Anisotropic electro-osmotic flow over super-hydrophobic surfaces

Abstract

Surface conduction in presence of slip is characterized by full Dukhin number, which is given by [1]: Du = 4(1 + m) κL sinh 2 zeζ 2k B T + 2mb L sinh 2 zeζ 4k B T , (1) where m = 2ε(k B T /ze) 2 /(ηD), and D is the ion diffusiv-ity, which is assumed equal for both types of ions. The first term in (1) is the Dukhin number for the no-slip surface, Du b=0 , while the second one, Du b , is due to hy-drodynamic slip. In the Debye-Hückel model zeζ k B T ≪ 1. (2) This restriction (2) allows the simplification: Du b=0 ≈ 4(1 + m) κL zeζ 2k B T 2 , (3) Du b ≈ 2mb L zeζ 4k B T 2. (4) The parameter m ≈ 100/z 2 , and κL ≫ 1 since EDL is thin. Whence Du b=0 ≈ 1/(z 2 κL) ≪ 1 provided the potential is low. For a "slippery part" of Du we evaluate Du b ≈ 0.5b/L if eζ/(k B T) ≈ 0.1, and Du b ≈ 5 · 10 −3 b/L for eζ/(k B T) ≈ 0.01. Therefore for increasing surface charge (potential) and b/L, the conductivity of the diffuse layer can become comparable to the bulk, and surface condition must be considered. The Péclet number, P e = U L/D , in presence of slip can be evaluated as P e = q 2 E t (1 + bκ)L κηD. (5) Typically, electroosmotic velocity is of order is of order micrometers per second for no-slip surfaces [2]. For nano-scale patterns L < 1 µm and typical ion diffusivities D ≈ 10 −6 cm 2 /s this gives P e b=0 < 0.01 ≪ 1. The slip implies a correction factor (1 + bκ) , which suggests that the convective ion transport can safely be neglected only for bκ < 10. Larger values of bκ should relax this standard approximation of small P e. ELECTRO-OSMOTIC VELOCITY IN EIGENDIRECTIONS Longitudinal stripes.-In this configuration only x−velocity component remains, and the Stokes equation takes the form ∂ 2 y + ∂ 2 z u = εκ 2 ψE t (6) We expand surface charge density in a Fourier series, and the potential is then ψ(y, z) = q εκ e −κy + ∞ n=1 q n εξ n e −ξny cos λ n z, (7) where ξ n = κ 2 + λ 2 n , λ n = 2nπ/L , q = q 1 φ 1 + q 2 φ 2 is the mean surface charge, and q n = 2(q 2 − q 1) πn sin πnδ L. (8) The general solution to (6) for u(y, z) has the form u(y, z) = U + ∞ n=1

Electro-osmosis on Anisotropic Superhydrophobic Surfaces

“…(12) and (18) indicates that the EO flow is generally anisotropic, so that our results do not support an earlier conclusion [4] that the electro-osmotic mobility tensor is isotropic in the thin EDL limit. This inconsistensy [4] (due to an erroneous expression for a transverse electro-osmotic velocity, where factor of 2 was lost) has been corrected for a case b 2 = ∞ in [5].…”

“…Here I is the unit tensor, and we keep notations, q 1 and q 2 , to characterize the surface charge density at the no-slip and slip regions, as above. This expression indicates negligible flow enhancement in case of an uncharged liquid-gas interface (which has been confirmed by later studies [17,18]), and shows that surface anisotropy generally leads to a tensorial EO response.…”

“…This result coincides with expected for hydrophilic slip sectors. In the limit of b/L ≫ 1 this inhibition becomes negligibly small, and we obtain the simple result of [16,18], where EO mobility becomes equal to M 1 regardless of the orientation or area fraction of the slipping stripes. These results suggest that although the absence of the screening cloud near the gas region tends to inhibit the effective EO slip, the hydrodynamic slip acts to suppress this inhibition.…”

“…The EO mobility is represented by 2 × 2 matrices diagonalized by a rotation [18]. By symmetry, the eigendirections of M correspond to longitudinal (θ = 0) and transverse (θ = π/2) alignment with the applied electric field (Fig.…”

Regimes of wetting transitions on superhydrophobic textures conditioned by energy of receding contact lines

“…15,16 Dense SH grooves generate transverse hydrodynamic phenomena 17 and can be successfully used to separate tiny particles 18,19 or enhance mixing rate 20,21 in microfluidic devices. These striped textures amplify electrokinetic pumping 22,23 and are employed for sorting droplets. [24][25][26] We study a wetting transition by monitoring the evaporation of a drop.…”