Modeling of Droplet Impact onto Polarized and Nonpolarized

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Modeling of Droplet Impact onto Polarized and Non-polarized Dielectric Surfaces Vitaliy Yurkiv, Alexander L. Yarin, and Farzad Mashayek Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01443 • Publication Date (Web): 31 Jul 2018 Downloaded from http://pubs.acs.org on August 3, 2018

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Modeling of Droplet Impact onto Polarized and Non-polarized Dielectric Surfaces

Vitaliy Yurkiv,* Alexander L. Yarin, Farzad Mashayek* Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA *

Corresponding authors:

E-mail: [email protected], Phone: +1 (312) 996-8357, Fax: +1 (312) 413-0447 (Vitaliy Yurkiv) E-mail: [email protected], Phone: +1 (312) 996-1154, Fax: +1 (312) 413-0447 (Farzad Mashayek)

Keywords: dynamic electrowetting-on-dielectric (DEWOD), Cahn-Hilliard-Navier-Stokes approach, droplet impact, droplet oscillation

Highlights: A coupled phase-field and Navier-Stokes model describing drop impact onto a solid dielectric surface is developed. The phase-field model is based on the advective Cahn-Hilliard equation. The model is validated through the prediction of the equilibrium contact angle, the droplet oscillations as well as charge density estimation. The simulation results of the droplet impact onto a dielectric surface compare well with the available prior literature reports. The predicted body forces are commensurate with the droplet shape evolution.

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ABSTRACT This paper is concerned with simulation of the droplet impact on a dielectric surface, referred to as the dynamic electrowetting-on-dielectric (DEWOD). In particular, we seek to shed more light on the fundamental processes occurring during the impact of an electrically-conducting droplet onto a dielectric surface with and without an applied voltage. The liquid in the droplet is an ionic conductor

(a

leaky

dielectric).

This

work

employs

an

approach

based

on

the

Cahn‒Hilliard‒Navier‒Stokes (CHNS) modeling. The simulations are validated by predicting the equilibrium contact angle, droplet oscillations and charge density estimation. Then, four cases of droplet impact are studied, namely, the impact onto a surface with no voltage applied, and the impacts onto the surfaces with 2 kV, 4 kV and 6 kV applied. The modeling results of water droplet impact allow for direct comparison with the experimental results reported by Lee et al. [Langmuir, 29, 7758, 2012]. The results reveal the electric field, the body forces acting on the droplet, the velocity and pressure fields inside and outside the droplet as well as the free charge density and the electric energy density. The model predicts the droplet shape evolution (e.g., the spreading distance over the surface and the rebound height) under different conditions that are consistent with the experimental observations. Thus, our findings provide new qualitative and quantitative insights into the droplet manipulation that can be used in novel applications of the DEWOD phenomenon.

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INTRODUCTION It is widely recognized that electric field can be used as a driving force in the actuation methods for microfluidic systems which employ leaky dielectric droplets.1,2 Among the broad variety of the actuation methods3,4, electrowetting-on-dielectric (EWOD)5 or dynamic electrowetting-on-dielectric (DEWOD)6 is an important method of the electrical control of fluids (e.g., shape, position, etc.)7. For example, the contact angle between the liquid and the solid phases changes with the applied voltage6,8, morphology of the surface9, etc. Besides these fundamental scientific concerns, the EWOD or DEWOD phenomena hold great promise for technological and industrial applications in microfluidics (e.g., adaptive lenses10,11, pixelated optical filters12, etc.). In particular, the possibility of manipulation of the motion and shape of fluid in technological devices without altering the design or introducing additional parts (e.g., valves, pumps, etc.) is of interest. A separate class of problems related to the EWOD and DEWOD arises in relation to droplet impact onto different surfaces.13,14 From the theoretical viewpoint, several methods have been proposed to model droplet impacts onto surfaces.15,16 These methods can be classified into two groups: (i) atomistic models, where an investigating object is resolved to the level of atoms creating molecules and clusters for which the laws of statistical mechanics are applied to find the corresponding thermodynamic and kinetic parameters; and (ii) mesoscale/continuum models, where an investigating object is regarded as a continuum, and classical macroscopic laws (e.g., the Cahn–Hilliard (CH) equation, the Navier-Stokes (NS) equations etc.) are applied to characterize the evolution of the object’s shape. The mesoscale/continuum models such as the boundary element method (BEM)14–17, the spine-flux method (SFM)18–20, the volume of fluid

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(VOF) method21, the lattice Boltzmann method (LBM)21 and phase-field modeling (PFM)22–24 are the most reliable and robust. Among the above-mentioned models, one of the most promising techniques to model two– phase flow (e.g., liquid/air or liquid/liquid) is the PFM, since it facilitates a thermodynamic treatment of the phase interfaces, rendering it more physically consistent for the direct simulation of two-phase flow. In the literature, the PFM of two-phase flow is referred as coupled Cahn– Hilliard–Navier–Stokes (CHNS) modeling, since the PFM binds the solution of the CH equation with the NS equations, where the necessary terms are exchanged between the two sets of equations. The foundation of the CHNS modeling was laid by Jacqmin22,23 (and independently in Ref.24), where the proper coupling between the equations were thoroughly described. In particular, Jacqmin showed how to modify the NS equations by the addition of the continuum forcing from the CH equation solution, and vice versa, by modifying the CH equation to take into account the velocity field obtained from the solution of the NS equation. Following Jacqmin’s work, many models have been established to simulate two–phase flow (including a droplet impact onto a surface). For example, Zhang et al.25 have implemented the CHNS model, where an efficient gradient stable scheme is used to solve the system in axisymmetric coordinates. They numerically investigated the mechanisms leading to different impact phenomena such as adherence, bouncing, partial bouncing, and splashing. In particular, they studied how various processes are affected by the relevant dimensionless parameters: the Reynolds number, the Weber number, the density ratio, the viscosity ratio, and the wettability of the solid surface. In the same work of Zhang et al., the accurate description of the dynamics of the contact line was adopted following the work of Ding et al.26,27. The latter proposed a geometrical formulation of wetting conditions with respect to the microscale contact angle to

