Article pubs.acs.org/Langmuir
Neither Lippmann nor Young: Enabling Electrowetting Modeling on Structured Dielectric Surfaces Nikolaos T. Chamakos, Michail E. Kavousanakis, and Athanasios G. Papathanasiou* School of Chemical Engineering, National Technical University of Athens, 15780 Greece S Supporting Information *
ABSTRACT: Aiming to illuminate mechanisms of wetting transitions on geometrically patterned surfaces induced by the electrowetting phenomenon, we present a novel modeling approach that goes beyond the limitations of the Lippmann equation and is even relieved from the implementation of the Young contact angle boundary condition. We employ the equations of the capillary electrohydrostatics augmented by a disjoining pressure term derived from an effective interface potential accounting for solid/liquid interactions. Proper parametrization of the liquid surface profile enables efficient simulation of multiple and reconfigurable three-phase contact lines (TPL) appearing when entire droplets undergo wetting transitions on patterned surfaces. The liquid/ambient and the liquid/solid interfaces are treated in a unified context tackling the assumption that the liquid profile is wedge-shaped at any three-phase contact line. In this way, electric field singularities are bypassed, allowing for accurate electric field and liquid surface profile computation, especially in the vicinity of TPLs. We found that the invariance of the microscopic contact angle in electrowetting systems is valid only for thick dielectrics, supporting published experiments. By applying our methodology to patterned dielectrics, we computed all admissible droplet equilibrium profiles, including Cassie−Baxter, Wenzel, and mixed wetting states. Mixed wetting states are computed for the first time in electrowetting systems, and their relative stability is presented in a clear and instructive way. deposited on a dielectric layer coating a flat base electrode. When voltage between the base electrode and the conductive droplet is applied, electric charge accumulates in the liquid/ solid interface. The resulting decrease in the corresponding interfacial energy is observed as a decrease of the apparent contact angle (i.e., as an enhancement of the wettability of the solid by the liquid). A simple mathematical description of the apparent contact angle, θa, dependence on the applied voltage, V, is provided by the Lippmann equation7,8
1. INTRODUCTION 1,2
The miniaturization of devices renders electrowetting (EW) as the most convenient tool for manipulating microdroplets in a variety of contemporary applications (from liquid lenses3,4 and lab-on-a-chip5 systems to energy harvesting6). In a typical electrowetting-on-dielectric (EWOD) setup (see Figure 1), a droplet of a conductive liquid (usually an aqueous solution) is
cos θa = cos θY +
ε0εrV 2 ⇒ cos θa = cos θY + η 2dγLA
(1)
where εο is the vacuum permittivity and εr is the dielectric constant of the solid. The wettability of the solid dielectric material is quantified by the Young contact angle value, θY. The liquid/ambient (LA) interfacial tension is denoted by γLA, with the ambient medium being an insulating fluid and d being the thickness of the solid dielectric. The dimensionless EW number, η, expresses the relative strength of the electrostatic over the surface tension forces in the system, assuming a uniform electric field at the liquid/solid interface (ideal parallel Figure 1. Schematic of the electrowetting-on-dielectric (EWOD) setup illustrating a drop at zero voltage (solid line) and its new profile when voltage is applied (dashed line). © 2014 American Chemical Society
Received: January 30, 2014 Revised: April 1, 2014 Published: April 3, 2014 4662
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longer form wedges, allowing for a more accurate computation of the curvature of the liquid surface and of the field distribution close to the TPL. This article is organized as follows: We first present the mathematical formulation of the electrocapillary augmented YL equation for cylindrical liquid droplets (see Figure 2). The
