# Thermocapillary Flow on Superhydrophobic Surfaces

Microtextured surfaces have mainly received attention due to their wetting properties. A liquid drop placed on a suitably structured hydrophobic surface will only be in contact with the material on protruding tips, while gas is trapped in the valleys in between. In this so called Cassie-Baxter state nearly perfect hydrophobicity can be obtained, reflected in contact angles close to 180°. A liquid in Cassie-Baxter state above a structured superhydrophobic surface is ideally suited for surface driven transport due to its large free surface fraction in close contact to a solid.

In our group we analyze temperature induced Marangoni convection as a driving force for fluid transport along microtextured surfaces. In particular, we focus on finned surfaces as sketched below, with fins oriented along or transverse to a temperature gradient, and the liquid being in the Cassie-Baxter state. An integral relation for the Stokes equation allows deriving an analytical formula for the macroscopic flow velocity observed at some distance above the surface [1]. Our analysis focuses on substrates of high thermal conductivity, such as silicon and shows that even moderate temperature gradients of the order of 10 K/cm can lead to fluid velocities of several mm/s for water based systems on superhydrophobic surfaces. Thermocapillary convection may thus add to the portfolio of actuation principles in microfluidic settings and may even enable larger flow velocities than typically achieved with electroosmosis.

Sketch of the geometry. A liquid column is assumed to be in contact with the fin surface only and gas pockets are trapped between the fins leading to reduced friction (Cassie-Baxter state). In this example we assume a temperature gradient along the fins (longitudinal case).

Longitudinal (filled symbols) and transverse (open symbols) thermocapillary slip coefficients β_{th} for different free surface fractions a = B/L and temperature gradients parametrized by ∂_{z}T=-10^{-f} 60 K / cm. The bulk flow velocity in a channel lined by two planar superhydrophobic surfaces is related to the thermocapillary slip coefficient by u_{M,l}= β_{th} (L/η) (∂σ/∂T) (∂_{z}T), where η is the viscosity, σ the surface tension and (∂_{z}T) is the average temperature gradient on the liquid-gas surface [1].

[1] Baier, Steffes, Hardt, “Thermocapillary Flow on Superhydrophobic Surfaces”, Phys. Rev. E 82, 037301 (2010); http://arxiv.org/abs/1003.4161v1;

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This research is support by the German Research Foundation (DFG) through the Cluster of Excellence 259 and grant number HA 2696/12-1.