Variational approach to droplet transport via bendotaxis: Thin film dynamics and model reduction

Bendotaxis has recently been theoretically proposed and experimentally observed by Bradley et al. [Phys. Rev. Lett. 122, 074503 (2019)] as a mechanism for self-transport of droplets at small scales. As a result of the coupling between bending elasticity and capillarity in a thin channel, a pressure gradient is induced within a liquid droplet and leads to its spontaneous movement, regardless of whether the droplet is wetting or nonwetting the channel walls. In the present work, using Onsager's variational principle, we present a variational framework for systematically studying the mechanism of bendotaxis. We derive a full model for the thin film dynamics of bendotaxis with thermodynamical consistency. Based on the full model, we specify the conditions under which the simplified model used by Bradley et al. can be validated. Furthermore, we use Onsager's principle as a tool for approximation to derive a reduced model. This model reduction leads to a description for droplet self-transport using a few slow state variables by which a clear physical picture is presented for the mechanism of bendotaxis. The reduced model is validated by comparing its predictions with the numerical results from solving the full model. Finally, we investigate the mechanism of bendotaxis for active droplets capable of self-propelled motion. We find that wettability and activity can jointly operate to enhance or weaken the self-transport effect of bendotaxis, depending on the sign of wettability (hydrophobic or hydrophilic) and the sign of activity (contractile or extensile).

Figure 1

A schematic illustration of the system in which the mechanism of bendotaxis works. Here the upper half of the system is shown ($z\ge 0$), and the beam surface is hydrophobic ($cos\theta <0$).

Figure 2

Time evolution of the beam shown by its shape at four different stages: (a) initial stage (dotted), (b) bending stage (dashed), (c) moving stage (solid), and (d) final stage (dash-dotted). The part of the beam exposed to the air is marked in blue, while the part of the beam in contact with the liquid is marked in red. The upper row shows the results for a large droplet initially positioned between $x=0.3$ and $x=0.7$, while the lower row shows the results for a small droplet initially positioned between $x=0.6$ and $x=0.7$. The left column shows the results for hydrophilic droplets with $\theta =\frac{\pi }{3}$, while the right column shows the results for hydrophobic droplet with $\theta =\frac{2\pi }{3}$.

Figure 3

The asymptotic results for the left contact line position ${\tilde{x}}_{-}\left(\tilde{t}\right)$ are compared with the numerical results from the full model. The upper row is for the hydrophilic case with $\theta =\frac{\pi }{3}$, while the lower row is for the hydrophobic case with $\theta =\frac{2\pi }{3}$. The left column shows the contact line position ${\tilde{x}}_{-}$ as a function of $\tilde{t}$ for ${\tilde{x}}_{-}\left(0\right)=0.6$, with the asymptotic results compared to the full model results for $\tilde{V}=0.1$. The right column shows the convergence test for $(\frac{1}{{\tilde{x}}_{-}\left(0\right)}-\frac{1}{{\tilde{x}}_{-}\left(\tilde{t}\right)})/\tilde{V}$. The full model results are computed for a diminishing set of values $\tilde{V}=0.1$, 0.05, and 0.025. The red line is the asymptotic result given by $\frac{\nu {cos}^2\theta }{6{\left(\text{Ca}A\right)}^2}\tilde{t}$.

Figure 4

A schematic illustration of the forces acting on the droplet. Left panel: In the hydrophobic case, the lower and upper beams are pushed away from each other by the droplet. As a result, the bending elastic force exerted by the upper beam on the droplet is downward pointing with a positive horizontal component. Right panel: In the hydrophilic case, the lower and upper beams are pulled toward each other by the droplet. As a result, the bending elastic force exerted by the upper beam on the droplet is upward pointing with a positive horizontal component. In either case, the vertical component of the bending elastic force is in balance with the interfacial force, and this vertical force balance determines the beam deformation, as expressed in Eqs. (38, 39, 40). The positive horizontal component of the bending elastic force, which is given by $-\frac{d{F}_{\text{tot}}}{d{x}_{-}}=-\frac{\partial F_b}{\partial x_{-}}$ in Eq. (41), always drives the droplet to the right end.

Figure 5

The upper row shows the time evolution of the left contact point position ${\tilde{x}}_{-}$, computed from both the full model Eq. (55) and the asymptotic result in Eq. (61). The droplet is positioned between ${\tilde{x}}_{-}\left(0\right)=0.6$ and ${\tilde{x}}_{-}\left(0\right)=0.7$. Different activity parameters are considered for the hydrophilic case with $\theta =\frac{\pi }{3}$ (upper left) and the hydrophobic case with $\theta =\frac{2\pi }{3}$ (upper right). The lower left panel illustrates how the bendotaxis factor varies with the activity. The lower right panel shows the convergence test for $(\frac{1}{{\tilde{x}}_{-}\left(0\right)}-\frac{1}{{\tilde{x}}_{-}\left(\tilde{t}\right)})/\tilde{V}$. Numerical results from the full model are computed for a diminishing set of values $\tilde{V}=0.1$, 0.05, and 0.025. The red line is the asymptotic result given by $\frac{\nu {(cos\theta +\lambda )}^2}{6{\left(\text{Ca}A\right)}^2}\tilde{t}$.

Figure 6

The left panel shows the time evolution of droplet width ${\tilde{x}}_{+}-{\tilde{x}}_{-}$, computed from the full model Eq. (55). The droplet is initially positioned between ${\tilde{x}}_{-}\left(0\right)=0.6$ and ${\tilde{x}}_{+}\left(0\right)=0.7$. Different activity parameters are considered for the hydrophilic and hydrophobic cases with $\theta =\frac{\pi }{3}$ and $\theta =\frac{2\pi }{3}$ respectively. The right panel shows the time evolution of beam deformation $K_0^{-}-1$ for the hydrophilic case. The dashed lines represent the asymptotic result given by $-\frac{\nu {\tilde{x}}_{-}^3\tilde{V}}{3\text{Ca}A}(cos\theta +\lambda )$ in Eq. (59). It is observed that a very short transient time period is needed by the full model to establish the quasistatic state for vertical deformation.