What x rays can tell us about the interfacial profile of water near hydrophobic surfaces

The free surface of water and the interface between water and a hydrophobic surface both have positive interface energies. The water density near a free surface drops below the bulk density, and thus it is expected that water near a hydrophobic surface will also show a density depletion. However, efforts by multiple groups to detect and characterize the predicted gap at water-hydrophobic interfaces have produced contradictory results. We have studied the interface between water and fluoroalkylsilane self-assembled monolayers using specular x-ray reflectivity and analyzed the parameter-space landscapes of the merit functions being minimized by data fitting. This analysis yields a better understanding of confidence intervals than the customary process of reporting a unique best fit. We conclude that there are unambiguous gaps at water-hydrophobic interfaces when the hydrophobic monolayer is more densely packed.

Figure 1

Schematic diagram of a fluoroalkylsilane molecule and a representation of the slab model of its density. The semi-infinite SiO${}_2$ is at far left, and the semi-infinite water layer is at far right.

Figure 2

(a) Reflectivity data (open circles), best fit (solid line), and density profile (inset) for the dry FAS25 monolayer. The dashed line in the inset shows the density profile if the interfaces were not rounded, i.e., it shows the slabs more clearly (cf. Fig. 1). (b) Merit function landscape in the $D_c$-${\rho }_c$ plane (with $D_h$ $=$ 0 for simplicity). The merit function has been scaled and its minimum value subtracted in order to more clearly display the contours.

Figure 3

Reflectivity data (open circles) and best fits (solid lines), for SAMs of FAS13, FAS17, FAS21, and FAS25 in contact with water. The best-fit parameters are given in Table .

Figure 4

Fitting of reflectivity data from the water-FAS25 SAM interface (the experimental data are in Fig. 3). (a) Best-fit density profile (dashed line shows the slabs without interface rounding); (b) merit function landscape in the $D_{\mathrm{gap}}$-${\rho }_{\mathrm{gap}}$ plane when all other parameters are fixed at the best-fit values; (c) merit function landscape when all other parameters are allowed to float. Note that in (c) the axis extends to negative values, indicating the possibility of an “inverse gap” with density larger than water. The merit function ${\gamma }^2$ [Eq. (3)] is shown with its best value ${\gamma }_{\min }$${}^2$ subtracted and then scaled by a factor of 1000. The visible color range is from ${\gamma }^2$ $=$ ${\gamma }_{\min }$${}^2$ (dark) to γ${}^2$ $=$ 1.50${\gamma }_{\min }$${}^2$ (light).

Figure 5

Merit function landscapes in the $D_{\mathrm{gap}}$-${\rho }_{\mathrm{gap}}$ plane for fitting of reflectivity data from water-FAS13/FAS17/FAS21 interfaces (the experimental data are in Fig. 3). All other parameters were allowed to float. Note that the $y$ axes extend to negative values, allowing for “inverted gaps” with density larger than water. The merit function ${\gamma }^2$ [Eq. (3)] is shown with its best value ${\gamma }_{\min }$${}^2$ subtracted and then scaled by a factor of 1000. The visible color range is from ${\gamma }^2$ $=$ γ${}_{\min }$${}^2$ (dark) to ${\gamma }^2$ $=$ 1.50${\gamma }_{\min }$${}^2$ (light).

Figure 6

Effective gap width $D_{\mathrm{eq}}$ [Eq. (4)] vs SAM chain density ${\rho }_c$ (from dry SAM fits) for the four SAMs studied. The points show the $D_{\mathrm{eq}}$ values for the best possible fits (as discussed in the text, there are many almost degenerate fits). The error bars show the $D_{\mathrm{eq}}$ range corresponding to the merit function ranging from its minimum value to 50$\%$ above that value. It can be seen that the two higher density, longer chain SAMs have $D_{\mathrm{eq}}>0$ (positive definite gap)${}_{.}$