Dynamic Response in Nanoelectrowetting on a Dielectric Jyoti Roy Choudhuri,† Davide Vanzo,† Paul Anthony Madden,‡ Mathieu Salanne,§,∥ Dusan Bratko,† and Alenka Luzar*,† †
Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284, United States Department of Material Science, Oxford University, Park Road, Oxford OX1 3PH, United Kingdom § Sorbonne Universités, UPMC Université Paris 06, CNRS, UMR 8234 PHENIX, 75005 Paris, France ∥ Maison de la Simulation, CEA, CNRS, Université Paris-Sud, UVSQ, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France ‡
S Supporting Information *
ABSTRACT: Droplet spreading at an applied voltage underlies the function of tunable optical devices including adjustable lenses and matrix display elements. Faster response and the enhanced resolution motivate research toward miniaturization of these devices to nanoscale dimensions. The response of an aqueous nanodroplet to an applied field can differ significantly from macroscopic predictions. Understanding these differences requires characterization at the molecular level. We describe the equilibrium and nonequilibrium molecular dynamics simulations of nanosized aqueous droplets on a hydrophobic surface with the embedded concentric electrodes. Constant electrode potential is enforced by a rigorous account of the metal polarization. We demonstrate that the reduction of the equilibrium contact angle is commensurate to, and adjusts reversibly with, the voltage change. For a droplet with O(10) nm diameter, a typical response time to the imposition of the field is of O(102) ps. Drop relaxation is about twice as fast when the field is switched off. The friction coefficient obtained from the rate of the drop relaxation on the nonuniform surface, decreases when the droplet approaches equilibrium from either direction, that is, by spreading or receding. The strong dependence of the friction on the surface hydrophilicity points to the dominance of the liquid-surface friction at the drop’s perimeter as described in the molecular kinetic theory. This approach enables correct predictions of trends in dynamic responses associated with varied voltage or substrate material. KEYWORDS: nanoscale, dynamic electrowetting, dielectric, molecular dynamics, nanopixel
E
supporting an apolar drop under an aqueous electrolyte. Berge and Peseux7 presented an improved design where the apolar drop is localized on the hydrophobic domain of a film separating two aqueous compartments. Using multiple electrodes, Krupenkin et al.8 showed the lens position can be controlled as well. Principles used in the lens manipulation can also be applied to an array of cells acting as electrowettingbased pixels in an electronic display.9 Reduction of a droplet size to nanoscale10,11 opens possibilities of designing optoelectronic devices with an unprecedented speed and resolution. In view of the experimental challenges associated with these small length
lectrowetting is a method used to control surface wettability through the application of an electric field.1 The prevalent experimental setup involves electrowetting-on-dielectric (EWOD), where a liquid drop is separated from the supporting electrode by an insulator layer preventing electric current through the system.2−5 For macroscopic drops with nonzero conductivity, the electric capacitance, C, is proportional to the area of the insulator layer beneath the drop. The net change in electric energy, comprised of the capacitive energy and the energy at voltage source, 1 Ec = − 2 CU 2 , therefore decreases with wetted area, providing an incentive for drop’s spreading under the applied voltage U. This mechanism finds applications in switchable optical devices such as tunable lenses and reflective displays. Gorman et al.6 obtained a variable focus lens by the application of voltage across a self-assembled monolayer on a transparent electrode © 2016 American Chemical Society
Received: June 7, 2016 Accepted: August 24, 2016 Published: August 24, 2016 8536
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Since the spreading of the drop determines the electric capacitance, these dynamics are associated with nonlinear changes of the electrode charge with the varied voltage. Our in silico droplet can be viewed as a microscopic analogue of tunable liquid lenses8 and pixels9 with a high contact angle sensitivity and an accelerated response in a subnanosecond regime.
