Drop Behavior Influenced by the Correlation Length on Noisy Surfaces

Article Cite This: Langmuir XXXX, XXX, XXX−XXX

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Drop Behavior Influenced by the Correlation Length on Noisy Surfaces Rodica Borcia,*,† Ion Dan Borcia,‡ Michael Bestehorn,† Olga Varlamova,§ Kevin Hoefner,§ and Juergen Reif§ Lehrstuhl Statistische Physik und Nichtlineare Dynamik, ‡Lehrstuhl Computational Physics, and §Lehrstuhl Experimentalphysik II/Materialwissenschaften, Brandenburgische Technische Universität, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany

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ABSTRACT: We investigate numerically the role of the correlation length in drop behavior on noisy surfaces. To this aim, a phase field tool has been used. Theoretical results are confirmed by experiments of distilled water drops sitting on stainless steel and silicon surfaces textured by laser-induced periodic self-organized structures: an increase of the noise amplitude results in an amplification of the original behavior (i.e., hydrophobic is getting more hydrophobic, hydrophilic is getting more hydrophilic). Furthermore, computer simulations in two and three spatial dimensions allow for predictions of drop behavior on noisy sloped substrates under a gravitational force, a problem of large interest in controlled motion in micro- and nanofluidics.

1. INTRODUCTION Dynamics of water drops has been intensively studied during the last century on different materials: from natural and artificial porous clothing surfaces (to avoid the falling rain drops to penetrate the structure)1−4 to heterogeneous/ chemically patterned substrates,5−14 and, more recently, on gradient and soft surfaces.15−18 The main aim is to create at the liquid−air, liquid−solid interfaces the physico-chemical conditions in such a way that, tending to minimize its surface energy, the droplet experiences an already anticipated motion. This droplet manipulation has large applications in microfluidic and eventually nanofluidic devices without a power supply, laboratory-on-a-chip, ink-jet printing, or surface cleaning technologies. Altering the surface wettability by an external stimulation has received great attention over the past years. Smart materials were created, which are adaptive, reconfigurable, and switchable under different external parameters, such as temperature, pH, or light (see refs 19−22 and references therein). These substrates change adaptively the wettability properties of the liquid droplet and are valuable for control and manipulation of droplet motion. Dependence of the wetting characteristics of a solid on surface roughness and surface morphology is inherent in the everyday life. In particular, they can strongly influence the characteristic time scales of fluid flow in thin films during spinoidal dewetting and of spreading drops.23,24 Thus, the control of surface roughness and morphology would allow regulated motion of small portions of liquids (or drops) on structured substrates. In this article, we present a systematic investigation of liquid droplets on noisy surfaces using a phase field approach. We study the modifications of the drop © XXXX American Chemical Society

wettability induced by the noise at the solid boundary. The computer simulations are supported by experiments on distilled water drops sitting on (stainless steel and silicon) surfaces covered by laser-induced periodic self-organized structures (LIPSS). The article is organized as follows: Section 2 deals with the theoretical formulation of noisy substrates in the frame of a phase field model. Numerical results in two spatial dimensions delivered here are compared with the experimental results realized on laser-structured surfaces. Numerical predictions, in two and three spatial dimensions, of drops on noisy sloped substrates under gravitation are presented in Section 3. We gather the conclusions in Section 4.

2. NOISY SURFACES: THEORY AND EXPERIMENT Depending on the correlation length at the solid substrate, several distinct drop behaviors can be achieved. We study these phenomena using a phase field model intensively validated earlier for describing static and dynamic contact angles,25 dewetting phenomena, shaping of liquid films, controlled pattern formation,26 coalescence of drops,27 and drop motion under vibrations.28 Phase field models adopt a continuum thermodynamic description of multiphase systems: they introduce an order parameter ρ (phase field variable), which is in every bulk region nearly constant and varies continuously and rapidly from one phase to the other. Our order parameter is the density ρ (scaled to the liquid density ρliq), which controls the composition of the system. So, ρ = 1 designates Received: November 19, 2018 Revised: January 4, 2019

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DOI: 10.1021/acs.langmuir.8b03878 Langmuir XXXX, XXX, XXX−XXX

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Langmuir the liquid phase and ρ ≈ 0 the vapor phase. The position of the interface is controlled by the gradients of ρ. For a two-phase system with diffuse interfaces and without evaporation phenomena, the Helmholtz free-energy functional is given by (see, e.g.,25,29) ÄÅ ÉÑ Å Ñ 2 Å 2Ñ Å - [ρ ] = ÅÅf (ρ) + (∇ρ) ÑÑÑdV Å ÑÑÖ (1) 2 V Å Ç

