Complete Wetting of Nanosculptured Substrates

Complete wetting of geometrically structured substrates by one-component fluids with long-ranged interactions is studied theoretically. We consider periodic arrays of rectangular or parabolic grooves and lattices of cylindrical or parabolic pits. We show that the midpoint interfacial heights within grooves and pits are related in the same way as for complete wedge and cone filling. For sufficiently deep cavities with vertical walls and small undersaturation, an effective planar scaling regime emerges. The scaling exponent is $-1/3$ in all cases studied, and only the amplitudes depend on the geometrical features. We find quantitative agreement with recent experimental data for such systems.

Figure 1

Top view of the substrate geometries. (a) Quadratic lattice of identical pits (cylinders or paraboloids). (b) Periodic array of grooves (rectangular or parabolic). All cavities have finite depths $D$.

Figure 2

Typical interfacial heights $l_{p,g}^{(0)}$ (measured from the plane $z=0$) at the middle of the cylindrical and parabolic pits, $l_p^{(0)}$ (dashed lines), and of the rectangular and parabolic grooves, $l_g^{(0)}$ (solid lines), as a function of $\Delta \mu$ with $C_p=1$ and $C_g=2$. Rescaling the variables according to Eq. (1) leads to data collapse. The inset shows the same data unscaled. The depth of the cavities is $D=500$; $pp(g)$ denotes parabolic pits (grooves). All lengths are measured in units of $\sigma$.

Figure 3

Wetting film thickness $l_g$ above grooves for small $\Delta \mu$. The symbols represent the midpoint interfacial height $l_g^{(0)}$. The solid lines are obtained by minimizing Eq. (3) assuming $l(\mathbf{x})=l_g$. On the present scales they are indistinguishable from those for film thicknesses $l_{\pi }^e(\Delta \mu )$ on layered and flat ersatz substrates, with a top layer of height $D$ and an effective Hamaker constant $A^e=A{\Phi }_g$, where $A$ is the Hamaker constant of the solid without grooves. In the inset the vertical axis is rescaled according to Eq. (6) leading to data collapse within an intermediate regime, the width of which increases as $\Delta {\mu }_{\pi }^e-\mathrm{const}\times D^{-3}$ for $D\to \infty$. The groove depth is $D=500$.

Figure 4

Liquid adsorption at parabolic pits. Inset: ${\Gamma }_t(\Delta T)$. Red line—theory; symbols and black lines—experimental data from Ref. 4 (where ${\Gamma }_p$ is denoted as ${\Gamma }_c$). $\Delta T$ is proportional to $\Delta \mu$ (see main text).