scales, molecular modeling is ideally suited to provide preliminary insights toward the design of nanosized optical devices. We use molecular dynamics (MD) simulations to examine the electric actuation of a sessile nanodroplet of water on an insulating layer above the electrode. As the Debye screening length in water exceeds the nanodrop dimensions, the system behaves as a miniature version of a dielectric drop atop a conducting solid (see system B of ref 4, for example). The applications of electrowetting in nanosized devices, from microelectronic and nano-optoelectronic technologies, such as inkjet printing,12 electrostatic painting and spraying,13 nanoimprinting and nanomanufacturing14,15 benefit from understanding the interactions between liquid molecules and solid surfaces in the presence of an electric field. The effect of an applied voltage, U, on the surface spreading of a drop can be described by the Young−Lippmann equation4 cos θc = cos θ0 +
U 2 dC 2γlv dA sl
RESULTS AND DISCUSSION Simulation Setup. We consider an aqueous nanodrop placed on a hydrophobic insulator layer covering a pair of concentric electrodes. A similar design has been tested experimentally.8,20 The electrodes are separated by a ring of the identical insulator material (Figure 1). This way, the field at
(1)
with cos θ0 =
γsv − γsl γlv
(2)
Above, θc = θc(U) is the equilibrium contact angle at voltage U and θ0 corresponds to U = 0. γαβ represents the interfacial free energies of the solid (s), liquid (l), and vapor (v) phases and C is the total capacitance associated with a given area of a solid/ liquid interface, Asl. When the drop contains conducting liquid, C is usually proportional to Asl and the derivative dC in eq 1 dA sl
can be replaced by the areal capacitance C/Asl.1,2,4,5 The differential form of eq 1 is, however, appropriate for nanodrops where a portion of electric energy is stored inside the liquid phase.4 According to eq 1, the wettability depends only on absolute voltage, and not on the field direction. Molecular simulations, however, reveal a significant coupling between the angular bias of interfacial molecules and their alignment with the external field.16−18 This coupling is reflected in a strongly anisotropic polarization response19 of the interfacial water. An especially important consequence for nanoelectrowetting is a significantly amplified effect on wetting when the field is directed parallel with, rather then perpendicular to, the surface.16−18,21,22 Experimentally, a precise contact angle regulation can only be achieved trough the electric potential. Simulation studies of electrowetting, however, typically have been performed in analogy with the experimental setup of the drop in a capacitor,23,24 characterized by a fixed strength of the electric field. Alternatively, an imposed charge has been used as the control variable in modeling EWOD by a coarse grained beadchain liquid.25 To implement an explicit voltage control in molecular simulation, we rely on MD integration at fixed electrode potential.26−28 The method treats fluctuating charges of metal atoms as additional degrees of freedom by perpetually adjusting them to maintain the desired potential through the metal polarization. This way we capture the generic electrode behavior absent in the simplified constant charge models.28,29 We demonstrate the viability of the constant voltage MD as a demanding but accurate technique to rigorously model nanoeletrowetting-on-dielectric. We study changes of a nanodroplet contact angle under preset voltage between electrodes, and dynamic responses of drop’s shape to the voltage change.
Figure 1. Layered representation of the EWOD system with the concentric electrodes. The pale green atoms under the electrode constitute a hydrophobic insulator monolayer with the area 9.8 nm × 9.8 nm. The inner and outer Pt electrodes, shown in magenta and blue, are separated by an insulator ring. Rb is the base radius of the drop, Ri the diameter of the inner Pt electrode, and Ro the external diameter of the insulator ring. The layers are in direct contact. Vertical shift of the bottom layers is used in the plot for a better visualization.