We start from a flat thin liquid layer with ρ = 1 in a gas atmosphere with ρ ≈ 0. The initial height of the liquid film is h0 = 0.4 mm. The whole system is at rest (v⃗ = 0, everywhere). The thin film breaks up in small droplets. Finally, by coalescence, only one single drop remains. By changing ρS, ρ dx dz one obtains droplets of the same mass m = ∬

The first term in the free-energy functional (eq 1) denotes the free-energy density for the homogeneous phases. For a system in equilibrium and without interfacial mass exchange, the freeenergy density has the form of a double-well potential with two local minima corresponding to the two coexisting phases: liquid and vapor. We choose the free-energy density given by

with different contact angles at the bottom plate. The droplet radius in two spatial dimensions is calculated as R = m/π . For the numerical results presented in this article, we have circular drops of radius R = 0.78 mm, ρliq = 1000 kg/m3, η = 0.001 kg/ms, and σ = 0.05 N/m. The substrate is assumed to be noisy around a given ρS

cos θ = −1 + 6ρS2 − 4ρS3

∫

f (ρ) ∼ ρ2 (ρ − 1)2

ρ≥ 0.99

ρ(x)|z = 0 = ρS + Aξ(x)

(2)

ξ(x) = kn =

S=

+∞

σ=∫ 2(∂ρ /∂z)2 dz . The numerical code is based on −∞ the momentum equations with the Korteweg stress and the mass conservation equation

M 0 /2

∑

exp(iknx) exp(iϕn)

n =−M 0 /2, n ≠ 0

2π n , ϕn = −ϕ−n x1

(8)

2x1 M0

(9)

which can be used as a control parameter in our simulations (x1 denotes the lateral length of the substrate). One changes the correlation length by changing M0. For M0 → ∞, one obtains a Gaussian white noise. Defining ρ = 0.5 as the liquid−solid interface, a variation of ρ(x) at z = 0 [according to the relation (7)] can been seen as a variation of the interface elevation: higher ρ means higher elevation. In this way, a noisy distribution of ρ can model a corresponding rough surface. For very small correlation lengths, the liquid droplet sees the textured substrates as being flat (Figure 1a). By increasing S , when the correlation length becomes comparable with the length of the diffuse interface, the drop starts to feel the noise at the solid boundary. After the liquid thin film breakup, the small droplets try to minimize their free energy on the noisy surface. This leads to less binding energy (Figure 1b−d), i.e., increase of the liquid repellency (Cassie−Baxter regime3). The surface becomes more hydrophobic as in the absence of roughness at the solid boundary. When the correlation length becomes comparable with the drop size, the drop can be pinned by the hydrophilic irregularities of the surface (Figure 1e). When the correlation length becomes larger than the droplet diameter, the final droplet comes at rest on the heterogeneous randomly distributed hydrophilic region (Figure 1f). For both situations illustrated in Figure 1e,f, that means more interaction at the solid boundary, less contact angle (Wenzel regime1). The surface becomes more hydrophilic as in the absence of noise at the substrate. The depth of the surface roughness also plays an important role in the drop behavior on noisy substrates. We present the

→

→ ∂ρ + ∇·(ρ v ) = 0 ∂t

1 M0

(with ϕ uniformly random-distributed in the interval [0, 2π], M0 an even natural number larger than zero, and A the noise amplitude). With this choice, the random distribution has the correlation length

where δij is the Kronecker symbol and 2 a parameter connected to the surface tension coefficient:

⎯→ ⎯ → dv = −∇p + ∇· T + ∇·(η∇v ) dt

(7)

with ξ a zero-mean noise in the interval 0 ≤ x ≤ x131

The second term in the free-energy functional (eq 1) is a “gradient energy”, which is a function of the local composition. As already shown in25 or,29 minimizing the free-energy functional (eq 1), one can derive the nonclassical phase field terms, which have to be included in the Navier−Stokes equation for assuring the shear stress balance at the droplet interface. These terms are incorporated in the Navier−Stokes equation with the help of the Korteweg stress tensor27,29,30 ÅÄÅ ÑÉÑ ∂ρ ∂ρ 1 Tij = 2ÅÅÅÅρΔρ + (∇ρ)2 ÑÑÑÑδij − 2 ÅÇ Ñ 2 ∂xi ∂xj Ö (3)

