Numerical simulation of the coalescence-induced polymeric droplet jumping on superhydrophobic surfaces

Self-propelled jumping of two polymeric droplets on superhydrophobic surfaces is investigated by three-dimensional direct numerical simulations. Two identical droplets of a viscoelastic fluid slide, meet and coalesce on a surface with contact angle 180 degrees. The droplets are modelled by the Giesekus constitutive equation, introducing both viscoelasticity and a shear-thinning effects. The Cahn-Hilliard Phase-Field method is used to capture the droplet interface. The simulations capture the spontaneous coalescence and jumping of the droplets. The effect of elasticity and shear-thinning on the coalescence and jumping is investigated at capillary-inertial and viscous regimes. The results reveal that the elasticity of the droplet changes the known capillary-inertial velocity scaling of the Newtonian drops at large Ohnesorge numbers; the resulting viscoelastic droplet jumps from the surface at larger Ohnesorge numbers than a Newtonian drop, when elasticity gives rise to visible shape oscillations of the merged droplet. The numerical results show that polymer chains are stretched during the coalescence and prior to the departure of two drops, and the resulting elastic stresses at the interface induce the jumping of the liquid out of the surface. This study shows that viscoelasticity, typical of many biological and industrial applications, affects the droplet behaviour on superhydrophobic and self-cleaning surfaces.


Introduction
When two droplets coalesce, the total surface area decreases. Hence, surface energy is released during this process. If the two droplets are far from a wall, the new bigger drop oscillates symmetrically until the released surface energy has been dissipated by viscosity. However, when two drops of micro-or nanometer size coalesce on a superhydrophobic surface, the presence of a repellent wall breaks the vertical symmetry and the resulting droplet propels in the direction perpendicular to the wall [2]. Coalescence-induced jumping has been reported on a variety of natural repellent surfaces such as cicada, lacewings [30] and gecko skin [29], and can be exploited in a variety of applications such as anti-icing [36] and self-cleaning surfaces [29,30], and to control heat transfer [10]. Several researchers have studied the different aspects of the coalescence-induced droplet jumping numerically and experimentally, including the basic mechanism of the two equal-sized drop self-propelled jumping [19], and the effects of droplet size mismatch [28,27], droplet initial velocity [14,18], surface topology [26,24,34,20,23], surrounding gas properties [11,32,33], and surface wettability [6]. A few main results are outlined in the following.
When two equal-sized static drops coalesce on a superhydrophobic surface, their total surface area decreases. This implies that surface energy is released and converted into viscous dissipation and kinetic energy, in a proportion determined by the Ohnesorge number, which represents the ratio between viscous and capillary-inertial forces. At large Ohnesorge numbers, corresponding to the viscous regime, the kinetic energy is completely absorbed by viscous forces, preventing the jumping of the merged droplet [19]. Even at small Ohnesorge numbers, corresponding to the capillary-inertial regime, only less than 4 % of the released surface energy converts to vertical translational kinetic energy, which nevertheless causes the jumping of the merged droplet. The conversion rate of surface energy into kinetic energy reduces when the droplets are of unequal sizes, due to the strong asymmetric flow [28,27]. The merged droplet attains an asymmetric shape and jumps with an oblique angle when one of the two droplets has an initial velocity; moreover, the jumping velocity of the merged droplet increases significantly above a critical initial velocity [18].
In addition, macrostructures on the surface affect the jumping velocity and energy transfer rates significantly. The jumping velocity and the conversion efficiency of surface energy to kinetic energy decrease if the lower contour of the merging drop falls between the gap of two rectangular grooves, whereas both jumping velocity and energy conversion increase when the liquid bridge expands on a triangular prism structure [26]. The critical Ohnesorge number for droplet jumping depends on both the surface wettability and the ambient fluid properties such as density and viscosity; in particular, a larger density contrast between the ambient and drops will cause the merged droplet to jump higher [11].
In very recent experiments, the effect of the drops' elasticity on the coalescence process was studied for both freely suspended drops and sessile drops with radius O(1) micrometer on the hydrophobic surfaces [8]. They found that elasticity enhances the curvature of connecting bridge between two merging drops, and polymer stresses remain confined in a small region around the liquid bridge between the coalescing drops. However, the induced elastic stresses were found to be insufficient to alter the temporal evolution of the bridge in the capillary-inertial regime, and hence elasticity did not change the flow regime.
In the present work, we perform numerical simulations to study the effects of the non-Newtonian viscoelastic properties of two equal-sized static droplets on the coalescence-induced droplet jumping at large Ohnesorge numbers. Our studies extend from the viscous-capillary to the inertial-capillary regime, and in the former case we do observe prominent changes due to elasticity. We use the Cahn-Hilliard Phase-Field method for capturing the interface between the two phases, and the Giesekus constitutive equation to model the viscoelasticity of the drops. First, the role of elasticity is investigated by comparing the vertical velocity and different components of energy for a Newtonian and Oldroyd-B droplet at the same Ohnesorge number based on the same zero shear viscosity, while the influence of the liquid shear-thinning rheology is examined by using Giesekus model.