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solve the CH equation to describe the moving contact line problem. They showed that compared with the widely used surface–energy formulation, the geometric formulation can successfully enforce a wetting condition for a constant microscale contact angle that corresponds to a smooth and chemically homogeneous solid surface, while the surface-energy formulation actually would result in a deviated value. Similarly to the Zhang et al.25 work, various approaches have been implemented in the literature28–33. Another set of models21,34,35 has been developed to investigate drop motion under the electrowetting conditions. Lu et al.34 have proposed a model for droplet motion due to electrowetting in a Hele–Shaw geometry. In the limit of small interface thickness, the asymptotic analysis shows that the model is equivalent to the Hele–Shaw flow with a voltage–modified Young–Laplace boundary condition at the free surface. They showed that the details related to the contact angle significantly affect the time scale of motion in the model. In addition, they compared their predictions with their own experimental data showing that the shape dynamics and topological changes in the model agree fairly well with the experiment, down to the length scale of the diffuse-interface thickness. A similar model has been proposed by Huang et al.21, who have developed a phase–field based hybrid lattice-Boltzmann finite–volume model and applied it to study droplet motion under electrowetting control. They focused their study on the effects of the key control parameters (e.g., the Reynolds number, the gradient energy coefficient, etc.) on the induced droplet oscillations and provided useful guidelines for mixing enhancement within the droplet by the EW control. To the best of our knowledge, there are no CHNS studies devoted to the investigation of electrically-conducting droplet impact onto dielectric surfaces. Thus, in this work an extended CHNS model for water droplet impact onto dielectric surfaces is developed. This is a step

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forward toward the development of the physically sound, predictive, quantitative models required for fundamental understanding of EWOD and DEWOD. Furthermore, in the present work, the CHNS results are directly compared with the available experimental data. The results of this study can be further used to assist the design and the improvement of the DEWOD systems operation. It should be emphasized that in the present work only DEWOD is in focus, whereas dielectrophoresis (DEP) is not considered.

MODEL FORMULATION In the present study, the modeling domain is axisymmetric and the solid surface is assumed to be ideally smooth, with roughness being neglected. The droplet impact is assumed to be normal to the surface. Figure 1 illustrates the computational domain, mathematical description of the main physical properties and the boundary conditions used in the present modeling work. Specifically, a drop impact onto a polarized or non-polarized surface of a dielectric layer is being considered. Although, the present work deals with droplet impact onto a dielectric surface, the model is built in the general way such that any types of surfaces might be simulated. The modeling approach builds upon the work of Anderson et al.36, Jacqmin22,23 and Ding et al.26. As mentioned above, the computational domain is related to the axisymmetric problem, and the medium consists of two phases. The first phase is the droplet (liquid), and the second phase is the surrounding gas‒phase (air). The Navier‒Stokes equations are applied for simulating the two-phase flow (water-air), and the coupled Cahn‒Hilliard equation is used to model the droplet shape evolution and to track the interface between water and air. The usage of the Cahn-Hilliard equation is often referred to as the phase-field method, which is a diffuse-interface approach. The PFM diffuse-interface method in its present formulation was introduced by Cahn and Hilliard.

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However, the origin of the method can be traced back to the density-gradient theory described by van der Waals and Reynolds36. In the PFM, the interface has a finite thickness, which is a key difference comparing to the traditional two-phase fluid mechanics methods that consider a sharp interface between the two fluids37. It should be emphasized that the PFM often applies both the Cahn-Hilliard and the AllenCahn (AC) equations to model the evolution of conserved and non-conserved order parameters, respectively. Both approaches could be used to model the droplet impact onto a surface; however, based upon the formulation, only the CH equation enforces the mass conservation. Due to the fact that both models minimize the total free energy of the system, leading to the same equilibrium solution, albeit along different kinetics pathways, there have been few attempts31,38 to apply the AC equation to model a two-phase flow. In particular, Ben Said et al.38 have used the AC equation in the multi-phase PFM by applying a Lagrange multiplier approach. They argued that the AC equation is easier to handle numerically, and therefore, the simulations will take fewer numerical time steps to reach the equilibrium state. This is an important statement in favor of the AC equation to be used in the two-phase PFM simulations, as the usage of the CH equation gives rise to a fourth-order partial differential equation, which is often challenging to solve numerically especially for multiphase system (e.g., for three and more phases). Similarly to Ben Said et al.38, Jeong et al.31 used a conservative AC equation with a space-time dependent Lagrange multiplier for a two-phase system. As described above, we use the PFM to track the phases and the interface, and the NS equation to model the two-phase flow of the liquid and gas phases. Below, first, the PFM approach is described in brief, followed by the description of the NS model. In the framework of

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the PFM, the free energy functional is comprised of the contributions of all types of energy existing in the system and is described by the following equation22,28 1   F ( c , ϕ ) = ∫  f ch ( c ) + ε c γ c α (∇ c ) 2 + f elst ( ρ V , ϕ ) d V , 2  V 

(1)

where, f ch (c ) is the chemical free energy density, with the variable c being used to model the two phases with two minima at c = ‒1 and c = 1. The second term on the right–hand side in eq 1 accounts for the excess free energy due to the inhomogeneous distribution of volume fraction ( c ) in the interfacial region. In addition, ε c is the measure of interface thickness, and γ c is the surface tension coefficient. The constant α takes a value of 6 2 following the work of Ding et al.27 This term originates from the integration of the excess free energy per unit surface area across the droplet interface. The function f elst ( ρ V , ϕ ) describes the electrostatic energy density. The chemical free energy density is expressed by the double-well function as follows

f ch (c ) =

1 γ cα c (c 2 − 1) , 2 εc

(2)

The evolution of the conserved order parameter (phase variable), c , is expressed by the extended CH equation with the advective term22

 ∂c  ∂F (c ,ϕ )  + ∇⋅ ( uc ) = ∇⋅  M c ∇   , ∂t  ∂c  

(3)

where, u is the fluid velocity vector (cf. eq 9), which is evaluated by the NS equations coupling these two sets of equations. The advection term ( ∇ ⋅ ( uc ) ) in eq 3 changes the free energy locally; however, it has to keep the total free energy of a fluid unchanged. It is of imperative importance to note the form of the second term ( ∇ ⋅ ( uc ) ) in eq 3. This formulation corresponds to the so-called conservative

