Thermal Marangoni Flow Impacts the Shape of Single Component Volatile Droplets on Thin, Completely Wetting Substrates

Despite surface energies dictating complete wetting, it has been classically observed that volatile alkanes do not spread completely on glass substrates, and faster evaporation rates lead to higher contact angles. Here we investigate how substrate thickness influences this behavior. For sufficiently thin substrates, we find alkanes evaporate slower and display higher apparent contact angles, at odds with the typical explanations involving just evaporation, capillarity, and viscous dissipation. We derive the droplet temperature distribution and use it as part of a criteria to show that thermal Marangoni contraction plays a significant role in establishing droplet shape on thin substrates.

Figure 1

Single component volatile liquids on a completely wetting surface. (a) Nonuniform evaporation is known to prevent droplet spreading through “direct evaporation-driven stabilization" by removing liquid fastest from the droplet perimeter. Here we ask, might Marangoni contraction from an inward surface tension-driven flow along the droplet-vapor interface play a role? (b) Apparent contact angle ${\theta }_{\mathrm{app}}$ decreases with alkane chain length for hexane (), heptane (), octane (), and nonane () on glass (triplicate measurements shown).

Figure 2

Apparent contact angle ${\theta }_{\mathrm{app}}$ of evaporating alkane droplets over time for different glass thicknesses $L$. The direct evaporation-driven mechanism proposed to stabilize droplets predicts higher angles for faster evaporation rates. C6–C9 alkanes on thinner glass show larger apparent contact angles and slower evaporation, suggesting the need to consider an additional effect determining droplet shape. Darker shades represent thicker glass substrates (0.063, 0.138, 1, 3, and 6 mm).

Figure 3

Marangoni flow within evaporating alkane droplets on glass. (a) Time-lapse microscopy of tracer particles indicates flow direction and magnitude. Flow is fastest for hexane. Flow in heptane on thin glass is faster than heptane on thicker glass and, in octane tracer particles, sediment before fully circulating. (b) To formulate the governing equation, energy conservation is applied on a circular elements of radius $r$, thickness $dr$, and height $L$. We balanced energy by: conduction $q_r$ through the cross-sectional area $A_c$, free convection $d{q}_c$ through surface area $d{A}_s$, and removal by evaporation $d{q}_e$ within the droplet existence region. (c) Predicted radial substrate [$T_s$, dashed, Eq. (4)] and droplet [$T_d$, solid, Eq. (5), inset] temperature profiles, for heptane droplets on different glass thicknesses. (d) Surface tension gradients $d\gamma /dr$ are caused by temperature gradients $d{T}_d/dr$ (inset) for heptane droplets on different glass thicknesses. (e) Velocities $u$ of thermal Marangoni-induced flow for heptane droplets on different glass thicknesses. Experimental velocity was measured by tracking particles that followed streamlines near the droplet-vapor interface (indicated in green in inset), and theoretical values are from Eq. (6).

Figure 4

Impact of Marangoni flows on droplet shape depends more on the substrate thickness than the evaporation rate. (a) Larger alternative Marangoni number ${\mathrm{Ma}}_Q$ indicates stronger Marangoni flow relative to potential capillary flow, which leads to a larger ratio in apparent contact angle to minimum apparent contact angle $\theta /{\theta }_{\min }$. Here ${\theta }_{\min }$ is the angle on the thickest glass, where the thermal Marangoni is the weakest. Shade indicates substrate thickness as in Fig. 2. (b) Schematic depiction showing evaporation of a droplet on thinner glass generating a larger temperature gradient that induces a stronger thermal Marangoni flow and a higher apparent contact angle despite slower evaporation.