Sound waves, diffusive transport, and wall slip in nanoconfined compressible fluids

Although continuum theories have been proven quite robust to describe confined fluid flow at molecular length scales, molecular dynamics (MD) simulations reveal mechanistic insights into the interfacial dissipation processes. Most MD simulations of confined fluids have used setups in which the lateral box size is not much larger than the gap height, thus breaking thin-film assumptions usually employed in continuum simulations. Here we explicitly probe the long-wavelength hydrodynamic correlations in confined simple fluids with MD and compare to gap-averaged continuum theories as typically applied in, e.g., lubrication. Relaxation times obtained from equilibrium fluctuations interpolate between the theoretical limits from bulk hydrodynamics and continuum formulations with increasing wavelength. We show how to exploit this characteristic transition to measure viscosity and slip length in confined systems simultaneously from equilibrium MD simulations. Moreover, the gap-averaged theory describes a geometry-induced dispersion relation that leads to overdamped sound relaxation at large wavelengths, which is confirmed by our MD simulations. Our results add to the understanding of transport processes under strong confinement and might be of technological relevance for the design of nanofluidic devices.

Figure 1

Wall slip leads to an effective gap height $h_{\mathrm{eff}}=h\sqrt{\kappa }$. The parameter $\kappa$ is found by equating the $z$ average of the original velocity profile $u\left(z\right)$ with that of $u^{*}\left(z\right)$, which is $u$ shifted to yield zero slip.

Figure 2

Dispersion relation for a height-averaged continuum formulation of confined fluids. The solid line describes the angular frequency using the magnitude of the phase velocity, given by Eq. (20). The dashed line describes the bulk reference. The wave number is normalized by $k_{\mathrm{crit}}=6\nu /h^2\kappa c_{\mathrm{T}}$, and the angular frequency is normalized by the characteristic relaxation time of longitudinal modes in the underdamped limit ${lim}_{k\to k_{\mathrm{crit}}}{\tau }_{\parallel }=h^2\kappa /6\nu$. The critical transition from underdamped to overdamped dynamics occurs with diverging group velocity.

Figure 3

Fourier coefficients for density and longitudinal momentum in the underdamped and overdamped case, respectively.

Figure 4

Simulation setup for molecular dynamics simulations of confined fluids. One lateral dimension is much larger than the gap height.

Figure 5

Autocorrelation functions of the real part of density and momentum modes. The first column (a–d) corresponds to transverse momentum ($\vec{j}\perp \vec{k}$) with fits to Eq. (25a), the second column (e)–(h) corresponds to longitudinal momentum ($\vec{j}\parallel \vec{k}$) with fits to Eq. (25b), and the third column (i)–(l) corresponds to density correlations with fits to Eq. (25c). Panels within the same row share the same wavelength. Solid lines show results from MD calculations, and dashed lines are the corresponding fits. For simulations of the confined system, all wavelengths shown here are in the underdamped regime ($470.6\sigma <{\lambda }_{\mathrm{crit}}$).

Figure 6

Relaxation times obtained from fits to the autocorrelation functions of momentum and density fluctuation as a function of the wavelength for bulk and confined systems with $\alpha =0.75$. For all three subfigures, simulation results for the bulk system are shown as blue disks, and results for the confined system are shown as green diamonds. Error bars signify the error corresponding to two standard deviations of the fit parameters and are smaller than the symbol size in most cases. Panels (a) and (b) show the theoretical prediction for bulk and confined systems $[{\tau }_{\perp }\left(k\right)$, Eq. (24a), and ${\tau }_{\parallel }\left(k\right)$, Eq. (24b)] as dash-dotted and dashed lines, respectively. In panel (b) the relaxation times according to the isothermal theory [Eq. (24b)] are modified for short wavelengths by considering thermal effects, which is illustrated as a dotted line. Panel (c) shows thermal relaxation times obtained from density autocorrelation functions, which have quadratic wavelength scaling for both bulk and confined systems, as long as walls are rigid. Open diamonds in panel (c) show results from simulations with thermal walls, which indicate a similar transition as in (a) and (b).

Figure 7

Effective sound period and velocity of sound as obtained from longitudinal momentum correlations for bulk and confined systems from simulations with $\alpha =1.5$. Error bars signify the error corresponding to two standard deviations of the fit parameters and are smaller than the symbol size in most cases. In panel (a) we observe an overall shift to longer sound periods for the confined fluids, which increases with wavelength, and panel (b) compares the sound velocities directly. For the confined system, the prediction based on the dispersion relation with $s_{\mathrm{T}}\left(k\right)$ [Eq. (20)] adequately describes the deviation from the bulk. The critical wavelength marks the transition from underdamped to overdamped behavior at ${\lambda }_{\mathrm{crit}}=675\sigma .$

Figure 8

Transition from underdamped to overdamped dynamics. The last four sound relaxation times in the underdamped regime are shown as green diamonds. After reaching the critical wavelength, we obtain two real eigenvalues from Eq. (19). One leads to a converging relaxation time for momentum perturbations (lower branch of the dashed line), the other one to a diverging relaxation time for density perturbations (upper branch of the dashed line). For $\alpha =1.5$ and box lengths of $941.2\sigma$ and $1411.8\sigma$, we characterize three modes in the overdamped regime with our MD calculations. Error bars are smaller than the symbols size. The fits to the simplified (exponential) autocorrelation functions are shown in the inset, and the resulting relaxation times of density and momentum modes are shown as upward and downward pointing triangles, respectively.

Figure 9

Time evolution of a small, Gaussian-shaped density perturbation in a slit geometry. The gap height in panel (a) is large enough to accommodate sound waves. Panel (b) shows the propagated distance $\mathrm{\Delta }x$ over time $t$, rescaled by the critical wave number $k_{\mathrm{crit}}$ and the long-wavelength sound attenuation time $\tau ={lim}_{k\to k_{\mathrm{crit}}}{\tau }_{\parallel }$, respectively. Therefore, we recover the sound speed $c_{\mathrm{T}}=1/\tau k_{\mathrm{crit}}$ from the slope of the curve. In panel (c) the gap height is one order of magnitude smaller, which leads to a purely diffusive behavior. Panel (d) shows the variance ${\sigma }^2$ over time, rescaled by the squared critical wave number and sound attenuation time, respectively. Hence, the slope is proportional to the corresponding diffusion coefficient $D_{\rho }=h^2{c}_{\mathrm{T}}^2/12\nu$. Only the r.h.s. of the symmetric profiles is shown in (a) and (b).

Figure 10

Shear attenuation time for varying wall-fluid interaction and comparison to the bulk (a). The dash-dotted blue line illustrates a fit to the bulk relaxation time, and dashed lines show the wavelength-dependent shear relaxation time ${\tau }_{\perp }\left(k\right)$ [Eq. (24a)] fitted to the data from confined systems. Error bars in (a) are always smaller than the symbols size. The long-wavelength limit of the relaxation time determines the effective gap height, and since the geometric gap height is known, we can directly compute the slip length $b$ using the definition of $\kappa$. Results of this fit procedure are shown in (b) for varying fluid wall interactions $\alpha$ and are compared to nonequilibrium calculations to obtain the slip length. The dashed lines in (b) are a guide to the eye.

Figure 11

Same as Fig. 3, but with the effective long-time expression for $\tilde{\rho }(k,t)$. The expansion shown in Eq. (C5) is around $t_0=2/a$, and except for the critically damped mode, we obtain a good agreement with the original form for large $t$.