ρ

(6)

(4)

(5)

where p = ρ ∂f(ρ)/∂ρ − f(ρ) is the thermodynamical pressure and η the dynamic viscosity. A second-order central finitedifference scheme for the spatial derivatives and an Euler method for the time integration is applied. Periodic boundary conditions in horizontal direction and no-slip condition for the velocity field v⃗ at the top and bottom walls are assumed. The mesh is of 400 × 200 points, the distance between two mesh points is δx = δz = 2, and the integration time step is δt = 0.1. The lengths are scaled to 10−5 m and the time to 10−5 s. That means the computational domain is 8 × 4 mm2. A good convergence of the numerical code is achieved for meshes with more than 100 points in one direction. In this way, one assures more than 10 lattice points in the diffuse interface. The density field is ρ = 10−3 at the top boundary and ρ = ρS at z = 0, simulating in this way at z = 0 a solid boundary with the associated van der Waals long-range interactions at the liquid−solid interface.29 ρS is a free parameter between 0 and 1, which represents the density at the substrate and describes the wettability properties at the bottom plate. 0 ≤ ρS ≤ 0.5 denotes hydrophobic surfaces and 0.5 < ρS ≤ 1 hydrophilic ones. The static contact angle θ is related to the substrate density ρS through29 B


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Figure 3. Numerical simulations in two spatial dimensions showing the influence of the noise amplitude: (a) A = 0; (b) A = 0.1; and (c) A = 0.25 on drops sitting on hydrophilic substrates with the random distribution correlation length S = 400. (On a flat homogeneous surface, the static contact angle of the drop is θ = 55°, ρS = 0.7.)

several tens to several hundred of nanometers were produced on the ablated area. By irradiation of stainless steel 316L at a fluence of 1.5 J/cm2 with the pulse number in the range of 10−100, we observed complex microstructures with periods from 1.7 μm up to 3.5 μm (Figure 4a). It is worth to note that fine nanoripples of a

Figure 1. Numerical simulations in two spatial dimensions of drop topology on surfaces with different correlation lengths: (a) S = 2 ; (b) S = 7 ; (c) S = 8; (d) S = 10; (e) S = 40 ; and (f) S = 400 for the same noise amplitude A = 0.1. (On a flat homogeneous surface, the static contact angle of the drop is θ = 55°, ρS = 0.7.)

influence of the noise amplitude [A in relation (7)] on liquid droplets sitting on hydrophobic substrates having ρS = 0.5, θ = 90°, S = 8 (Figure 2) as well as on hydrophilic ones with ρS =

Figure 2. Numerical simulations in two spatial dimensions showing the influence of the noise amplitude: (a) A = 0; (b) A = 0.08; and (c) A = 0.16 on drops sitting on hydrophobic substrates with the random distribution correlation length S = 8. (On a flat homogeneous surface, the static contact angle of the drop is θ = 90°, ρS = 0.5.)

Figure 4. Photographs of sessile water drops on laser-treated samples reveal a change of wettability in dependence on the laser-induced surface morphologies (scanning electron microscopy images): (a) to a high-hydrophobic regime for stainless steel 316L irradiated at F ≈ 1.5 J/cm2, correspondingly from top to bottom, with 10, 40, and 100 pulses/spot; (b) to a high-hydrophilic regime for silicon irradiated at F ≈ 1.1 J/cm2, correspondingly from top to bottom, with 5, 400, and 2000 pulses/spot. The static contact angle is shown in the upper-left corner.

0.7, θ = 55°, S = 400 (Figure 3). Thus, a contact angle of more than 90° is increased by the surface roughening and one of less than 90° is diminished by roughening. Experimentally, controlled noise on solid surfaces has been realized by generation of laser-induced periodic self-organized structures (LIPSS) upon multipulse femtosecond laser ablation.32−35 This is a reliable technique to modify the surface topography of ablated targets and, correspondingly, their statistical roughness and wettability. We use pulses of 100 fs @ 800 nm from an amplified Ti/ sapphire laser system operating at 1 kHz repetition rate to generate extended areas (10 × 10 mm2) of LIPSS.36 Depending on the applied irradiation dose (fluence × number of pulses), surface morphologies with lateral periods at both nano and microscales with roughness features ranging from

size of about 625 nm hierarchically cover these micropatterns, resulting in a significant increase of the surface roughness Sa from 60 nm at low irradiation doses (1.5 J/cm2 × 10 pulses/ spot) up to 385 nm at higher irradiation doses (1.5 J/cm2 × 100 pulses/spot). Periods of micropatterns generated on silicon at increasing irradiation doses (1.1 J/cm2 × 5, 400, and 2000 pulses/spot) change in the range of 1.5 μm up to more than 5 μm (Figure 4b). Here, the character of micropatterns differs from the structures observed on the steel samples; we C


