## Capillary filling inside subnanometric carbon nanotube (CNT).

Capillary filling in macroscopic channels is a well understood phenomena. However, when considering subnanometric channels, and particularly subnanometric carbon nanotubes, several surprises can be expected. Indeed, the confinement offered to the fluid by this structures is such that the molecular nature of fluids come into play. Moreover, as the dimensions of the channel are reduced, the interaction between the liquid and the surfaces becomes more and more important in comparison with the bulk (liquid-liquid) interaction. Finally, deviations from the classical no-slip boundary condition will impact significantly the dynamics of the fluid at this scale. Each of these phenomena are expected to lead to a failure of the continuum predictions, with the possibility of the appearance of new interesting regimes of capillary filling.

We start by a description of the classical case that considers friction along the tubes' wall and negligible entrance dissipation. Then, we extend this description to the specific case of CNTs, for which most of the dissipation comes from the entrances, due to the large slip length in comparison to the tube radius.

###### The Lucas-Washburn law

Let us consider a cylindrical pore of radius $a$ and length $H$ in contact with a reservoir of liquid (density $\rho$, viscosity $\eta$), see figure 1:

Figure 1: schematic of the system.

The liquid/vapor surface tension is written $\gamma$, and the contact angle of the liquid on the solid surface is noted $\theta$. We call $L$ the length of the liquid column inside the tube. The velocity of the meniscus is related to $L$ through: $v_c=\text{d}L/\text{d}t$. The equation of motion for the liquid inside the pore can be written $$\dfrac{\text d(M v_c)}{\text d t} = F_c + F_v,$$ where $M=\rho \pi a^2 L$ is the mass of liquid inside the pore, $F_c$ the capillary force and $F_v$ the friction force. The capillary force can be written as $$F_c = 2 \pi a \Delta \gamma,$$ where $\Delta \gamma = \gamma_\text{SG} - \gamma_\text{SL}= \gamma \cos \theta$ is the capillary force per unit length at the contact line. Note that this force results from the integration of the pressure force on the meniscus. One can show that it is equivalent to a force per unit length acting at the contact line. $\gamma_\text{SG}$ and $\gamma_\text{SL}$ are respectively the solid-gas and the solid-liquid surface tensions. The friction force is given by the Poiseuille law, modified to take into account liquid/solid slip: $$F_v = - \dfrac{8 \pi \eta L v_c}{1 + 4 b / a},$$ which leads to the following equation of motion $$\dfrac{\text d(\rho \pi a^2 L v_c)}{\text d t} = 2 \pi a \Delta \gamma - \dfrac{8 \pi \eta L v_c}{1 + 4 b / a}. \label{eq:eq_of_motion}$$ The solution of this nonlinear equation can be written as: $$L(t) = L_c \left( \dfrac{t}{\tau_c} + \dfrac{e^{-2 t/ \tau_c} -1 }{2} \right)^{1/2}, \label{eq:solnonlin}$$ with $$\tau_c = \dfrac{\rho a^2}{4 \eta} \left( 1 + \dfrac{4 b}{a} \right),$$ and $$L_c = \dfrac{(2 \gamma \rho a^3)^{1/2}}{4 \eta} \left( 1+ \dfrac{4 b}{a} \right).$$ In the long time limit, where the liquid inertia can be neglected compared to the viscous friction inside the tube, the previous equation simplifies to the Lucas-Washburn law: $$L^2(t) = \dfrac{\Delta \gamma a}{2 \eta} \left( 1 + \dfrac{4 b}{a} \right) t. \label{eq:Washburn}$$ In this regime $L \propto \sqrt{t}$, so $v_c \propto \mathrm{d}L/\mathrm{d}t \propto 1/\sqrt{t}$: the filling slows down with time, which is logical since there is friction between the increasingly long water column and the tube. In the short time limit, the viscous friction can be neglected in comparison to the inertia, and one gets an equation that does not depend on the liquid friction nor the slip length $b$ : $$L(t) = \left( \dfrac{2 \Delta \gamma}{\rho a} \right)^{1/2} t.$$ Note that a pre-inertial regime appears for $L < a/2$, but we will always ignore it here due to the dimension of the considered system (i.e. $a < 3$ nm). The Lucas-Washburn law is a standard equation in capillary filling study, but it is not suitable for describing the dynamics of the present work. Indeed, due to the extremely low friction at the CNT wall, one has to consider the viscous dissipation at the entrance, see references Joly2013 for more details. When the entrance dissipation dominates the overall dissipation, the length of the column does not matter and the filling velocity is constant over the time. The equations of motion of this particular situation are described now.