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form of the advective CH equation, whereas the non-conservative formulation of the second term would be u ⋅∇c . Due to the continuity equation, the two formulations are formally identical, albeit numerically, they make a significant difference. Specifically, the conservative form requires smaller time step and an a priori knowledge of the mass loss of the droplet due to the finite thickness of the interface, as explained below. If the mass loss is significant, there is no possibility to obtain numerical solution with the conservative form. Thus, one approach is to initially use the non-conservative form to decrease the mass loss and then switch to the conservative form. The elaborate description of the usage and application of these two forms is provided in the Results section. As mentioned above, a mass loss could occur due to the finite thickness of the interface in the PFM. Yue et al.39 have investigated this phenomenon and have found that the mass loss is proportional to the interface thickness, and a critical radius exists below which the droplet will eventually disappear. Following the work of Yue et al.39, the final shrinkage of the droplet is described by

δ r2 = −

2V ε , 24π r02

(4)

where, V is the volume of the modeling domain, ε is the measure of the interface thickness as appeared in eq 1, and r02 is the initial droplet radius. However, it should be emphasized that eq 4 merely applies to non-colliding droplets that do not impact onto any surface. In the case of droplet impact onto a solid hydrophilic surface, other effects play significant role. In particular, velocity field resulting from the impact may accelerate mass loss, since it will alter the energy level revealed from the CH equation.25 The parameter Mc in eq 3 is the phenomenological mobility (often referred to as the Onsager

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transport coefficient) M c = χε 2 ,

(5)

where, χ is the mobility tuning parameter. According to eq 5, the mobility is active at the interface only. Analyzing eq 3‒5, one observes that the droplet shrinkage can be made very slow by properly adjusting mobility tuning parameter, and thus such a process will be much slower than the physics of interest. The electrostatic free energy density is calculated as follows

f elst ( ρV , ϕ ) =

1 ρVϕ , 2

(6)

where, ρ V is the volumetric charge density, and ϕ is the potential. To find the potential, the Poisson’s equation is used

∇⋅ ( −ε0εr∇ϕ ) = ρV ,

(7)

where, ε 0 is the permittivity of the vacuum (i.e. the SI system of units is used), and ε r is the relative permittivity. In order to evaluate the total interfacial charge (Qd), the integration of the surface charge density along the common interface between the droplet and the dielectric surface (S) is performed Qd = ∫ σ s dS ,

(8)

S

where, σ s is the surface charge density being calculated using the boundary relation

σ s = n ⋅ ( ε 0E ) with the outward-pointing surface normal unit vector n representing the local spatial orientation of the droplet surface. This condition implies that the characteristic impact time scale, τimpact, is much longer than the charge relaxation time scale, τC, of water, and thus, the

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electric field inside the droplet is fully screened by the surface charges (which is true in reality, since τimpact ~ 1 ms, whereas for water τC ~ 1 µs). In other words, it is assumed that the surface charge distribution reaches its equilibrium (due to the admixture or the original water ion mobility) practically instantly, meaning that all the charges are immediately present at the surface of the dielectric at the moment the droplet touches the surface. Note that in general for a leaky dielectric droplet the full Nernst-Planck-Poison’s model40 should be employed to describe spatial and temporal redistribution of ions, which in the present case happens practically instantaneously due to the inequality τimpact >>τC. The NS equations are used to simulate the fluid flow and are coupled with the advective CH equation (cf. eq 3). The NS equations comprised of the momentum and mass (the continuity equation) balances read  ∂u  + u ⋅ ∇u  = −∇p + ∇ ⋅  µ ∇u + ∇uT  + Fst + Felef + ρ F g ,  ∂t 

(

ρF 

)

∇⋅ u = 0 .

(9) (10)

As mentioned above, u is the velocity vector (here and hereafter all boldfaced symbols denote vectors), p is the pressure, ρ F is the density of fluid (note that subscript F is used here and hereinafter simply to distinguish between the charge density parameter ( ρ V ) that appears in eq 6 and fluid density in the NS equation), µ is the viscosity and g is the gravity acceleration. Similarly to the relative permittivity (cf., eq 7), the density and viscosity are extrapolated between the two phases using the tilting function. The CH and the NS equations are coupled via the surface tension force ( Fst )22,28 Fst =

∂F ( c , ϕ ) ∇c . ∂c

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(11)

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This surface stress term22 arises from the compositional gradient at the surface, where the term ∂F (c , ϕ ) / ∂c corresponds to the chemical potential. Since the chemical potential ( ∂F (c , ϕ ) / ∂c )

of the droplet changes depending upon the interfacial thickness, the surface tension presented as a body force is not constant during the droplet shape evolution. Thus, the surface tension-related body force variation in time should be balanced by the phenomenological mobility ( Mc in eq 3). The electrostatic force in the NS equation ( Felef ) couples the flow equations with the equations of the electrostatics, where the general form for this body force is ρ V E , with the electric field strength being E = −∇ ϕ . The electrostatic force per unit volume is found as the divergence of the Maxwell stress tensor

Felef = ∇T ,

(12)

where, the Maxwell stress tensor is given by

T = EDT −

1 (E ⋅ D) I . 2

(13)

Here, D is the electric field displacement ( ε 0ε r E ) and I is the unit tensor (cf. Appendix A).

Boundary conditions. Due to the symmetry of the problem, only one half of the entire domain is considered, with the symmetry boundary conditions used to account for the rest. As shown in Figure 1, the horizontal axis corresponds to the r-axis and the vertical axis is used as the z-axis. The wetting (contact angle) boundary condition for the advective CH equation is expressed in the form of the geometrical formulation26. In this formulation, for a given prescribed contact angle, θ s , the wetting condition is given by26

n ⋅ ε∇c = −ε cos(θ s ) ∇c .

(14)

In addition, the non-penetration boundary condition at the dielectric surface is necessary for the

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CH equation (eq 3)

n ⋅∇

∂F (c , φ ) =0. ∂c

(15)

This equation implies that there is no flux of the chemical potential through the solid surface. At the solid dielectric surface, the no-slip boundary condition holds,

u = 0.