the solid/liquid interface is essentially parallel to the surface, corresponding to the optimal direction to maximize the surface wetting by water.16−18,30 Unlike the setup of the drop-incapacitor,17 the field between concentric electrodes acts in the radial direction, thus a centrally positioned drop will spread more or less symmetrically over the surface. To keep the drop’s perimeter above the strong-field region between the electrodes, the electrode dimensions are chosen according to the surface hydrophobicity and the droplet size as described next. The droplet contains 2.2 × 103 water molecules. Previous simulation studies of sessile drops17,20,31 confirmed that this size was sufficient to secure accurate statistical representation and showed only minor line tension effects.31,32 The model substrate consists of three layers of atoms at positions of a facecentered cubic Pt crystal with lattice parameter 3.9 Å along the (111) crystallographic plane. The specific crystallographic plane is chosen because it has lower surface energy compared to the (100) and (110) planes and is prevalent in polycrystalline surfaces.33 For the sake of simplicity, the insulator atoms in the top layer and between Pt electrodes are of the identical size as Pt atoms in the electrodes, however, only Pt atoms carry 8537
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ACS Nano fluctuating charges to capture the conductor behavior. Two systems are used to check for any dependence of the effects of electrowetting on the initial contact angle, and to explore the dynamics of spreading and retraction at varied liquid/substrate frictions. The target contact angles of water at 111° in System A and 149° in System B are captured by adjusting the shortranged attraction between water and insulator atoms, which was essentially weaker than for the Pt atoms. The circular electrodes beneath the hydrophobic layer are separated by an insulator ring of width 2.6 nm in System A and 2 nm in System B. The electrode under the center (inner electrode) has the shape of a disk of radius 1.4 nm in System A and 1.0 nm in System B, and the inner radii of the ring-like outer electrode in Systems A and B are 4 and 3.6 nm, respectively. The width of the outer electrodes is 0.6 nm in both systems. The remainder of the substrate is made of the insulator atoms. With the selected dimensions, the perimeter of the drop’s base, Rb, is located above the region between the electrodes both in the presence and absence of the electric field, that is, Ri < Rb < Ro. The simulation box is a cube of 98 Å side with the periodic boundary conditions (PBC) imposed along the lateral directions. From above, the box is closed by a purely repulsive wall to prevent the escape of vapor molecules along the nonperiodic z direction. Below, we report the magnitude and dynamics of the response of simulated nanodrop’s shape to the applied voltage, as done macroscopically in EWOD experiments. Static Response. By attracting dipolar water molecules, the local electric field generally enhances materials’ wetting propensity. The field effect on wetting can be characterized through nanodrop contact angle calculations.17 Our results for a set of interelectrode potential differences in the interval 0−4 V show consistently enhanced spreading under the applied field, in analogy to experimental34 and continuum-simulation35,36 results. Figure 2 presents the fitted droplet contours at varied
Table I. Equilibrium Contact Angle θc at Different Values of Potential Difference, U, between the Electrodes in Systems A and Ba U/V
θAc /°
θBc /°
0 1 2 3 4
111 106 99 90 84
149 144 135 125 119
The error bar of θc, estimated as the standard deviation of the mean for 12-15 subaverages in each of the 1.2 ns runs, is rounded up to δθc ∼ ± 1°. (See Supporting Information). a
squared. Simulation results for cos θc(U2) in systems A and B, Figure 3, however, reveal a sublinear dependence on U2
Figure 3. Variation of the cosine of contact angle with square of the applied potential difference (U) between the electrodes for System A (black) and System B (blue). Solid circles are simulation results, with the dotted lines showing a likely interpolation between them. Error bars correspond to the estimated ±1° uncertainties in θc (See Supporting Information). Long-dashed lines show predictions from eq 1.
reminiscent of the saturation behavior observed in macroscopic experiments. In the absence of charge carriers in the drop, the saturation cannot be attributed to charge redistribution effects discussed in ref 25. To explore possible relation to changes in the system’s capacitance C, we determined C = q/U as a function of wetted area, Asl(U), in both systems studied. Here, q is the absolute value of the charge on the electrodes, required to maintain preset voltage U at the specified extent of the droplet spreading. Area Asl equals πR2b, where Rb = Rs sin θ and Rs is the radius of curvature of the drop with volume V ⎞1/3 ⎛ 3V R s(θ ) = ⎜ ⎟ ⎝ π (1 − cosθ )2 (2 + cosθ ) ⎠
Figure 2. Vertical cross section of the water droplet on the insulator surface, showing drop’s contours (solid red line) fitted through simulation data points for different electric fields. The solid line in magenta represents the position of the surface reference plane. Left, System A; right, System B.