plate capacitor). Despite its simplicity, the Lippmann equation provides accurate predictions in many EW configurations9,10 for low applied voltages and flat dielectrics;1 however, when higher voltages are applied, the Lippmann equation fails to predict the apparent contact angle accurately since it neglects the electric fringe fields close to the three-phase contact line (TPL), where the three different phases meet. In addition, the Lippmann equation predicts a steep monotonic decrease of the apparent contact angle (up to complete wetting) as the voltage increases. This contradicts the observed contact angle saturation phenomenon,1 which limits the electrostatic enhancement of the contact angle, and its onset is closely connected to the electric field distribution close to the TPL.11 Thus, for a more accurate description of the EW effect, modeling approaches need to incorporate droplet shape coupled with field distribution computations. Accurate electrowetting modeling for droplets sitting on solid dielectric horizontal planes has been demonstrated by solving the equations of capillary electrohydrostatics.12 The Young− Laplace (YL) equation13 is augmented by an electric stress term14 to account for the balance of the electrostatic and capillary forces at the free surface of the droplet. The Young contact angle boundary condition applied at the TPL accounts for the wetting properties of the solid dielectric material. Although this modeling approach can accurately compute the equilibrium droplet shape on flat, solid dielectric topology, this is not the case when droplets are deposited on geometrically patterned solid surfaces, which can admit wetting states with an a priori unknown cardinality of TPLs. Patterned solid dielectric surfaces exhibit a wide range of admissible apparent wettabilities15−17 (ranging from superhydrophobic Cassie− Baxter18,19 to superhydrophilic Wenzel19 states), where the distinct equilibrium states correspond to droplet shapes with multiple TPLs; the implementation of the Young contact angle boundary condition, at each TPL, is not a trivial task. We recently demonstrated16 that wetting states with multiple TPLs are well predicted by employing an augmented Young− Laplace equation with a disjoining pressure term,20,21 modeling the microscale liquid/solid (LS) interactions.22 By parametrizing the liquid surface profile in terms of its arc length (of the effectively 1D droplet surface), the liquid/ambient and the liquid/solid interfaces are treated in a unified context (one equation for both interfaces). In this way, the Young contact angle emerges implicitly from the combined action of liquid/ solid, and liquid/ambient microscale interactions rendering the implementation of the Young contact angle boundary condition not necessary. In this work, we utilize the augmented YL formulation16 to model the electrowetting phenomenon on any kind of patterned solid dielectric surface. The electric field effect is incorporated into the augmented YL equation through an electric stress term accounting for the electric forces exerted on the droplet surface. In this new formulation of the capillary electrohydrostatics, no pinning boundary condition (necessary when studies considering computations in a unit cell are performed23,24) is required at any TPL, thus enabling the computation of wetting states with multiple and reconfigurable contact lines. In addition, the proposed YL formulation does not suffer from the electric field singularities that arise in electrocapillary systems when the Young contact angle boundary condition is imposed, assuming a sharp wedgeshaped conductor of liquid at the TPL. The droplet profile solutions obtained from the augmented YL formulation no
Figure 2. Electrowetting of a cylindrical drop on a stripe-patterned solid dielectric. The droplet shape is obtained by solving the augmented Young−Laplace equation (eq 3).
proposed formulation is initially validated with the predictions of the electrohydrostatics equations (i.e., using the conventional YL,12 where the Young contact angle boundary condition is explicitly imposed). In order to investigate the local deformations of the droplet surface in the vicinity of the TPL, we perform accurate local mean curvature and electric field distribution computations for different dielectric thicknesses. Next, we perform computations of the entire droplet surface shape (as opposed to unit cell computations) on a multistriped solid dielectric and demonstrate the efficiency of our methodology to deal with multiple and reconfigurable TPLs. In particular, we present computations of Cassie−Baxter wetting states, where the surrounding medium is trapped beneath the droplet, as well as mixed and Wenzel wetting states, where the droplet has partially and fully penetrated the solid roughness, respectively, by varying the applied voltage.
