On well-fabricated self-cleaning substrates, droplets roll off at a small substrate tilt angle or bounce up easily when dropping on the substrates. The self-cleaning property is usually termed as superhydrophobicity, the researches on which have been focusing on various aspects, including the liquid wetting behaviors on chemically or topologically modified substrates,^[1,2] the droplet contact angle (CA) advancing and receding,^[3,4] the biomimetic fulfilment of the lotus-leaf effect via various novel surface roughing techniques,^[5,6] and so on. One of the most widely adopted superhydrophobic substrates is the periodic microscopic stripe pillar array. Numerous experimental studies and computer-assisted simulations have investigated the relation between the pillar repeating mode and the droplet repellency.^[7,8] What is more, it is revealed that the surface modification (physically roughing or chemically decorating) of individual pillar top facets and sidewalls efficiently improves the pillar array’s superhydrophobicity.^[5,9]

There are two requirements that superhydrophobic substrates should satisfy: the droplet CA should be larger than 150° in general;^[1] and the difference between the advancing CA and the receding CA, i.e. CA hysteresis, should be small. On pillar array substrates, generally, droplets are in two states: the Cassie state^[11], and the Wenzel state.^[12] The droplet CA is smaller for the Wenzel state, and CA hysteresis in the Wenzel state is larger than that in the Cassie state. There exists an apparent Gibbs pinning energy for the Cassie–Wenzel state transition,^[13] and once the Wenzel state is approached, the droplet rolling off on the pillar array is prohibited since the droplet adheres to the substrate bottom rather than the pillar top.^[14,15]

At high temperatures, the particles on the droplet surface evaporate and then re-condense on the pillar sidewalls, which lowers the Cassie–Wenzel state transition energy barrier,^[15,16] resulting in the Wenzel state prevailing, and undermining the self-cleaning performance of the pillar array. Therefore, it is necessary to explore the droplet vapor–liquid–solid triple-phase contact line (TPL) deforming behavior at the pillar edges during the Cassie–Wenzel state transition process in a more detailed way taking the droplet evaporation effect into consideration. As the first step, we explore the droplet critical wetting behavior at the edge defect of a single stripe pillar by molecular dynamics (MD) simulation, and the droplet evaporation effect is included by adopting the Lennard–Jones (LJ) droplet (simple liquid) as the research candidate, with the particles on the droplet surface unstable above the evaporation temperature.

Specifically, the droplet critical wetting behavior is where the droplet TPL will deform and cross over the pillar edge defect to wet the pillar sidewall at a certain critical droplet size V_c, which has been explored experimentally with different kinds of liquids by increasing the droplet volume through a center hole drilled in the pillar.^[17] Correspondingly, the critical wetting angle θ_c is depicted in figure 1, which is the acute dihedral angle between the droplet tangential plane and the plane parallel to the pillar sidewall at a certain point of the TPL. The critical wetting angle θ_c equals the droplet equilibrium CA θ_e on the planar substrates with the same surface properties as the pillar sidewall, and this equivalence relation was first discussed geometrically by Gibbs.^[18]

Fig. 1.

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	Fig. 1. Side view of a schematic representation of the droplet critical wetting process on a single stripe pillar.

In Fig. 1, α is the angle between the pillar top facet and pillar sidewalls, which indicates the pillar sharpness (a sharper pillar corresponds to a smaller α). In this work, α is 90°, and we believe that the calculations with other pillar tilt angles will lead to the same conclusions.

The equilibrium CA θ_e on the planar substrates can be computed from the extended Young’s equation:^[19] γ_LVcos θ_e = γ_SV − γ_SL − τ/r, where γ_LV, γ_SV, and γ_SL are the liquid–vapor, solid–vapor, and solid–liquid interfacial tensions, respectively, while τ is the line tension of the droplet contact line with 1/r. as its curvature. Since the curvature of a cylindrical droplet contact line on the stripe pillar is zero, the droplet size effect included in the line tension term is eliminated, and we then use cylindrical droplets to measure θ_e. For the same reason, the droplet on the pillar top is also cylindrical along the y direction (the axes are shown in Fig. 2) for the measurement of the critical wetting angle θ_c. We need to mention that the planar substrates and pillars are infinite along the y direction to support the cylindrical droplets.

Fig. 2.

