were taken.

Figure 4

A sequence of images taken over time of a 10 000 cSt silicone oil sessile drop immersed in 5 cSt silicone oil and spreading across a hydrophilic glass surface. In each frame, the upper, light region is the ambient liquid and the lower, dark region is the substrate. The sessile drop begins to spread by developing an elevated leading edge that leads the advancement of the drop across the surface. Refer to the Supplemental Material [24] for the full movie from which these frames were taken.

Figure 5

A sequence of images taken over time of a corn syrup sessile drop immersed in water and spreading across a hydrophilic glass surface. In each frame, the upper, light region is the ambient liquid and the lower, dark region is the substrate. The sessile drop begins to spread by developing an elevated leading edge that leads the advancement of the drop across the surface. Refer to the Supplemental Material [24] for the full movie from which these frames were taken.

Figure 6

Image from a PTV experiment of a corn syrup sessile drop immersed in water and spreading across a hydrophobic glass surface. The corn syrup contains 6-$\mu \mathrm{m}$ microspheres at a concentration of ${10}^{-3}$ g/ml, which scatter the incident laser light. The arrows indicate velocity vectors obtained from individual particle movement. Motion was largely restricted to the corn syrup–water interface and particles within the drop and away from the liquid-liquid interface remain stationary. Refer to the Supplemental Material [24] for the full movie from which this frame was taken.

Figure 7

Reconstructed, side view image sequence from a confocal microscopy experiment of a corn syrup sessile drop spreading across a hydrophobic glass surface while immersed in water. A frustum of the sessile drop, its base coinciding with the glass surface, was scanned along a radial slice, as displayed in the top frame. The corn syrup contains 0.2-$\mu \mathrm{m}$ fluorescent polystyrene microspheres at a concentration of ${10}^{-3}$ g/ml, while the water contains no fluorescent material. Thus, the corn syrup phase exists as the bright regions of the images and the water phase appears dark. One can see the development of the leading edge, first visible at 0.9 s here, and indicated by white arrows thereafter; it is elevated approximately 35 $\mu \mathrm{m}$ above the substrate at its tip after approximately 4.5 s. Refer to the Supplemental Material [24] for a movie of this image sequence.

Figure 8

Plots of leading-edge radius measurements as a function of time for homologous silicone oil pairs on hydrophilic and hydrophobic glass. (a) Unscaled data. (b) The leading-edge radius data are scaled by the initial radius of each sessile drop and time is scaled by a characteristic drainage time, $\tau ={\mu }_a/{\rho }_{s}g{h}_{\text{LE}}$, where ${\mu }_a$ is the viscosity of the ambient liquid, ${\rho }_s$ is the density of the spreading liquid, $g$ is the acceleration due to gravity, and $h_{\text{LE}}$ is the thickness of the elevated leading edge. The dashed line in each plot represents a square-root power law, which is a time dependence indicative of a diffusion process. Symbol shape and color are coded for the ambient and spreading liquids, respectively. Symbol shapes $▾$ and $▸$ indicate the ambient liquids are 5 cSt and 10 cSt silicone oils, respectively. Symbol colors indicate the spreading liquids are 10, 20, 50, 100, 200, 500, 1000, 5000, and 10 000 cSt silicone oils, respectively. Solid and open symbols $▾$ and $▿$ represent experiments performed with 10 cSt silicone oil as the spreading liquid and 5 cSt silicone oil as the ambient liquid on hydrophilic and hydrophobic surfaces, respectively. One experimental pair exhibits an initial retraction of the radius prior to spreading outward following a square-root power law in time. Refer to the Supplemental Material [24] for a series of images and movie for this particular experiment.

Figure 9

Plot of leading-edge radius measurements as a function of time for pairs of nonhomologous liquids on hydrophilic and hydrophobic glass surfaces. The leading-edge radius data are scaled by the initial radius of each sessile drop and time is scaled by a characteristic drainage time, $\tau ={\mu }_a/{\rho }_{s}g{h}_{\text{LE}}$, where ${\mu }_a$ is the viscosity of the ambient liquid, ${\rho }_s$ is the density of the spreading liquid, $g$ is the acceleration due to gravity, and $h_{\text{LE}}$ is the thickness of the elevated leading edge. The dashed line is plotted alongside the data representing a square-root power law, which is a time dependence indicative of a diffusion process. Symbol shape and color are coded for the ambient and spreading liquids, respectively. Symbol shapes, $●$, $⧫$, and $\blacksquare$ indicate the ambient liquid as water, ethanol, or isopropanol, respectively. Symbol colors indicate the spreading liquid as corn syrup, glycerol, or tricresyl phosphate, respectively. Solid and open symbols $●$ and $○$ represent experiments performed with corn syrup as the spreading liquid and water as the ambient liquid on hydrophilic and hydrophobic surfaces, respectively.

Figure 10

Plot of contact-line radius measurements as a function of time for liquid pairs not involving silicone oils on hydrophilic and hydrophobic glass surfaces. The contact-line radius data are scaled by the initial radius of each sessile drop and time is left unscaled. The dashed line is plotted alongside the data representing a power law of $t^{1/8}$. Symbol shapes $●$, $⧫$, and $\blacksquare$ indicate the ambient liquids are water, ethanol, and isopropanol, respectively. Symbol colors indicate the spreading liquids are corn syrup, glycerol, and tricresyl phosphate, respectively. Solid and open symbols $●$ and $○$ represent experiments performed with corn syrup as the spreading liquid and water as the ambient liquid on hydrophilic and hydrophobic surfaces, respectively.

Figure 11

Image sequence taken in time of a corn syrup pendant drop immersed in water. A strand emanates from the apex of the drop and continues to flow as the entire drop descends and elongates. Refer to the Supplemental Material [24] for the full movie from which these frames were taken.

Figure 12

Image from a PTV experiment of a corn syrup pendant drop immersed in water. The corn syrup contains 6-$\mu \mathrm{m}$ microspheres at a concentration of ${10}^{-3}$ g/ml, which scatter the incident laser light. The arrows indicate velocity vectors obtained from particle movement. Motion was largely restricted to the corn syrup–water interface. Particles within the pendant drop and away from the liquid-liquid interface move with the drop as it descends and elongates; within the reference frame of the pendant drop, these interior particles do not move significantly. Refer to the Supplemental Material [24] for the full movie from which this frame was taken.