2. MATHEMATICAL FORMULATION The dimensionless Young−Laplace equation accounting for the electric stress term (and neglecting the effect of gravity) reads12 C=K+
NeE2 2
(2)
When augmented by the disjoining pressure term, it yields16 4663
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Langmuir R o LS N E2 p +C=K+ e γLA 2
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action of the surface tension and the disjoining pressure (which is practically active only at the liquid/solid interface). The wetting parameter can be correlated to a specific Young’s contact angle (θY = θY(wLS)) through an effective interface potential, ω, which expresses the cost of free energy per unit area to maintain a distance, δ, between the solid and the liquid phases (ω → 0, when δ → ∞) and is related to the disjoining pressure, pLS, according to16,29,30
(3)
where C is the dimensionless local mean curvature of the droplet surface, K is a reference pressure which is constant along the droplet surface, and Ro is a characteristic length (radius of a sphere with volume equal to the droplet volume or radius of a circular disc with area equal to the droplet crosssectional area for spherical or cylindrical drops, respectively). For the sake of clarity, we refer to eqs 2 and 3 as the conventional and augmented forms of the Young−Laplace equation, respectively. The augmented Young−Laplace equation is endowed with microscale interactions between the solid and the liquid phases22 using a disjoining (or Derjaguin)25 pressure term, pLS. The disjoining pressure expresses essentially the excess pressure close to the solid surface and in dimensionless form reads16 ⎡⎛ σ ⎞C1 ⎛ σ ⎞C2 ⎤ R o LS ⎟ ⎟ ⎥ −⎜ p = w LS⎢⎜ ⎝δ + ε⎠ ⎦ γLA ⎣⎝ δ + ε ⎠
p LS = −
dω dδ
(5)
In addition, the effective interface potential, ω, reaches its minimum value, ωmin, in the vicinity of the TPL, and the following relation holds16,28−30 ωmin = γSA + γLA − γLS (6) where γSA and γLS are the solid/ambient and liquid/solid interfacial tensions, respectively. By combining eq 6 with the Young equation22
γLA cos θY + γLS = γSA (4)
(7)
the correlation between the Young contact angle and the wetting parameter, wLS, can be derived:16,28−30 ω cos θY = min − 1 γLA (8)
The wetting parameter, wLS, controls the wettability of the solid material (increasing wLS enhances the liquid/solid affinity). Parameters C1 and C2 define the range of microscale interactions: large C1 and C2 reduce the range within which microscale interactions are active (see also SI); σ and ε regulate the minimum distance between the liquid and the solid phase. The functional form of the disjoining pressure term resembles a Lennard-Jones potential26 modeling strong repulsion exerted on the liquid phase when in close proximity and attraction at an intermediate Euclidean distance, δ, from the solid material. The above formulation approximates the two main components of the disjoining pressure: the molecular or van der Waals component, created by the fluctuating electric field of the liquid and solid molecules, and the electrostatic component originating from the overlapping of the electrical double layers.22 Here, for simplicity, the disjoining pressure isotherm type corresponds to a nonwetting case (i.e., liquid film absorption on the solid surface is not taken into account16). Despite this rough consideration, the employed model can capture the basic physical features of wetting. Alternative disjoining pressure isotherms incorporating nontrivial effects (e.g., precursor film formation) can be also applied; however, this analysis is beyond the scope of this work. The microscale interactions are active only within a short distance, δ, from the solid boundary. For flat, solid boundaries, δ is the dimensionless vertical distance of the droplet surface from the solid surface; for nonflat, curved, solid surfaces, the definition of wall distance, δ, requires special attention. Here, δ is defined as the signed distance from the solid boundary (arbitrarily shaped in general), which is given by the solution of the Eikonal equation27 (see section 3 in the SI). This particular selection of δ has already been successfully tested in our previous study concerning droplet equilibrium on patterned surfaces.16 Note that even on the scales in which surface forces are active (i.e., from 30 Å to 1 μm) the continuum-level modeling is still applicable.28 Here, the augmented Young−Laplace equation (eq 3), contrary to its conventional form (eq 2), governs the entire droplet surface (i.e., both the liquid/ambient and the liquid/ solid interfaces). Therefore, the droplet shape and consequently the Young contact angle emerge “naturally” by the combined
Such types of equations, commonly known as Frumkin− Derjaguin formulas,22,31 are used for the theoretical evaluation of the contact angle, given the disjoining pressure isotherm. Apart from the capillary forces, the electric field affects the shape of the droplet surface by modifying the apparent wettability of the solid material. The contribution of the electric field to the force balance along the droplet surface is incorporated into the augmented YL equation (eq 3) with the electric stress term, (NeE2)/2, where Ne is the electric Bond number:
Ne =
εoV 2 γLAR o
(9)
The dimensionless electric field strength, E, is calculated along the droplet surface by solving the equations of electrostatics (Gauss’ law for electricity) for both the ambient phase and the dielectric material (see Figure 2)
∇·(εr∇u) = 0
(10)
where u is the dimensionless electric potential and E ≡ ∥∇u∥. The permittivity, εr, for the ambient phase (insulating medium) is εs, and it is εd for the solid dielectric. Here we assume that the droplet is perfectly conductive; however, in the case of a dielectric liquid or a liquid with finite conductivity, the equations of electrostatics should also be solved in the interior of the droplet, and the electric stresses in eq 3 should also be modified in order to account for the nonzero electric field inside the droplet. In our computations, we study droplets with translational symmetry along the direction perpendicular to the xy plane (cylindrical droplets). In order to capture both the liquid/solid and the liquid/ambient interfaces, the droplet profile, which is defined in cylindrical coordinates (r, φ), is parametrized in terms of the arc length, s, of the effectively 1D droplet surface (see Figure 2). The mathematical problem to be solved is a nonlinear free-boundary problem, with the unknown droplet 4664
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shape affecting the electric field distribution and vice versa. The coupled electrohydrostatic problem is solved numerically by the finite element method. The details of the numerical scheme, implemented here, are provided in the Supporting Information (SI) section.