(16)

The Neumann (zero gradient) boundary conditions are used at all other boundaries. In the case of a certain voltage applied to the dielectric surface, the contact angle will be affected by it, which is expressed using the Young‒Lippmann equation41,42

CDφ 2 cos(θ ) = cos(θ ) + , 2γ c V S

0 S

(17)

where, θ S0 is the contact angle between the liquid and solid phases in the absence of the applied voltage, θ SV is the contact angle for a certain applied voltage, φ is the applied voltage, and γ c is the surface tension. The parameter C D , in eq 17, is the capacitance formed between the conducting electrode (see Figure 1 for details) and the droplet. In order to estimate the capacitance we follow the approach developed by Lee et al.6 The Dirichlet type boundary conditions for potential at the dielectric interfaces are used. In particular, certain constant voltage values (e.g., 2 kV, 4kV and 6 kV) are applied on the bottom of the 100 µm dielectric and a ground (i.e., 0 V) on the top surface of the dielectric. Computational implementation. The model is implemented in the COMSOL Multiphysics software employing phase-field, laminar flow and electrostatics modules. The adaptive mesh algorithm, as well as the adaptive time stepping are used as available in COMSOL. The adaptive meshing is further discussed in this work. MUMPS solver is used to solve both stages of the

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initialization and transient problem.

RESULTS The study presented here is concerned with a droplet impact onto a solid dielectric surface. In order to validate the model, we first present the comparison between our model predictions and the available results in literature, followed by the actual droplet impact study. The modeling investigation elucidates the effect of the applied voltage on the droplet impact by the detailed analysis of the acting forces and the change in physical parameters, in particular, droplet evolution.

Parametrization. The model parameters used in the CHNS simulations are comprised of the interface thickness, the phenomenological mobility, density and viscosity of phases. Ideally, the interface thickness should be as small as possible to minimize the energy dissipation across it. Also, as it is revealed by eq 4, the interface thickness directly influences droplet shrinkage; thus, it should be relatively small to minimize the mass loss of the droplet. However, with the decrease of the interface thickness, the computational cost rises significantly. Moreover, in order to keep the correct interface development there should be at least two mesh elements across the diffuse interface. Thus, the general approach for the selection of the interface thickness should be as follows: (i) the numerical results in regard to parameter c should be smooth enough to express the diffuse interface, where c changes between one and zero, correctly; (ii) the simulation should lead to numerical convergence with reasonable time step size. To ensure these criteria are satisfied, the CHNS model should be run with an adaptive time step and an appropriate adaptive mesh algorithm. Without an adaptive mesh, a very fine structured mesh throughout the domain could be used.

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Another important parameter, which influences the convergence and the shape of a droplet is the phenomenological mobility M c . Depending on the problem, the mobility has different physical meaning, for example, in the case of species diffusion across a solid phase (e.g., Na+ diffusion through a grain boundary of a secondary battery electrode43) the mobility directly reflects concentration‒dependent Na+ diffusion coefficient. However, in the CHNS modeling, the mobility is used to control the shape of the droplet29 and to balance the surface tension body force variation in time. Thus, the value of the phenomenological mobility in the CHNS model should be chosen judiciously. The best practice in choosing the mobility value is through comparison to experimental data, if available. A more detailed discussion of the mobility and the interface thickness parameters in the CHNS modeling and their influence on the results is given in the recent work by Bai et al.29. It is important to ensure that the results are independent of the mesh resolution. When the mesh resolution is inadequate, first, the physical parameters across the interface are significantly altered, and second, the interface width is larger, which leads to energy dissipation. Thus, for the discretization of the modeling domain (meshing), and to ensure reliable finite element solutions for the conserved variable ( c ) and other dependent parameters, at least two quadratic triangular mesh elements are used across the droplet/air interface. However, such a fine mesh is not needed within the rest of the domain, thus slightly coarser mesh is used away from the interface. To ensure enough resolution across the interface, the adaptive meshing algorithm is used. In the present work, the Reynolds (Re) and the Weber (We) numbers are used after the problem is rendered dimensionless. They are defined as follows

ρFUR ρU 2 R Re = , We = µ γ where, U is the reference velocity (defined later) and R is the radius of the droplet. 15


(18)

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Model validation. Validation in the absence of electric field. For validation of the model, two physical problems are selected: the equilibrium contact angle problem and the droplet oscillation problem. These are well established classical problems for the two-phase flow and they have been studied in several publications28. These validation problems are designed to distinguish between different fundamental multi-phase processes by studying them independently. The first problem is the test of correct determination of the angle between the liquid and the solid phases leading to the validation of the wetting (wall) boundary condition implementation. The second problem investigates the oscillation of a droplet, in which a liquid droplet is subjected to different initial surface perturbations. This problem is designed to test the exchange of the appropriate terms between the CH and NS equations (i.e., the surface tension term and those responsible for the velocity field). The first example of our developed model demonstrates the equilibrium shape of a droplet at the surface, which depends upon the contact angle. For two cases with different contact angles imposed as the equilibrium wetting boundary condition (eq 14) at the bottom wall, the predictions of the present model are shown in Figure 2. The initial conditions for both cases were identical; in particular, the initial geometry for the wetting conditions determination is a cylinder with its height equal to the radius of 2 mm. The initial velocities in both phases are set to zero. From the physical point of view, the contact angle is considered as a measure of the wettability of a homogeneous surface, thus for the angle less than 900 the surface is considered to be wettable, i.e. hydrophilic. In contrast, for the angle larger than the 900 the surface is nonwettable, i.e. hydrophobic. The left-hand column in Figure 2 reveals the droplet evolution on the hydrophilic surface ( θ S0 = 63 0 ) and the right-hand column in Figure 2 depicts the droplet