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	Fig. 2. The ρ* isodensity contours of the droplet on the planar substrate. From outer to inner, ρ* equals 0.4, 0.5, and 0.6, respectively.

Considering that the droplet resting on the soft substrates introduces a complex elastic displacement field,^[20] which deforms the TPL and brings uncertainties to the CA measurement, the pinning force calculation would be more complex for the droplets on the flexible pillars.^[21] To exclude the TPL deformation effect, the planar substrates and pillars are kept rigid during the MD simulation in this work to avoid influences from the asymmetric strain distribution of the substrates.

In the following, θ_c and θ_e are compared for the droplets with different LJ potential parameters. In this way, the critical wetting properties of the droplets at sharp edge defects are explored.

The width of the stripe pillar is 51 platinum face-centered cubic crystal (fcc) layers, the pillar length is 21 fcc layers, and the height is 20 fcc layers, which is thick enough to avoid the interactions of particles from the vertical neighboring simulation cells. For the same purpose, the thickness of the planar substrate is also 20 fcc layers.

The interactions between all the particles are described by the LJ 12–6 potential

All droplets with different LJ solid–liquid parameters are stabilized after 3 × 10⁶ time steps, and then the droplets reach the equilibrium state on the planar substrate. The particle number density field is computed over 1.5−3 × 10⁶ time steps by time and space averaging until the density profile converges. In the computing procedure, the mass centers of the droplets are fixed to accelerate the density profile’s convergence.

To measure the equilibrium CA θ_e, we use a circular fit of the droplet Gibbs dividing surface between the liquid and vapor phases^[23] at ρ* = 0.5. ρ* is the post-treated density contour, defined as . Here, ρ is the untreated density at a certain point of the density field, while ρ_V and ρ_L are the bulk vapor density and bulk liquid density, respectively. To obtain the measurement deviation, the circular fitting of the droplet profiles with ρ* equal to 0.4 and 0.6 is also performed. These three typical density contours are depicted in Fig. 2.

The liquid density oscillation near the substrate surface results in the droplet layering as depicted in Fig. 2. To measure θ_e, the laying part is ignored, and the circular line is extrapolated to the substrate surface as done in Refs. [23] and [24]. The measurement results of θ_e will be presented in Section 3.

In Fig. 3, the droplet density oscillation pattern along the z direction from the substrate surface is depicted. It is computed by time and space averaging along the cylindrical droplet midline at which the droplet mess center is fixed. From Fig. 3, the bulk liquid density ρ_L is obtained at a height of 7σ_LL (2.38 nm) from the substrate surface along the cylindrical droplet midline, where the droplet density is stabilized. The bulk vapor density ρ_V is obtained away from the droplet.

Fig. 3.

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	Fig. 3. The density oscillation along the droplet midline as a function of the distance z from the substrate surface.

To obtain the droplet critical wetting angle θ_c at a stripe pillar edge defect, the droplet is increased by adding particles onto it to approach the critical volume V_c:

The particle deposition region is a cuboid and the distance from its lower surface to the droplet top is within the cut-off radius of the LJ interaction (2.5σ_LL). To reduce the artificial disturbance to the minimum, the mass centers of the deposition region and the droplet are set to along the pillar midline and the net vertical velocity of the particles deposited is set to be zero. After 300000–600000 timesteps, the droplet profile is checked. If the droplet slides down to wet the pillar sidewall (Fig. 4(b)), we then redo the deposition process with half the particle number. We need to mention that the depositing particle number in the first trial increases as the droplet equilibrium CA θ_e increases (a large θ_e corresponds to a large droplet critical volume with a small σ_SL and small ε_SL). One snapshot of the MD evolution after particle deposition is shown in Fig. 4(a).

Fig. 4.

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	Fig. 4. Snapshots of the particle deposition and the droplet TPL deforming across the pillar edge defect.

Finally, given that the density contour is difficult to converge since the droplet on the pillar top is highly unstable near its critical size, the critical CA θ_c defined in Fig. 1 is obtained by a circular fit of the droplet side view on the xz plane^[24] as in Fig. 5. The measurement results of θ_c will be presented together with θ_e in the following section.

Fig. 5.

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	Fig. 5. Circular fitting of the side view of the critical droplet on the stripe pillar top.

The results of the droplet equilibrium CAs θ_e on the planar substrates and the critical wetting angles θ_c at the stripe pillar edge defects are listed in Table 1 for different LJ solid–liquid interaction parameters.