applied) is modified accordingly and is given by a Young-type equation: γLA cos θa + γLSeff = γSA
By combining eqs 12 and 13, we can determine the apparent contact angle from the minimum effective interface potential value, ω*min:
3. RESULTS AND DISCUSSION 3.1. Computation of the Apparent Contact Angle: Frumkin−Derjaguin Approach versus Circular Fitting. In this section, we present computations of the dependence of the apparent contact angle, θa, on the applied voltage, V, for a cylindrical aqueous droplet in ambient silicone oil (AK 5, Wacker). Such insulating oils are commonly used as ambient phases in experimental practice.32,33 For the selected ambient phase, the Young contact angle of the aqueous droplet on a PTFE34 flat surface is approximately 170° (corresponding to a wetting parameter, wLS = 84, from eqs 4, 5, and 8). The disjoining pressure parameters used here are σ = 9 × 10−3, ε = 8 × 10−3, C1 = 12, and C2 = 10. The selected characteristic length, Ro = 0.62 mm, corresponds to a droplet with cross-sectional area Adroplet = πRo2 = 1.21 μm2. The same configuration is used for the derivation of all results presented in this work. In Figure 3, we show the dependence of the apparent contact angle value on the applied voltage, as computed from the
cos θa =
ω*(δ) =
2R o 2 2
2
=−
dω* dδ
(11)
2
where (εoV E )/(2Ro ) = Pel is the electrostatic pressure acting on the liquid surface with a negative contribution to the total pressure.7 The electric field strength, E, is computed along the droplet surface and can be expressed as a function of the distance, δ (i.e., E = E(δ)). The effective interface potential, ω*, reaches its minimum value, ω*min, in the vicinity of the TPL, and eq 6 is modified as follows ω*min = γSA + γLA − γLSeff
γLSeff
(14)
⎡ ε V 2[E(δ′)]2 ⎤ ⎥ dδ ′ ⎢p LS (δ′) − o 2R o 2 ⎦ min ⎣
∫δ
δ
(15)
where δmin is the minimum value of the liquid/solid distance. In order to compute the apparent contact angle from eq 14, apart from the droplet shape computation, the computation of the electric field strength along the droplet surface is also required. The computations are performed as indicated for a PTFE substrate with thickness d = 10 μm; similar findings also hold for different thicknesses. The comparison, in Figure 3, shows excellent agreement. (The two curves are optically indistinguishable.) The effective interface potential, employed in eq 14, can describe with remarkable accuracy the shape of the droplet on a macroscopic scale given the electric field distribution along the droplet surface. Simplifications regarding the electric field distribution (e.g., neglecting the effect of fringe fields around the contact line) will lead to erroneous estimations of the apparent contact angle value. Next, we validate our proposed methodology by comparing the augmented YL equation with the conventional one, which explicitly imposes the Young contact angle boundary condition.12 3.2. Validation with the Predictions of the Conventional Young−Laplace Equation. The results obtained from the electrocapillary augmented YL equation (see eq 3) are compared to those obtained from the conventional electrohydrostatics formulation (see eq 2, also described in detail in ref 12) for flat, solid dielectrics (PTFE foils) of variable thickness (d = 10, 50, and 150 μm). The apparent contact angle presented in Figure 4 is obtained by circular fitting at the liquid/ambient interface. The comparison between the results obtained from the two different formulations shows excellent agreement for all dielectric thicknesses tested. However, the predicted contact angle values obtained by the Lippmann equation are systematically smaller. This deviation is expected and has been reported in the literature,35 since the Lippmann equation accounts only for the electrostatic energy stored within the dielectric layer between the droplet and the flat electrode, neglecting the corresponding energy of the ambient medium. Indeed, if we neglect the surrounding medium permittivity (i.e., assuming εs = 1), then our computations agree with the Lippmann equation (see Figure 4). In Figure 5, the droplet profiles and the electric potential distribution are shown in the TPL region. In the case of the conventional YL equation, the Young contact angle is explicitly imposed as a boundary condition by assuming a wedge-shaped liquid/ambient interface profile at the TPL. This geometric assumption is not necessary when solving the augmented YL equation. In this case and in order to obtain the Young contact