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evolution on the hydrophobic surface ( θ S0 = 117 0 ). The pictures in between (second row) the initial and the final states depict intermediate shape of the droplet for both prescribed angles. The next validation problem considered here is that of a droplet oscillating in the absence of gravity (a levitating drop). This type of problems has been well studied and intensively investigated in the prior literature.44,45 Figure 3 shows the oscillations of a water droplet of the volume-equivalent radius of 13.44 µm. Following eq 18, the Reynolds number is estimated to be 30 using the reference velocity calculated by setting the Weber number to 1. Initially, the droplet is deformed to an ellipsoid shape with the radii ratio of s = a / b = 2 (a = 17.42 µm and b = 8.71 µm). The results are compared to those of Mashayek et al.46, who used the sharp-interface, spineflux method (SFM) to investigate droplet evolution and collision under different conditions. In particular, we compare the shape of the oscillating droplet, the oscillation time and the amplitude of the oscillations predicted by the two models. The evolution in time is shown in Figure 3 (the upper panel), whereas the lower panel in Figure 3 depicts the velocity magnitude and vectors. In Figure 4, the axes ratio versus time during droplet oscillations predicted by the two models is compared. It reveals an excellent agreement between the present simulation and that of Mashayek et al.46 over the entire investigated time range (0.092 ms). The simulation results could be continued until a practically complete decay of the oscillations at 0.15 ms. Figure 5A illustrates the oscillating droplet shape at the time moment of 0.001 ms. In particular, it visualizes the adaptive meshing scheme for the interface (the right-upper quarter) with the corresponding droplet part without the mesh the (left-upper quarter). Figure 5A also depicts the corresponding velocity vectors and magnitude (the right-lower quarter), and the pressure field (the left-lower quarter). As discussed in the model formulation section, there is a significant difference between the conserved and non-conserved CH equation. This difference in

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the case of the droplet oscillation problem is addressed in Figure 5B. The conservative CH equation leads to a 0.096% mass loss over the time period of 0.15 ms, whereas the nonconservative form results in an 8.5% mass loss for the same time period. These results signify the importance of using the conservative form of the CH equation in the CHNS approach. Validation of polarized dielectric model. It is generally difficult to measure the movement

of charges in the droplet during the impact. In the case of the droplet impact onto the surface of the polarized dielectric layer a charge separation in the droplet occurs. This situation is particularly complicated, since both convection and electro-migration mechanisms may equally contribute during the impact. Simulation, on the other hand, can quantitatively describe this situation, shedding more light on the actual mechanism behind globally observed droplet impact. In order to validate our polarized dielectric model, we compare our prediction to the analytical model results reported by Lee at al.6. In particular, Lee et al. calculated potential distribution in the Teflon dielectric layer (100 µm) and the surface charge density as a result of the applied voltage to the bottom of the dielectric. Figure 6 shows our simulation result depicting the potential distribution within the dielectric layer (Figure 6A) together with the surface charge density (Figure 6B). It should be noted that for this case, we use the 2D Cartesian model (not 2D axisymmetric model as in all other calculation results in this work). Following the conditions reported by Lee et al., the dielectric (Teflon) layer is set to be 100 µm thick and 600 µm long, where the positive and the negative terminals, each 150 µm wide, are applied on the bottom. At all other boundaries, zero voltage boundary conditions are used. As a result of different polarization (i.e., positive and negative) the corresponding charge separation occur. Figure 6B shows the surface charge density of anions and cations at the appropriate location at the surface

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of the dielectric, which stands in good agreement with the reported value of 1.3·10‒7 C·m‒2 by Lee et al.6. Following this approach, we model the droplet impact on the surface of the dielectric considering different potentials applied on the bottom of the dielectric as further described below.

Droplet impact. This sub-section presents the results of the droplet impact onto a Teflon surface. The interface thickness ( ε c ) of the droplet is set to 5 µm. Based upon our comparisons to the experimental results of Lee et al.6, it was determined that the mobility tuning parameter ( χ ) is equal to 10 m·s·kg‒1. A detailed analysis of the influence of the interface thickness and the mobility tuning parameter on the CHNS modeling results is given by Bai et al.29. In the present work, the same trend was observed, that is, small changes in the values of ε c or χ do not lead to significant changes in the results. In addition, separate changes in the mobility tuning parameter have a more significant influence on the results than changes in the interface thickness considering a similar percentage variation of the parameters. In the following sub-sections, the droplet impact onto a solid Teflon surface without the applied voltage is discussed first. Then, three cases with the voltages of 2 kV, 4 kV and 6 kV applied will be presented. For all cases, the volume-equivalent water droplet diameter is 1.92 mm and the initial droplet velocity is chosen as the reference velocity, U, leading to Re = 371, with the density being 1000 kg·m‒3 and the viscosity being 1 mPa·s. The droplet is initialized right above the solid surface and a uniform vertical velocity of 0.62 m·s‒1 toward the surface is imposed on the entire droplet. Due to a significant difference in the densities of the water droplet and air, the initial velocity of air is set to zero. As described above, the model predictions are

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compared with the experimental results of Lee et al.6. However, it should be mentioned that although the physical principles of the impact process on a polarized dielectric is same in the experiments and in the present work, the model configuration is slightly different. In particular, here a constant voltage value is applied at the entire bottom surface of the dielectric and a zero voltage at the surface of impact. In the Lee et al.6 experiments two electrodes with the equal opposite voltage were attached at the bottom of the dielectric layer. The numerically explore situation is close to the cases where drop impacts were not along the centerline between the electrodes, but rather with an offset toward the positive electrode. In the experiments and theory the setup is surrounded by a grounded faraday cage, which implies that the dielectric surface is sustained at zero potential. Water droplet impact onto dielectric surface without applied voltage. We first investigated

the droplet impact on a dry solid dielectric surface without the applied voltage. The modeling results of an impact, spreading and rebound of water droplet on Teflon surface are shown in Figure 7. The upper panel in this figure depicts the evolution of the droplet shape in time, with the red color representing the water droplet and blue color corresponding to the gas-phase (air). The second panel in Figure 7 depicts the velocity magnitude together with the velocity vectors and the lower panel shows the evolution of the pressure field during the impact. The results were directly compared to the experimental results of Lee at al.6 in terms of the spreading distance (smax) and the maximum rebound height (hmax). In particular, the maximum spreading distance reached at the time of 0.004 s is equal to 1.8 mm compared to 1.75 mm reported by Lee et al.6. The maximum rebounding height observed in the experiments was 4.16 mm versus the current prediction of 4.08 mm. The numerical predictions showed that the spreading tip bumps up as a rim, which is typical of the experimental observations6. Thus, we can conclude that the present