Governing equations and Numerical methods
The numerical method used in this work has been described in detail in Bazesefidpar et al. [3], so we only give a brief outline here. We consider two immiscible fluids with different densities and viscosities. The outer fluid is Newtonian with viscosity µ n , whereas the droplets consist of a Giesekus fluid with solvent viscosity µ s , polymeric viscosity µ p , and the other non-Newtonian rheological properties as below. To distinguish between the phases, we introduce a phase-field variable, where φ = ±1 in the bulk fluids and φ = 0 at the fluid/fluid interface. This problem can be modelled with the following coupled equations [35,1]: the Cahn-Hilliard model: and the Giesekus constitutive model: In the above equations, u(x, t) is the velocity vector, p(x, t) is the pressure, and τ (x, t) is the extra stress due to the polymers, equal to τ p inside the droplet and 0 outside (see eq. 6). In the Cahn-Hilliard equation, G is the chemical potential, M is the mobility parameter, and η is the capillary width of the interface. In the Giesekus model, τ p is the polymer stress, λ H is the polymer relaxation time, α is the Giesekus mobility parameter, and the polymeric retardation time can be related to the polymeric relaxation time by λ r = µs µs+µp λ H . In Eq. (4), λ is the mixing energy density, and it is related to the surface tension in the sharp-interface limit [35] by: Fluid 1 indicates the droplet phase and fluid 2 represents the surrounding fluid (air). The density ρ and the dynamic viscosity µ fields are expressed using the phase-field variable as: The total viscosity of the non-Newtonian phase is µ t = µ s + µ p . The density satisfies the following relation [1] ∂ρ ∂t where J = − (ρ1−ρ2) 2 M ∇G. Boundary conditions imposed on the substrate are, following Jacqmin [16], Qian et al. [25], the no-slip boundary condition for the velocities: and the static contact angle θ s for the phase-field variable: where n is the outward pointing normal vector to the boundary, and f w (φ) is a function describing the fluid-solid interfacial tension.
A second-order accurate scheme is employed for the temporal discretization of Eq. (3) and (1) while a semi-implicit splitting scheme is used to treat the linear parts implicitly and the non-linear parts explicitly [9]. To avoid the High-Weissenberg number problem (HWNP), the log-conformation reformulation (LCR) of equation Eq. (5) [12,13] is used and advanced in time by a second-order total variation diminishing (TVD) Runge-Kutta method [15]. Finally, we use second-order central differences to approximate spatial derivatives, except for the advection terms in Eq. (3) and (5), where the fifth-order WENO-Z is used to improve stability and accuracy [4]

