Nanoroughness, Intrinsic Density Profile, and Rigidity of the Air-Water Interface

From large-scale molecular dynamics simulations of the free water surface the capillary wave spectrum is obtained and analyzed in terms of a wave vector dependent surface tension. The bending rigidity is positive but suggested to change sign in the presence of long-range dispersive interactions. Nanoroughness and capillary waves give rise to a scale-dependent density profile that displays minimal broadening at a nanometer scale which defines the intrinsic density profile.

Figure 1

(a) Simulation snapshot for lateral box size $L=3.0\mathrm{nm}$, containing 1807 water molecules. The enlarged slice to the right exhibits pronounced surface roughness. (b) Average density profile (circles, $L=3.0\mathrm{nm}$) compared with a hyperbolic tangent fit (solid line). Vertical dashed lines show the positions of the global GDS. (c) Density profiles centered around the GDS for varying $L$; lines are guides to the eye.

Figure 2

Intrinsic density profiles ${\rho }_{\mathrm{int},n}(z)$ for $\alpha =3$ and block factor $n=30$ (circles) and $n=1$ (squares, corresponding to the global laterally averaged density profile) with fits ${\rho }_{\mathrm{int},n}(z)=({\rho }_l+{\rho }_v)/2+(\Delta \rho /2)\tanh (2z/w)$ (solid lines). The inset shows the width $w$ as a function of $n$ for $\alpha =2$, 3, 4. Here $L=24\mathrm{nm}$.

Figure 3

(a) Capillary wave spectra $⟨|{\tilde{h}}_n(q){|}^2⟩$ for $n=1024$ and $\alpha$ ranging from 1 to 5 (from bottom to top, full lines). For comparison $S_{\parallel }^b(\overrightarrow{q})$, the lateral bulk structure factor for a slab thickness $5w$, scaled by an arbitrary factor, (dashed line) and the CWT result with $\gamma ={\gamma }_p$ (dash-dotted line) are included. (b) Bare capillary wave spectra $(\sigma q{)}^2⟨|{\tilde{h}}_n(q){|}^2⟩$ (upper data set, left scale) and $(\sigma q{)}^2⟨|h_n(q){|}^2⟩$ according to Eq. (5) (lower data set, right scale) for $n=64$ and varying $\alpha$. The dashed straight lines show the surface tension contribution determined independently. (c) Rescaled effective surface tension ${\gamma }_{\mathrm{eff},n}/{\gamma }_p$ for $\alpha =3$ and varying $n$. Full squares for $n=30$ correspond to a LJ cutoff $r_c=1.6\mathrm{nm}$, all other data are for $r_c=0.8\mathrm{nm}$. Dashed lines are fits of the form ${\gamma }_{\mathrm{eff},n}/{\gamma }_p=1+{\kappa }_n/{\gamma }_p{q}^2$. In all plots $L=24\mathrm{nm}$ and $q$ is rescaled by the water LJ diameter $\sigma =3.166\AA$.

Figure 4

Bending rigidity ${\kappa }_n$ from fits in Fig. 3c as a function of the inverse squared block factor $1/n^2$ for different $\alpha$. The line denotes a fit of Eq. (6) to the data for $\alpha =2–5$ yielding a value of $\kappa =0.04k_{B}T$ as $n\to \infty$. The inset shows the temperature dependence of $\kappa$ (circles), the simulated surface tension (squares, including a tail correction) and the experimental surface tension (triangles, [16]).