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modeling results of the droplet impact profile match the experimental results well. Note that a direct comparison of the predicted droplet shapes is impossible, since the present simulations are axisymmetric, whereas in the experimentally observed droplets5 there are visible deviations from the axial symmetry. Analyzing the velocity magnitude and velocity vectors presented in the middle panel of Figure 7, one observes the highest values corresponding to the initial moment of the droplet impact onto the surface, as expected. The velocity magnitude decays significantly as the droplet spreads and rebounds off the surface. In order to achieve an additional insight into the situation with the droplet impact onto the surface without an applied voltage, the pressure evolution during the impact (the lower panel in Fig. 7) was analyzed. A significant pressure increase can be observed at the droplet bottom at the initial moments of impact, which evolves to a more graduate pressure variation at the later stages. A significant pressure change causes high droplet rebound heights. Following the work of Bai et al.29 we have conducted a sensitivity analysis in regard to the interface thickness and the mobility tuning parameter, in particular, how the macroscopic droplet behavior (e.g., maximum spreading distance) depends on the change of those parameters. In this analysis, we change first the interface thickness by 10% and the impact on the maximum spreading distance is quantified numerically. The change of the interface thickness from 5 µm to 5.5 µm (10 %) leads to a tiny increase of the maximum spreading distance from 1.8 mm to 1.82 mm. The increase of the mobility tuning parameter by 10 % (from 10 m·s·kg‒1 to 11 m·s·kg‒1) leads to a slightly bigger increase of the maximum spreading distance of 1.85 mm. It should be noted, that such a small change of maximum spreading distance in regard to change of the

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modeling parameters is almost negligible, however, if both the parameters are changed by 10 % simultaneously the effect would be noticeable. Water droplet impact onto dielectric surface at 2 kV voltage. Water droplet impact onto

hydrophobic Teflon surface can be altered by introducing an external force to the system. For example, an electric potential can be imposed to the dielectric layer that separates the electrode and the droplet. Then, an almost instantaneous ion redistribution through the droplet to the liquid/solid interface will occur in concert with the droplet shape change. In the following, the modeling results for the water droplet impact onto a 100 µm-thick dielectric Teflon surface polarized by the applied DC voltage of 2 kV at the bottom are presented. Figure 8 shows the results for the droplet impacting, spreading and rebounding off the dielectric surface at 2 kV. The meaning of all colors in Figure 8 is the same as in the case without the applied voltage. Red color in the upper panel depicts the droplet shape evolution and the blue color shows the gas-phase (air). The small layer on the bottom depicts 100 µm Teflon dielectric layer, where voltage gradient between 2 kV and 0 V is present. In the middle panel of Figure 8 velocity vectors and magnitude are shown, where different colors represent different velocity magnitude corresponding to the scale bar on the right. The lower panel illustrates the pressure field evolution during the impact. Once again, the predictions are compared with the experimental data of Lee et al.6 In particular, the maximum spreading distance of 2.1 mm and the maximum rebound height of 3.1 mm were predicted, which compare well with the experimental results of 2.1 mm and 3.2 mm, respectively. In addition, similarly to the experimental observations of Lee et al.6, the droplet breakup was predicted as it rebounds off the surface. There is a significant pressure variation during the impact, as the droplet changes shape and direction of motion. In the simulations the uniform air pressure can be subtracted from all

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pressure values, which means that the air pressure is zero. Accordingly, positive pressure corresponds to an increase in pressure in the droplet, whereas negative values indicate a decrease in pressure. The formation of the satellite droplet directly relates to the pressure field evolution inside the droplet (the lower panel in Fig. 8). As it is indicated in Fig. 8, the pressure scale for two pictures in the last panel, where the satellite droplet forms, is different to better visualize the pressure field evolution. In particular, a significant pressure builds up in the regions of high curvatures, which causes the droplet to split. For this 2 kV case, a dimple formation is observed (the fourth image in Figure 8). The dimple is formed at the bottom of the drop since the lubrication pressure at the center of the gas layer significantly increases47. Figure 9 shows the comparison between the present simulations and the experiments by Lee et al.6 in terms of the droplet configurations. The comparison reveals that the numerical simulations predict satellite droplet formation similar to the one observed experimentally. As a result of the droplet touching the surface, a thin double layer at the interface is formed by the ions migrating through water. Since the charge relaxation time in water is shorter than the characteristic impact time by at least three orders of magnitude, water in the droplet behaves as a perfect conductor. Thus, the thickness of the electric double layer is infinitesimally thin, and the charge redistribution in the droplet can be treated as a quasi-static process. Therefore, we do not consider electric double layer at the free surface of the droplets in the present study dealing with the water droplet. Figure 9 shows the predicted charge at the droplet bottom in contact with the polarized Teflon surface corresponding to the results of Figure 8. In particular, the charge buildup during the impact is shown in Figure 10A and the charge density at the maximum spreading is depicted in Figure 10B. The surface density of free charges at the droplet bottom reaches value

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of 3.72·10‒4 C·m‒2, which corresponds to the total electric energy per unit area of the droplet bottom of 16.04 J·m‒2. Water droplet impact onto dielectric surface at 4 kV voltage. We next present the results

for an impact of the water droplet onto the surface of the dielectric layer at 4 kV applied voltage at the bottom of the dielectric. Figure 11 depicts the evolution of the droplet shape (the upper panel), the velocity magnitude and vectors (the middle panel) and the pressure field evolution (the lower panel) during the impact. The maximum spreading distance is 2.2 mm and the maximum rebound height is 3.05 mm. Comparing with Fig. 8 (2 kV impact), the most pronounced difference is in the droplet shape after the impact. Although in both cases the droplet is breaking up into two parts, for the case of 4 kV the droplet splits into two parts leaving about half on the surface. Such a behavior is driven by an increased pressure inside the droplet comparing to the 2 kV case. This complex interplay between the impact velocity and the curvature of the interface (driven by the applied voltage), which leads to the evolving pressure field inside the droplet can potentially yield intriguing consequences. For example, different impact conditions can yield similar or even identical spread and rebound distances, albeit also greatly different droplet shapes and behavior between the maximum spread and the maximum rebound points. The charge density plot (Fig. 12) shows that more charges are accumulated at the interface between the droplet and the dielectric layer compared to the results shown in Fig. 10A (the 2 kV impact). Since, after the impact only a part of the droplet remains on the surface, a small amount of charge would be left at the surface when the other part of the droplet is already in air. Water droplet impact onto dielectric surface at 6 kV voltage. Figure 13 shows the results of

the droplet impact on the dielectric surface for the case of the 6 kV voltage applied. The upper

