Fibrotaxis: gradient-free, spontaneous and controllable droplet motion on soft solids

TL;DR

Fibrotaxis introduces a gradient-free mechanism for passive droplet transport on soft, anisotropic solids by leveraging elastocapillary interactions at the fluid–solid interface. The authors develop a high-fidelity fluid–structure model that couples Navier–Stokes–Cahn–Hilliard fluids with a transversely isotropic hyperelastic solid, capturing the role of fiber orientation and anisotropy. They show that four controllable parameters—fiber orientation $eta$, wettability $ heta$, anisotropy strength $k_1$, and elastocapillary number $ ext{zeta}$—govern droplet velocity and trajectory, with maximum speeds achieved when $ heta oughly eta$ or $ heta oughly eta+90^ ext{o}$ and direction set by $eta$. Compared to gradient-based mechanisms like durotaxis, fibrotaxis offers gradient-free, long-range transport with tunable speed, with implications for microfluidics, self-cleaning surfaces, water harvesting, and diagnostics; the work also outlines extensions to more complex anisotropy and interfaces. Overall, the study provides design rules for anisotropic soft substrates to achieve controlled, actuator-free droplet transport.

Abstract

Most passive droplet transport strategies rely on spatial variations of material properties to drive droplet motion, leading to gradient-based mechanisms with intrinsic length scales that limit the droplet velocity or the transport distance. Here, we propose droplet {\it fibrotaxis}, a novel mechanism that leverages an anisotropic fiber-reinforced deformable solid to achieve spontaneous and gradient-free droplet transport. Using high-fidelity simulations, we identify the fluid wettability, fiber orientation, anisotropy strength and elastocapillary number as critical parameters that enable controllable droplet velocity and long-range droplet transport. Our results highlight the potential of fibrotaxis as a droplet transport mechanism that can have a strong impact on self-cleaning surfaces, water harvesting and medical diagnostics.

Figures (8)