Table 1.

Table 1.

Table 1. The results of θe and θc for the LJ liquids with different solid–liquid interaction parameters. . σSL = 0.9σLL θe/(°) θc/(°) εSL = 0.50εLL 52.4±0.6 44.5 εSL = 0.55εLL 40.5±0.7 32.5 εSL = 0.60εLL 37.1±0.5 17.0

Table 1. The results of θe and θc for the LJ liquids with different solid–liquid interaction parameters. .

The measurement accuracies of θ_e and θ_c are chosen as 0.1° and 0.5°, respectively. The different measurement accuracies stem from the fact that the circular fitting of the side view of the cylindrical droplet on the pillar is less accurate than the circular fitting of the droplet isodensity contour on the planar substrate.

From Table 1, for each group of ε_SL and σ_SL, it is obvious that the critical wetting angle of the LJ liquid droplet is always smaller than the corresponding equilibrium CA. As a result, the droplet TPL will deform and slide down from the pillar edge even when the droplet volume is smaller than the critical value. This phenomenon is termed as pre-wetting here, and for droplets with a smaller θ_e, the pre-wetting phenomenon is more significant according to Table 1. We need to mention that, with the CA approaching 90°, the droplet volume tends to be infinite (as drawn in Fig. 6); for this reason, in this work only the hydrophilic droplets are simulated.

Fig. 6.

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	Fig. 6. The dependence of the critical droplet volume on the CA according to Eq. (2) with normalized a and L.

From the work of Mayama,^[25] it is suggested that the droplet TPL should overcome a pinning energy barrier E_P at the pillar edge to wet the pillar sidewall. The pinning energy is derived from the free energy minimum difference between two equilibrium states (droplet on the pillar top and droplet wetting of the pillar sidewall), which arises from the droplet shape deformation during the sidewall wetting process. Considering the pinning energy, to wet the pillar sidewall, the droplet volume should be larger than the critical value, i.e. θ_c should be larger than θ_e. Through combining Mayama’s formulating and the results in this work together, it is straightforward that the pinning energy barrier at the pillar edge defect for the LJ droplet is lowered, and we attribute it to the moisture layer (ML) effect at the pillar sidewall. The ML here has a similar nature as the precursor layer in Ref. [26]; the formation reason is that the Gibbs free energy of the molecular adhered to the solid surface is lower than that in the gaseous state, which will be discussed in detail in the following sections.

From the side view of the droplet on the stripe pillar top in Fig. 5, we find that the liquid particles wet the pillar sidewall and form a an ML. Besides, the isodensity contour of ρ* = 0.4 of the droplet on the pillar top spreads over the pillar sidewall. Both results prove the existence of the ML on the pillar sidewall, and the underlying reason for this is that the moist sidewall is energetically favorable over the dry sidewall without any particle adsorption.^[27] The ML formation has two main resources: one is the direct adsorption of the particles from the gaseous atmosphere, and the other is from the droplet particle evaporation–condensation process near the pillar edge.

The molecular kinetic theory^[28] is adopted here to qualitatively explain the pillar pinning energy reduction ΔE_P caused by the ML effect. We define κ₀ as the equilibrium attempting frequency for the droplet particles at the TPL to escape and to be trapped by the ML, and κ₀ is related to the pinning energy barrier E_P, . The existence of the ML on the pillar sidewall greatly enhances κ₀, which indicates that the pinning energy E_P is evidently lowered.

The ML thickness and ML particle density are two key factors for the pinning energy reduction. We need to mention that the ML thickness depends on how to define the ML–vapor interface.

These ML properties are directly related to the sidewall potential field V (d)

Fig. 7.

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	Fig. 7. The pillar sidewall potential field.

However, the ML should be multi-layered. For the second ML, the position of its particle density peak should be determined by both the pillar sidewall potential field and the first ML. We assume that only the particles in the first ML condensed at the distance d_Q from the pillar sidewall have influence on the second ML formation. From Ref. [29], the potential field of the first ML is

After calculation, we find that at point Q′ the sidewall potential is only about 0.06 of the first ML with σ_M equal to σ_LL; then, the influence of the sidewall on the second ML formation can be safely ignored, and the particle density peak of the second ML appears at the distance Z = σ_LL from the first ML where the potential energy valley of the first ML appears, or equivalently, the second ML thickness is σ_LL. We need to point out that one important underlying simplification is that the influences of the second and other MLs on the first ML formation are ignored, which are much weaker than the influence of the pillar sidewall.