Frumkin−Derjaguin theoretical formula and by performing a circular fitting on the computed liquid/ambient (here water/ oil) interface. The theoretical formula is derived according to the methodology presented in the previous section (eqs 5−8). The effective interface potential is now modified (ω*) in order to incorporate both the microscale (disjoining pressure) and the electric field (electrostatic pressure) effects εoV 2E2
ω*min −1 γLA
The effective interface potential is computed from the integral
Figure 3. Apparent contact angle dependence on the applied voltage; θa is calculated following the Frumkin−Derjaguin approach (eq 14) as well as by circular fitting on the liquid/ambient interface (εd = 2.1 (PTFE), εs = 2.58 (AK 5, Wacker), θY = 170°, γLA = 0.038 N/m, and d = 10 μm).
p LS −
(13)
(12) 7
where is an effective liquid/solid interfacial tension, with γLSeff < γLS. The apparent contact angle (when voltage is 4665
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thickness of the solid dielectric is small. Below, we examine whether a change in the dielectric thickness has an effect on the Young contact angle value, which in the augmented YL formulation arises implicitly from the combined action of microscale and capillary forces. 3.3. Investigation of the Dielectric Thickness Effect on the Young Contact Angle. In this section, we perform computations of the Young contact angle and the liquid surface curvature distribution for different dielectric thicknesses in order to investigate a possible link between the Young contact angle value and the electric field strength in the vicinity of the TPL. In Figure 6, we present the liquid surface curvature and the disjoining pressure distribution along the droplet interface
Figure 4. Dependence of the apparent contact angle on the applied voltage, V, for d = 10, 50, and 150 μm PTFE, calculated from the conventional YL (eq 2), the augmented YL (eq 3), and the Lippmann equation (eq 1).
Figure 6. Dimensionless local mean curvature and disjoining pressure distribution along the droplet surface (d = 50 μm PTFE, η = 0.5, V = 319.7 V).
Figure 5. Electric potential distribution in the vicinity of the TPL computed from the solution of (a) the augmented and (b) the conventional YL equation. The Young contact angle boundary condition in the conventional formulation creates wedge-shaped geometry at the TPL. The equipotential lines are depicted in light yellow (d = 10 μm PTFE, η = 0.67, V = 165 V).
profile for η = 0.5. For a computed droplet equilibrium profile, the curvature distribution is calculated from eq SI-2 (see the SI section). One can observe the existence of three distinct regions corresponding to (I) the liquid/ambient interface where the curvature is constant, (II) the region close to the TPL where the curvature increases sharply up to a maximum finite value, and (III) the liquid/solid interface where the curvature is equal to that of the solid surface (zero for flat, solid dielectrics). The dimensionless disjoining pressure (see Figure 6) is negligible along the liquid/ambient interface and drops abruptly close to the TPL, reaching a minimum value. The disjoining pressure is constant along the liquid/solid interface: RopLS/γLA = K + NeE2/2 (from eq 3). Thus, the droplet shape is mainly determined by the capillary forces along the liquid/ambient interface; the microscale forces (disjoining pressure) prevail along the TPL and the liquid/solid interface. Finally, electrostatic forces are mainly concentrated within a region close to the contact line, the extent of which depends on the dielectric thickness32 (smaller for thin dielectrics). In Figure 7, we present important shape features of the droplet surface for different solid dielectric thicknesses of PTFE (d = 10, 50, and 150 μm) over a range of electrowetting numbers (η ∈ [0, 1]). In particular, we compute the dependence of the Young contact angle, θY, and the dimensionless maximum local mean curvature of the surface
