Filling of charged cylindrical capillaries

We provide an analytical model to describe the filling dynamics of horizontal cylindrical capillaries having charged walls. The presence of surface charge leads to two distinct effects: It leads to a retarding electrical force on the liquid column and also causes a reduced viscous drag force because of decreased velocity gradients at the wall. Both these effects essentially stem from the spontaneous formation of an electric double layer (EDL) and the resulting streaming potential caused by the net capillary-flow-driven advection of ionic species within the EDL. Our results demonstrate that filling of charged capillaries also exhibits the well-known linear and Washburn regimes witnessed for uncharged capillaries, although the filling rate is always lower than that of the uncharged capillary. We attribute this to a competitive success of the lowering of the driving forces (because of electroviscous effects), in comparison to the effect of weaker drag forces. We further reveal that the time at which the transition between the linear and the Washburn regime occurs may become significantly altered with the introduction of surface charges, thereby altering the resultant capillary dynamics in a rather intricate manner.

Figure 1

Schematic of an electrolytic solution filling a charged capillary. The schematic also shows the different forces on the filling liquid column. The circular boundary of the half-shaded region is the three-phase contact line.

Figure 2

(a) Variation of the magnitude of the dimensionless streaming potential $|E_S/E_0|$ with dimensionless EDL thickness $\overline{\lambda }$ with different values of the ionic Peclet number $S$. In the inset of the figure, we show the variation of the magnitude of the dimensionless charge $|q_p/q_0|$ with $\overline{\lambda }$. $|q_p/q_0|$ is independent of $S$. (b) Variation of the dimensionless ratio $A^{\prime }=q_p{E}_S/\pi \gamma R$ with $\overline{\lambda }$ for different values of $S$. For both plots, except for the case where $|{\overline{\psi }}_0|=ez|{\psi }_0|/k_{B}T$ is specified, we always take $|{\overline{\psi }}_0|=1$. For the case where $|{\overline{\psi }}_0|=4$, we obtain the results numerically.

Figure 3

Variation of the dimensionless ratio $K_2^{\prime }/K_2$ with the dimensionless EDL thickness $\overline{\lambda }$ for different values of the dimensionless ionic Peclet number $S$. The ratio $K_2^{\prime }/K_2$ is equal to the ratio $\eta /{\eta }_{\text{app}}$, where ${\eta }_{\text{app}}$ is the apparent viscosity characterizing the electroviscous effect. Except for the case where $\left|{\overline{\psi }}_0\right|=ez\left|{\psi }_0\right|/k_{B}T$ is specified, we always take $|{\overline{\psi }}_0|=1$. For the case where $\left|{\overline{\psi }}_0\right|=4$, we obtain the results numerically.

Figure 4

Variation of $\overline{\ell }\text{-vs-}\overline{t}$ for uncharged and charged capillaries. In the plot, we indicate the initial inertial regime [23, 30] (where $\overline{\ell }\sim \overline{t})$ and the Washburn regime [2, 3, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 37, 59] (where $\overline{\ell }\sim \sqrt{\overline{t}})$. $\ell \text{-vs-}t$ behavior for these two regimes is separately magnified in insets 1 and 2 of the plot. The results for charged capillaries are provided for different values of $S$. We choose dimensionless EDL thickness $\overline{\lambda }$ values, corresponding to these $S$ values, such that the magnitude of the electric force on the liquid column [see Fig. 2] is maximum. In the main plot and the plots in the inset, we use a bold line for the uncharged capillary and a dashed line for the charged capillary $(S=0.3$ and $\overline{\lambda }=0.2178)$. For the plots in the inset, dashed-dotted lines represent the case for the charged capillary $(S=3$ and $\overline{\lambda }=0.2911)$, whereas the plot with square markers represents the case corresponding to larger $\left|{\overline{\psi }}_0\right|$ $(S=3$ and $\overline{\lambda }=0.1265)$. All the results are shown for $\text{Oh}=1,A=2K_{1}cos\theta =2$, and $|{\overline{\psi }}_0|=1$ (except for the case of larger $\left|{\overline{\psi }}_0\right|$, where we take $\left|{\overline{\psi }}_0\right|=4)$.

Figure 5

Variation of the ratio ${\overline{\ell }}_{c,\text{ch}}/{\overline{\ell }}_c={\ell }_{c,\text{ch}}/{\ell }_c$ with the dimensionless EDL thickness $\overline{\lambda }$ for different values of the dimensionless ionic Peclet number $S$. The plotted ratio is independent of $\text{Oh}$, although ${\overline{\ell }}_c$ and ${\overline{\ell }}_{c,\text{ch}}$ depend on it. For example, we have ${\overline{\ell }}_c=0.2964$ (for $\text{Oh}=1)$ and ${\overline{\ell }}_c=2.964$ (for $\text{Oh}=0.1)$, so that the linear regime can be defined for ${\overline{\ell }}_c\times \text{Oh}\ll 1$ [30] (and the transition from a linear to Washburn regime would occur for ${\overline{\ell }}_c\times \text{Oh}\sim 1$ [30]). In the inset of the figure, we plot the ratio ${\overline{t}}_{c,\text{ch}}/{\overline{t}}_c=t_{c,\text{ch}}/t_c$ with $\overline{\lambda }$ for different values of $S$. Here both ${\overline{t}}_{c,\text{ch}}$ and ${\overline{t}}_c$ are independent of $\text{Oh}$, and we get ${\overline{t}}_c=t/{\tau }_c=0.2883$, i.e., the linear regime is witnessed for $t\ll {\tau }_c$ and the transition from the linear to Washburn regime occurs for $t\sim {\tau }_c$ [30]. For these plots, except for the case where $|{\overline{\psi }}_0|=ez|{\psi }_0|/k_{B}T$ is specified, we always take $\left|{\overline{\psi }}_0\right|=1$. For the case where $\left|{\overline{\psi }}_0\right|=4$, we obtain the results numerically.

Figure 6

(a) Schematic of the capillary filling. The control volume denoted with the dotted line is the filling liquid column that we analyze. (b) The volume of the liquid at the interface $\left(V_{\text{int}}\right)$ that we neglect in our analysis. $V_{\text{int}}$ is equal to the volume of the cylinder minus the volume of the spherical cap. (c) Wedge at the three-phase contact line indicated in (a).