Dynamics of capillary absorption of droplets by carbon nanotubes

We consider the capillary absorption of liquid metal droplets by carbon nanotubes using molecular dynamics simulations and the steady-state flow model due to Marmur [A. Marmur, J. Colloid Interface Sci. 122, 209 (1988)]. We find an exact solution to Marmur’s evolution equation for the height of the absorbed liquid column as a function of time, and show that this reproduces the dynamics observed in the simulations well. The simulations show that the flow of the metal exhibits a large degree of slippage at the tube walls, with slip lengths of up to $10\mathrm{nm}$ depending on the wettability of the nanotube. The results support the use of the Lucas-Washburn approach for modeling capillary absorption at the nanoscale.

Figure 1

Geometrical representation of the capillary tube and the penetrating droplet in cylindrical coordinates $(\widehat{r},\widehat{\theta },\widehat{z})$. There are four parameters ($r$, $r_t$, $h$, and ${\theta }_c$) describing the geometry, but only three of them are independent due to the constant volume constraint. Velocity field of the encapsulated fluid has a parabolic profile with slip length $b$. $P_L$ and $P_C$ represent the Laplace pressure and the pressure difference across the meniscus, respectively.

Figure 2

(Color online) Plots of $H\left(t\right)$ (top panel) and $R\left(t\right)$ (bottom panel) comparing our exact solution with Marmur’s approximation and MD simulations for three wetting angles. The time scale in the analytic solutions has been fitted to the $R\left(t\right)$ data. The exact solution is in much better agreement with MD simulations, particularly in the final stages of uptake. The arrow in each panel points to the relevant axis.

Figure 3

(Color online) (Top) A snapshot of the simulated system, showing capillary absorption of the palladium nanodroplet by a fixed CNT. The encapsulated liquid column height $h\left(t\right)$ was measured from the entrance end of the CNT to the contact line of the meniscus. The scaling distance $r_t$ was measured by finding the maximum radius of the encapsulated liquid column. (Bottom) Pd nanodroplet supported on a flat graphene sheet.

Figure 4

Averaged velocity (top) and density (bottom) profiles of the encapsulated liquid metal column during the uptake. The density profiles (in arbitrary units) indicate ordering of the fluid near the walls of the CNT. This corresponds to the well-defined concentric radial layers of palladium atoms, and the separation between these layers is around $2.2\mathrm{\AA }$. The velocity profile was calculated by averaging the atomic velocities in each radial layer, and then fitted with a quadratic expression of the form $A{x}^2+B$.

Figure 5

(Color online) Plots of $H\left(t\right)$ for the model and MD data, where the column height predicted by the model has been scaled to match the observed density.