###### The special case of subnanometric CNTs

Viscous dissipation - When studying transport inside a subnanometric nanochannel, one has to take into account a viscous entrance dissipation. The pressure drop at the entrance of the tube $\Delta p$ is linked to the total flow $Q$ as $$\dfrac{\Delta p}{Q} = R_\text{out} = \dfrac{1}{2} \times \dfrac{C \eta}{a^3},$$ where the factor 1/2 comes from the fact that there is only one entrance in the situation of capillary filling, and $C \sim 3$. The competition between entrance and inner dissipations leads to the appearance of a critical pore length $L_0 = \pi C b / 2$ which separates a regime dominated by entrance dissipation and a regime dominated by inner dissipation, see reference Gravelle2014 for details. Since $b$ is in the range of several tens of nanometers in our case, and since the considered tube length $L$ is equal to 10 nm, we made the assumption that CNTs filled by water are in the regime $L \ll L_0$.

Neglecting the inertia - Hereafter, inertial effects will always be neglected. An argument based on the dissipated power confirms the limited importance of inertial effects, in comparison to viscous ones. Indeed, since most of the dissipation is expected to come from entrances, the power dissipated by viscosity $\mathcal P_v$ can be roughly estimated as : $$\mathcal P_v = \dfrac{\eta}{2} \int_V (\partial_i v_j + \partial_j v_i)^2 \mathrm{d}V \sim \eta a v^2,$$ where $V$ is the volume of the system and where we used that the only length scale is the tube radius $a$. On the other hand, the kinetic power of the liquid can be written as $$\mathcal P_k \sim \rho a^2 v^3.$$ In the present study, we consider nanotubes with radii below 3 nm, and typical fluid velocities are around 10 m/s. One finds $P_k/P_v \sim 0.03$ (using $a=3$ nm, $v=10$ m/s, $\rho$ = 1000 kg/m$^3$ and $\eta = 1$ mPa.s), which indicates that inertial effects can be neglected in comparison to viscous ones.

Capillary filling velocity - After neglecting both inner viscous dissipation and inertial effects, two contributions to the equation of motion are remaining. The first one is the capillary force $F_c = 2 \pi a \Delta \gamma$, which leads to the following pressure jump through the meniscus: $$\Delta p^\text{men} = \dfrac{2 \Delta \gamma}{a}.$$ The second one is the viscous entrance dissipation, which leads to the following pressure jump at the entrance of the nanochannel: $$\Delta p^\text{ent} = \dfrac{Q}{2} \times \dfrac{C \eta}{a^3}.$$ Using that $Q=\pi a^2 v$ and writing $\Delta p^\text{men}=\Delta p^{ent}$, one finds the following expression for the capillary velocity $v_c$: $$v_c = \dfrac{4 \Delta \gamma}{\pi C \eta}. \label{eq:vc}$$ Hence in this particular regime, the filling velocity is the result of a competition between surface tension and entrance dissipation, and is constant along time.

Disjoining pressure correction - Structuring effects can be described in terms of a disjoining pressure $\Pi_d$ that can be added to the continuum Laplace pressure $\Delta p^\text{Laplace}$. The total pressure drop across the meniscus, previously defined as $\Delta p^\text{men} = p_0 - p_L$, writes now: $$\Delta p^\text{men} = \Delta p^\text{Laplace}+ \Pi_d . \label{eq:pl}$$ $\Pi_d$ is a function of the ratio between the fluid molecule diameter $\sigma$ and the tube radius $a$. While the expression of $\Pi_d$ as a function of the radius $a$ is not trivial for a cylinder, $\Pi_d$ can be expressed analytically for a fluid of hard spheres confined in a slit (2D) pore as: $$\Pi_d^\text{2D} (h) = - \rho_\infty k_B T \cos \left( 2 \pi h / \sigma \right) e^{- h / \sigma} , \label{eq:disj_pres}$$ with $\sigma$ the fluid particle diameter, $\rho_\infty$ the fluid bulk density, and $h$ the distance between the walls. More details in Gravelle2016.

Coming soon

###### References

Joly2011: Laurent Joly. Capillary filling with giant liquid / solid slip : Dynamics of water uptake by carbon nanotubes J. Chem. Phys., 214705 (2011).

Gravelle2013 : Simon Gravelle, Laurent Joly, François Detcheverry, Christophe Ybert, Cécile Cottin-Bizonne and Lydéric Bocquet. Optimizing water permeability through the hourglass shape of aquaporins PNAS, 110 (41), 16367–72 (2013)

Gravelle2014 : Simon Gravelle, Laurent Joly, Christophe Ybert and Lydéric Bocquet. Large permeabilities of hourglass nanopores: from hydrodynamics to single file transport J. Chem. Phys. 141 (2014)

Gravelle2016 : Simon Gravelle, Christophe Ybert, Lydéric Bocquet and Laurent Joly,. Anomalous capillary filling and wettability reversal in nanochannels PRE 93 (2016)