Static wetting of a barrel-shaped droplet on a soft-layer-coated fiber

Abstract

A droplet can deform a soft substrate due to capillary forces when they are in contact. We study the static deformation of a soft solid layer coated on a rigid cylindrical fiber when an axisymmetric barrel-shaped droplet is embracing it. We found that elastic deformation increases with a decreasing rigid fiber radius. Significant disparities of deformation between the solid–liquid side and the solid–gas side are found when their solid surface tensions are different. When the coated layer is soft enough and the rigid fiber radius is less than the thickness of the coated layer, pronounced displacement oscillations are observed. Such slow decay of deformation with distances from the contact line position suggests a possible long-range interaction between droplets on a soft-layer-coated fiber.

---

1. Introduction

A layer of a soft solid material such as gels and elastomers coated on rigid solid substrates can be significantly deformed by capillary forces, when a droplet is in contact with it.^1,2 The shape of deformation has been explored extensively in the last few decades; however, those studies have mainly focussed on situations of a droplet on a planar substrate.^1–25 Although there have been numerous studies of droplets on rigid fibers,^26–31 wetting on a rigid fiber coated with a soft layer has been far less investigated.³² How this geometry modifies the deformation of the coated elastic layer remains unclear, which is the focus of this study.

The wetting of droplets on fibers is ubiquitous in both industrial and natural phenomena, and examples range from the modulation of the mechanical properties of spider silk,³³ to the droplet transport on spines of cacti,³⁴ to water collection by fog nets. Compared to a planar surface, the slender geometry of fibers modifies the droplet shape,^27,31 and could induce directional motion of droplets.^26,35–37 When it comes to droplet wetting on soft fibers, most studies have focused on the bending and buckling of flexible fibers which result in, for instance, the winding around a droplet,³⁸ the coiling inside a droplet³⁹ or the modification of droplet morphology by multiple fibers.⁴⁰ On the other hand, a soft fiber with a rigid core, i.e. a soft-layer-coated fiber, can hardly bend. Hence, the interaction of droplets with a soft-layer-coated fiber is expected to be different from that of a totally soft fiber.

Unlike droplets on planar surfaces, which appear in a spherical-cap shape given that the Laplace pressure is the dominant force (e.g. effects of gravity are neglected), the shape of a droplet on a fiber is far more complex. Extensive studies in the literature^31,41,42 have shown that stable droplets can appear in an axisymmetric barrel-shape or a non-axisymmetric clam-shell shape. For small equilibrium contact angles θ and large droplet sizes relative to the fiber radius, axisymmetric barrel-shaped droplets tend to be more stable. When the droplet size (V^1/3, here V is the droplet volume) is smaller than the fiber radius, barrel-shaped droplets exist only for θ ≲ 10°.³¹

In this study, we investigate the axisymmetric deformation of a soft solid layer coated on a rigid cylindrical fiber when a barrel-shaped droplet is embracing it. Previous studies of elastic deformation by wetting often assume that θ = 90°, and thus the solid surface tension of the soft-solid/liquid interface γ_sg is the same as that of the soft-solid/gas interface γ_sl.^12,14 For stable barrel-shaped droplets on a fiber, the contact angle has to be smaller than 90°. We hence consider generic situations where the two solid surface tensions can be different, i.e. γ_sg ≠ γ_sl. Moreover, it is known that the solid surface tensions not only determine the contact angle at the contact line, but also play a crucial role in suppressing the deformation of a planar soft solid layer.^8,12 How the solid surface tensions affect the soft solid deformation in fiber geometries is studied in detail below.