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panel in this figure depicts the droplet evolution during the impact, the middle panel illustrates the corresponding velocity magnitude and vectors and the lower panel shows the pressure field evolution during the impact. In this 6 kV case, the water droplet cannot detach from the hydrophobic Teflon surface anymore. The predictions were compared with the experimental results of Lee at al.6. In particular, the maximum spreading distance reported by Lee et al.6 for this case was 2.25 mm and the maximum rebound height was 2.8 mm. In the present simulations, the value of 2.3 mm was predicted for spreading on the dielectric surface, and the value of 2.9 mm ‒ for the maximum rebound height. Figure 14 reveals the results for the charge accumulation at the droplet bottom in the case of water droplet impact at 6 kV. The total accumulated charge is higher than that in the 2 kV and 4 kV cases, due to a higher applied voltage in the present case. Since there is no droplet detachment from the dielectric surface in the 6 kV case, the accumulated ions would always be present at the wetted Teflon surface.

Conclusions A 2D axisymmetric model of water droplet impact onto a dielectric hydrophobic Teflon surface has been developed. Local and global characteristics of the droplet impact were taken into account by using the combined Cahn-Hilliard and Navier-Stokes formulations. The model includes the description of the droplet bulk, as well as of the smooth interface using the phasefield model and the NS equations coupled with the electric field predicted using the Poisson’s equation. The appropriate couplings between the different phenomena have been established using the following steps: (i) The velocity field governed by the NS is passed to the phase field model (the advective CH equation). (ii) The electric body force in the NS equations is coupled

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with the electric field via the Maxwell stress tensor. (iii) The surface tension force from the PFM is coupled to the NS equations. (iv) The electrostatic energy density originating from the electric field is coupled to the PFM via the Young-Lippmann equation and the Maxwell stress tensor. The model is successfully validated using the model case of a free droplet oscillation about its spherical form. Also, the wetting boundary conditions are validated by predictions of the droplet evolution to a steady state with a prescribed contact angle with a solid surface. The third validation is performed for the case of certain applied voltage to the dielectric layer evaluating surface charge density. The main aim of this work is the droplet impact onto solid hydrophobic (Teflon) dielectric surface with and without applied DC voltage. The results of the simulations are in good agreement with the experimental data of Lee et al.6 for water droplet impact onto Teflon surface with and without an applied voltage. In particular, the maximum spreading distance and the rebound height for different conditions were predicted rather accurately. The present results reveal that by changing the voltage applied to the dielectric layer, the physical properties of the surface can be effectively changed from hydrophobic to hydrophilic due to the charge accumulation at the water/Teflon interface. The results also reveal a significant local variation of the pressure field inside the droplet and at the interface between the droplet and air. In addition, charge density of the droplet bottom under different conditions is evaluated. The present work also discusses some details of convergence issue of the numerical model and sheds light on the mass conservation issue in the CHNS approaches. In particular, details of conservative and non-conservative CH equations are presented and analyzed emphasizing the importance of the mesh resolution in the bulk, as well as across the interface. In addition,

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sensitivity towards two main parameters (i.e., the interface thickness and the mobility tuning parameter) is quantified revealing their importance and significance.

Appendix A: Maxwell stress tensor in cylindrical coordinates According to eq 12 the Maxwell stress tensor in cylindrical coordinates r, φ and z, reads  E 2r − E φ2 − E 2z ε 0ε r  2E φ E r 2   2E z E r 

2E r E φ E φ − E r2 − E 2z 2

2E z E φ

  , E 2z − E r2 − Eφ2  2E r E z 2E φ E z

(A1)

where, E with different subscripts denote the cylindrical components of the electric field strength. The problem is axially symmetric about the z axis. The corresponding stress components are denoted as

Fr =

Fz = ε 0ε r E z E r ,

(A2)

Fφ = ε 0 ε r E φ E r ,

(A3)

ε 0ε r 2

(E

2 r

)

− Eφ2 − E 2z .

(A4)

In the axisymmetric case Fφ vanishes, thus only the Fr and Fz components are non-zero.

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List of figures

Figure 1. Illustration of the computational domain, quantities of physical interest, and main conservation equations and boundary conditions employed in the present study. The upper right corner inset shows an isometric view of droplet impact onto a dielectric surface.

Figure 2. The evolution of a droplet on the hydrophilic (left) and the hydrophobic (right) surfaces. The initial geometry is a cylinder with its height equal to the radius (2 mm) and the initial velocities are zero in both cases.

Figure 3. Time-resolved droplet shape evolution of an oblate spheroidal droplet with the initial radii ratio of 2 released levitating at zero gravity. U is the velocity magnitude, and arrows depict vectors.

Figure 4. Comparison of the predicted axes ratio versus time (solid black line) with the prior literature46 (dashed line with red circles). The results are for water droplet oscillation at Re = 30 and an initial axes ratio of s = a / b = 2

Figure 5. (A) Example of a free triangular (quadratic) mesh used for the drop oscillation problem. The zoomed-in insert highlights the generated mesh over the interface. (B) Mass change of a droplet (Re = 30 and s = 2) for the case shown in Figures 3 and 4. Black curve presents the results for the conservative CH equation and blue curve shows the results for the non-conservative CH equation (cf. Model Formulation section).

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Figure 6. (A) potential distribution across the 100 µm dielectric layer; (B) surface density of the free charges at the surface of the dielectric.

Figure 7. Predicted droplet evolution (the upper panel), with red color being the water droplet and blue color being air. Velocity magnitude together with the velocity vectors (the second panel) and the pressure development during the impact (the lower panel). The droplet diameter is 1.92 mm and the initial droplet velocity is 0.62 m·s‒1.