(3)
Drop dimensions for different voltages are included in Table 1 of the SI. Charge q and capacitance C depend on the shape of the drop, which requires a finite time to respond to the imposition of the voltage. The charge equilibration is illustrated in Figure 4. The solid symbols in Figure 5 represent the equilibrium capacitances for the four values of the droplet base area Asl corresponding to U = 1, 2, 3, or 4 V. The empty circles correspond to capacitances at zero voltage, obtained by an extrapolation to respective Asl. The capacitances at the origin (hypothetical states with Asl = 0, θc = π) differ between the two
voltages for both System A and B. In Table I, we present simulated contact angles as functions of the applied voltage for both systems. We observe no hysteresis in contact angles provided at least ≃150 ps equilibration is used after each voltage change. For constant areal capacitance, C/Asl, the Young-Lippman eq (eq 1) predicts cos θc to rise in proportion to the voltage 8538
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dynamic contact angle as a function of time. A statistically meaningful estimate of θ by contour averaging requires sampling time of O(0.1) ns or more. The process of the droplet spreading or receding is too fast to allow a direct determination of θ(t). Instead, we quantify the relaxation rate by monitoring the height of the center of mass of the drop, h(t), which provides an indirect route to study the dynamic changes in the drop geometry. Presuming the drop retains the shape of a truncated sphere, data for h(t) enable estimates of the dynamic contact angle, θ = θ(t) and the associated radius of curvature of the droplet, Rs(θ), as solutions of the implicit equation πR s2(H − h) −
π V [(R s3 − (h − H + R s)3 ] − =0 3 2 (4)
Figure 4. Variation of electrode charge q with time for individual voltages. e0 is the elementary charge 1.6 × 10−19 C. Abrupt charging at essentially constant initial capacitance, which occurs during the first fraction of a picosecond after the voltage is turned on, is not shown. The time interval on the x axis spans the range from 1 to 250 ps.
for the entire relaxation process. To solve eq 4 for cos θ, we express Rs by eq 3 and the height of the drop H(t) by the equation H = Rs(1 − cos θ). Figure 6 illustrates the temporal dependence of the height h(t) in Systems A and B as the drop spreads after connecting
Figure 5. Variation of the capacitance with the base area (Asl) of the drop for System A (black) and System B (blue). Capacitance is expressed in attofarad (aF). Solid symbols: simulation results for U = 1, 2, 3, and 4 V. Open symbols: extrapolation to zero voltage. Error bars are indicated by the size of the symbols.
systems because of different electrode sizes, the ratio of 1.6 being equal to the ratio of combined electrode areas in the two systems. In both systems, the capacitance features an approximately linear increase with Asl within the range covered in the simulations. The linear dependence reflects the predominant contribution of the first hydration layer to the total polarization of the nanoscale aqueous phase, consistent with observations in nanofilms.37,38 The slopes dC are equal in both systems. Approximating this derivative
dA sl dC in dA sl
eq 1 by
Figure 6. Variation of the height of the center of mass of the drop spreading under voltage 4 V (solid lines), and the reverse process (dashed lines), when the electric field is switched off. Black, System A; blue, System B.
the electrodes to the voltage source at U = 4V, or retracts when the voltage is turned off. Figure S2 in the Supporting Information illustrates the corresponding change in the contact angle θ(t), deduced by using eqs 3−4 and the smoothed data of h(t). The relaxation times for wetting and retraction processes are estimated from the time correlation function R(t)
C A sl
overestimates the predicted increase in cos θc by a factor of 2−3 (see Figure S1 in the Supporting Information). Using the simulated values of dC from Figure 5 reduces the deviation dA sl
R (t ) =
(dashed lines in Figure 3) but does not capture the observed trend toward the saturation. The differences between simulated cos θ and predictions of eq 1 are rationalized by partial distortion of the droplet’s hemispherical shape and the neglect of the influence of the electric field on the surface tension, γlv,16 of the droplets. Dynamic Response. The estimation of the rate of droplet’s response to a voltage change requires the knowledge of the
h(t ) − h(∞) h(0) − h(∞)
(5)
Here, h(0) is the height of the drop at time t = 0 and h(∞) is the equilibrated height. Results for R(t) for the two systems presented in Figure 7 show that the retraction process with voltage turned off is considerably faster than spreading under the applied voltage. The difference can be quantified in terms of the correlation time, obtained by integrating R(t) from t = 0 8539
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ACS Nano ⎡ γ λ2 ⎤ v = 2k 0λ sinh⎢ lv (cos θ0 − cos θ)⎥ ⎢⎣ 2kBT ⎥⎦
(6)
kB is the Boltzmann constant and T temperature. For small arguments of sinh, eq 6 can be linearized to v=
γlv(cos θ0 − cos θ) ζ
ζ=
kBT k 0λ 3
(7)
or FW = ζv, where FW = γlv(cos θ0 − cos θ) is the driving force (per unit length) and ζ is the friction coefficient, which determines the dissipation rate due to the perimeter motion. ζ has the dimension of the shear viscosity and can be compared with the bulk viscosity η of the liquid.42 Figure 8 illustrates the dependence of the MKT friction coefficient on the perimeter velocity v in Systems A (top) and B
Figure 7. Variation of R(t) with time for spreading in the presence of 4 V applied potential (solid line) and retraction at 0 V (dashed line), when electric field is switched off. Black, System A; blue, System B.