Although the position of the second ML particle density peak should be determined by the valley point of the summation of the sidewall potential and the first ML potential, since there is no analytical expression of the sum potential, we use the above method for simplicity. In this way, the influence of the MD parameter on the simulation results is highlighted.

Similarly, the properties of the other MLs would only be determined by the potential field of their previous MLs. Straightforwardly, the total ML thickness is d_Q + nσ_LL, where d_Q is the first ML thickness, σ_LL is the thickness of the other MLs, and n is the number of MLs except the first ML, which depends on how we define the ML–vapor interface. No matter where the ML–vapor interface is, d_Q + nσ_LL (equals to 0.86σ_SL + nσ_LL and σ_LL is fixed during the simulations) only varies with σ_SL in our simulations. Then, the total ML would be thicker for larger σ_SL, and the escape of the droplet particles at the TPL to the pillar sidewall would be easier, which actually corresponds to a larger κ₀. Eventually, the pinning energy at the pillar edge defect is reduced more significantly. This conclusion is in good accordance with the data in Table 1.

What is more, the ML with larger particle density causes easier particle escape at the TPL, and thus corresponds to a larger κ₀ and larger pinning energy reduction. In the first ML, the particle density is determined by the sidewall potential energy valley depth, which is linearly related to ε_SL as indicated in Eq. (3); in the other MLs, the particle densities are directly determined by the potential valley depths of the previous MLs, as discussed above.

The fact is that, when the sidewall potential valley depth is deeper (i.e. larger ε_SL), the particle density in the first ML is larger, and then the equilibrium inter-particle distance σ_M in the first ML is smaller. This results in a deeper potential valley of the first ML according to Eq. (6), and then further results in a denser second ML. In the same manner, the overall particle density in the MLs is determined by ε_SL, no matter where the ML–vapor interface is defined.

Briefly speaking, when σ_SL or ε_SL is larger, the interaction between the particles at the TPL and the MLs on the pillar sidewall is stronger. Then, the pinning energy E_P is easily overcome by LJ droplets with a volume smaller than the critical value ignoring the ML effect, which agrees well with the results in Table 1.

The critical mechanical balance conditions of the TPL at the pillar edge defect along the x and z directions are constructed by adopting the notion of static contact line friction^[24]:

The variations of the TPL static friction along the x and z directions caused by the evaporation effect are

Table 2.

Table 2.

Table 2. Total reduction of static contact line friction by MD calculation. . σSL = 0.9σLL Δf(γLV σSL = 0.8σLL Δ f(γLV εSL = 0.50εLL –0.194 εSL = 0.60εLL –0.216 εSL = 0.55εLL –0.203 εSL = 0.65εLL –0.304 εSL=0.60εLL –0.470 εSL=0.70εLL –0.332

Table 2. Total reduction of static contact line friction by MD calculation. .

From the results in Table 2, for the same σ_SL, a larger ε_SL corresponds to a larger absolute value of Δf; for the same ε_SL (ε_SL = 0.60ε_LL), a larger σ_SL results in a larger absolute value of Δf. The total reduction of the pinning energy ΔE_P at the pillar edge consists of two parts: ΔE_x along the x direction, and ΔE_z along the z direction. ΔE_x and ΔE_z are assumed to be the increasing functions of the friction reductions Δf_x and Δf_z, respectively. Then, we know that the large absolute value of Δf (corresponding to a large σ_SL or ε_SL) indicates that the pinning energy barrier reduction ΔE_P is large, which agrees well with the discussion in Subsection 3.1.

From the MD simulation of the LJ liquid wetting behaviors at the edge defect of a single stripe pillar, the pre-wetting phenomenon is observed. It is revealed that the existence of the quasi-liquid ML on the pillar sidewall reduces the edge pinning energy barrier, which prevents the droplet from sliding down. The degree of pre-wetting is directly related to the particle interaction parameters, which is explained in both atomic and classical ways. The results in this work are useful to understand the Cassie–Wenzel state transition process on the stripe pillar array at high temperatures when the droplet evaporation–condensation effect should be considered and the existence of the ML on the pillar array sidewalls significantly lowers the Cassie–Wenzel state transition energy barrier.