Validating the Transition Criteria from the Cassie–Baxter to the Wenzel State for Periodically Pillared Surfaces with Lattice Boltzmann Simulations

Abstract

Microfabrication techniques allow the development and production of artificial superhydrophobic surfaces that possess a precisely controlled roughness at the micrometer level, typically achieved through the arrangement of micropillar structures in periodic patterns. In this work, we analyze the stability and energy barrier of droplets in the Cassie–Baxter (CB) state on such periodic patterns. In addition, we further develop a transition criterion using the CB equation and derive an improved version which allows predicting for which pillar geometries, equilibrium contact angles, and droplet volumes the CB state switches from a metastable to an unstable state. This enables a comparison with existing experiments and three-dimensional multiphase Lattice Boltzmann simulations for different pillar distances, two contact angles, and two droplet volumes, where a good agreement has been found.

Introduction

Superhydrophobic surfaces have garnered significant attention because of their outstanding wetting properties and various potential technological applications.¹ Their production has significantly advanced due to improved microfabrication techniques that allow for controlled local roughness at the micrometer scale. Often, these roughnesses are created by periodically arranged micropillar structures.²⁻⁷ Among other applications, this is especially interesting for membrane distillation; Xiao et al.⁷ showed that micropillars and hydrophobic coatings have the capability to decrease scaling and extend the operational duration of a membrane. While the development of artificial superhydrophobic surfaces has made progress, certain fundamental aspects regarding the wetting behavior of these surfaces still raise debate. One particular unresolved issue concerns the transition from the superhydrophobic state,¹ known as the Cassie–Baxter (CB) state,⁸ for which gas is trapped within the surface roughness, to the completely wet state, known as the Wenzel (W) state.⁹ During the transition, the apparent contact angle is decreased, and most of the desirable superhydrophobic characteristics of the surface are lost.^4,7

It is generally accepted that for many common geometries, the CB state is metastable or unstable. If an energy barrier between the two states is overcome, the transition to the energetically stable Wenzel state occurs.¹⁰⁻¹³ This energy barrier can already be overcome due to small perturbations such as surface waves created by gravitational or capillary forces at the liquid–air interface.⁴ The transition behavior is often approached by describing the Gibbs free energy for the problem. Based on the function for Gibbs free energy, different transition criteria have been formulated.^{3,6,10,12,14,15} An intermediate state, where the liquid wets the pillar walls but does not reach the bottom, also exists.⁶

Jung and Bhushan^4,5 observed in experiments that beyond certain pillar distances, droplets transition from the CB into the Wenzel state. Jäger et al.¹⁶ compared Lattice Boltzmann (LB) simulations to the experimental study from Jung and Bhushan⁴ and observed a qualitatively similar behavior for the transition from CB to Wenzel state but observed a much lower critical pillar distance. Zheng et al.¹² and Patankar¹¹ already developed a transition criterion based on a critical pressure difference and the Young–Laplace pressure difference. In this work, we try to answer the question of which quantities determine the critical pillar distance and aim to derive a formula that allows us to predict the critical pillar distance for a given droplet volume. This allows a comparison to experiments and simulations. In wetting experiments where a droplet is placed on a periodically pillared surface, one can often control the droplet volume; therefore, it is beneficial to know if and how the droplet volume affects the critical pillar distance.

Two mechanisms were identified for this transition: (a) Jung and Bhushan⁴ argued that the transition occurs if the droop/sag (δ) at the bottom side of the droplet is much greater than the pillar height (h_p). (b) Patankar¹¹ derived a transition criterion based on the pressure differences: even before the droop touches the pillar bottom, the Young–Laplace pressure difference Δp between the inside of the liquid droplet and the surrounding air pressure p₀ can result in penetration of liquid between the pillars, ultimately leading to wetting of the bottom between the pillars. Murakami et al.⁶ also observed that the Young–Laplace pressure difference is much greater than the hydrostatic pressure on the bottom side of the droplet because of gravity. For small droplets, this suggests that the Young–Laplace pressure difference is an important aspect for the transition from the CB to the Wenzel state. The two aforementioned transition mechanisms can both independently of each other lead to a transition from a CB to the Wenzel state.¹¹