2. Formulation

A schematic of a droplet of volume V resting on a soft-layer-coated fiber is shown in Fig. 1. We consider the droplet bond number Bo ≡ ρgV^2/3/γ ≪ 1, where ρ is the liquid density, g is the gravitational acceleration and γ is the liquid–air surface tension. Hence, the effects of gravity on the droplet shape are negligible. The rigid fiber has a radius r₀. A soft solid layer of uniform thickness H at an undeformed state is coated on the surface of the rigid fiber. Due to the axisymmetry of the problem, we will use the cylindrical coordinate system (r, ϕ, and z) and the corresponding unit vectors are denoted as r̂ and [small phi, Greek, circumflex], ẑ. The center of mass of the droplet is at the origin, i.e. r = 0 and z = 0. The soft layer is deformed by the droplet. The displacement of the material of the soft layer is denoted as U(r, z) = U_rr̂ + U_ϕ[small phi, Greek, circumflex] + U_zẑ. Due to symmetry, the displacement has a property that U(r, z) = U(r, −z). Hence, we consider only z ≥ 0 in the following. The equilibrium contact angle θ is defined as the angle of the droplet liquid–air interface at the contact line position z = R with respect to the z-axis as shown in Fig. 1. In this study, we assume that θ is determined by Young's law,¹⁴ which reads

γ cos θ = γ_sg − γ_sl. (1)

Denoting the elastic stress tensor as σ, the deformation of the soft layer is governed by the force balance equation:

∇·σ = 0. (2)

As we only consider small slopes of the deformed soft-solid/fluid interface, we employ a linear elastic constitutive model for determining the relationship between the stress tensor and the displacement, for which the tensor components are as follows:where the isotropic part of the stress tensor (or the pressure) is:where E is the Young modulus and ν is the Poisson ratio. Note that U_ϕ = 0 due to axisymmetry.

Fig. 1 Schematic diagram of an axisymmetric barrel-shaped droplet embracing a rigid cylindrical fiber of radius r₀ coated with a soft (elastic) solid layer of thickness H. The droplet makes an equilibrium contact angle θ with the fiber. The soft solid layer is deformed by the droplet due to capillary forces.

The boundary conditions for governing eqn (2) are as follows. Consider that the length of the fiber 2L is much larger than the droplet radius R, we impose the condition

U(r, z = L) = 0. (8)

Due to symmetry, at z = 0, we have

U_z(r, z = 0) = 0 (9)

andAt the interface where the soft layer is in contact with the rigid solid, i.e. r = r₀, the soft material is undeformed, so we have

U(r = r₀, z) = 0. (11)

For the soft-solid/fluid boundary, i.e. r = r₀ + H, we impose a force balance condition. Here, we introduce all the tractions (force per unit area) acting on the interface as the following. At the contact line, the localized capillary traction f^l = γδ(R − z)(sin θr̂ − cos θẑ) is pulling the soft layer,^9,16 where δ(z) is the Dirac delta function. Second, the Laplace pressure inside the droplet can also deform the soft layer. The traction f^La due to the Laplace pressure is given as f^La = −γκ_lH_s(R − z)r̂, where κ_l is the curvature of the droplet's liquid–air interface and H_s(z) is the Heaviside step function. Third, the elastic traction due to soft solid deformation is f^el = −(σ_rrr̂ + σ_rzẑ)|_r=r0+H. Fourth, the contribution from the soft solid surface tension gives a traction f^s = −γ_sκ_sr̂ + ∂γ_s/∂zẑ, where γ_s is the soft solid surface tension and κ_s is the curvature of the deformed soft-solid/fluid interface. We assume that the solid surface stress is the same as the solid surface energy, and call them the solid surface tension. In other words, we neglect the Shutter worth effect.^21,43,44 Note that the assumption of a small soft-solid/fluid interface slope has been used to obtain the expression of f^s. For small interface slopes, the curvature can be simplified as κ_s = [1/(r₀ + H + U_r) − ∂²U_r/∂z²]|_r=r0+H. For the soft solid surface tension, there is a jump across the contact line, meaning that γ_s = γ_sl + (γ_sg − γ_sl)H_s(R − z). Thus, f^s = −γ_sκ_sr̂ + (γ_sg − γ_sl)δ(R − z)ẑ. Balancing all the tractions we have the following boundary conditions at r = r₀ + H,

f^l + f^La + f^el + f^s = 0. (12)

Regarding the shape of the droplet's liquid–air interface, described by r = g(z), the profile is obtained by solving the equation of uniform Laplace pressure, which implies κ_l has a constant value, i.e. κ_l = k_c. We describe the details of the governing equation and the corresponding boundary conditions at the later part of this section.