angle implicitly, we perform high-order (sixth) polynomial fitting to the droplet profile close to the TPL.34 The Young contact angle emerges within the range of action of the microscale interactions (disjoining pressure) (see Figure 5c). The augmented YL equation also governs the liquid/solid interface, defined as the part of the droplet interface which is at a minimum distance, δmin, from the solid boundary. This minimum distance, δmin, corresponds to the thickness of the liquid/solid diffuse interface. Indicatively, when the characteristic length is Ro = 0.62 mm, δmin = 3.25 × 10−4 mm. It should also be noted that the proposed augmented YL formulation can provide a more detailed picture of the droplet surface shape and of the field distribution close to the TPL, compared to the conventional formulation; in the latter case, the field strength theoretically reaches an infinite value due to the singularity induced by the wedge-shaped droplet profile at the TPL. Theoretical studies of Buehrle et al.,36 based on the local balancing of electrostatic and capillary forces at the TPL, suggest that the Young contact angle is independent of the applied voltage. Experimental studies,32 however, show a slight variation of the Young contact angle, especially when the 4666
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Figure 7. Dependence of (a) the maximum curvature of the droplet surface and (b) the Young contact angle, θY, on the electrowetting number, η, for different dielectric thicknesses (d = 10, 50, and 150 μm).
as a function of the electrowetting number, η. For the computation of the Young contact angle (see also Figure 4), a sixth-order polynomial fitting is applied to the droplet surface in the vicinity of the TPL. One can observe that the maximum curvature and Young contact angle appear to be insensitive to the electrowetting number only for thick dielectrics (d = 50, 150 μm). Variations in both the maximum curvature value and the Young contact angle value are observed for the thin dielectric substrate case (d = 10 μm). A sensitivity analysis regarding the validity of this claim for different disjoining pressure types (different C1 and C2 parameters) is presented in the SI section. We also note that for all three cases the change in the apparent contact angle, θa, due to the electrowetting effect ranges from 170 to approximately 125°. Similar trends have been presented in the experimental work of Mugele and Buehrle.32 There, the variation of the Young contact angle value measured for thin dielectrics was attributed to the finite and limited optical resolution of their setup to resolve the details of the droplet profile; measuring curvature values in experimental practice involves high-order polynomial fitting of the droplet interface, a process that is highly sensitive to the optical resolution. According to our perspective, the Young contact angle variation is not really a resolution issue but is related to the electric field distribution along the droplet surface, which strongly depends on the thickness of the dielectric. The investigation of the dielectric thickness effect on the liquid surface curvature distribution and on the electric stress term of the augmented YL equation provides more evidence on the possible link between the Young contact angle and the electric field strength at the TPL. In Figure 8, we present the variation of the curvature and the electric stress term of the augmented YL, along the droplet surface for different PTFE thicknesses and for a fixed electrowetting number (in this case η = 0.5). We observe that for the case of the thin dielectric (d = 10 μm) the strength of the electric stress term increases sharply at the TPL and then reaches a constant value along the liquid/solid interface. For thicker dielectrics (see Figure 8, d = 50 and 150 μm), the maximum electric stress value decreases and its effect extends over a wider region around the TPL (as compared to the case of the thin dielectric with d = 10 μm in Figure 8). The observed variations of the computed maximum curvature value for thin dielectrics can be attributed to the higher electric stresses developed within the action range of the microscale forces. In Figure 9, we present the relative strength, λF, of the electric stress term of the augmented YL equation over the disjoining pressure in the vicinity of the TPL for different dielectric thicknesses and electrowetting numbers. The
Figure 8. Variation of the dimensionless local mean curvature and the electric stress term of the augmented YL equation along the droplet surface for different dielectric thicknesses (η = 0.5).
