Tunable superapolar Lotus-to-Rose hierarchical nanosurfaces via vertical carbon nanotubes driven electrohydrodynamic lithography† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c6nr08706j Click here for additional data file.

The development of an advanced technique that enables fabrication of superhydrophobic structured surfaces, easily tuneable from lotus-leaf to rose-petal state is essential to realise the full applied potential of such architectures.

To evaluate the site-densities, liquid-induced compaction method was used. Isopropanol was added to the samples and been allowed to dry leading to collapse of the CNT forests as shown in the Figure S1.
The percentage of the aggregated CNTs was the calculated by the fraction of number of pixels of the aggregated forest areas divided by that of the entire image. CNTs with various densities (10 10 -10 12 cm -2 ) and coverage of 7%, 15% and 30%, respectively.
Electronic Supplementary Material (ESI) for Nanoscale. This journal is © The Royal Society of Chemistry 2017

S2: Young's Equation and Classical Wetting Theory
On an ideal flat surface the contact angle is described by the equation, formulated by Young, which demonstrates that contact angle is a property directly dependent on the surface tension between the three interfaces (  According to this equation a hydrophobic material has an inherent water contact angle above 90° and a hydrophilic material has an inherent water contact angle below 90°. The wetting behaviour on rough surfaces however, is more complex than described by the Young's theory. It has been established that the surface roughness enhances the inherent wetting properties of the materials. [Ind. Eng. Chem., 1936, 28, 988;Trans. Faraday Soc.1944, 40, 546] While Wenzel's theory is established on complete contact between the liquid and the surface and for hydrophobic materials there is no gap between the droplet and the surface with the contact angle > 90°, in the Casie-Baxter's model the small contact area between the liquid and the solid phase and the resulting small contact angle hysteresis allow drops to easily 'roll' off a surface due to the presence of air-pockets under the droplet.

S2.1: Wenzel's Model
Wenzel's wetting is strongly dependent on the surface roughness, which is a measure of the true against the projected surface area:

S2.2: Cassie-Baxter Model
In the case when the contact angle is above 90° and there is a thin air-filled gap between the drop and the surface with high surfaces roughness, it is energetically more favourable for the hydrophobic Cassie-Baxter regime to prevail.
Cassie-Baxter wetting is characterised by the drop located on the peaks of the surface structures which does not penetrate into the deeper holes. This air-pocket state corresponds to partial non-wetting of the surface, which is the most favourable solution for the energy equation for very rough hydrophobic materials: where,  Figure S4. Schematic representation of Cassie-Impregnating wetting state.
The impregnating state, depicted in Figure S4 can be determined using the following equation: where, is the solid fraction underneath the liquid drop. However, for this Cassie impregnating state to be viable, it must satisfy: A classic example in nature of a Cassie impregnating regime is the rose petal, also known as the 'Rose Petal Effect'`. The water film is able to impregnate the surface by seeping into the larger crevices, but not the smaller ones. Therefore, the water drop depending on its size can attach to the surface due to the very high adhesive forces.

S2.4: Combined Model
An additional possibility exists when a liquid droplet may not bridge the gap on a rough surface completely. Unlike in Wenzel's model, where liquid film must be present before droplet is placed on the surface, there is an initially dry rough surface. Without a liquid film, the drop might spread below the flat surface, as occurs on surfaces such as sol-gels and soils, where internal spreading can occur rapidly, resulting in a combined Wenzel's and the Cassie-Baxter model., which gives the effective macroscopic water contact angle for Cassie-Baxter wetting [Trans. Faraday Soc.1944, 40, 546] cos θCB = f cos θγ +f -1.
The roughness of the fraction of the surface in contact with the liquid, rf also controls the surface wetting properties, and a more specific definition of the air-pocket state can be given by: where, θr is the apparent contact angle of the micro-and nano structured surfaces, fs is the fraction of the areas occupied by the solid-water interface and fv is the fraction that correspond to the vapour gaps, θ is the Young's contact angle and ρ is the roughness factor, which can be simplistically calculated from triadic curve for fractal geometry [Advanced Materials, 2002, 14, 1857: For a hexagonal array of cones, the area fraction of the solid surface that is in contact with the liquid is given by fs ≈ π 4√3 (r/R) 2 .

S3: Casie-Baxter Equation in terms of Structures' Geometrical Parameters
For the hexagonal packing geometries, typically generated during the EHL process, the Casie-Baxter equation: can be reformulated taking into account the geometrical parameters: .
Resulting in: Plotting the dependence of the cones' surface roughness in Figure S5 reveals that the triple-phase contact length has a linear dependence on the contact angle hysteresis.