Lucas-Washburn theory revisited for capillary rise in nanotubes*

In the 1920s, Edward Wight Washburn derived an equation, known as the Lucas-Washburn formula, based on the Hagen-Poiseille (HP) law to describe capillary flow or fluid filling dynamics in circular tubes. However, the HP law, which was incorporated in the Lucas-Washburn theory, only works for flow in infinitely long tubes where end effects are neglected. The entrance effects are important for large-slip flows such as water flow in carbon nanotubes (CNTs). With the recent experimental advances in water filling of isolated CNTs and the fact that CNTs have recently drawn a great deal of attention in a variety of applications, such as water desalination, power generation and biosensing, there is a need for an accurate theory describing flow in CNTs. Here, we revisit the Lucas-Washburn theory to account for the hydrodynamic entrance effects as well as the variation of capillary pressure and key hydrodynamic properties with the radius and length of CNTs. We show that our modified Lucas-Washburn theory is able to accurately describe filling dynamics in CNTs when compared to the data from molecular dynamics simulations. Using our modified theory, slip lengths in CNTs could potentially be estimated from experimental data in isolated CNTs. We believe that our findings are a leap forward in the field of nanofluidics.