until a time characterized by a negligibly small R(t). The correlation times we obtain in System A using the upper integration limit of 500 ps are 115 ± 5 ps for spreading at voltage 4 V and 78 ± 5 ps for the reverse process. The standard deviations are obtained by running multiple independent simulations. In System B, the respective times are 56 ± 5 ps and 30 ± 3 ps. To put these times in perspective, we compare them with the experimental spreading time of about 5 ms, determined8 in a similar setup with an aqueous microlens of volume V = 6 μL (Rs ≃ 1 mm). The observed dynamic hysteresis can be explained in terms of the interaction between water molecules and the solid surface. These interactions are much stronger in System A than in the highly hydrophobic System B. The attraction is further strengthened in the presence of the electric field, which draws water molecules toward the substrate. The drop therefore experiences the strongest friction force during spreading on Surface A in the presence of an electric field. During retraction on Surface A at zero field, the friction coefficient is close to that observed on Surface B with the voltage turned on. Changes of the friction coefficient conform to equilibrium contact angles collected in Table I, with a smaller contact angle corresponding to a stronger friction and slower relaxation. In general, friction opposing the sessile droplet relaxation can be attributed to viscous forces inside the moving liquid (hydrodynamic friction),39,40 and to the molecular friction between the moving liquid and the substrate.41,42 In contrast to our results shown in Figures 6 and 7, the hydrodynamic mechanism implies only a slight dependence on material’s properties and the applied voltage. The liquid/substrate friction mechanism, which strongly depends on both properties,41,42 is therefore likely to dominate in our system. The latter mechanism can be interpreted in terms of the molecular kinetic theory, MKT,43 which concerns thermally activated displacements of liquid molecules at the three phase contact line. Individual molecular jumps occur with a characteristic frequency k0 and length λ. The bias of displacement probability, introduced by the capillary forces, results in the perimeter propagation in the direction of the force. The relation between the dynamic contact angle and the perimeter velocity (v) is given by
Figure 8. MKT friction coefficient ζ as a function of the perimeter velocity v in System A (top) or B (bottom graph) during spreading at 4 V (insets, solid line) and retraction at 0 V (main graphs, dashed lines).
(bottom) during spreading under voltage 4 V (insets), or retraction at zero voltage (main graphs). The friction coefficients were calculated under the assumption that the MKT mechanism dominates dissipation. The perimeter dR velocity v = dt s was determined from the temporal dependence of the height of droplet’s center of mass (Figure 6) using eqs 3-4. Significant changes of the calculated coefficients ζ with velocity could reflect a coexistence of the two mechanisms, hydrodynamic and MKT.39,42 Though our calculations for a purely hydrodynamic friction mechanism indicate it cannot play a major role (see SI), we attempted to interpret the observed relaxation velocities in terms of additive contributions39,42 from both mechanisms. The hydrodynamic friction is quantified in terms of the friction coefficient μ, which is expected to vary in proportion to the inverse of the dynamic contact angle θ. No combination of non-negative friction coefficients could, however, reproduce the observed behavior while keeping the values of μθ and ζ approximately constant. This finding, along with the observed influence of the properties of the substrate and the applied voltage, as expected with the liquid/substrate friction,42,44 confirms that the latter represents the dominant dissipative mechanism in our system. 8540
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ACS Nano The MKT friction coefficient ζ can be related to the liquid adhesion to the substrate surface, γlv(1 + cos θ). Assuming that molecular displacements are associated with the kinetic barrier δ Wa ≃ λ2γlv(1 + cos θ), the displacement frequency ko can be approximated41 by ko ≃
⎡ λ 2γ (1 + cos θ ) ⎤ kBT ⎥ exp⎢ − lv ⎥⎦ Vmη ⎢⎣ kBT
(8)