The paper is structured as follows: we first give a theoretical background to the problem, particularly regarding the work of Zheng et al.¹² and Patankar,¹¹ who derived a condition for the critical pressure for an arbitrary periodically pillared surface. We then give a brief overview of the LB multiphase model that is being used to characterize the stability of the CB state. We further develop the condition from Patankar¹¹ and Zheng¹² using the CB equation⁸ and derive a condition which allows predicting for which pillar geometries, equilibrium contact angles, and droplet volumes the CB state switches from a metastable to an unstable state. This condition additionally allows comparison of the theoretical predictions to multiphase LB simulations and existing experiments for multiple pillar geometries.

Theoretical Background

A periodically pillared surface is characterized by A, the cross-sectional area of the pillar, the rectangular unit cell area A_c which is repeated periodically in x- and y-directions, the pillar perimeter l, and the pillar heights h_p. For a quadratic unit cell with pillar distance P (see Figure 1), A_c = P × P holds. For pillars with a circular cross section, l = 2πr_p and A = πr²_p and for pillars with quadratic cross section, l = 4d_p and A = d²_p with pillar side length d_p.

Figure 1.

(a) Aerial view (reprinted with permission from Jäger et al.¹⁶ CC BY 4.0) and (b) corresponding 3D structure of a periodically pillared surface with a quadratic unit cell and round pillars.

Bond number (Bo) is a dimensionless quantity that characterizes the balance between gravitational and capillary forces¹⁷ and is expressed through the subsequent equation: Bo = ΔρgL²/γ, where g denotes the gravitational acceleration, Δρ is the difference in the density of liquid and gas, γ denotes the surface tension, and L represents the characteristic length. In the context of liquid droplets, the characteristic length L corresponds to the droplet radius. When the bond number is much smaller than 1 (Bo ≪ 1), gravitational effects can be considered negligible.

Wenzel and CB Equation

The equilibrium contact angle α₀ on a flat surface of the same material and the apparent contact angle β_W for a droplet in the Wenzel state (Figure 2) are connected through the Wenzel equation.⁹ For details, see the Appendix.

Figure 2.

(a) Droplet in the Wenzel state and (b) droplet in an intermediate state. α is the contact angle between the liquid and the vertical pillar wall.

In the case of a droplet in the CB state,⁸ in contrast, the equilibrium contact angle α₀ for a similar surface is connected to the apparent contact angle β_CB through the subsequent equation

1

If gravitation is negligible, the shape of a droplet in the CB state can be approximated by a spherical cap (see Figure 3), and the droplet radius r_d is given by

2

where V_d is the volume of the droplet (spherical cap). Equation 1 can be used to eliminate the apparent contact angle of the droplet β_CB in the equation for the droplet radius.

Figure 3.

Droplet in the CB state. Full droplet in (a) and the droplet bottom in (b).

Stability of the CB and Wenzel States

Based on an equation for the Gibbs free energy by Patankar¹⁰ (shown in the Appendix), Zheng¹² derived a condition to evaluate which of the two droplet states (CB or Wenzel) is the stable state

3

If that is true, the Wenzel state is stable, whereas the CB wetting mode is meta stable or even unstable.^12,13 In this study, the emphasis is directed toward situations in which the Wenzel state is the stable state with the analysis focusing on which geometries the droplet in the CB state switches from a meta stable to an unstable state.

Young–Laplace Equation

According to Laplace,¹⁸ the pressure difference across a liquid–gas interface Δp is linked to the mean curvature of the interface and the liquid surface tension γ. In a static scenario, where external forces such as gravity are negligible, the pressure difference at the interface is independent of the spatial location. Consequently, the pressure difference Δp is constant everywhere along the liquid–gas interface, including both the liquid–gas interface on top of the droplet and the one between the pillars.

