Irregular, nanostructured superhydrophobic surfaces: Local wetting and slippage monitored by fluorescence correlation spectroscopy

Superhydrophobic surfaces used in various applications to enhance flow and reduce drag typically consist of irregular nanostructures. However, many theoretical models and most laboratory microscale experiments dealing with these phenomena are limited to structures consisting of regular microarrays and cannot explain the macroscopic flow enhancement observed in applications. Here, we investigated microscopically the wetting and flow over fluorinated silicon nanofilaments as an example for an application-relevant, irregularly nanostructured, superhydrophobic surface. Using fluorescence correlation spectroscopy with an improved evaluation method, we found that velocity profiles are still nonlinear at distances below 1 $\mu \text{m}$ to the surface. Furthermore, we observed that the air layer in between and on the nanofilaments is not continuous on a micrometer length scale. First, there are regions with homogeneous wetting, where the air-water interface regularly touches all uppermost fibers. These regions possess a low slip length ($<5\mu \mathrm{m}$). Both the wetting and the slip length match with expectations from microarray or homogeneous porous surfaces. Second, there are large patches with air inclusions, which present two orders of magnitude higher slip lengths. Our results contribute to the understanding of the drag reduction observed in applications and can help in designing new, optimized surfaces.

Figure 1

(a) Scanning electron microscopy image of the nanofilament coating. (b) Schematic of the flow setup.

Figure 2

(a) Schematic of velocity variations due to changing boundary conditions. (b) Schematic of the detection volume. (c) Definition of the parameters for Eq. (2). (d) Comparison of autocorrelation curves calculated by the full numerical solution of the autocorrelation function for a partly immersed detection volume and their fits with Eq. (2). (e) Comparison of velocity profiles recorded with FCS close to a bare glass surface and evaluated with the theory for a fully immersed volume (open circles) and with Eq. (2) (filled circles).

Figure 3

Confocal imaging of the superhydrophobic coating (reflection channel). The water-air interface appears as a bright line. (a) Cut planes from a 3D image showing a very large air inclusion; (b) exemplary cut sections with typical sizes of air inclusions; (c) interruption in the reflection signal of the air-water interface; (d) analysis of the wetting areas (image width: $225\mu \mathrm{m}$).

Figure 4

Velocity profiles perpendicular to the interface (red circles) obtained by fitting the corresponding experimental FCS autocorrelation curves with Eqs. (2) and (3): (a) on the homogeneous porous region and (b) at the center of an air inclusion of about $22\times 28\mu \mathrm{m}$. Average fluorescence intensity curves (black, open circles) are fitted with a cumulative distribution function of a Gaussian distribution to detect the position of the interface.

Figure 5

(a) Sketch of the model of an air inclusion. (b) Exemplary velocity profile at the center of an air inclusion (numerically $D=50\mu \mathrm{m}$, experimentally $43\times 47\mu \mathrm{m})$. In the calculations, the pressure gradient was fitted to reproduce the slope of the velocity profile far from the surface

Figure 6

Schematic of the detection volume. (a) Fluorescent signal can originate only from the part immersed in water. (b) For mathematical convenience, this part can be divided in two segments ($V_1$ and $V_2$) separated by the centerline C.

Figure 7

Comparison between full numerical solution and approximate analytical solution. $L$ is -1, $-0.8$, $-0.6$, -0.4, $-0.2$, 0, 0.2, 0.6, 1 ($\mu \mathrm{m}$), spanning the range from not immersed to fully immersed. The half height of the detection volume is $z_0=1\mu \mathrm{m}$.

Figure 8

Motion of particle in detection volume. One particle enters detection volume at $z$ with an initial velocity $V\left(z_1\right)$. $z$ is irrelevant to $V\left(z_1\right)$. During the motion process in the detection volume, the particle does not change its velocity.

Figure 9

(a) Position of the detection volume relative to the interface (detection volume is defined to be a normalized Gaussian function); (b) detected fluorescence intensity described by a CDF of Gaussian function. The $xyz$ coordinate system corresponds to the same coordinate that were used for the derivation of the autocorrelation function. The $x^{\prime }{y}^{\prime }{z}^{\prime }$ coordinate system is located at the interface as shown in the figure. The $x^{\prime }{y}^{\prime }{z}^{\prime }$ coordinates are shifted until the $x^{\prime \prime }{y}^{\prime \prime }{z}^{\prime \prime }$ coordinates are obtained, which are the reference coordinates for the experimental values.

Figure 10

Example for a small inclusion: (a) classical method, values $<1\mu \mathrm{m}$ to the surface are discarded, evaluated slip length $b=7.5\mu \mathrm{m}$; (b) with method for partial immersion, evaluated slip length $b=12\mu \mathrm{m}$.

Figure 11

Example for a medium size inclusion: (a) classical method, values $<1\mu \mathrm{m}$ to the surface are discarded, evaluated slip length $b=14.8\mu \mathrm{m}$; (b) with method for partial immersion, evaluated slip length $b=37\mu \mathrm{m}$.

Figure 12

Top view of the surfaces that define the limiting cases for the calculation of the slip length of the homogeneously wetted areas of the nanofilaments (not to scale).