where Vm is molecular volume of the liquid. This estimate of ko, combined with eq 7, captures qualitative changes we observe. Specifically, it predicts about 2-fold increase in the friction coefficients during spreading and retraction when increasing the surface hydrophilicity by switching from System B to System A, or by applying the interelectrode voltage of 4 V. As simulated friction coefficients show velocity dependence, for a trend comparison we choose values at the highest velocities, where the determination of ζ from MD simulation is most accurate. Addressing the four representative situations in the order of increasing hydrophilicity, we compare System B at zero voltage (θ = 149°), at 4 V (θ = 118°), System A at 0 V (θ = 111°) and at 4 V (θ = 84°). The simulated values of ζ are 0.5, 1.36, 1.25, and 5.4 mPa·s, respectively. Equations 7 and 8 with Vm = 30 Å3, η ≃ 0.9 mPa·s, and displacement length λ at 4 Å yield the friction coefficients ζ for the same situations at 0.53, 1.39, 1.83, and 5.7 mPa·s. The associated displacement frequencies ko are 112, 46, 35, and 11 GHz. The above choice of λ compares favorably to the molecular size of water, σ ≃ 3.17 Å. A good agreement between the trends predicted by eqs 7−8 and those observed in direct simulation reinforces the conclusion that the liquid/substrate friction dominates dissipation associated with droplet relaxation. The variation of the observed friction coefficients with velocity v can be attributed to the nonuniform properties of the region (Figure 1) traversed by the drop’s perimeter during spreading or retraction. Because of the greater hydrophilicity of Pt atoms compared to those of the insulator, the surface hydrophilicity under the drop perimeter is increased in the proximity of the electrodes. By design, at zero voltage, the equilibrium position of the perimeter is above the area roughly between the electrodes, hence in the field-free cases (retraction) friction coefficient is smallest at small velocities close to the equilibrium. If the voltage is turned on, the hydrophilicity gradient changes because the electric field under the perimeter decays with the radial distance (Figure 9). The stronger field translates to the enhanced attraction of water molecules to the substrate and hence the bigger friction coefficient.44 Moreover, the viscosity becomes anisotropic in a strong electric field and shows a field-induced increase of the component along the field direction.45 This increase fades as the perimeter of a spreading droplet moves to regions with decreased field strength (Figure 9), contributing to further reduction of the friction coefficient as the droplet approaches the equilibrium. Like in the process of retraction, the friction coefficient is reduced at small perimeter velocities.
Figure 9. Electric field E under the drop perimeter, and the friction coefficient ζ (insets) as functions of the perimeter radius during spreading under voltage 4 V. Black lines describe System A (top) and blue ones system B (bottom).
analogues4 are the strong dominance of the interfacial layer of water in the total polarization of the droplet, which surprisingly preserves the linear dependence of the capacitance on the area of the drop base; the greatly increased sensitivity of the drop’s contact angle to the voltage because the electric potential varies over short distances; and orders of magnitude faster actuation. The study demonstrates a reversible switching of the droplet’s contact angle through the voltage applied between electrodes under the droplet base. About 30° change in the contact angle is obtained at potential difference of 4 V. The changes are reversible and show no hysteresis. The analysis of the droplet relaxation rates shows that the retraction at zero voltage is considerably faster than the field-assisted spreading. This is explained in terms of the voltage-enhanced liquid/solid friction under the propagating droplet perimeter as formulated in the molecular kinetic theory. The approximate validity of the theory allows extrapolations to computationally prohibitive but experimentally more feasible length scales. Since the perimeter friction coefficient proves independent of the droplet size, the shape relaxation time will scale in proportion to the drop’s linear dimension. Using our nanoscale droplet as a reference shows that a remarkably fast response at microsecond time scale can be achieved even with a microdroplet of O(10) μm size at which a device sketched in Figure 1 would be easier to manufacture. The study can benefit the design of miniature liquid lenses that can be tuned9 by electrowetting. In a practical setting, the combination of water and gas phases can be replaced by oil and water for better mechanical stability and to prevent evaporation. Both, the use of air,8 or oil,7,8 proved viable in macroscopic experiments. Addition of a dye can turn the microlenses to ultrafast pixel elements in a display with extreme resolution (nanopixel).10,11 The study also invites applications in the dynamic control of permeation in hydrophobic nanopores46−48 for nanofluidic devices.