If gravitation is negligible, a droplet in the CB state can be approximated by a spherical cap of radius r_d, and therefore one can calculate the pressure difference according to Laplace¹⁸ by

4

Transition Criterion

Critical Pressure and Energy Barrier for Periodically Pillared Surfaces

Zheng et al.¹² used a force balance approach to derive a general expression to calculate the critical pressure for a liquid film on an arbitrary periodically pillared structure and found good agreement with numerical simulations. Jäger et al.¹⁹ were able to confirm their results with multiphase LB simulations. In the critical state, according to¹²

5

Rearranging gives¹²

6

Patankar¹¹ concluded that the CB state transits to the Wenzel state if the Young–Laplace pressure difference (Δp = 2γ/r_d) exceeds the critical value Δp_crit (shown in eq 6) and showed that the same condition can be derived when considering the energy barrier and the work done when the liquid moves down the pillar.

If the pressure difference between the liquid above the pillars and the air below exceeds the critical pressure difference Δp_crit, the liquid starts to move down the pillars. Based on the work needed to move down the liquid, Patankar³ and Zheng¹² calculated the magnitude of the energy barrier per area from CB to Wenzel state by

7

Sag Transition

Jäger et al.⁴ and Patankar¹¹ estimated the droop/sag δ by a spherical approximation of the liquid gas interface and obtained . Here, is the diagonal distance between two pillar centers, and r_p is the pillar radius. For quadratic pillars, the previous equation can be modified as

8

A transition to the Wenzel state occurs when δ ≥ h_p.^4,11 Patankar¹¹ concluded that the sag transition is likely to occur only when the Wenzel state leads to lower energy than the CB state.

Numerical Method

To determine the interface of a droplet on top of a rough hydrophobic surface, this study uses a LB multiphase model with additional interactions between fluid and solid. This enables the adjustment of the equilibrium contact angle (α₀) between a flat solid surface and a liquid droplet and allows simulating the liquid–gas flow. Mesoscopic methods, like LB, provide viable predictions for intricate 3D structures, both on the micro- and nano-scales. These methods can be categorized into particle- and lattice-based approaches.²⁰ The LB framework shows promise due to its ability to handle complex boundary conditions, including rough surfaces.²¹ Moreover, it is well-suited for parallelized calculations on GPUs,^20,22,23 enabling simulations in both the continuum and the slip flow regime.²⁴ In contrast to classical continuum based solvers, the fluid is described by a density distribution function f(x, v, t) within the LB framework. f(x, v, t) can be seen as an extension to the mass density, also containing information about the velocity distribution.

Similar to previous works, e.g.,^16,21 an isothermal multiphase LB method is employed in this study to predict the shape and state of liquid droplets on micropillar structures and to validate our theoretical findings. A D3Q27 lattice is used, which employs 27 discrete velocities (c_i, i = 0....26) at every lattice node within a three-dimensional (3D) spatial framework.²⁵ During one time step of length Δt, the discrete distribution function f_i(x, t) is streamed to its neighboring nodes according to

9

Typically, the LB method uses the Bhatnagar–Gross–Krook approach to approximate the collisions between the fluid particles.²⁶ The viscosity is associated with relaxation time τ. Macroscopic quantities like velocity and density are derived by a summation over the discrete distribution function f_i and velocities c_i at a given point in space.²⁵ The LB method employs a dimensionless approach, where time step (Δt = 1 ts), lattice spacing (Δx = 1lu), and particle mass (m = 1 mu) are set to one unit each, providing stability and faster convergence as commonly done for LB simulations.²⁷ The method reproduces the Navier–Stokes equation in the hydrodynamic limit.²⁵

10

11

In this study, the guided equilibrium model^25,28 was employed to calculate the equilibrium distribution f^eq_i. Additional implementation details are given in refs (16), (25), and (28). The selected multiphase model for this research was the Shan–Chen model,^27,29,30 which incorporates a pseudo potential to account for the interactions between fluid particles

12

G represents the fluid–fluid interaction strength and controls the surface tension of the fluid. The fluid–solid interaction is modeled by including an additional interaction force F_ads, where s(x) is a step function that takes the value 0 for fluid voxels and 1 for solid voxels.^27,31

13

The model parameter G_ads exhibits a linear correlation with the equilibrium contact angle α₀ measured on a flat solid surface, as demonstrated in a benchmark presented in ref (16) Therefore, this enables the manipulation of the hydrophobicity by adjusting G_ads. Detailed information is given in refs (16), (27), and (29).