Physical model and computational domain
We consider two equal-sized initially static viscoelastic drops touching a homogeneous surface (Fig. 1) with a static contact angle of 180 o . When two adjacent droplets coalesce on a superhydrophobic surface, the formed liquid  bridge impinges on the substrate, and the merged droplet may jump above the substrate. Here, gravity is neglected, because the droplet radius is assumed to be much smaller than the capillary length and therefore capillary forces are expected to dominate.
The capillary-inertial velocity is chosen as the velocity scale [2] u ci = σ/ (ρ 1 r 0 ), and the droplets initial radius as the length scale. This gives rise to seven nondimensional numbers. Firstly, the Ohnesorge number Oh = µ 1 / √ ρ 1 σr 0 representing the relative importance of viscous to capillary-inertial forces; the Weissenberg number W i = (λ H u ci /r 0 ) representing the ratio between elastic and viscous forces; the Peclet number P e = 2 √ 2u ci r 0 η / (3M σ) representing the ratio between the advection and diffusion in the Cahn-Hilliard equation.
Furthermore, the Cahn number Cn = (η/r 0 ) is the ratio between the interface width and the characteristic length scale; β = µ s / (µ s + µ p ) is the ratio between the polymeric viscosity and total viscosity; k µ = (µ 2 /µ 1 ) is the ratio between the ambient and droplet viscosities; k ρ = (ρ 2 /ρ 1 ) between the ambient and droplet densities. The different components of the energy are scaled by σr 2 0 . In what follows, all quantities will be nondimensional unless indicated otherwise.
To quantify the role of the fluid elasticity on the droplet jumping, we will measure the mass-averaged velocity of the droplet, defined as: where Ω is the computational domain (see Fig. 1), and z the direction perpendicular to the solid substrate. We also analyze the different components of the energy during the coalescence and jumping. The total energy E T of an Oldroyd-B fluid is the sum of the surface energy E s , kinetic energy E k , and elastic energy E e , defined in phase-field framework as [5,22]: where the relationship between polymer stress τ p and the conformation tensor c is The part of the kinetic energy associated to the vertical velocity component is most relevant here as it can be associated with the jumping motion, while the rest is related to interface oscillatory motions [19]. We will therefore consider a translational kinetic energy, defined as, where v is the droplet mass-averaged velocity in z-direction, and E k, Cn , according to the guidelines in Magaletti et al. [21], Xu et al. [31] to approach the sharp-interface limit.   Thus, we choose Cn = 0.025, corresponding to a grid with N x × N y × N z = 760 × 760 × 608 grid points. This satisfactory convergence is achieved adopting the scaling between the Peclet number and Cn number suggested by [21,31].

Results
For the results presented here, the density ratio and viscosity ratio are kept constant ro k ρ = 1 839 and k µ = 1 58.8 ., following the values from the experiments in Yan et al. [33].

Newtonian droplets -comparison with experiments
The solver has been validated against several Newtonian and viscoelastic two-phase flow benchmarks in 2D and 3D [3]. Here, we compare the spontaneous coalescence and jumping motion of a Newtonian drop on a superhydrophobic surface with the experimental data of Yan et al. [33]. We choose the same physical parameters as in the experiment, see Table 1. The influence of the solid-liquid adhesion on the self-propelled jumping is negligible when the contact angle hysteresis (∆θ app ) is less than 10 o [32,7], so we ignore the contact angle hysteresis and impose a static contact angle θ s = 180 o on the bottom wall. Fig. 4 presents the experimental data [33] for the coalescence of two Newtonian drops on a superhydrophobic surface and the corresponding numerical results; the visualization shows that the numerical simulation is able to capture the coalescence and jumping process accurately in time.
The jumping velocity of the merged water (Newtonian) drop on a superhydrophobic surface is constant in the capillary-inertial region (i.e., Oh 0.1); [2] reported v j ≈ 0.2 for the water drop at 19 o C on a textured superhydrophobic surface. Later, [33] reduced the level of undesired external disturbances and measured a velocity v j ≈ 0.26 for self-propelled jumping of water drops upon  confirming that a small fraction of the released surface energy is converted into transitional kinetic associated to the jumping motion.

Viscoelastic droplets -elasticity effect
Most of the existing studies are restricted to experiments with water droplets and numerical simulation of Newtonian drops; [33] newly investigated the effect of the liquid internal hydrodynamics by conducting experiment for the self-propelled jumping upon coalescence on a superhydrophobic surface with ethanol-water and ethylene-glycol solutions. Their experiment shows that the properties of the droplet affect the coalescence and jumping process significantly. A very recent experimental study, however, addressed the effect of the drops' elasticity on the coalescence process [8], and their findings will be referred to later in this section. In the following, we investigate numerically how the droplet elasticity (Weissenberg number) influences the jumping process for different values of the Ohnesorge number. One way to vary the Ohnesorge number in experiments is to keep the physical properties constant and vary its radius, and we adopt this approach in our simulation. It should be noted that other parameters change also with the droplet radius; the parameters for each case are summarized in Appendix A.