Next, we non-dimensionalize the variables as the following. We rescale the coordinates and displacements by H and the pressure by E, namely

For the elastic deformation, the dimensionless forms of the governing eqn (2) written in vector components arein the r-direction andin the z-direction. The dimensionless forms of the boundary conditions (8)–(11) are

Ũ(r̃, z̃ = L̃) = 0, (16)

Ũ_z(r̃, z̃ = 0) = 0, (17)

and

Ũ(r̃ = r̃₀, z̃) = 0, (19)

respectively, where r̃₀ = r₀/H and L̃ = L/H. In this study, we consider L̃ ≫ 1.

For the soft-solid/fluid interface, we first define ũ_r ≡ Ũ_r(r̃ = r̃₀ + H̃, z̃) and ũ_z ≡ Ũ_z(r̃ = r̃₀ + H̃, z̃). The dimensionless form of the force balance condition (12), written in vector components, is given as follows. For r-components, we obtain

where the dimensionless parameters are defined asFor the z-components of the force balance (12), after using Young's law (1), −γ cos θδ(R − z) cancels out (γ_sg − γ_sl)δ(R − z); hence, we obtain the vanishing elastic shear stress conditionFor the droplet profile, the dimensionless form of the uniform curvature equation iswhere g̃ =g/H. The boundary conditions at the contact line are as follows:

g̃(z̃ = R̃) = r̃₀ + 1 + ũ_r(r̃ = r̃₀ + 1, z̃ = R̃) (24)

andThe droplet volume is given by:In this study, we consider the soft material to be incompressible, which means ν = 0.5. The incompressibility condition impliesThe governing eqn (14), (15), (23) and (27) are solved together with the conditions (16)–(20), (22), (24)–(26) using the finite element method for which the details are given in the Appendix. The dimensionless control parameters are r̃₀, Ẽ, [small gamma, Greek, tilde]_sl, [small gamma, Greek, tilde]_sg and Ṽ.

3. Results

3.1 Large droplet limit

First we look at situations in which the droplet length scale (R) is much larger than the other length scales: H, r₀, γ/E, γ_sl/E and γ_sg/E. The Laplace pressure f^La is neglected in this limit, and thus, the only external capillary force acting on the soft solid is the localized force f^l at the contact line. We will demonstrate how the solid deformation varies with the change of r̃₀, Ẽ and [small gamma, Greek, tilde]_sg for both cases of γ_sg = γ_sl (i.e. θ = 90°) and γ_sg ≠ γ_sl (i.e. θ ≠ 90°).

3.1.1 Cases of γ_sg = γ_sl (i.e. θ = 90°). We look at two different cases of softness, namely Ẽ = 1 and Ẽ = 10. In Fig. 2, the displacements at the soft-solid/fluid interface ũ_r and ũ_z are plotted as a function of z̃−R̃ for different values of r̃₀. To validate our computations, we first compare our results for a large rigid fiber radius r̃₀ = 10 with the analytical solution of the 2D plain strain case.¹² In the insets of Fig. 2, we show that our numerical results collapse with the analytical 2D solutions. Next, we look at how the rigid fiber radius modifies the deformation. When the layer is stiff (Ẽ = 10), as shown in Fig. 2(a), the out-of-plane displacement ũ_r is insensitive to the value of r̃₀. Relatively, there is a stronger dependence of the in-plane displacement ũ_z on r̃₀ as shown in Fig. 2(b). In contrast, for the softer case of Ẽ = 1 shown in Fig. 2(c) and (d), the magnitude of displacements increases significantly with decreasing r̃₀ when r̃₀ ≤ 1. Moreover, unlike the stiff or large rigid fiber radius cases, in which the magnitude of displacement decays quickly with distances from the contact line, the oscillations are significant for the case of Ẽ = 1 when r̃₀ ≪ 1.