Figure 9. Dependence of the relative strength, λF, of the electric stress term (NeE2/2) over the disjoining pressure (RopLS/γLA) of the augmented YL (eq 3), at the TPL, on the electrowetting number for different dielectric thicknesses: d = 10, 50, and 150 μm.
electric stress term is of lower magnitude compared to the disjoining pressure in all studied cases of electrowetting numbers when the dielectric thickness is 50 and 150 μm. However, in thinner dielectric thickness cases (d = 10 μm) the electric field effect is significantly enhanced especially at high electrowetting numbers; the magnitude of the developed electric stresses exceeds the disjoining pressure, causing an increase in the local curvature and a concomitant decrease in the Young contact angle. Similar results can be obtained using different forms of the disjoining pressure isotherm (see the SI section). Apart from the ability to perform accurate curvature and electric field computations, the main advantage of the proposed methodology is its ability to model electrowetting on geometrically patterned solid dielectric surfaces, where multiple and reconfigurable TPLs arise. Existing fine-scale electrowetting modeling approaches (e.g., molecular dynamics17,37 and mesoscopic lattice Boltzmann models15,38,39) suffer from severe computational limitations (especially when real millimeter-sized droplets are studied), whereas continuum-level models are based on significant simplifications regarding the actual shape of the droplet and the field distribution40 at the TPL. 4667
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3.4. Electrowetting on a Geometrically Patterned Dielectric Surface. In our example, we model electrowetting on a multistriped dielectric (see Figure 2). The height (hd) and the width (wd) of the protrusions are 62.03 and 93.05 μm, respectively. The minimum thickness of the dielectric (PTFE) is dmin = 25 μm, and the maximum thickness is dmax = hd + dmin = 87.03 μm. For the computation of the electrowetting number (see eq 1), we select d = dmax = hd + dmin in order to account for the nonuniformity of the dielectric thickness. As reported above, geometrically patterned dielectrics can admit multiple droplet equilibrium profiles ranging from Cassie−Baxter to Wenzel wetting states. By utilizing parameter continuation techniques (see the SI section), all of the admissible wetting states (stable and unstable) can be traced. In particular, in Figure 10, we present the dependence of the
By decreasing the applied voltage, the droplet dewets the solid surface following a different route: D → E → F (see Figure 10a). The range of hysteresis is determined by the range of the intermediate unstable branches (dashed lines in Figure 10). We also note that the initial fakir droplet with θa = 177° cannot be recovered by switching off the applied voltage; the wetting transition is irreversible, and the highest apparent contact angle that can be recovered through voltage reduction is 164°. Interestingly enough, the intervening stable solution branches between points E and D in Figure 10 correspond to wetting states exhibiting nearly identical macroscopic characteristics (the same apparent wettability for the same applied voltage), yet with significantly different fractions of the solid surface area wetted by the liquid. In Figure 11, we present the
Figure 10. Dependence of (a) the apparent contact angle, θa, and (b) the dimensionless half-perimeter length, smax, on the electrowetting number, η, for a patterned solid dielectric surface.
apparent contact angle (calculated by fitting a circle to the liquid/ambient interface above the solid surface asperities) and the dimensionless half-perimeter length of the droplet, smax, on the electrowetting number. The different solution branches correspond to distinct wetting states. In particular, the stable solution branch AB corresponds to droplets wetting a single stripe, while the stable solution branch FC corresponds to Cassie−Baxter states wetting three stripes (taking into account the reflection symmetry about the vertical plane (φ = 0)). Several solution branches intervene between points E and D, exhibiting approximately the same apparent wettability for the same electrowetting number, and correspond to (I) Cassie− Baxter states wetting five stripes (see Figure 10b, η = 1 (Cassie−Baxter)), (II) fully collapsed Wenzel states wetting the same number of stripes (see Figure 10b, η = 1 (Wenzel)), and (III) mixed wetting states where the liquid has partially penetrated the solid roughness (see Figure 10b, η = 1 (mixed)). We also observe that the electrostatically induced wetting transitions on the patterned dielectric under study are hysteretic. When no voltage is applied, a “fakir” droplet sits on top of protrusions (see Figure 10, point A for η = 0, θa = 177°). By quasi-statically increasing the applied voltage, the apparent wettability is enhanced (the apparent contact angle decreases from 177 to 128°) and the liquid gradually penetrates the solid surface roughness, following the route A → B → C → D along the stable solution branches (solid lines in Figure 10a).