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Langmuir do not have any hierarchical overstructuring and the surface roughness Sa is changing in a narrow interval from 36 nm at 1.1 J/cm2 × 5 pulses/spot up to 64 nm at 1.1 J/cm2 × 2000 pulses/spot. We note that the lateral periods of ablated structures and the resulting statistical roughness of the patterned substrates are described in the theoretical model by the correlation length S and the noise amplitude A [see eqs 7 and 8]. In contrast to other theoretical works, where the regular heterogeneous substrates have a well-defined wettability period,10−13 formula (8) assures a multiscale roughness, which still maintains a correlation length S given by formula (9). Sessile drop method has been used to investigate the dependence of the wetting properties of LIPSS-structured surfaces on their roughness. We work with distilled water drops. The drop volume was controlled automatically via software. For the experimental results presented in this article, we have drops of radius R = 0.85 mm. The experimental results qualitatively confirm the findings from the numerical simulations. As illustrated in Figure 4a, for distilled water drops on structured stainless steel surfaces, larger feature size of the multiscale hierarchical structures and deeper surface roughness accentuate the wettability changes toward a more hydrophobic regime (as in Figures 1a−c and 2). The transition to the enhanced hydrophilicity (as in Figure 1d−f) is demonstrated on the structured silicon surfaces at the increasing feature size of the ablated patterns, as shown in Figure 4b. Here, the slightly increasing statistical surface roughness seems again to amplify the hydrophilic properties (as illustrated in Figure 3). For a quantitative comparison, we consider the case of a water drop sitting on a silicon surface. Its static contact angle on the flat homogeneous nonstructured surface (before irradiation) is θ = 55°. This case can be achieved from the theory for the model parameters: ρS = 0.7 and A = 0. By theory/numerics and experiment, we plot the contact angles of the water drop for increasing roughnesses at the silica substrate. From numerical simulations, the curvature of the droplet (the static contact angle θ) has been measured using contour plots (level 0.5) of the phase field variable in which tangent lines are drawn at the macroscopic foot of the liquid droplet. The curve plotted in Figure 5 shows an almost linearly monotonic decrease with the surface roughening, from θ = 55 to 19°. Figure 5 also shows very good agreement between the contact angles estimated from numerics and experiment, having as measure of the roughness, respectively, the normalized noise amplitude A/Amax in the simulation and the normalized deflection at the substrate Sa/Sa,max in experiment (where the maximal values Amax and Sa,max correspond to θ = 19°).

Figure 5. Quantitative comparison of contact angles for different roughnesses for a silicon substrate with S = 400. The mean correlation length in experiment has been scaled to the thickness of the boundary layer in the vicinity liquid−air and liquid−solid interfaces (10−7 m, the range of surface forces’ action37). As measure of the surface roughness, we use the normalized noise amplitude A/ Amax in the simulation and the normalized deflection at the substrate Sa/Sa,max in experiment, where the maximal values Amax and Sa,max correspond to θ = 19°. Very good agreement has been achieved.

Figure 6. Sliding drop on a noisy sloped surface for S = 2 , sin α = 0.035, and A = 0.1: (a) t = 2000; (b) t = 4000; (c) t = 200 000; and (d) t = 300 000. After the liquid film breakup [case (b)], the small droplets coalesce to one single drop moving to the right in the direction of the tangential component of the gravitational force [cases (c) and (d)].