Fig. 2 Rescaled displacements ũ_r and ũ_z as a function of z̃ − R̃ for different values of r̃₀ and Ẽ = 10 in (a) and (b) and Ẽ = 1 in (c) and (d). Insets: The analytical solutions of the 2D plain strain case (solid lines)¹² and our numerical results for r̃₀ = 10 (dotted lines). Other parameters are as follows: [small gamma, Greek, tilde]_sl = [small gamma, Greek, tilde]_sg = 5.

How do we understand the results? For r̃₀ ≫ 1, the deformation behaves the same as the 2D solution as one might expect. The more interesting regime is when r̃₀ ≲ 1. When the material is stiff, namely Ẽ ≫ 1 (i.e. γ/E ≪ H), the dimensional out-of-plane displacement u_r scales as/E and weakly depends on H,^7,12 meaning that the deformation is small compared to the thickness of the soft layer. The displacements vanish at a short distance from the soft-solid/fluid interface. Hence, the no-displacement condition at r̃ = r̃₀ only plays as a small correction to the displacements. On the other hand, when Ẽ ≲ 1, the bulk material deforms more significantly; hence, the boundary conditions at r̃₀ have a stronger effect. Furthermore, not only do the bulk elastic stresses play a significant role, but also the traction f^s involving the azimuthal curvature of the soft solid interface, i.e. 1/(r̃₀ + 1 + ũ_r), is crucial for inducing strong displacement oscillations. We found that removing the azimuthal curvature term in eqn (20) results in the disappearance of displacement oscillations. In Section 3.1.4, we will look at how the displacement oscillations depend on the solid surface tension.

3.1.2 Cases of γ_sg ≠ γ_sl (i.e. θ ≠ 90°). We take [small gamma, Greek, tilde]_sl = 4.13, [small gamma, Greek, tilde]_sg = 5, and thus θ = 30° using Young's law (1). In Fig. 3, we plot the displacements ũ_r and ũ_z as a function of z̃−R̃. For Ẽ = 10, although the maximum of ũ_r is almost independent of r̃₀, the dimple (minimum) depends significantly on r̃₀ as shown in Fig. 3(a). When the rigid fiber radius is large, for example, for r̃₀ = 1000, the dimple on the solid–liquid side (left) is slightly larger than the solid–gas side (right). This is consistent with the finding of previous studies for plate cases that the solid surface tension suppresses deformation.⁸ Note that [small gamma, Greek, tilde]_sl < [small gamma, Greek, tilde]_sg. However, when decreasing r̃₀, ũ_r increases on the solid/liquid side and decreases on the solid/gas side. Hence, the dimple of the solid/liquid interface decreases and disappears. On the other hand, the dimple of the solid/gas interface becomes larger. This disparity can be observed clearly for r̃₀ = 0.1 in Fig. 3(a).

Fig. 3 Rescaled displacements ũ_r and ũ_z as a function of z̃−R̃ for different values of r̃₀ and Ẽ = 10 in (a) and (b) and Ẽ = 1 in (c) and (d). Other parameters are as follows: [small gamma, Greek, tilde]_sl = 4.13 and [small gamma, Greek, tilde]_sg = 5.

The in-plane displacement ũ_z shown in Fig. 3(b) also demonstrates specific features when varying r̃₀. For large r̃₀, the material around the contact line at the interface displaces towards the contact line for both sides. When decreasing r̃₀, ũ_z decreases. For r̃₀ ≲ 1, ũ_z is negative for the whole interface, meaning that all the material at the interface displaces towards the side of interface with a smaller solid surface tension (left).

For a softer material (Ẽ = 1) shown in Fig. 3(c) and (d), the behavior of ũ_r and ũ_z when decreasing r̃₀ is similar to that of Ẽ = 10. Remarkably, when r̃₀ is reduced to r̃₀ = 0.1, ũ_r around the contact line on the solid–liquid side increases to values larger than ũ_r at the contact line. Hence, the maximum of ũ_r is not at the contact line position but shifts to the solid/liquid side as shown in Fig. 3(c). Similar to that we have seen for the case of θ = 90°, pronounced oscillations of ũ_r and ũ_z appear when r̃₀ ≪ 1.

