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DOI: 10.1039/C9SM01663E
(Paper)
Soft Matter, 2019, Advance Article

Arvind Arun Dev‡,
Ranabir Dey§ and
Frieder Mugele*

Physics of Complex Fluids, MESA + Institute for Nanotechnology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. E-mail: f.mugele@utwente.nl

Received
16th August 2019
, Accepted 5th November 2019

First published on 6th November 2019

We study here the microscopic deformations of elastic lamellae constituting a superhydrophobic substrate under different wetting conditions of a sessile droplet using electrowetting. The deformation profiles of the lamellae are experimentally evaluated using confocal microscopy. These experimental results are then explained using a variational principle formalism within the framework of linear elasticity. We show that the local deformation profile of a lamella is mainly controlled by the net horizontal component of the capillary forces acting on its top due to the pinned droplet contact line. We also discuss the indirect role of electrowetting in dictating the deformation characteristics of the elastic lamellae. One important conclusion is that the small deflection assumption, which is frequently used in the literature, fails to provide a quantitative description of the experimental results; a full solution of the non-linear governing equation is necessary to describe the experimentally obtained deflection profiles.

Owing to such a wide range of technological applications, extensive research has been performed to understand the wetting characteristics of liquids on both natural and artificial superhydrophobic surfaces.^{18–23} However, these studies focus on the wetting behaviour on rigid structured substrates; studies pertaining to wetting characteristics on substrates with soft/flexible structures are relatively scarce despite the wide range of applications. The existing studies in this regard reveal that the wetting characteristics on soft micro-structured substrates are definitely dependent on the elasticity of the underlying structures. The droplet rolling contact angle increases, and the receding contact angle decreases, with increasing softness of the elastic micro-structures.^{24} Furthermore, contact angle hysteresis often increases due to the deformation and collapse of the soft micro-structures underneath the drop.^{25} However, these studies do not examine in detail the microscopic deformation of the underlying soft structures due to capillary interactions during droplet wetting, specifically when the droplet is still in the Cassie state.^{18,26} In order to have a comprehensive idea of wetting and superhydrophobicity on soft structured substrates, it is imperative to understand the changes in shape of the underlying structures due to the involved capillary interactions.

The deformation of an array of soft micropillars, constituting a superhydrophobic substrate, during wetting was studied very recently.^{27} This work highlights the relationship between the deformation of the underlying soft micropillars and the macroscopic wetting characteristics of a glycerol/water drop as characterized by the apparent advancing and receding contact angles.^{27} However, the behaviour of the soft micropillars was analyzed assuming small deflections of the elastic pillars, which may not be the general case considering the wide range of applications involving different magnitudes of capillary forces and structure geometries. Therefore, a detailed analysis of the deformation characteristics of flexible superhydrophobic substrates over a wide range of wetting conditions, and hence, capillary force magnitudes, has remained unexplored.

In this paper we investigate the deformation of flexible stripes/lamellae, constituting a superhydrophobic substrate, under different wetting conditions of a sessile droplet using electrowetting (EW).^{28} During EW, the presence of surface charges results in a Maxwell stress distribution at the droplet liquid–vapour interface, which gives rise to a net electrical force pulling on it.^{29,30} This electrical force results in enhanced wetting and deformation of the drop on a macroscopic scale. Moreover, at equilibrium the balance of the Maxwell stress and the Laplace pressure results in large curvature of the liquid–vapour interfaces. So, during EW on a structured superhydrophobic substrate, provided the droplet is still in Cassie state, the enhanced wetting, macroscopic deformation of the drop, and the increasing curvature of the liquid–vapour interface in between the structures can result in the application of different magnitudes of capillary forces on the structures at the droplet contact line. Therefore, EW can provide a flexible way for probing the mechanics of the soft deformable structures over a range of capillary force magnitude. This is precisely what we try to exploit in the present work. Additionally, the present work also explores the feasibility of using such soft superhydrophobic substrates for the myriad of EW applications.^{31,32} We perform a systematic study of the deformation shapes of the soft lamellae during EW of a sessile drop for different structure geometry, material elasticity, and EW conditions using confocal microscopy. We also explain the observed deformation characteristics using the classical elastica theory which goes beyond the small deflection assumption.