Furthermore, a nonslip boundary condition (achieved by a bounce-back of the distribution function) was utilized between the fluid and solid surfaces. Unless specified otherwise, periodic boundary conditions were applied in all directions at the boundaries of the computational domain.

For all LB simulations conducted here, we employed G = 120.0, which leads to a surface tension of γ = 14.04 mu ts^–2 in LB units. This results in ρ_l = 524.4 mu lu lu^–3 (liquid density) and ρ_g = 85.7 mu lu^–3 (gas density) for a flat interface. To realize an equilibrium contact angle of 100°, we used G_ads = −174.52.

As the liquid–gas interface shape is solely influenced by the contact line of liquid gas and solid and the parameter q≔Δp/γ [lu^–1], it is enough to make sure that q corresponds to the physical problem to capture the correct interface shape. To dimensionalize the problem, the following relationship is used: Δp^SI/γ^SI = q^SI = q/Δx^SI, where Δx^SI represents the distance between two lattice nodes in SI units. As a consequence, for a droplet on top of the pillar structure, the interface shape is only determined by the droplet radius, the geometry of the pillars, and the equilibrium contact angle α₀. Equation 14 shows the convergence criterion for all simulations conducted in this work, fulfilled with ε = 1 × 10^–6.

14

All simulations were conducted using an adapted high-performance computing code from Safi.³² All 3D plots presented in this study were generated using ParaView³³

Results and Discussion

Energy Barrier and Transition Criterion for a Finite Droplet

If one places a finite droplet on top of a periodically pillared surface, the Young–Laplace pressure difference effectively lowers the energy barrier per area between the CB and Wenzel states, and a lower external pressure needs to be applied to achieve the transition. Therefore, based on the results from Patankar¹¹ and Zheng,¹² we propose a modified equation to calculate the energy barrier per area for droplets on periodically pillared surfaces

15

Compared to eq 7, which is only applicable if the Young–Laplace pressure difference can be neglected, e.g., for a film on an infinite periodically pillared surface, the new eq 15 takes into account the curvature and is therefore applicable for a finite droplet. Equation 6 was similarly derived for a film on an infinite periodically pillared surface but still remains applicable if the droplet is in contact with multiple pillars. Therefore, it is still used for the calculation of Δp_crit. However, a droplet only in contact with a single pillar in the absence of gravity will always remain in a stable state; therefore, eq 6 is not applicable in this case.

It is crucial to keep in mind that this is only valid in the case where gravitation can be neglected, since in this case the pressure difference at the droplet interface does not depend on the direction and the droplet has the shape of a spherical cap with a radius r_d. As the bond number approaches 1, the assumption of a spherical cap is no longer reasonable. And an additional term accounting for gravity would enter the equation for the energy barrier and further lower the magnitude of the barrier.

Therefore, we can conclude that in the absence of gravity the energy barrier for a droplet depends not only on h_p, γ, l, A, A_c, and α₀ but also on r_d, the radius of the droplet. The smaller the droplet becomes, the lower the barrier gets. According to the criterion from Patankar¹¹ (Δp > Δp_crit), an increase of the pressure difference, Δp = 2γ/r, increases the angle α (see Figure 3) until α₀ is reached and eq 4 equals eq 6. When this critical point is reached, the energy barrier vanishes completely, and the CB state switches from a metastable to an unstable state. From this, one can derive a condition for the critical radius of the droplet

16

It immediately becomes clear that when gravity is absent, the condition for the critical droplet radius in eq 16 is independent of the surface tension.

In experiments, one can often control the volume of the droplet. Therefore, we chose to use the relation for the droplet volume and the droplet (spherical cap) radius in eq 2 to reformulate eq 16.

17

Using the CB (1 we can eliminate the apparent contact angle β_CB. With A_c = P × P for a quadratic periodically pillared surface, we then gain a condition (eq 18) for the critical volume which only depends on l the perimeter of the pillar, P the distance between two pillars, A the top area of the pillars, and the equilibrium contact angle α₀.