3.1.3 Dependence on the contact angle. We look at how the features of deformation change when the contact angle is varied from θ = 90° to θ = 0° for a fixed value of r̃₀ = 0.1. According to Young's law, changing θ also means that [small gamma, Greek, tilde]_sg − [small gamma, Greek, tilde]_sl is varied. We keep [small gamma, Greek, tilde]_sg = 5 for all cases and change the value of [small gamma, Greek, tilde]_sl. We again consider two different values of Ẽ. The displacements ũ_r and ũ_z as a function of z̃−R̃ for different θ (or [small gamma, Greek, tilde]_sl) are plotted in Fig. 4.

Fig. 4 Rescaled displacements ũ_r and ũ_z as a function of z̃−R̃ for different values of [small gamma, Greek, tilde]_sl and Ẽ = 10 in (a) and (b) and Ẽ = 1 in (c) and (d). Other parameters are as follows: [small gamma, Greek, tilde]_sg = 5 and r̃₀ = 0.1.

In Fig. 4(a), we can see that the maximum of ũ_r decreases when θ reduces due to the fact that the pulling contact line force scales as γ sin θ. The sharp tip at the contact line disappears for θ = 0° as the pulling force vanishes. Interestingly, there is still a significant deformation for θ = 0°. The deformation then comes from the traction f^s generated by the solid surface tensions. When [small gamma, Greek, tilde]_sl and [small gamma, Greek, tilde]_sg are different, the traction f^s involving the azimuthal curvature, i.e. 1/(r̃₀ + 1 + ũ_r), gives different stresses at the liquid side and the gas side.

As θ is decreased, the asymmetry of displacements between the solid–liquid side and the solid–gas side becomes more apparent. When compared with the results in Fig. 3, we observe that the enhancement of asymmetry when reducing θ (for a fixed value of r̃₀) is similar to that of decreasing r̃₀ (for a fixed value of θ). In the following, we point out some of these similar trends. As shown in Fig. 3(a) or in Fig. 4(a) for Ẽ = 10, when reducing r̃₀ or θ, the dimple of ũ_r on the solid–liquid side (with a smaller solid surface tension) decreases and the dimple on the solid–gas side becomes larger. For Ẽ = 1, the maximum of ũ_r shifts from the contact line position to the solid–liquid side when r̃₀ or θ is reduced to below a certain value as shown in Fig. 3(c) or in Fig. 4(c). For the in-plane displacement shown in Fig. 3(b) and (d) or Fig. 4(b) and (d), ũ_z around the contact line decreases when reducing r̃₀ or θ. This means that the material at the interface displaces more to the side of the interface with a smaller solid surface tension (left).

3.1.4 Dependence on the solid surface tension. In Fig. 5, we plot the dimensionless displacements ũ_r and ũ_z as a function of z̃−R̃ for θ = 90° in (a) and (b) and θ = 30° in (c) and (d). We see that when [small gamma, Greek, tilde]_sg is reduced, ũ_r at the contact line enhances. This is because the traction f^s term involving the longitudinal curvature, i.e. ∂²ũ_r/∂r̃², suppresses the sharp tip deformation at the contact line.^8,12 On the other hand, the displacement oscillations and the displacement disparity (for θ = 30°) become less pronounced when [small gamma, Greek, tilde]_sg is reduced. This can be understood as follows. Away from the contact line, the azimuthal curvature term, i.e. 1/(r̃₀ + 1 + ũ_r), dominates over the longitudinal curvature term. The displacement oscillations and displacement disparity are due to the azimuthal curvature term in the traction f^s. Decreasing [small gamma, Greek, tilde]_sg thus reduces the effects of this traction. In Fig. 5, we also add the results for a smaller value of Ẽ. As we demonstrate previously, reducing the value of Ẽ enhances the displacement oscillations and displacement disparity.