3. EFFECTS INDUCED BY GRAVITATION ON DROPS ON NOISY SLOPED SUBSTRATES: NUMERICAL PREDICTIONS Next, we include gravitation ρg⃗ in the momentum eq 3. On sloped noisy substrates, three different situations can be distinguished: sliding drops (Figure 6), pinned drops by surface irregularities (Figure 7) for the same inclination as in Figure 6, and hopping drops over the small obstacles for a substrate with the same correlation length as in Figure 7 but much more sloped (Figure 8). We extend the numerical simulations to three-dimensional (3D) with a mesh of 400 × 200 × 100 points under periodic

Figure 7. Pinned drop on a surface with a larger correlation length as in Figure 6: S = 9, sin α = 0.035, and A = 0.1: (a) t = 2000; (b) t = 4000; (c) t = 200 000; and (d) t = 300 000. Following the same time moments as in Figure 6, one can observe the pinning effect at the substrate by comparing the last two snapshots from the second row.

boundary conditions in the horizontal plane and randomly distributed noise at the bottom boundary represented by31 D


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with x1 and y1 denoting, correspondingly, the lateral and the transversal length of the substrate. Figure 9a,b shows twodimensional (2D) patterns produced with (eq 10) for two different correlation lengths; Figure 9c shows a striped surface combining the patterns illustrated in Figure 9a,b. We incline the patterned substrate (from Figure 9c) along the stripe (in y direction). The simulations in three spatial dimensions reveal that on inclined patterned substrates the liquid film starts to break up in the regions with larger S (Figure 10). The final drop has the preference to slide along the stripe Figure 8. Hopping drop over small obstacles for a substrate with the same correlation length as in Figure 7 (S = 9, A = 0.1) but much more sloped, sin α = 0.5: (a) t = 2000; (b) t = 4000; (c) t = 200 000; and (d) t = 300 000.

ξ(x , y) =

1 M 0N0

M 0 /2

N0 /2

∑

∑

exp(ikmx)

m =−M 0 /2, m ≠ 0 n =−N0 /2, n ≠ 0

exp(ikny) exp(iϕm , n) km =

(10)

2π 2π m , kn = n , ϕm , n = −ϕ−m , −n x1 y1

Figure 10. Liquid film starts to break up in the regions with a larger correlation length S (ρS = 0.7, A = 0.1, sin α = 0.5; the stripes have the same size with S = 2 on the lateral sides and S = 8 in the middle, and the isodensity of the surface snapshots follows the liquid−vapor interface ρ = 0.5).

For an isotropic distribution at the substrate, we have S=

2y 2x1 = 1 M0 N0

Figure 9. Random patterns with different correlation lengths and A = 0.1: (a) S = 2 ; (b) S = 8; and (c) striped substrate combining the patterns illustrated in panels (a) and (b). E


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Figure 11 (3D simulations) pave the way for more studies devoted to controlled drop motion on micro- and nanochannels.

with a larger correlation length (Figure 11a). This preference becomes stronger for larger contrasts between the correlation

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Rodica Borcia: 0000-0002-0261-1273 Notes

The authors declare no competing financial interest.

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REFERENCES

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Figure 11. (a) Final drop has the preference to slide along the stripe with a larger correlation length; (b) for bigger drops, one can achieve the situation when the liquid drop covers the stripe with a lower correlation length with its edges leaning on the stripe with higher S (ρS = 0.7, A = 0.1, sin α = 0.5; the stripes have the same size with S = 2 on the lateral sides and S = 8 in the middle, and the isodensity of the surface snapshots follows the mean density at the solid substrate ρS = 0.7).

lengths. For bigger drops, one can achieve the situation illustrated in Figure 11b, when the liquid drop covers the stripe with a lower correlation length with its edges leaning on the stripe with higher S .

4. CONCLUSIONS We have systematically studied the role of the correlation length in drop behavior on noisy substrates using phase field simulations. The correlation length is introduced as a control parameter for describing modifications of drop wettability induced by the noise at the solid boundary. By increasing the correlation length, the drop behavior changes from the Cassie−Baxter to Wenzel regime. Regarding the role of the noise amplitude at the rough substrate, theory is confirmed by experiments of water drops sitting on stainless steel and silicon surfaces textured by laser-induced periodic structures: an increase of the noise amplitude results in an amplification of the original behavior (i.e., hydrophobic is getting more hydrophobic, hydrophilic is getting more hydrophilic). Furthermore, computer simulations in two and three spatial dimensions make predictions of drop behavior on noisy sloped substrates under a gravitational force. In 2D, the “contact line” is a point and pinning can be easily achieved. In 3D, the contact line is a real line and may show an irregular form on the scale of surface fluctuations. Despite this aspect, 2D simulations are in very good agreement with the experiments qualitatively and quantitatively. The situations illustrated in F


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Drop Behavior Influenced by the Correlation Length on Noisy Surfaces

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