18

Iterative root-finding procedures can be used to solve eq 18 numerically for the other variables (l, P, A, α₀). We used this procedure to calculate P_crit(α₀, l, A, V) numerically and compare the predictions of condition 18 to experimental results from Jung and Bhushan⁴ and multiphase LB simulations from Jäger et al.¹⁶ (see Figure 4a) and new multiphase LB simulations for additional geometries (see Figure 4b,c). In Jung and Bhushan’s⁴ experiments, a droplet of 5 μL was placed on a periodically pillared surface; then they observed whether the static droplet ended up in either the CB or Wenzel state and measured the apparent contact angle. The experiment was repeated for different pillar distances P. Therefore, Jung and Bhushan⁴ were able to determine a range of P for which a transition from CB to Wenzel state takes place.

Figure 4.

Multiphase simulation results LB1 (Adapted with permission from Jäger et al.¹⁶ CC BY 4.0) in (a). Multiphase simulation results LB2 in (b) and LB3 in (c). The apparent contact angle β for the CB or Wenzel state as a function of the pillar distance P. The blue area marks the CB regime and the red area the Wenzel regime.

Numerical and Experimental Validation

Previously, Jäger et al.¹⁶ compared the results from Jung and Bhushan⁴ (Exp. 1 and 2) to multiphase LB simulations (LB1) with a similar geometry and equilibrium contact angle but a smaller droplet of only 1 μL volume due to computational limitations. Simulation results from Jäger et al.¹⁶ (LB1) are shown in Figure 4 (a). They observed the same qualitative behavior in the simulations, but the transition from CB to Wenzel state occurred for a smaller P compared to the experimental study by Jung and Bhushan⁴ (see Table 1). We also added the theoretical predictions for P_crit based on eq 18 in Table 1.

Table 1. Observed and Theoretically Predicted Critical Pillar Distance P_crit for Different Experimental and Numerical (LB1, LB2, LB3) Setups.

	Exp. 1 Jung4	Exp. 2 Jung4	LB116	LB2	LB3
α0 [°]	109	109	109	100	100
l [μm]	15.708	43.982	49.62	66.16	49.62
A [μm2]	19.635	153.938	153.88	273.57	153.88
Vd [μL]	5	5	1	1	1
hp [μm]	10	30	33.08	33.08	33.08
stable state	Wenzel	Wenzel	Wenzel	Wenzel	Wenzel
δ [μm]	2	4	5	3	2
Bo	0.24	0.24	0.083	0.083	0.082
observed Pcrit [μm]	44–60	125–167	70–83	54–58	45–54
[lu]			17–20	13–14	11–13
predicted Pcrit [μm]	52.27	88.03	71.87	61.96	53.17

In the LB simulations, the droplet transitions to a different state even in the absence of gravity and with zero initial velocity because, after initializing a spherical droplet just above the pillar structure, small oscillations occur until the liquid and gas density converged to their equilibrium values. This is enough perturbation to make the transition happen if the critical state is reached. These dampened oscillations are typical for LB multiphase simulations and vanish over time.

In this paper, we investigate two additional pillar geometries (LB2 and LB3) with the same multiphase LB method. The corresponding results for the droplet state and the contact angle are shown in Figure 4b,c. An example of a droplet in the CB or the Wenzel state is given in Figure 5. For all LB simulations in this paper, we used rectangular pillars of either A = 3 × 3 lu³ or A = 4 × 4 lu³. Here, Δx = 1 lu ≡ 4.135 μm was chosen to match the geometries in the experimental study from Jung and Bhushan.⁴ A spherical droplet of volume 1 μL is initialized right above the pillar structure with zero velocity according to eq 19

19

for every simulation with r = 150lu. This leads to a volume of V_d ≈ 1.0 μL.

Figure 5.

Droplet in contact with geometry LB2. In (a) is shown a droplet in the CB state for P = 54 μm and in (b) is shown a droplet in the Wenzel state P = 58 μm.

Total domain size was about 370 × 370 × 370 lu³ ≡ 3.58 mm³ for each simulation. For all the setups which we investigated numerically, we ensured full contact with the droplet in the Wenzel state for at least 9 pillars.