Fig. 5 Rescaled displacements ũ_r and ũ_z as a function of z̃−R̃ for θ = 90° in (a) and (b) and θ = 30° in (c) and (d).

3.2 Dependence on the droplet size

For a droplet of the finite size, the traction term due to the Laplace pressure f^La is included in our computations. As small stable barrel-shaped droplets exist only when the contact angle is small, we choose two small contact angles, i.e. θ = 30° and θ = 0°, and investigate how the deformation depends on the droplet size. We fix r̃₀ = 0.1. In Fig. 6 and 7, the displacements ũ_r and ũ_z are plotted as a function of z̃ − R̃ respectively for θ = 30° and θ = 0°. We can see that for Ṽ^1/3= 40, the results are the same as those for the large droplet limit. When Ṽ^1/3 decreases, ũ_r decreases on the liquid side and increases on the gas side. This is because the Laplace pressure pressing the soft layer on the liquid side is stronger for smaller droplets. For ũ_z, the Laplace pressure term reduces the magnitude of ũ_z on both the liquid and gas sides. The magnitude of displacement oscillation is also reduced when the droplet size decreases. Note that we assume the contact angle follows Young's relationship and is independent of the droplet size. Hence, the contact angle transition observed for small droplets on planar surfaces^14,17 is not considered in our present study.

Fig. 6 Rescaled displacements ũ_r and ũ_z as a function of z̃−R̃ for different values of Ṽ^1/3 and Ẽ = 10 in (a) and (b) and Ẽ = 1 in (c) and (d). Other parameters are as follows: [small gamma, Greek, tilde]_sl = 4.13 (θ = 30°), [small gamma, Greek, tilde]_sg = 5 and r̃₀ = 0.1.

Fig. 7 Rescaled displacements ũ_r and ũ_z as a function of z̃−R̃ for different values of Ṽ^1/3 and Ẽ = 10 in (a) and (b) and Ẽ = 1 in (c) and (d). Other parameters are as follows: [small gamma, Greek, tilde]_sl = 4 (θ = 0°), [small gamma, Greek, tilde]_sg = 5 and r̃₀ = 0.1. Insets: Zoom in view of Ṽ^1/3 = 0.5.

4. Conclusions

We study the elastic deformation of a soft layer coated on a rigid cylindrical fiber when an axisymmetric barrel-shaped droplet is embracing it. For a droplet contact angle of θ = 90°, and thus γ_sl = γ_sg according to Young's law, we found that the magnitudes of both displacements ũ_r and ũ_z increase with a decreasing r̃₀. For θ ≠ 90° (i.e. γ_sl ≠ γ_sg), the deformations on the solid–liquid side and the solid–gas side are different. This disparity of deformation is enhanced when decreasing r̃₀ (for a fixed value of θ) or decreasing θ (for a fixed value of r̃₀). The dimple of ũ_r on the side with a smaller solid surface tension becomes smaller while the dimple becomes larger on the other side. Interestingly, significant deformation is observed even for θ = 0° for which the pulling force at the contact line vanishes and the Laplace pressure is neglected for large droplets. The deformation is due to the part of traction f^s involving the azimuthal curvature term, i.e. 1/(r̃₀ + 1 + ũ_r), which generates different stresses on the liquid and the gas sides when the solid surface tensions are different.

Pronounced oscillations of displacements are observed for the cases of Ẽ < 1 and r̃₀ ≲ 1. This slow decay of deformation with distances from the contact line position suggests a relatively long-range interaction between droplets on a soft-layer-coated fiber. Hence, it is expected that droplet migration and interactions are significantly different from those observed on a planar soft substrate or a purely rigid fiber, for example, the inverted Cheerios effect⁴⁵ and coalescence,³⁷ which remain open questions to be explored.

Conflicts of interest

There are no conflicts to declare.