Small Ohnesorge numbers
To isolate the effect of the droplet elasticity on the coalescence and jumping on a superhydrophobic surface in the inertial-capillary region, Oh 0.1, we set The total energy is dissipated more rapidly in the Newtonian droplet after departure, while there is less dissipation in the merged viscoelastic droplet, see Fig. 6 (b). The droplet kinetic energy is presented in Fig. 6 (c): the viscoelastic droplet has more kinetic energy so that it undergoes larger shape oscillations than the Newtonian droplet. The oscillations of the viscoelastic droplet are due to its elasticity and independent of the surface tension, see [17]. The viscoelastic droplet oscillates even at the large Ohnesorge numbers corresponding to a highly viscous drop [17].An extensional flow occurs when the two viscoelastic drops are coalescing, so polymer chains stretch and store elastic energy during the coalescence process, see Fig. 6 (c).
Summarizing, the average and jumping velocity are not considerably affected by the elasticity of the drops in the inertial-capillary regime. This result is in line with the experiments of Dekker et al. [8], where elasticity did not considerably influence the coalescence process in the inertial-capillary regime. However, quantitatively we found that the oscillations are promoted by elasticity, and that the average velocity decays less rapidly after the droplet departure from the superhydrophobic surface. The results from the simulations are reported in Fig. 7. First, we note that the merged viscoelastic droplet gains much larger average velocity than the Newtonian one during the coalescence process, as seen by comparing the To gain a better understanding on the effect of elasticity on the self-propelled jumping, the flow field is visualized on the central XZ and Y Z planes inside the merging drops, see Fig. 9 (a). As concern the Y Z plane, shown in the top row, we depict the interface, identified by the φ = 0 value of the order parameter (red contour), velocity field (arrows), and trace of conformation tensor (colormap) for merging viscoelastic drop at three times, together with the the interface of merging Newtonian drop (white contour). This illustrates how the shape of Newtonian and viscoelastic droplets differ, and confirms the localisation of polymeric stresses at the merging cross-section. Moreover, the bottom row of the same figure displays contoues of the trace of the conformation tensor for the viscoelastic drops on the XZ plane at the same dimensionless times as in Fig.   9 (b). The data indicate that polymers are most elongated prior to jumping, near the bottom wall. A possible physical explanation for the polymer effect can be as follows.
When the two initially static drops start to merge, the liquid moves driven by the capillary pressure towards the center of the expanding bridge. Then, due to the conservation of mass, the liquid is forced to move in the transverse XZ plane, see the velocity field of the merging viscoelastic drops in Fig. 9 (b) at t = 0.24.
This flow causes the polymer molecules to stretch in the XZ plane and produce extra elastic stresses, which push the liquid bridge connecting the two polymeric drops to move faster. When the liquid bridge interacts with the substrate at t ≈ 1, the liquid is induced to move upwards due to the impermeability of the surface. This upward flow converges toward the XZ plane and causes the polymers to stretch mainly in the vicinity of the substrate, as shown by the trace of conformation tensor at t = 1.96 in Fig. 9

Conclusions and outlook
In the present study, three-dimensional direct numerical simulations have been performed to study the self-propelled jumping of two equal-sized polymeric drops on a superhydrophobic surface with contact angle of 180 o . The results demonstrate that the viscoelastic properties of the droplets have a significant These results are obtained with a typical value in the literature for the polymeric viscosity ratio, i.e. β = 0.1; here, we also observe that the merged viscoelastic drop behaves like a Newtonian drop when β 0.7. The larger β corresponds to larger retardation times in our simulation, so that the polymer molecules do not have enough time to stretch. Finally, the shear-thinning effect is found to be negligible in the coalescence and jumping process of two equal-sized drops on a superhydrophobic surface.
Our results indicate that the elasticity of the droplet can change the viscous cutoff radius (for example 30µm for water) for the self-propelled jumping of drops on superhydrophobic surfaces. Thus, it is expected that polymeric drops jump from a superhydrophobic surface upon their coalescence with radii below the viscous cutoff radius for Newtonian drops at the same Ohnesorge number.
Moreover, the merged polymeric drop oscillatory motion is promoted by the elasticity of the drop in both inertial-capillary and viscous-capillary regimes.
In this study, we have neglected the contact angle hysteresis, assuming it to be smaller than 10 o ; however, superhydrophobic surfaces may have large contact angle hysteresis, which might play an important role in the case of polymeric drops. Studying the effect of the contact angle hysteresis is one of the possible extensions of this work. Appendix A.

Acknowledgments
In this Appendix, we report the values of the non-dimensional numbers used in the simulations.

Case
Droplets