## 3 Results and Discussion
With seven and nine layers of coating, the contact angle values reached to a high of \(150\pm 1^{\circ}\) and \(151\pm 1^{\circ}\), respectively. These values show that a critical number of seven layers was required to attain superhydrophobicity when using dilute solution as compared to only three layers that was sufficient to obtain this property with the as-prepared solution of the fluorinated silica nanoparticles. Although superhydrophobic properties could be obtained using both as-prepared and diluted solutions, the difference in the critical number of layers required in obtain these properties is attributed to the difference in the surface roughness obtained on the corresponding surfaces. Figure 3(B) shows the variation in the root mean square (rms) roughness measured on the surfaces prepared with varying numbers of layers coated using the two different solutions deposited on aluminum substrates. As an example, a three dimensional surface profile obtained during the rms measurements of aluminum surface prepared by coating three layers using as prepared solution has been presented in the inset of Figure 3(B). These graphs show that the contact angle and the roughness vary proportionally as it is clear that the water contact angle increases with increase in surface roughness until a critical roughness value is reached. Figure 3(C) shows a plot of water contact angle as a function of rms roughness on the coatings prepared using as prepared solution. We have previously reported a detailed mathematical model to show the relationship between the roughness and water contact angle in reference to Cassie model [6]. [PERSON] _et al._[40] have also shown similar behavior on their electrochemically deposited copper state surface. On the coatings prepared using the as-prepared solution, an rms value of \(0.60\pm 0.10\)\(\upmu\)m on a one layer coating increased to \(0.69\pm 0.01\)\(\upmu\)m when coated with three layers where superhydrophobic properties were obtained with a contact angle above \(150^{\circ}\). With further increase in the number of layers to five, seven and nine, the rms roughness remained unchanged (\(\sim\)0.71 \(\upmu\)m). The similar rms values on the coatings prepared with more than three layers shows that increasing the number of layers further does not cause any change in the surface roughness and therefore provides the same contact angle values on these surfaces. Therefore, an rms roughness of \(0.69\pm 0.01\)\(\upmu\)m obtained with three layer coating is found to be a critical value in obtaining superhydrophobic properties. Similarly, in case of the coatings prepared using diluted solution, the rms values show that superhydrophobic properties were only reached when the rms values reached \(0.70\pm 0.02\)\(\upmu\)m when coated with seven layers on the substrate providing a water contact angle value of \(150\pm 1^{\circ}\). These values were lower on the surfaces coated using less layers, that is, \(0.57\pm 0.10\)\(\upmu\)m,\(0.61\pm 0.03\)\(\upmu\)m and \(0.67\pm 0.03\)\(\upmu\)m, respectively on the surfaces coated with one, three and five layers, which provided water contact angles lower than 150\({}^{\circ}\). With nine layers of coating, the rms value slightly increased to 0.72 \(\pm\) 0.05 \(\upmu\)m, maintaining superhydrophobic properties. Therefore, a critical rms roughness value of \(0.69\pm 0.01\)\(\upmu\)m in case of as-prepared solution which is not very different from the rms values of \(0.70\pm 0.02\)\(\upmu\)m in case of diluted solution is required for obtaining superhydrophobic properties, however, the presence of low surface energy fluorinated components is another key factor in obtaining these properties. In our case, the presence of a low surface energy fluorinated component is inherent as the silica nanoparticles are functionalized with FAS-17 molecules prior to deposition on the substrate surfaces.