Appendix: finite element method

The non-dimensional deformation of the soft layer of the fiber and the droplet profile are computed by solving the governing eqn (14), (15), (23) and (27) together with the conditions (16)–(20), (22), and (24)–(26) by using a finite element method (FEM). We discretize our non-dimensional variables with linear elements and solve the coupled equations using a Newton solver from the FEM library FEniCS.⁴⁶

For the Dirac delta function, we approximate it with a Gaussian function as , where l_m can be interpreted as a microscopic length (e.g. interface thickness) such that l_m ≪ H. In the limit that l_m → 0, δ(R − z) = W(R − z). In this study, l_m/H = 10⁻⁵ is set for all the cases. We have used the adaptive mesh sizing such that the smallest mesh size near the contact line is dx₁/H = 10⁻⁷ and the largest mesh size far away from the contact line is dx₀/H = 0.05.

The mesh convergence of the numerical solver is tested with the case of the large droplet limit and the condition of r̃₀ ≫ 1. In this limit, the deformation of the soft layer will converge towards the 2D soft plate case. In Fig. 8, we compare our numerical results with the analytical solution for the 2D soft plate case by plotting the deformation of the soft solid for three different mesh resolutions, namely dx₀/H = 0.1, 0.05 and 0.025. For all three mesh sizes, the numerical solutions are in good agreement with the analytical solution.

Fig. 8 Convergence of the mesh. The rescaled displacements ũ_r and ũ_z as a function of z̃ − R̃ for different mesh sizes. Other parameters are as follows: θ = 90°, [small gamma, Greek, tilde]_sl = [small gamma, Greek, tilde]_sg = 5, and r̃₀ = 1000.

Acknowledgements

The authors gratefully acknowledges financial support from the Research Council of Norway (project no. 315110). BXZ thanks Stephane Poulain, Jarle Sogn and Miroslav Kuchta for the discussions of the numerical method.