CONCLUDING REMARKS We provide the first molecular insight into the strongly nonuniform nanodroplet electrostatics and the actuation on a polarized electrode. We achieve this by monitoring the dynamics of an aqueous nanodrop spreading on the dielectric in a constant potential MD simulation. The significant features that distinguish the nanodrop behavior from its macroscopic 8541
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0.452 kJ mol−1 in system A and 0.052 kJ mol−1 in system B. We use Lorentz−Berthelot mixing rules. Nonelectrostatic interactions are truncated at cutoff distance Rcut = 10 Å. In analogy with macroscopic liquid lens devices, where the drop is localized by a hydrophobicity gradient on the substrate,9 in the simulation, we apply a harmonic constraint with the spring constant of 10 kcal mol−1 Å−2 to counter an eventual shift of the drop through thermal fluctuations. Our results confirm this suffices to maintain a centered drop’s position with its perimeter restricted to the region between the inner and outer electrode at all field strengths we consider. Constant Voltage Simulation. To implement the simulation at preset interelectrode potential, we model the conducting electrodes using adjustable atom charges.26−28,54 The condition of constant potential is enforced at all metal sites of both electrodes. Each metallic atom at site ri carries a Gaussian charge distribution
So far, we considered neat water on a molecularly smooth surface that optimizes the rate of the drop’s response to the voltage change. It is of interest to discuss possible effects of the drop salinity and surface roughness. Because of the nanoscale droplet dimensions, only a limited ionic screening can be achieved by the addition of salts. The presence of ions is expected to increase the capacitance,49 the zero-field value of the contact angle,31 and its sensitivity to the voltage. Their presence should, however, not change the linear dependence of the capacitance on the droplet’s base area, Asl, because the buildup of the electric double layer further localizes the effects of the field in the interfacial layer. In the presence of substantial surface corrugations, we anticipate a stronger perimeter friction and hence a somewhat slower actuation. The computational analyses of such effects will be presented in future work.
METHODS Force Fields. We model water using the SPC/E force field,50 which is known to perform successfully in similar contexts.17,20,21,31,32,51 For Pt−water interactions, we use the potential of Siepmann and Sprik,26 modified by Reed et al.,27 to account for a two-dimensional PBC correction for the Ewald summation. The potential comprises two- and three-body Stillinger-Weber52 terms. The two-body term is U2(rij , ϕij) = Arij−α − Crij−6 − Df2 (rij)rij−3Φij(ϕij)
ρi (r) = qiGexp( −|r − ri|2 η−2)
where qi is the total integrated charge, r represent positions inside the metal electrode, η is the distribution width and G = η3π3/2 is the normalization constant. The amplitudes of individual charges qi represent additional degrees of freedom adjusted variationally at every step of the simulation. This is achieved by minimizing the energy functional U
(9)
where f 2(rij) is the cutoff function
U=
⎛ B ⎞ ⎟⎟ f2 (rij) = exp⎜⎜ ⎝ rij − rcut ⎠
(11)
Fi = −
The three-body potential needed to simulate the attraction of water molecules for top metal sites is ⎡
⎤
⎢⎣
⎝ 2 ⎠⎥⎦
⎛ θijk ⎞ Ef2 (rij)rij−β exp⎢F cos2⎜ ⎟⎥
⎣
j
2
+
⎤ − Ψ 0⎥ ⎥⎦ 2π η
qi(t )
(14)
with respect to all variable charges while keeping the electrode potential difference at a desired value. Ψi({qj(rj)}) is the electrostatic potential at the position i due to all electrode and water atom charges qj, the second term inside the rectangular brackets determines the self-interaction of electrode atom charges, and Ψ0 is the (fixed) input electrode potential. The electrostatic forces exerted on mobile charges in the solution are determined as
(10)
and Φij(ϕij) is a function depending on the angle between the water dipole vector and oxygen−metal bond vector rij favoring orientations where hydrogen atoms point away from the surface
U3(ri, rj, rk) =
⎡ Ψi({q (rj)})
∑ qi(t )⎢⎢ i
⎡ ⎛ cos(ϕ ) − 1 ⎞4 ⎤ ij ⎟⎟ ⎥ Φij(ϕij) = exp⎢ −8⎜⎜ ⎥ ⎢ 4 ⎝ ⎠⎦ ⎣
(13)
∂Utot ∂U =− c ∂ri ∂ri
(15)
