^{1}and Tobias Baier

^{2,a)}

### Abstract

It has been suggested that superhydrophobic surfaces, due to the presence of a no-shear zone, can greatly enhance transport of surface charges, leading to a considerable increase in the streaming potential. This could find potential use in micro-energy harvesting devices. In this paper, we show using analytical and numerical methods, that when a streaming potential is generated in such superhydrophobic geometries, the reverse electro-osmotic flow and hence current generated by this, is significant. A decrease in streaming potential compared to what was earlier predicted is expected. We also show that, due to the electro-osmotic streaming-current, a saturation in both the power extracted and efficiency of energy conversion is achieved in such systems for large values of the free surface charge densities. Nevertheless, under realistic conditions, such microstructured devices with superhydrophobic surfaces have the potential to even reach energy conversion efficiencies only achieved in nanostructured devices so far.

We thank Mathias Dietzel, Clarissa Schönecker, and Steffen Hardt for fruitful discussions. G.S. kindly acknowledges support by the German Academic Exchange Service (DAAD) through the WISE program. T.B. kindly acknowledges support by the German Research Foundation (DFG) through the Cluster of Excellence 259.

I. INTRODUCTION

II. GENERAL FORMULATIONS

A. Bi-layer potential distribution

B. Streaming velocity and current

C. Electro-osmotic flow and current

D. Energy efficiency and streaming potential

III. RESULTS AND DISCUSSION

A. Approximate analytical solutions

B. Numerical methods

C. Graphical analysis

IV. CONCLUSION

### Key Topics

- Free surface
- 38.0
- Double layers
- 34.0
- Surface charge
- 34.0
- Electric currents
- 22.0
- Electroosmosis
- 18.0

## Figures

Schematic diagram of the superhydrophobic surface. Within the grooves air is entrapped and we assume the air-liquid interface to remain ideally flat. The length of the channel is assumed to be l and consists of two parallel surfaces at distance 2h. The surfaces are periodically structured with groves of width wδ and periodicity w.

Schematic diagram of the superhydrophobic surface. Within the grooves air is entrapped and we assume the air-liquid interface to remain ideally flat. The length of the channel is assumed to be l and consists of two parallel surfaces at distance 2h. The surfaces are periodically structured with groves of width wδ and periodicity w.

Output power density for varying free surface charge density, σ s , for different values of the free surface fraction, δ. The solid line is calculated based on our analytical expression (34) , and the crosses (×) are calculated based on the more exact semi-analytic expressions (11), (14), (21), and (23) . Pluses (+) and circles (o) are numerical results using the nonlinear Poisson-Boltzmann equation, differing in the evaluation of the conduction current by, respectively, a constant and variable charge density according to Eqs. (24) and (38) . The dotted line shows the behavior when the electro-osmotic flow is not accounted for (at δ = 0.7). The figure was plotted assuming w = h = 100 μm, σ = 0.1 μS/cm, λ D = 1 μm, μ = 1 mPa s and σ s /σ ns = 2.

Output power density for varying free surface charge density, σ s , for different values of the free surface fraction, δ. The solid line is calculated based on our analytical expression (34) , and the crosses (×) are calculated based on the more exact semi-analytic expressions (11), (14), (21), and (23) . Pluses (+) and circles (o) are numerical results using the nonlinear Poisson-Boltzmann equation, differing in the evaluation of the conduction current by, respectively, a constant and variable charge density according to Eqs. (24) and (38) . The dotted line shows the behavior when the electro-osmotic flow is not accounted for (at δ = 0.7). The figure was plotted assuming w = h = 100 μm, σ = 0.1 μS/cm, λ D = 1 μm, μ = 1 mPa s and σ s /σ ns = 2.

Relative drop of streaming potential for increasing free surface charge density, σ s , for different values of the free surface fraction, δ. The solid line is calculated based on our analytical expression (34) , and the crosses (×) are calculated based on the more exact expressions (11), (14), (21), and (23) . The figure was plotted assuming w = h = 100 μm, σ = 0.1 μS/cm, λ D = 1 μm, μ = 1 mPa s and σ s /σ ns = 2.

Relative drop of streaming potential for increasing free surface charge density, σ s , for different values of the free surface fraction, δ. The solid line is calculated based on our analytical expression (34) , and the crosses (×) are calculated based on the more exact expressions (11), (14), (21), and (23) . The figure was plotted assuming w = h = 100 μm, σ = 0.1 μS/cm, λ D = 1 μm, μ = 1 mPa s and σ s /σ ns = 2.

Variation of efficiency with free surface charge density, σ s , both for varying values of bulk conductivity σ and free surface fractions δ = 0.4 and δ = 0.9. Lines are calculated based on the analytical expressions (29) and (35) . Circles (o) are numerical results obtained using the nonlinear Poisson-Boltzmann equation. The dashed lines shows the maximum theoretical efficiency predicted for such systems. Here w = h = 100 μm, λ D = 1 μm, μ = 1 mPa s and σ s /σ ns = 2.

Variation of efficiency with free surface charge density, σ s , both for varying values of bulk conductivity σ and free surface fractions δ = 0.4 and δ = 0.9. Lines are calculated based on the analytical expressions (29) and (35) . Circles (o) are numerical results obtained using the nonlinear Poisson-Boltzmann equation. The dashed lines shows the maximum theoretical efficiency predicted for such systems. Here w = h = 100 μm, λ D = 1 μm, μ = 1 mPa s and σ s /σ ns = 2.

(a) Variation of maximum efficiency with free surface charge density, σ s , for values of λ D /h between 0.01 and 0.1, fixing the geometry length scale h and varying λ D . Lines are calculated based on the analytical expressions (29) and (35) . Circles (o) are numerical results obtained using the nonlinear Poisson-Boltzmann equation. Here w = h = 100 μm, δ = 0.9, μ E = 5.43 × 10−7 m2/(Vs), μ = 1 mPa s and σ s /σ ns = 2. (b) Variation of maximum efficiency as a function of λ D /h at fixed λ D = 1 μm, σ = 0.1 μS/cm and variable length-scale h for several values of surface charge density σ s . Circles (•) are numerical results obtained using the nonlinear Poisson-Boltzmann equation and the dotted line is the analytical expression (29) and (35) . Here w = h, δ = 0.9, μ = 1 mPa s, and σ s /σ ns = 2.

(a) Variation of maximum efficiency with free surface charge density, σ s , for values of λ D /h between 0.01 and 0.1, fixing the geometry length scale h and varying λ D . Lines are calculated based on the analytical expressions (29) and (35) . Circles (o) are numerical results obtained using the nonlinear Poisson-Boltzmann equation. Here w = h = 100 μm, δ = 0.9, μ E = 5.43 × 10−7 m2/(Vs), μ = 1 mPa s and σ s /σ ns = 2. (b) Variation of maximum efficiency as a function of λ D /h at fixed λ D = 1 μm, σ = 0.1 μS/cm and variable length-scale h for several values of surface charge density σ s . Circles (•) are numerical results obtained using the nonlinear Poisson-Boltzmann equation and the dotted line is the analytical expression (29) and (35) . Here w = h, δ = 0.9, μ = 1 mPa s, and σ s /σ ns = 2.

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