• Figure 1: Schematic of a droplet wetting on (A) rigid solid and a (B) fiber-reinforced deformable solid. In (B), gray colored arrow depicts the orientation of fibers. This depiction does not indicate that we model the fibers discretely; rather, the model assumes that there are infinite fibers continuously distributed.
• Figure 2: Three-dimensional simulation of droplet fibrotaxis. (A) shows the positions of the droplet at different times. The initial position of the droplet is indicated by a semi-transparent spherical cap, while the droplet in blue is at time $t = 1.38$ s. The droplets are represented by isosurfaces of the phase field. The black solid arrow depicts the fibers' orientation. Note that we do not model the fibers discretely; rather, the model assumes that there are infinite fibers continuously distributed. (B) shows the solid deformation (magnified $3 \times$) colored by the dimensionless vertical solid displacement $u_\mathrm{y}$. We use a non-wetting droplet with $\theta = 120^{\circ}$ and a radius of $160 \ \mu \mathrm{m}$. The fluids properties are $\gamma_{\mathrm{LA}} = 46 \ \mathrm{mN/m}$, $\rho = 1260 \ \mathrm{kg/m^3}$, $\eta_1 = 1.412 \ \mathrm{Pa \cdot s}$ and $\eta_2 = 1.85 \times 10^{-5} \ \mathrm{Pa \cdot s}$. The solid properties are $\rho_0^s = 1000 \ \mathrm{kg/m^3}$, $E = 5$ kPa, $\nu = 0.25$, $k_1 = 40$ kPa, $k_2 = 7$ and $\beta = 60^{\circ}$. The initial solid geometry is a rectangular prism of size $1000 \ \mu \mathrm{m} \times 500 \ \mu \mathrm{m} \times 50 \ \mu \mathrm{m}$. The numerical parameters are $M = 2 \times 10^{-11} \ \mathrm{m^3s/kg}$, $\epsilon = 20 \ \mu \mathrm{m}$ and $\Delta t = 100 \ \mu \mathrm{s}$. The values we choose correspond to the following dimensionless parameters: $Oh = 14.67$, $\hat{\eta} = 7.6 \times 10^4$, $Cn = 0.125$, $Pe = 906.52$, $\zeta = 0.058$ and $\Upsilon = 3.77 \times 10^4$. The initial solid geometry is a rectangular prism of size $1000 \ \mu \mathrm{m} \times 50 \ \mu \mathrm{m} \times 500 \ \mu \mathrm{m}$. We perform computations in a box of size $1000 \ \mu \mathrm{m} \ \times 400 \ \mu \mathrm{m} \ \times 500 \ \mu \mathrm{m}$ that includes the solid and fluid domains.
• Figure 3: Mechanism of droplet fibrotaxis (A-B) show droplet positions at two times. The insets in (A) show the solid deformation ($10 \times$ magnified) and the orientation of the capillary forces acting at the wetting ridges. The black solid arrows in (A-B) depict the fibers' orientation. (C) shows the vertical displacement of the solid $u_y$ along the fluid-solid interface. (D) shows the time evolution of droplet's velocity ($v_\mathrm{d}$) and the apparent contact angles at the left ($\alpha_{\mathrm{L}}$) and right ($\alpha_{\mathrm{R}}$) contact lines. We plot the droplet velocity using the data of droplet position over time. The variation of apparent contact angles here is a moving average that filters out noise in the data resulting from inaccurate measurements of the contact line due to the diffuse interface. We use a droplet of radius $160 \ \mu \mathrm{m}$ surrounded by air. The droplet is non-wetting with $\theta = 105^{\circ}$. The fluids properties are identical to those used in Fig. \ref{['fig:droplet_motion_3d']}. For the solid, we choose $\rho_0^s = 1000 \ \mathrm{kg/m^3}$, $E = 5$ kPa, $\nu = 0.25$, $k_1 = 50$ kPa, $k_2 = 7$, $\beta = 60^{\circ}$ and a thickness of $50 \ \mu \mathrm{m}$. We also use $M = 2 \times 10^{-11} \ \mathrm{m^3s/kg}$, $\epsilon = 20 \ \mu \mathrm{m}$ and $\Delta t = 25 \ \mu \mathrm{s}$. The values we choose correspond to $Oh = 14.67$, $\hat{\eta} = 7.6 \times 10^4$, $Cn = 0.125$, $Pe = 906.52$, $\zeta = 0.058$ and $\Upsilon = 4.71 \times 10^4$. We perform our computations in a box of size $1000 \ \mu \mathrm{m} \times 500 \ \mu \mathrm{m}$.
• Figure 4: Droplet motion driven by durotaxis. (A) shows the droplet positions at two times. The initial position of the droplet is indicated by a black spherical cap, while the droplet in blue is at time $t = 3.1$ s. (B) shows the vertical solid displacement $u_\mathrm{y}$ at the fluid-solid interface. (C) shows the time evolution of the droplet velocity. We use a non-wetting droplet with $\theta = 120^{\circ}$ and radius of $180 \ \mu \mathrm{m}$, and place it in contact with a heterogeneous solid of thickness $50 \ \mu \mathrm{m}$. The fluids properties are identical to those used in Fig. \ref{['fig:droplet_motion_3d']}. We assume the solid is isotropic and its stiffness varies linearly from $E = 5 \ \mathrm{kPa}$ to $E = 50 \ \mathrm{kPa}$. We also use $\nu = 0.25$, $M = 2 \times 10^{-11} \ \mathrm{m^3s/kg}$, $\epsilon = 25 \ \mu \mathrm{m}$ and $\Delta t = 25 \ \mu \mathrm{s}$. The values we choose correspond to $Oh = 13.82$, $\hat{\eta} = 7.6 \times 10^4$, $Cn = 0.14$, $Pe = 1147.4$, $\zeta \in \left[0.05, 0.005\right]$ and $\Upsilon = 0$. We perform our computations in a box of size $1000 \ \mu \mathrm{m} \times 500 \ \mu \mathrm{m}$.
• Figure 5: (A-B) Droplet positions at two different times. The insets in (A) show the solid deformation ($10 \times$ magnified) at the wetting ridges. The black solid arrows in (A-B) depict the fibers' orientation. (C) shows the vertical displacement of the solid $u_y$ along the fluid-solid interface. We use a droplet of radius $160 \ \mu \mathrm{m}$. The droplet is wetting with $\theta = 75^{\circ}$. The properties of fluid, solid, numerical and geometrical parameters are identical to those used in Fig. \ref{['fig:mechanism_1']}. The values we choose correspond to $Oh = 14.67$, $\hat{\eta} = 7.6 \times 10^4$, $Cn = 0.125$, $Pe = 906.52$, $\zeta = 0.058$ and $\Upsilon = 4.71 \times 10^4$. We perform our computations in a box of size $1000 \ \mu \mathrm{m} \times 500 \ \mu \mathrm{m}$.