In Figure 6 the dependence of the critical volume on the pillar distance P according to eq 18 is shown for the two experiments and the LB simulations. The intersection of V_d,crit(P) with a given droplet volume V_d gives a predicted value for P_crit. Table 1 contains the experimental and simulation parameters (experiment 1, experiment 2, and LB simulations LB1, LB2, and LB3), as well as the observed and theoretically predicted values for P_crit. For all analyzed geometries, according to eq 3 the Wenzel state is the stable configuration, and since the sag is much smaller than the pillar heights (δ < h_p) the transition occurs due to the Young–Laplace pressure difference (see Table 1). The bond number was calculated to be below 1 for all droplets.

Figure 6.

Dependence of the critical droplet volume V_d,crit on the pillar distance P according to eq 18 for different experimental⁴ and numerical (LB1,¹⁶ LB2, LB3) setups.

Comparing these results with the two experiments by Jung and Bhushan⁴ and LB simulations (see Table 1), we found very good agreement for experiment 1 and the LB simulations; the predicted critical values P_crit based on condition 18 lie within the observed range; only for LB simulation 2 the predicted value was slightly high. For the setup from experiment 2, the predicted value (P_crit = 88.03 μm) is below the range (125–167 μm) found in the experiment by Jung and Bhushan.⁴Figure 7b,d show the Wenzel and CB states for LB2, respectively.

Figure 7.

Slice of a 3D droplet (in red) on a periodically pillared surface from the present LB simulation results. For all setups shown in this figure, A = 4 × 4 lu³ holds, α₀ = 100°.

We also simulated cases with equilibrium contact angles larger than 109°. However, this results in a state in which the droplet is in contact with only a few pillars. Therefore, these results are not comparable to the theoretical transition criterion.

It is important to mention that for the present LB simulations, a change in the pillar height can also change the droplet state. For example, for the LB2 setup, if we increase the pillar height to h_p = 49.62 μm and rerun the simulations, no transition to the Wenzel state was observed for P = 58 μm (see Figure 7a). Whereas for pillars with a height h_p = 33.08 μm, as reported in Table 1, we observed a transition to the Wenzel state for P = 58 μm (see Figure 7b). Increasing P for h_p = 49.62 μm, we can also observe a transition to the intermediate state (see Figure 7c).

To compare the influences of the different variables on the critical droplet volume V_d,crit, we calculated the partial derivatives of V_d,crit in eq 18 with respect to the variables A, l, P, and α₀ and listed the magnitude of the derivatives in Table 2.

20

21

22

23

Table 2. Influence of the Different Variables on V_d,crit.

				A	l	P	α0	Vcrit
[μm–2]	[μm–1]	[μm–1]	[1/°]	[μm2]	[μm]	[μm]	[deg]	[μL]
–0.00020	–0.03431	0.02077	–0.76282	19.63	15.71	52.27	109	5
–0.000071	–0.01225	0.01249	–0.76285	153.94	43.98	88.03	109	5
–0.00011	–0.01086	0.01546	–0.76288	153.88	49.62	71.87	109	1

As shown in Table 2, we found that the equilibrium contact angle has by far the largest influence among all of the parameters. A 3° higher equilibrium contact angle for experiment 2 (α₀ = 112°) leads to a predicted P_crit = 94.3 μm. Therefore, an error in the equilibrium contact angle measurements alone can not explain the difference between observation and theoretical prediction for experiment 2, but it can explain the small difference observed in the LB2 simulations (see Table 1). A possible explanation is that in experiment 2 the droplet switched and maintained the intermediate state instead of fully transitioning to the Wenzel state when reaching P_crit. This hypothesis is supported by the equation for the energy barrier (eq 15), which increases linearly with h_p. Experiments 1 and 2 have the same droplet volume, but the pillar heights differ by a factor of 3, which could explain the difference in predictions between experiments 1 and 2.

We also observed in the LB simulations that the transition to the Wenzel state depends on h_p. Even if the energy barrier in eq 15 completely vanished when P_crit is reached, one still needs some perturbation to make the transition to the Wenzel state happen. In the absence of gravity, as for our simulation, the oscillation of the droplet after the initialization provides the perturbation needed to make the transition happen. If the pillars are too high the perturbations dampen out before the transition to the Wenzel state is completed, and the droplet remains in the Wenzel or intermediate state. During the transition and especially for high pillars, the droplet radius and therefore the Young–Laplace pressure difference could decrease and the critical pressure might not be reached anymore, and the droplet gets stuck in the intermediate state.