Sylgard 10:1 (E = 2.1 MPa) | |||
---|---|---|---|

Sample no. | Height (L) (μm) | Width (t) (μm) | Aspect ratio (α = L/t) |

1 | 34 | 11.3 | 3.0 |

2 | 34 | 15 | 2.2 |

3 | 34 | 25 | 1.3 |

Sylgard 15:1 (E = 0.9 MPa) | |||
---|---|---|---|

Sample no. | Height (L) (μm) | Width (t) (μm) | Aspect ratio (α = L/t) |

4 | 30.75 | 13.9 | 2.2 |

5 | 30.75 | 24 | 1.3 |

Sylgard 20:1 (E = 0.6 MPa) | |||
---|---|---|---|

Sample no. | Height (L) (μm) | Width (t) (μm) | Aspect ratio (α = L/t) |

6 | 30.75 | 13.9 | 2.2 |

7 | 30.75 | 24 | 1.3 |

For confocal imaging (Fig. 1(a)) we use a Nikon A1 inverted line scanning confocal microscope with excitation lasers at 488 nm (for DFSB-K175 in the substrate) and 638 nm (for Alexa Fluor 647 in the drop), and two objectives-60× water immersion with numerical aperture (NA) = 1.2 and 20× dry with NA = 0.85. The emissions from the two dyes are collected using band filters in the range 500 nm to 550 nm for DFSB-K175 and in the range of 663 nm to 700 nm for Alexa Fluor 647. The refractive indices of the different components involved are as follows – glass: 1.5, Sylgard 184: 1.42, air: 1.0, and water drop: 1.33. 3D confocal scans are performed in the immediate vicinity of the macroscopic droplet contact line such that lamellae wetted by the droplet as well as outside the droplet footprint are simultaneously visible (Fig. 2). The confocal scans are performed at two different locations (Fig. 2(a)) so that two different orientations of the macroscopic droplet contact line relative to a lamella can be investigated (Fig. 2(b) and (c)). xy projections of the two vertical planes along which xz slices are acquired for studying the deflection profiles of the lamella are schematically represented by A–A′ and C–C′ in Fig. 2(a); note that the planes for the xz slices are always perpendicular to the lamellae. Furthermore, C–C′ passes through the centre of the droplet footprint and A–A′ passes through a location where the macroscopic droplet contact line locally covers only one top edge of the lamella for the first time as one moves away from C–C′ (see the closeup of macroscopic droplet contact line in Fig. 2(b)). The 3D confocal scans are post-processed using ImageJ, and the xz slices are analyzed using an in-house MATLAB code to evaluate the deformation characteristics of the soft lamellae due to interfacial capillary interactions (see Section S4 in ESI† for image analysis details).

In Fig. 3, the image columns A–A′ and C–C′ show the configurations of the lamellae in the x–z planes A–A′ and C–C′ respectively for increasing magnitude (different rows) of the applied electrical voltage represented here by the so-called non-dimensional electrowetting number η.^{28} Note first that for the range of electrical voltage applied here the droplet progressively undergoes enhanced wetting while always maintaining the Cassie state (see the gradual progression of the macroscopic droplet contact line over the lamellae 2 to 5 with increasing η in Fig. 3(a) to (d) at both A–A′ and C–C′). Second, during EW only the lamella underneath the macroscopic droplet contact line deforms, while the other lamellae underneath the droplet remain undeformed (e.g. see lamellae 2, 3, 4, and 5 in rows (a), (b), (c), and (d) respectively in Fig. 3). Third, apparently the magnitude of deflection of the outermost lamella at A–A′ is more compared to that observed in C–C′ (compare the shapes of the lamella underneath the macroscopic droplet contact line in A–A′ and C–C′ corresponding to any value of η in Fig. 3). Finally, the curvatures of the liquid–air interfaces in-between the lamellae underneath the droplet increase with increasing η. Increase in the electric field between the menisci and the electrodes pulls the liquid menisci between the lamellae down until the additional curvature balances the electric(Maxwell) stress given by , where ε_{0} is the permittivity of free space and S is the magnitude of the local electric field.^{28,29} Furthermore due to applied frequency of AC voltage, the liquid interfaces underneath the droplet between the lamellae is expected to oscillate.^{36} Consequently, the lamellae deflections should also oscillate provided the response time of lamellae are comparable with liquid interfaces. However the confocal imaging is slow to capture these oscillations and therefore we do not observe any response of these liquid interfaces and lamellae deflections to applied frequency. Hence, we should ideally interpret the interface positions and deflections described here as time-averaged positions and deflections respectively. Note that in the experiments the change in curvature of the liquid–air interface between the lamella is weakly dependent on η and a clear detection of the same is somewhat restricted by the optical challenges of diffraction. However, comparison of the interfaces as outlined by the yellow dotted line (guide to the eye) in rows (a) and (d) in Fig. 3 makes the increased curvature due to the applied voltage (η = 0.19) still visible. Here, ψ(η) denotes the local angle at which the liquid–air interface meets the lamella edge underneath the droplet, while ϕ(η) denotes the apparent angle the droplet–air interface makes with the horizontal at the macroscopic droplet contact line (Fig. 3). It should be noted that with the decrease in the applied electric field, the curvature of the liquid–air interfaces underneath the drop is expected to decrease. In fact, this phenomenon of bending of the interface is reversible as long as the contact line remains pinned at the edge of the lamella, i.e. as long as the Maxwell stress does not exceed the critical stress for the transition from the Cassie to the Wenzel state.^{29}