Notes and references

1. S. J. Park, B. M. Weon, J. S. Lee, J. Lee, J. Kim and J. H. Je, Nat. Commun., 2014, 5, 1–7.
2. B. Andreotti and J. H. Snoeijer, Annu. Rev. Fluid Mech., 2020, 52, 285–308.
3. G. R. Lester, J. Colloid Sci., 1961, 16, 315–326.
4. A. I. Rusanov, Colloid J., 1975, 37(4), 614–622.
5. R. Pericet-Cámara, A. Best, H. J. Butt and E. Bonaccurso, Langmuir, 2008, 24, 10565–10568.
6. Y. S. Yu and Y. P. Zhao, J. Colloid Interface Sci., 2009, 339, 489–494.
7. R. Pericet-Camara, G. K. Auernhammer, K. Koynov, S. Lorenzoni, R. Raiteri and E. Bonaccurso, Soft Matter, 2009, 5, 3611–3617.
8. E. R. Jerison, Y. Xu, L. A. Wilen and E. R. Dufresne, Phys. Rev. Lett., 2011, 106, 1–4.
9. S. Das, A. Marchand, B. Andreotti and J. H. Snoeijer, Phys. Fluids, 2011, 23, 072006.
10. L. Limat, Eur. Phys. J. E: Soft Matter Biol. Phys., 2012, 35, 1–13.
11. Y. S. Yu, J. Appl. Math. Mech., 2012, 33, 1095–1114.
12. R. W. Style and E. R. Dufresne, Soft Matter, 2012, 8, 7177–7184.
13. R. W. Style, R. Boltyanskiy, Y. Che, J. S. Wettlaufer, L. A. Wilen and E. R. Dufresne, Phys. Rev. Lett., 2013, 110, 1–5.
14. L. A. Lubbers, J. H. Weijs, L. Botto, S. Das, B. Andreotti and J. H. Snoeijer, J. Fluid Mech., 2014, 747, R1.
15. C. Y. Hui and A. Jagota, Proc. R. Soc. A, 2014, 470, 20140085.
16. J. B. Bostwick, M. Shearer and K. E. Daniels, Soft Matter, 2014, 10, 7361–7369.
17. J. Dervaux and L. Limat, Proc. R. Soc. A, 2015, 471, 20140813.
18. R. W. Style, A. Jagota, C. Y. Hui and E. R. Dufresne, Annu. Rev. Condens. Matter Phys., 2017, 8, 99–118.
19. R. W. Style and Q. Xu, Soft Matter, 2018, 14, 4569–4576.
20. M. Zhao, J. Dervaux, T. Narita, F. Lequeux, L. Limat and M. Roché, Proc. Natl. Acad. Sci. U. S. A., 2018, 115, 1748–1753.
21. R. Masurel, M. Roché, L. Limat, I. Ionescu and J. Dervaux, Phys. Rev. Lett., 2019, 122, 1–6.
22. J. W. Choi, D. Ham, S. Han, D. Y. Noh and H. C. Kang, Nanomaterials, 2021, 11, 1–10.
23. Y. Yang, X. Li and W. Wang, Sci. Rep., 2021, 11, 1–17.
24. H. K. Khattak, S. Karpitschka, J. H. Snoeijer and K. Dalnoki-Veress, Nat. Commun., 2022, 13, 4436.
25. T. S. Chan, Soft Matter, 2022, 18, 7280–7290.
26. É. Lorenceau and D. Quéré, J. Fluid Mech., 2004, 510, 29–45.
27. E. Q. Li and S. T. Thoroddsen, Phys. Fluids, 2013, 25, 052105.
28. T. S. Chan, C. Pedersen, J. Koplik and A. Carlson, J. Fluid Mech., 2021, 907, A29.
29. T. S. Chan, C. L. Lee, C. Pedersen, K. Dalnoki-Veress and A. Carlson, Phys. Rev. Fluids, 2021, 6, 1–8.
30. C. Fournier, C. L. Lee, R. D. Schulman, É. Raphaël and K. Dalnoki-Veress, Eur. Phys. J. E: Soft Matter Biol. Phys., 2021, 44, 36–41.
31. C. Xu, Z. Lu and L. Li, Frontiers Energy Research, 2022, 9, 1–7.
32. D. Guan, E. Charlaix and P. Tong, Phys. Rev. Lett., 2020, 124, 188003.
33. F. Vollrath and D. Edmonds, Nature, 1989, 340, 305–307.
34. C. Liu, Y. Xue, Y. Chen and Y. Zheng, Sci. Rep., 2015, 5, 1–8.
35. B. J. Carroll, J. Colloid and Interface Sci., 1976, 57, 488–495.
36. T. S. Chan, F. Yang and A. Carlson, J. Fluid Mech., 2020, 894, A26.
37. C. L. Lee, T. S. Chan, A. Carlson and K. Dalnoki-Veress, Soft Matter, 2022, 18, 1364–1370.
38. R. D. Schulman, A. Porat, K. Charlesworth, A. Fortais, T. Salez, E. Raphaël and K. Dalnoki-Veress, Soft Matter, 2017, 13, 720–724.
39. H. Elettro, F. Vollrath, A. Antkowiak and S. Neukirch, Int. J. Non Linear Mech., 2015, 75, 59–66.
40. C. Duprat, S. Protière, A. Y. Beebe and H. A. Stone, Nature, 2012, 482, 510–513.
41. C. Bauer and S. Dietrich, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2000, 62, 2428–2438.
42. T. H. Chou, S. J. Hong, Y. E. Liang, H. K. Tsao and Y. J. Sheng, Langmuir, 2011, 27, 3685–3692.
43. R. Shuttleworth, Proc. Phys. Soc. A, 1950, 63, 444.
44. B. Andreotti and J. H. Snoeijer, EPL, 2016, 113, 66001.
45. S. Karpitschka, A. Pandey, L. A. Lubbers, J. H. Weijs, L. Botto, S. Das, B. Andreotti and J. H. Snoeijer, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, 7403–7407.
46. A. Logg, K.-A. Mardal and G. Wells, Automated solution of differential equations by the finite element method: The FEniCS book, Springer Science & Business Media, 2012, vol. 84.

---