where Uc is the Coulombic potential of the system and the derivatives of electrode-atom charges qi are omitted in analogy with the Car−Parrinello method in the ab initio MD.55 We use Gaussian charge distribution width η = 0.5052 Å from ref 27. The energy of the system shows a rather weak dependence on η with the optimal value varying in proportion to the inverse square root of the area density27 of metal atoms. Algorithms. Treating the electrodes as metal objects at constant potential, we performed molecular dynamics by adapting the code from refs 27, 28, and 54 which maintains a uniform electric potential within each electrode as appropriate for conductors. During the equilibriation in NVT ensemble for at least 250 ps, the temperature is kept at 300 K by the velocity rescaling using a time constant of 200 fs. We use NVE sampling (time step 2 fs) for the production of the typical length of 1.2 ns. Substrate layers are kept frozen and SHAKE algorithm56 is implemented to preserve the internal geometry of the water molecules. Perpetual iterative adjustments of electrode atom charges lead to high computational costs of the constantelectrode potential simulation. An additional increase in computation time is associated with the use of two-dimensional Ewald sums for long-range electrostatic interactions.27,57 About
(12)
where θijk is the angle between the oxygen−metal bond rij and a metal−metal bond between the top site atom j and its neighbor k. For our specific geometry, the force field is strongly simplified due to the presence of the insulating layer on top of the planar electrode. By increasing the minimal distance between water and metal atoms, this layer greatly reduces the short ranged top-site selectivity of the metal surface.53 The three body term given by eq 12 becomes negligible since the suitable cutoff distance for this term26 is just Rcut = 3.2 Å and the interaction described by eq 9 reduces to α − 6 LennardJones (LJ) potential with the following set of parameters: A = 1.7 × 105 kJ mol−1 Å11, C = 1.7 × 103 kJ mol−1 Å6, and α = 11. The σPt−Pt for the Pt atoms is considered to be 1.858 Å. In describing water/insulator short-range repulsion, we use conventional 12 − 6 Lennard-Jones (LJ) potential, instead of the 11 − 6. The size of insulator atoms is equal to that of Pt atoms. The LJ energy parameter ϵ for the insulator atoms is 8542
DOI: 10.1021/acsnano.6b03753 ACS Nano 2016, 10, 8536−8544
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ACS Nano
REFERENCES
30 000 CPU hours are needed for a 500 ps trajectory in a system with 659 metal atoms. The more efficient method relying on image charges58 cannot be used with the present geometries of the electrodes and the drop. The electrostatic potential difference between the electrodes, U, is set at 0, 1, 2, 3, or 4 V. A positive potential of U/2 is applied at the outer electrode and negative potential of equal magnitude at the inner one. The average electric field between electrodes reaches up to O(0.1) V·Å−1. With components of the molecular polarizability of water close to 1.5 Å3, the biggest fields we consider can induce up to a few percent change in the dipole moment of water. Significant polarization or even decomposition of water molecules,59,60 however, is not expected with fields up to 0.1 V· Å−1 provided any flow of current is prevented due to the insulation. First-principles calculations also show only fields above the threshold of 0.3 V·Å−1 are capable of dissociating water molecules,61,62 making the use of a rigid model of water an acceptable approximation. Data Analysis. We calculate the contact angle by a standard procedure63 that relies on fitting the contour of the drop to a truncated circle and determine the angle between the surface and the circle tangent at intersection.17,31 In small droplets, thermal motion results in visible shape fluctuations. However, these fluctuations remain statistically symmetric and, to a very good approximation, Young equation, eq 2, for the average contact angle continues to apply. Spherical shape is generally preserved when perturbations of the liquid remain restricted to thin interfacial layers. This condition is not always satisfied in electrowetting by nanosized drops where the field can penetrate deep into the core. Mild deviations from the spherical shape have been observed in simulations.17,20,31 and even in macroscopic experiments.23,24 In these cases, the droplet cross section is fitted to an ellipse instead of a circle.17 A couple of layers at the bottom of the drop are disregarded in contour fittings as rationalized in previous works17,51,63
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ASSOCIATED CONTENT S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b03753. Additional computational results including detailed drop dimensions, the contact angle change corresponding to a uniform area capacitance, dynamic contact angle evolution, hydrodynamic friction coefficients, and the error bar analysis. (PDF)
AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS We thank John Ralston for helpful discussions. This work was supported by the U.S. National Science Foundation through grant CHE-1213814. The supercomputing time was provided by the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF Grant No. OCI-1053575 and the National Energy Research Scientific Computing Center (NERSC), supported by the Office of Science of the U.S. Department of Energy (DEAC02-05CH11231). 8543
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