In Fig. 4, the left and right columns show the lamellae configurations for η = 0 and η = 0.18 respectively for decreasing E (different rows). Note that all the lamellae configurations in Fig. 4 are at A–A′ and the lamellae have identical aspect ratio α = 2.2. The deformation of the lamella at the macroscopic droplet contact line (i.e. lamella 2 for η = 0 and lamella 5 for η = 0.18) increases with decreasing value of E (Fig. 4) as qualitatively anticipated. Moreover, a comparison of Fig. 4(a)(i) and 3(a)(i) or Fig. 4(a)(ii) and 3(d)(i) shows that the deformation of the lamella decreases with decreasing α for a particular value of E. Furthermore, for a particular value of E, there is no significant difference in the deformation of the lamella underneath the macroscopic contact line for different values of η (compare images in columns (i) and (ii) corresponding to any row in Fig. 4). Similar observations were also recorded for soft substrates with lamellae of α = 1.3 (see Section S6 in ESI†).

In order to understand the EW-induced deformation characteristics of the soft lamellae for different values of α, E, and η, it is useful to first delineate the orientations of the capillary forces acting on a lamella due to the pinned contact lines. The orientations of the capillary forces acting at the top of the lamella underneath the macroscopic droplet contact line at A–A′ and C–C′ are schematically shown in the insets in Fig. 2(b) and (c) respectively. The red solid and dashed lines in the schematics represent the water–air interfaces without any applied voltage and under an applied electrical voltage respectively. The curvatures of these interfaces determine the angles ψ(η) and ϕ(η), and consequently, the magnitudes of the horizontal and the vertical components of the capillary forces acting on the top of the lamella (black arrows in the schematics shown in Fig. 2(b) and (c)). Note that at A–A′, the macroscopic droplet contact line crosses the lamella somewhere between the two top edges of the lamella and the droplet covers only one top edge. Moreover, the apparent contact angle for such a configuration of the macroscopic droplet contact line is always close to 90°. Accordingly at A–A′, there is an almost vertical force γ acting on the top of lamella besides the capillary force acting at the edge of the lamella covered by the drop (see Fig. 2(b) and left column of Fig. 3). Furthermore, with increasing η, ψ increases due to Maxwell stress induced enhanced curvature of the liquid–air interface underneath the droplet at A–A′ and C–C′; however, simultaneously ϕ decreases at C–C′ due to enhanced wetting with increasing η. The variation in ϕ between A–A′ and C–C′ for a given value of η is due to the anisotropy in the substrate, which makes the apparent contact angle dependent on the location of the macroscopic droplet contact line on the substrate.^{37} The overall η-dependence of ψ and ϕ makes the orientation of the capillary forces acting on the outermost lamella at the pinned droplet contact lines function of η, which at least theoretically makes the consequential deformation of the lamella dependent on η. So, at A–A′ we consider two capillary forces acting on the outermost lamella due to the pinning of the macroscopic droplet contact line – one acting somewhere between the two top edges of the lamella and the second acting on one edge of the lamella covered by the droplet (left edge in accordance with Fig. 2(b)); at C–C′, we consider the capillary forces on both the top edges of the outermost lamella (both left and right edges in accordance with Fig. 2(c)). The different components of these capillary forces can be written as following