Figure 11. Electric field distribution around coexisting wetting states on a stripe-patterned dielectric for η = 1.19 in the vicinity of the solid surface. The streamlines of the electric field are depicted in light yellow. The stable wetting states are denoted with “S”, and the unstable ones, with “U”.
electric field distribution in the vicinity of the solid surface for five coexisting wetting states (stable and unstable) with θa ≈ 139.3° and the same electrowetting number (η = 1.19). Apart from the stable Cassie−Baxter (Figure 11a) and Wenzel (Figure 11e) wetting states, the patterned dielectric can also admit mixed stable states (Figure 11c), where only two of the four oil pockets of the solid surface are filled with water. We remark that the intermediate unstable equilibrium states depicted in Figure 11b,d cannot be tracked experimentally. The sequence of equilibrium profiles in Figure 11b−d illustrates the intermediate states of a minimum-energy path connecting the Cassie−Baxter state (Figure 11a) and Wenzel state (Figure 11e); impalement originates from the outer side of the droplet and succeeds in the center for the particular solid surface geometry. 4668
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Further investigation of the effect of the surface topography on the wetting transition mechanism is required in order to realize surfaces with desirable wetting behavior (facilitating or preventing wetting transitions). Continuum-level studies of wetting transitions on geometrically patterned surfaces have been previously presented in the literature; however, they do not take into account the pinning/depinning effects of the droplet at the protrusion edges, performing the analysis on a unit cell of the texture.23,24 Under this assumption, no mixed states can be predicted (since the outer stripes are equivalent to the inner ones), and thus there is no information concerning the energetics of the mechanism of the wetting transition. The multiplicity of the wetting states, computed here for a relatively simple patterned geometry, is indicative of the wealth of equilibrium states expected to be computed when a smaller scale of arbitrary roughness (i.e., without any symmetry assumption) for solid surfaces is studied. Although the liquid surface parametrization presented is suitable only for cylindrical (translationally symmetric) or spherical (axially symmetric) droplets, proper parametrization/ discretization (e.g., unstructured discretization) of fully 2D surfaces (i.e., without any symmetry assumption) is required in order to deal with 3D electrowetting simulations.
The presented methodology can be readily applied to any kind of solid surface topography and could be proven to be a valuable tool for understanding the effect of geometric characteristics on wetting transitions. Such an analysis, paired with optimization algorithms, can suggest the proper surface roughness geometry, which facilitates (or prevents) the switching between Cassie−Baxter and Wenzel wetting states,41,42 especially when realistic modeling formulations of the disjoining pressure isotherm type are used (i.e., experimental force−distance isotherms obtained from atomic force microscopy (AFM)43,44).
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ASSOCIATED CONTENT
S Supporting Information *
Numerical algorithm for the solution of the electrohydrostatics equations. Effect of the disjoining pressure isotherm type. Eikonal equation. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
4. SUMMARY AND CONCLUSIONS In this work, we present a formulation of continuum modeling of electrowetting phenomena without using the Lippmann equation or the Young contact angle boundary condition. We apply our methodology by performing equilibrium droplet shape computations of electrowetting-on-(geometrically patterned)-dielectric surfaces. Upon validation of the presented methodology against the predictions of the conventional Young−Laplace equation (which imposes the Young contact angle boundary condition), we study the effect of the dielectric thickness on the liquid surface curvature distribution and on the Young contact angle for different electrowetting numbers. We found that increasing the applied voltage in adequately thin dielectrics results in variations in the droplet shape both macroscopically (apparent contact angle) and microscopically (Young’s contact angle). This slight dependence of the Young contact angle on the applied voltage has also been reported previously in the literature but not theoretically supported.32 The advantage of the proposed modeling approach is its ability to perform electrowetting computations on geometrically structured dielectric surfaces. To our knowledge, this is the first time that a continuum-level model can predict Cassie− Baxter, Wenzel, and mixed wetting states by simulating the entire droplet and not a small part of it, like a unit cell. We should also note that no predefinition of the cardinality and the position of TPLs is required, contrary to previous computational approaches. Cassie−Baxter and Wenzel wetting states can be computed by performing fine scale simulations (molecular dynamics,17,37 lattice Boltzmann models15,38,39). However, the required computational cost is considerably higher, especially in cases where real-life millimeter-sized droplets are simulated. Although the methodology is presented for cylindrical droplets, it can also be extended beyond one spatial dimension, preserving its computational merits. The simulation of 3D droplets requires proper parametrization/ discretization of their 2D surfaces and is the subject of ongoing research.
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ACKNOWLEDGMENTS We kindly acknowledge funding from the European Research Council under the Europeans Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 240710.
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REFERENCES
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