Full characterization of the hydrodynamic boundary condition at the atomic scale using an oscillating channel: Identification of the viscoelastic interfacial friction and the hydrodynamic boundary position

Flows in nanofluidic systems are controlled by the hydrodynamic boundary condition (BC), involving the friction coefficient and the hydrodynamic wall position. Here we considered a liquid nanoslab confined between two walls, where we derived, from the Stokes equation and the Navier slip BC, analytical expressions for the liquid response to an oscillatory tangential motion of the walls in terms of the wall shear stress and mean fluid velocity. By fitting these expressions to molecular dynamics simulation results, we could extract both the viscoelastic friction coefficient and hydrodynamic wall position for walls with three different wettabilities, hence fully characterizing the frequency-dependent hydrodynamic boundary condition. The proposed method could be applied to a variety of liquid-solid interfaces of interest, e.g., for flows of complex fluids or fluids at a low temperature. It should also support methodological developments on the characterization of the hydrodynamic slip in general.

Figure 1

Schematic of the considered system. The positions of the liquid molecules (light blue) and the solid atoms that compose the walls (orange) were taken from a snapshot of an MD simulation result described in Sec. 3. The domain boundaries normal to the $x$ and $y$ axes are connected with the periodic BC and therefore the effective liquid velocity is only a function of $z$ and $t$.

Figure 2

Left: Distribution of the local liquid density ${\rho }^{\prime }$ near the three walls investigated in the present study. The origin of the coordinate $z^{\prime }$ is on the innermost (001) plane of the left fcc crystal depicted in Fig. 1. Right: The parameters for the liquid-solid interaction potential and the channel height for each wall.

Figure 3

Response functions for walls A, B, and C. Symbols show results obtained by MD simulations (left, $-{\tilde{\tau }}_w/{\tilde{U}}_w$; right, $\tilde{\overline{u}}/{\tilde{U}}_w$) and black solid and dotted lines show the fitting curves to the MD results (left, ${\tilde{Z}}_{\tau }$; right, ${\tilde{Z}}_{\overline{u}}$). The vertical gray lines indicate the boundaries between frequency regimes classified based on the comparison of the penetration length $2\sqrt{\pi \eta /\left(\rho \omega \right)}$ to the characteristic lengths in the system. Predictions assuming the no-slip boundary condition are shown with dark gray lines for comparison. The error bars, which are smaller than the symbols, show the 95% confidence interval estimated from seven independent simulation runs. One simulation run was made of 4 oscillation cycles for the lowest frequency and $3\times {10}^5$ cycles for the highest frequency.

Figure 4

Velocity and temperature distributions in the channel (wall C) for three wall oscillation frequencies $\omega$ that belong to regimes I, II, and III, respectively. The velocity distributions (symbols) and the wall velocities (dotted lines) at four phases ($\theta =\pi /2, 9\pi /10, 11\pi /10$, and $3\pi /2$) are shown. The temperature was almost uniform in the channel and well controlled around the target temperature. The data were averaged over 100, 600, and 30000 oscillation cycles for $\omega =0.0090$, 0.054, and 2.7, respectively, which was the same condition to obtain the corresponding data plots in Fig. 3. Most of the error bars are smaller than the symbols.

Figure 5

Comparison of the velocity profiles given by the Stokes equation with the Navier BC (solid lines) and by the MD simulations (symbols). The origin of the $z$ axis is on the centerline of the channel and the dotted lines show the position of the hydrodynamic boundary for each wall. The error bars are smaller than the symbols.

Figure 6

The velocity profile of pressure-driven flows confined between walls A (left panel) and C (right panel), given by the Stokes equation and the Navier BC (solid lines) and by the MD simulation (symbols). The configurations are the same as in Fig. 5. The error bars are smaller than the symbols.

Figure 7

Dependence of the fit parameters in Eq. (4), the zero-frequency component of the FC ${\lambda }_0$, and the hydrodynamic system height shown in terms of $\mathrm{\Delta }$, on the frequency range below 0.54 to construct the fit. The table shows the fit parameters when the number of data points $n_{\omega }=20$, which corresponds to the frequency range $0<\omega \le 0.54$.