Sub-Nanometer Interfacial Hydrodynamics: The Interplay of Interfacial Viscosity and Surface Friction

Abstract

For an accurate description of nanofluidic systems, it is crucial to account for the transport properties of liquids at surfaces on sub-nanometer scales, where classical hydrodynamics fails due to the finite range of surface-liquid interactions and modifications of the local viscosity. We show how to account for both via generalized, position-dependent surface-friction and interfacial viscosity profiles, which enables the accurate description of interfacial flow on the nanoscale using the Stokes equation. Such profiles are extracted from non-equilibrium molecular dynamics simulations of water on polar, non-polar, fluorinated, and unfluorinated alkane and alcohol self-assembled monolayers of widely varying wetting characteristics. Power-law relationships among the Navier friction coefficient, interfacial viscosity excess, and depletion length are revealed, and these are each found to be exponential in the work of adhesion. Our framework forms the basis for describing sub-nanometer fluid flow at interfaces with implications for electrokinetics, biophysics, and nanofluidics.

Figures (3)

• Figure 1: (a) Snapshots of the SAM surfaces studied, along with close-ups of the SAM molecule head groups. The SAMs comprise either decane with the top eight carbons perfluorinated (F-SAM), plain decane (H-SAM), or decanol with the charges of the terminal OH-group varied ($\alpha$-SAMs). (b) Schematic illustrating the NEMD method used. The SAM is restrained at the bottom, while a constant force acts equally on each liquid atom. (c--j) Surface-friction and viscosity profile extraction for the low-friction F-SAM/water system (left column) and high-friction ($\alpha$=1)-SAM/water system (right column). The mass densities of the surface $\rho_{\rm surf}$ and liquid $\rho_{\rm liq}$ are plotted as shaded areas with arbitrary $y$-scaling in all plots. Plots are over $z$, relative to the water Gibbs dividing surface position $z_{\rm GDS}$. (c, d) Simulation snapshots for each respective system. (e, f) Plots of extracted surface--liquid friction force, $f_f(z)$, and liquid velocity, $u(z)$, profiles. The total applied driving stress $F_a$ (see eq \ref{['eq:applied_force_calc']}) is given in each plot. (g, h) Plots of the surface-friction profiles $\ell(z)$, and viscosity profiles $\eta(z)$, which are calculated from $f_f(z)$ and $u(z)$ profiles (see eqs \ref{['eq:ff_tdep_ss']} and \ref{['eq:visc_prof_calc']}, respectively). The force profiles $f_f(z)$ are also plotted schematically as translucent orange curves for comparison to $\ell(z)$. The viscosity profiles are calculated from driven-flow simulations with larger driving stresses: $F_a=11.2$ MPa for the F-SAM, and $F_a=21.6$ MPa for the ($\alpha$=1)-SAM. (i, j) Velocity profiles $u(z)$ extracted from driven-flow simulations with different applied driving stresses (points with error bars), compared with those calculated using $\ell(z)$ and $\eta(z)$ from the row above by solving eq \ref{['eq:stokes_ss_reconst1']} numerically (solid lines).
• Figure 2: Comparison of surface-friction profiles and viscosity profiles of water adsorbed on different SAM surfaces, plotted relative to the water Gibbs dividing surface position $z_{\rm GDS}$. As a positional reference, the mass densities of the surface $\rho_{\rm surf}(z)$ and liquid $\rho_{\rm liq}(z)$ for all systems are plotted in the upper panels as dotted and solid lines, respectively. (a) Water contact angles $\theta$, from droplet simulations for all surfaces. Also shown is a schematic illustrating the contact angle $\theta$, overlaid on a simulation snapshot of a cylindrical water droplet on an F-SAM. The ($\alpha$=1)-SAM was fully wetted by water, so we show it as having a contact angle of $0^\circ$. (b) Surface-friction profiles $\ell(z)$ for all systems, obtained via eq \ref{['eq:ff_tdep_ss']}. The inset shows the same data over a smaller range of $y$-values, so that the profiles for the low-friction surfaces are discernible. (c) Viscosity profiles $\eta(z)$ obtained via eq \ref{['eq:visc_prof_calc']}. The average of each viscosity profile in the bulk domain, $\eta_b$ is shown as a horizontal dotted line; these agree well among systems, with an average of $\overline{\eta_b} = 0.706$ mPa$\,$s, and with the viscosity extracted from an equilibrium bulk-water simulation, $\eta_{\rm eq}=0.698$ mPa$\,$s, which is shown as a solid black line. The viscosity dividing surfaces $z_\eta$ from each viscosity profile are shown as vertical dashed lines.
• Figure 3: Analysis of relationships among surface--liquid friction, viscosity excess, depletion, and wetting for water adsorbed on different SAM surfaces. (a) Schematic illustrating the depletion length, which is the distance between the Gibbs dividing surfaces of the liquid and solid, calculated using the packing density $\phi$ rather than the mass density $\rho$. (b) Viscosity excess distance, $d_\eta$, as extracted in Figure \ref{['fig:allprofs_fig']}(c), as a function of $k+1$, where $k=\cos\theta$ is the wetting coefficient. The data are fit with an exponential function plus a constant $d_\eta^0$, which is subtracted from $d_\eta$ in subsequent plots. Negative $d_\eta$ values indicate a viscosity deficit. (c--e) Quantities plotted over $k+1$, along with exponential fits of the data. The data for the ($\alpha$=1)-SAM is omitted here, as $k$ is undefined for full wetting. (c) The friction coefficient $\lambda$ (in units of Pa$\,$s$\,$m$^{-1}$ throughout the figure), extracted by integrating over each $\ell(z)$ in Figure \ref{['fig:allprofs_fig']}(b). (d) The viscosity excess distance $d_\eta$, less the constant $d_\eta^0$ (see (b)). The $d_\eta-d_\eta^0$ data and fit from (b) are replotted in log-linear to demonstrate that the data fall on a straight line and agree well with the fit. (e) The depletion length $\delta$, as illustrated in (b). (f--h) Relationships among $\lambda$, $d_\eta$, and $\delta$ plotted in log-log, alongside power law fits.