Figure 8. Predicted droplet evolution (the upper panel), with red color corresponding to the water droplet and blue color corresponding to air. The second panel shows the velocity magnitude together with the velocity vectors and the lower panel depicts the pressure development during the impact. The DC voltage of 2 kV is applied at the bottom of the dielectric layer.

Figure 9. Comparison between the model’s predicted droplet shapes (upper panel) and the experimentally measured shapes (lower panel, source Ref.6) for 2 kV impact.

Figure 10. (A) Time evolution of charge accumulated at the droplet/solid surface interface during the water droplet impact onto the dielectric surface at the 2 kV applied DC voltage, (B) surface charge density at the maximum spreading distance.

Figure 11. Predicted droplet evolution (the upper panel), with red color corresponding to the

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water droplet and blue color corresponding to air. The second panel shows the velocity magnitude together with the velocity vectors and the lower panel illustrates the pressure development during the impact. The DC voltage of 4 kV is applied at the bottom of the dielectric layer.

Figure 12. Time evolution of charge accumulated at the droplet/solid surface interface during the water droplet impact onto the dielectric surface at the 4 kV applied DC voltage.

Figure 13. Water droplet evolution (the upper panel), with red color corresponding to the droplet and blue color to air. The second panel depicts the velocity magnitude and vectors and the lower panel depicts the pressure development during the impact. The 6 kV DC voltage is applied at the bottom of the dielectric.

Figure 14. Charge accumulation at the droplet/solid surface interface during the impact onto Teflon at 6 kV applied DC voltage.

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Figure 1. Illustration of the computational domain, quantities of physical interest, and main conservation equations and boundary conditions employed in the present study. The upper right corner inset shows an isometric view of droplet impact onto a dielectric surface. 1117x782mm (96 x 96 DPI)


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Figure 2. The evolution of a droplet on the hydrophilic (left) and the hydrophobic (right) surfaces. The initial geometry is a cylinder with its height equal to the radius (2 mm) and the initial velocities are zero in both cases.

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Figure 3. Time-resolved droplet shape evolution of an oblate spheroidal droplet with the initial radii ratio of 2 released levitating at zero gravity. U is the velocity magnitude, and arrows depict vectors. 1219x444mm (96 x 96 DPI)


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Figure 4. Comparison of the predicted axes ratio versus time (solid black line) with the prior literature46 (dashed line with red circles). The results are for water droplet oscillation at Re = 30 and an initial axes ratio of 57x42mm (300 x 300 DPI)


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Figure 5. (A) Example of a free triangular (quadratic) mesh used for the drop oscillation problem. The zoomed-in insert highlights the generated mesh over the interface. (B) Mass change of a droplet (Re = 30 and s = 2) for the case shown in Figures 3 and 4. Black curve presents the results for the conservative CH equation and blue curve shows the results for the non-conservative CH equation (cf. Model Formulation section). 668x756mm (96 x 96 DPI)


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Figure 5. (A) Example of a free triangular (quadratic) mesh used for the drop oscillation problem. The zoomed-in insert highlights the generated mesh over the interface. (B) Mass change of a droplet (Re = 30 and s = 2) for the case shown in Figures 3 and 4. Black curve presents the results for the conservative CH equation and blue curve shows the results for the non-conservative CH equation (cf. Model Formulation section). 203x152mm (300 x 300 DPI)


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Figure 6. (A) potential distribution across the 100 µm dielectric layer; (B) surface density of the free charges at the surface of the dielectric. 866x261mm (96 x 96 DPI)


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Figure 6. (A) potential distribution across the 100 µm dielectric layer; (B) surface density of the free charges at the surface of the dielectric. 208x159mm (300 x 300 DPI)


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Figure 7. Predicted droplet evolution (the upper panel), with red color being the water droplet and blue color being air. Velocity magnitude together with the velocity vectors (the second panel) and the pressure development during the impact (the lower panel). The droplet diameter is 1.92 mm and the initial droplet velocity is 0.62 m·s‒1. 1374x637mm (96 x 96 DPI)


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Figure 8. Predicted droplet evolution (the upper panel), with red color corresponding to the water droplet and blue color corresponding to air. The second panel shows the velocity magnitude together with the velocity vectors and the lower panel depicts the pressure development during the impact. The DC voltage of 2 kV is applied at the bottom of the dielectric layer. 1353x642mm (96 x 96 DPI)


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Figure 9. Comparison between the model’s predicted droplet shapes (upper panel) and the experimentally measured shapes (lower panel, source Ref.6) for 2 kV impact. 347x462mm (96 x 96 DPI)


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Figure 10. (A) Time evolution of charge accumulated at the droplet/solid surface interface during the water droplet impact onto the dielectric surface at the 2 kV applied DC voltage, (B) surface charge density at the maximum spreading distance. 203x152mm (300 x 300 DPI)


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Figure 10. (A) Time evolution of charge accumulated at the droplet/solid surface interface during the water droplet impact onto the dielectric surface at the 2 kV applied DC voltage, (B) surface charge density at the maximum spreading distance. 208x159mm (300 x 300 DPI)


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Figure 11. Predicted droplet evolution (the upper panel), with red color corresponding to the water droplet and blue color corresponding to air. The second panel shows the velocity magnitude together with the velocity vectors and the lower panel illustrates the pressure development during the impact. The DC voltage of 4 kV is applied at the bottom of the dielectric layer. 1353x642mm (96 x 96 DPI)


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Figure 12. Time evolution of charge accumulated at the droplet/solid surface interface during the water droplet impact onto the dielectric surface at the 4 kV applied DC voltage. 203x152mm (300 x 300 DPI)


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Figure 13. Water droplet evolution (the upper panel), with red color corresponding to the droplet and blue color to air. The second panel depicts the velocity magnitude and vectors and the lower panel depicts the pressure development during the impact. The 6 kV DC voltage is applied at the bottom of the dielectric. 1353x642mm (96 x 96 DPI)


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Figure 14. Charge accumulation at the droplet/solid surface interface during the impact onto Teflon at 6 kV applied DC voltage. 203x152mm (300 x 300 DPI)


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TOC 1069x568mm (96 x 96 DPI)


Modeling of Droplet Impact onto Polarized and Nonpolarized

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