F_{xl} = γsin(π − ψ(η))
| (1a) |

F_{zl} = γcos(π − ψ(η))
| (1b) |

F_{zt} = γ
| (1c) |

F_{xr} = γcos(π − ϕ(η))
| (1d) |

F_{zr} = γsin(π − ϕ(η))
| (1e) |

(2) |

s = 0; θ = 0 | (2a) |

s = L; θ′ = 0 | (2b) |

(3) |

Fig. 5 shows comparisons between experimentally obtained (markers; see Section S4 in ESI† for details) and theoretically evaluated (lines) deformation profiles of the lamella underneath the macroscopic droplet contact line at A–A′ for different values of α, E, and η. The maximum deflection of the lamella observed in the present work for α ≈ 2.2 and E = 0.9 MPa is ≈3 μm (Fig. 5(b)(i) green circular markers), which is significantly larger than the maximum deflection of ≈1.8 μm reported in an earlier study^{27} for similar α and E on wetting induced deflection of soft pillars. It is important to note here that the theoretical deformation profile (dash-dotted line) based on the small deflection assumption given by eqn (3) fails to describe the experimental deformation profiles (Fig. 5(a)(i) and (a)(ii)); the small deflection assumption fails to predict the experimentally observed deflection of even about 3 μm by almost 1 μm. However, the experimental profiles are described reasonably well by the general solution (solid line) obtained using eqn (2). Hence, it can be concluded that the small deflection assumption must not be assumed a priori to describe the wetting induced deformation of micro-structures, as often done in the literature.^{27,45–47} A comparison of Fig. 5(a)(i) (or (ii)) and (b)(i) (or (ii)) shows that the smaller deformation of the lamella due to smaller value of α, for identical value of E, is also reasonably addressed by the solution of eqn (2). Furthermore, at a particular value of η, the increasing deformation of the lamella underneath the macroscopic droplet contact line with reducing value of E is also nicely captured (Fig. 5(b)(i) and (ii)); this is mainly due to the reducing value of B with decreasing E. Finally, Fig. 5(a(i)) and (a(ii)) show that the lamella deflection for η = 0 and η = 0.19 at z ≈ 30 μm is δ ≈ 3.28 μm and δ ≈ 2.87 μm respectively. Hence, the deflection of lamella decreases with η. This mainly stems from the lowering of the horizontal component of the capillary force (F_{xl} for A–A′; eqn (1a)) triggered by the enhancement in ψ due to the effect of Maxwell stress. It must be admitted here that the observable effect of η on the deformation is indeed weak. However, provided the drop stays in the Cassie State, it is possible to achieve significant reduction in the deformation of the lamella with increasing η (or increasing ψ) as predicted by the theoretical model (see Section S8 in ESI†). The solution of eqn (2) also nicely captures the smaller deformation of the lamella underneath the macroscopic droplet contact line at C–C′ for different values of η (Fig. 6). The smaller deformation is because at C–C′ there are capillary forces at both the top edges of the lamella (Fig. 2(c)) which reduce the resultant horizontal capillary force on the lamella (F_{x} = F_{xl} − F_{xr}; (see eqn (1))). Furthermore, at C–C′, the maximum deformation of the lamella is also slightly less for η = 0.18 than for η = 0. Based on the aforementioned discussion, it can be now intuitively concluded that the lamellae underneath the droplet but away from the macroscopic contact line do not deform because the horizontal components of the capillary forces acting on the two top edges of each are almost equal in magnitude but opposite in direction thereby resulting in zero net horizontal force.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sm01663e |

‡ Current address: Department of magnetism of nanostructured objects (DMONS), Institut de Physique et Chimie des Matèriaux de Strasbourg (IPCMS), CNRS UMR 7504, Universitè de Strasbourg, 23 Rue du Loess, 67034 Strasbourg, France. |

§ Current address: Dynamics of Complex Fluids, Max Planck Institute for Dynamics and Self-organization, Am Fassberg 17, 37077 Goettingen, Germany. |

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