Conclusions

Based on the work of Patankar¹¹ and Zheng et al.,¹² we have derived an equation for the energy barrier between CB and Wenzel states for droplets on periodically pillared surfaces. Due to the curvature of the droplets, there is a pressure difference between the droplet and the surrounding air, which lowers the transition barrier from the CB to the Wenzel state. The smaller the droplet becomes, the lower the transition barrier becomes. Furthermore, we derived a condition which allows to calculate the pillar geometries, equilibrium contact angles, and droplet volumes for which the CB state becomes unstable, which is especially useful if one wants to compare the theoretical findings from Patankar¹¹ and Zheng¹² to experimental results or simulations. The criterion in eq 16 is applicable for droplets on any periodically arranged roughness with vertical lateral surfaces and a flat horizontal top surface if gravitation is negligible and the area of the unit cell A_c is much smaller than the droplet cross section.

Based on theoretical considerations, we can conclude that the droplet size has a significant effect on the stability of the CB state. The surface tension plays a role only in the magnitude of the energy barrier. These findings are validated with experimental results and 3D multiphase LB simulations for two equilibrium contact angles (100 and 109°) and two droplet volumes (1 and 5 μL). The emphasis of this work was on the influence of pillar spacing, for which we found good agreement. To further validate the predictive power of 18 more experiments and LB simulations for different droplet volumes and different equilibrium contact angles are needed.

Glossary

Nomenclature

Δp_crit

critical pressure difference

γ

surface tension

q

Δp/γ

r_p

pillar radius

α

contact angles between the liquid and the pillar wall

α₀

equilibrium contact angle between liquid and solid

P

pillar distance (periodicity in x- and y-directions)

A_c

area of the periodically repeated cell (A_c = P × P)

A

top area of the pillar

l

perimeter of the pillar

LB

lattice Boltzmann

p

pressure

ρ

density

ρ_l

liquid density

ρ_g

gas density

F

force

Δx

lattice spacing

Δt

time step

m

fluid particle mass

G_ads

LB parameter to tune the equilibrium contact angle

G

LB parameter to control the fluid–fluid interaction strength

τ

relaxation time

ν

kinematic viscosity

c_s

speed of sound

g

gravitation acceleration

v

velocity

c_i

discrete lattice velocity

Appendix

Wenzel equation

The Wenzel equation⁹ links the apparent contact angle in the Wenzel state to the equilibrium contact angle α₀.

24

Here, R_f is the roughness factor defined by the ratio of rough to planar surface areas. For periodically repeated pillars, R_f = 1 + A_l/A_c, where A_l = l × h_p is the lateral surface of the pillar.

Gibbs Free Energy for a Droplet

Patankar¹⁰ derived an equation for the Gibbs free energy E_G of a droplet with the shape of a spherical cap in contact with the pillar structure in any state.

25

Since E_G is a strictly monotonically increasing function of −cos(β) in eq 25, the following relation needs to be true for the Wenzel wetting mode to be associated to a lower Gibbs free energy than the CB:^3,10,12,13 – cos β_W < – cos β_CB

Author Contributions

Conceptualization, T. Jäger; methodology, T. Jäger, N. I. Prasianakis, and S. Leyer; software, T. Jäger, A. Mokos, N.I. Prasianakis; validation, T. Jäger, A. Mokos, N. I. Prasianakis, and S. Leyer; formal analysis, T. Jäger; investigation, T. Jäger, A. Mokos, N. I. Prasianakis, and S. Leyer; resources, N. I. Prasianakis and S. Leyer; data curation, T. Jäger; writing–original draft preparation, T. Jäger ; writing–review and editing, T. Jäger, A. Mokos, N. I. Prasianakis, and S. Leyer; visualization, T. Jäger; supervision, A. Mokos, N. I. Prasianakis, and S. Leyer; project administration, T. Jäger, N. I. Prasianakis, and S. Leyer; funding acquisition, N. I. Prasianakis and S. Leyer. All authors have read and agreed to the published version of the manuscript.

University of Luxembourg, Paul Scherrer Institute. This research received no external funding.

The authors declare no competing financial interest.