Modified Lucas-Washburn theory for fluid filling in nanotubes

Ultrafast water transport in carbon nanotubes (CNTs) has drawn a great deal of attention in a number of applications, such as water desalination, power generation, and biomolecule detection. With the recent experimental advances in water filling of isolated CNTs, the Lucas-Washburn theory for capillary rise in tubes needs to be revisited for a better understanding of the physics and dynamics of water filling in CNTs. Here, the Lucas-Washburn theory is corrected for the hydrodynamic entrance effects as well as the variation of capillary pressure and hydrodynamic properties with the radius and length of CNTs. Due to the large slippage in CNTs, inclusion of the entrance effects is important particularly for the initial stages of filling where a $L\propto t$ scaling, as opposed to $L^2\propto t$, is observed in our molecular dynamics (MD) simulations. The corrected Lucas-Washburn theory is shown to predict the water filling dynamics in CNTs as observed in MD simulations. With the corrected theory, we achieve a better understanding of capillary rise and water filling dynamics in CNTs.

Figure 1

(a) Schematic of capillary filling of a CNT is shown. Radius (R), water contact angle (θ), and filling length (L) of the CNT are illustrated. $z$ is the axial coordinate along the CNT. The CNT and graphene walls are shown in cyan and water is shown in light blue. (b) A typical simulation system for a (26,26) CNT (R = 1.76 nm) is shown for the simulation time of $\sim 1\mathrm{ns}$. The carbon, oxygen, and hydrogen atoms are presented in cyan, red, and white, respectively. The simulation box is truncated for a better presentation.

Figure 2

(a) Cosines of water contact angles for the different CNTs considered in this study. The cosine of contact angle increases with increasing radius. It is expected to reach the cosine of contact angle on a flat graphene for very large CNT radii. (b) Water density distribution (normalized by the bulk water density) in the radial direction (normalized by the CNT radius) for the different CNTs considered in this study. (c) Axial velocity of water molecules as a function of the normalized radial distance from the center of CNTs. The water velocity is calculated for the CNT filling in the time interval of 1900–2100 ps. For both density and velocity calculations, we consider the water molecules that are 1 nm away from the pore entrance and the meniscus surface.

Figure 3

The length of the CNTs filled with water as a function of time from all-atom MD (AAMD) simulations is compared with the filling lengths predicted by the corrected Washburn (CLW) theory and the original Washburn (LW) theory for (a) (10,10), (b) (14,14), (c) (18,18), (d) (26,26), and (e) (34,34) CNTs. The corrected Washburn theory is able to describe the filling dynamics well compared to that of the AAMD simulations. The original Washburn theory, however, fails as it does not account for the hydrodynamic entrance effects, variations of slip length and viscosity with CNT length and radius, as well as the corrected capillary pressure in nanometer-diameter CNTs. In the original LW theory, viscosity is taken as 0.85 mPa s which is the bulk water viscosity. The slip length in the original LW theory is chosen such that we obtain the best possible match with the data from MD simulations. We note that slip lengths based on infinitely long CNTs lead to a massive overestimation of filling rates by the original LW theory (see Fig. S8a of the Supplemental Material).

Figure 4

Slip lengths as a function of radius of CNTs with different tube lengths predicted by different theories [corrected Hagen-Poiseuille theory (CHP, blue curve), corrected Lucas-Washburn theory (CLW, filled black symbols), and original Lucas-Washburn theory (LW, hollow black symbols)]. The CHP model, which is based on MD simulations, is used as the reference here. The LW theory fails to predict the slip lengths for short and small-radius CNTs. As both length and radius increase, the LW and CLW theories result in identical slip lengths since the effects of hydrodynamic entrance, viscosity variation and nanocapillary pressure vanish. Experimental [5, 6, 47] and computational [7, 35, 48, 49, 50, 51] data (shown by red and green symbols, respectively) show how wide the range of slip lengths are in the literature. In other notable experiments by Majumder et al. [5] and Du et al. [52] (not shown in the figure), very high slip lengths of 39 000–68 000 nm and 485 000 nm are reported, respectively.