Nonadditivity of van der Waals forces on liquid surfaces

We present an approach for modeling nanoscale wetting and dewetting of textured solid surfaces that exploits recently developed, sophisticated techniques for computing exact long-range dispersive van der Waals (vdW) or (more generally) Casimir forces in arbitrary geometries. We apply these techniques to solve the variational formulation of the Young-Laplace equation and predict the equilibrium shapes of liquid-vacuum interfaces near solid gratings. We show that commonly employed methods of computing vdW interactions based on additive Hamaker or Derjaguin approximations, which neglect important electromagnetic boundary effects, can result in large discrepancies in the shapes and behaviors of liquid surfaces compared to exact methods.

Figure 1

Schematic of fluid-grating geometry comprising a fluid (blue) of surface profile $\mathrm{\Psi }\left(\mathbf{x}\right)$ in close proximity (average distance $d$) to a solid grating (red) of height profile $h\left(\mathbf{x}\right)$, involving thin nanorods of height $H$, thickness $2P$, and period $\mathrm{\Lambda }$. (a) Representative mesh employed by a recently developed FSC boundary-element method [36] for computing exact vdW energies in complex geometries. (b) and (c) illustrate commonly employed pairwise-summation (PWS) and proximity-force approximations (PFA), involving volumetric and surface interactions throughout the bodies, respectively.

Figure 2

Maximum displacement $\mathrm{\Delta }\mathrm{\Psi }/d$ of a fluid-vacuum interface that is repelled from a grating (insets) by a repulsive vdW force, as a function of surface tension $\gamma /{\gamma }_{\mathrm{vdW}}$, obtained via solution of (3) using FSC (blue), PWS (red), and PFA (green) methods. Circles indicate results obtained through (1). Insets show the equilibrium fluid-surface profiles at selected $\gamma \in \{0.006,0.055,0.277\}{\gamma }_{\mathrm{vdW}}$, with the unperturbed $\mathrm{\Psi }=0$ surface denoted by black dashed lines.

Figure 3

Minimum surface-surface separation $\frac{h_{\min }-{\mathrm{\Psi }}_{\max }}{d}$ of a fluid-vacuum interface that is attracted to a grating (insets) by an attractive vdW force, as a function of surface tension $\frac{\gamma }{{\gamma }_{\mathrm{vdW}}}$, obtained via solution of (3) using FSC (blue), PWS (red), and PFA (green) methods. Circles indicate results obtained through (1). Contact transitions occurring at critical values of surface tension ${\gamma }^{\left(\mathrm{c}\right)}$, marked as “x.” The top-right inset shows the equilibrium fluid-surface profiles near ${\gamma }^{\left(\mathrm{c}\right)}$ while the bottom-left inset shows the relative changes in the equilibrium vdW energies from the energies of the unperturbed ($\mathrm{\Psi }=0$) plane-grating geometry (the limit of $\gamma \to \infty$).