Modelling of Droplet Dynamics in Strong Electric Fields

Abstract

We describe a modelling approach for the simulation of droplet dynamics in strong electric fields. The model accounts for electroquasistatic fields, convective and conductive currents, contact angle dynamics and charging effects associated with droplet breakup processes. Two classes of applications are considered. The first refers to the problem of water droplet oscillations on the surface of outdoor high-voltage insulators. The contact angle characteristics resulting from this analysis provides a measure for the estimation of the electric field inception thresholds for electrical discharges on the surface. The second class of applications consists in the numerical characterization of electrosprays. Detailed simulations confirm the scaling law for the first electrospray ejection and, furthermore, provide insight on the charge-radius characteristics for transient as well as steady state electrosprays.

1 Introduction

Applying electric fields to manipulate droplets, jets and sprays is an established technique used in various applications, including the development of microfluidic chips [1] and novel display technologies [2]. In more conventional fields such as electrostatic atomization, electric fields enable the generation of well-dispersed spray plumes with enhanced controllability of droplet size and spray angle as required, e.g., in industrial painting, electrowetting, ink-jet printing and coating applications.

Due to the vast range of applications, the electrohydrodynamic (EHD) modelling of droplet flows remains an important research subject. The interest in the topic is twofold. First, one is interested in the characterization of these flows for the purpose of design and optimization of specific technological processes. This includes, for example, the ability to control droplet positioning by electric fields generated by complex electrode arrangements in microfluidic devices or the optimization of electrospray properties for ink-jet applications. Second, the investigations aim at a better understanding of EHD phenomena. Analytical models are often not suitable for the description of practical problems, even for simplistic geometries and electric field configurations. A more appropriate means for investigating such problems is numerical simulation. However, also the numerical approach faces major difficulties. Despite the huge computational burden, numerical simulations often fail to capture all of the involved events due to neglecting or oversimplifying the underlying physics. This is the case, in particular, for droplet problems involving an interface between different fluid phases and fluid-solid phases. At such interfaces, the fluid and charge dynamics are determined by microscopic processes that on their turn strongly influence the behaviour of the macroscopic system [20].

Various EHD simulation models have been presented (cf. [29, 30]). An electroquasistatic (EQS) model using a sharp interface approach for the simulation of droplet oscillations in electric fields was described by Songoro et al. [28]. Recently, more sophisticated models based on the diffuse-interface description have been introduced, taking into account charge relaxation effects by explicitly solving the charge conservation equation [16, 24]. Most of these works, however, focus on the electrokinetics of charged species in electrolytic solvents rather than on capillary droplet dynamics. Furthermore, contact line effects at solid interfaces are typically not considered. In the following, a EHD modelling approach including these effects is described. The model is, furthermore, applied in the analysis of droplet induced partial discharges on the surface of outdoor high-voltage (HV) insulators as well as in the investigation of transient and steady state electrospray atomization processes.

2 The Electrohydrodynamic Model

The electric fields involved in most EHD applications are considered to be slowly varying, therefore, a quasistatic approximation for the Maxwell’s equations is sufficient. In the following, thermal diffusion, charge production in the bulk and interfacial charge layers are neglected. These assumptions are reasonable for typical applications with conductive droplets where the electrostatic screening length is much smaller than the characteristic size of the system [22]. Furthermore, the effect of Faradaic currents induced by electrochemical reactions on electrode surfaces is not taken into account. Under these assumptions, the EHD problem is well described by the Taylor-Melcher leaky dielectric model [18] that is summarized below.

2.1 Governing Equations

The fluid flow problem consists in the dynamics of an incompressible liquid (droplet) emerged in a gaseous phase (air), eventually interfacing a solid body (dielectric substrate or capillary). The Navier-Stokes equations read,

$$\begin{aligned} \frac{\partial \rho \textbf{u}}{\partial t} + \nabla \cdot {\rho \textbf{u}\textbf{u}}&= -\nabla p + \nabla \cdot \left( \mu \left[ \nabla \textbf{u}+\nabla \textbf{u}^\textrm{T}\right] \right) + \rho \textbf{g} + \textbf{f}_\textrm{s} + \textbf{f}_\textrm{e} \, , \end{aligned}$$

(1)

$$\begin{aligned} \nabla \cdot \textbf{u}&=0 \, , \end{aligned}$$

(2)

where $\textbf{u}$ denotes fluid velocity, $\rho$ density, $\mu$ dynamic viscosity and p pressure. The driving terms $\textbf{f}_\textrm{s}$ and $\textbf{f}_\textrm{e}$ are the surface tension force density and the electric force density, respectively. The former is derived from the surface force density given by the Young-Laplace equation, ${\hat{\textbf{f}}}_s = 2 \gamma H \textbf{n}$, where $\gamma$ is the surface tension, H is the mean curvature and $\textbf{n}$ the interface normal.

The total electric current in the liquid phase consists of a convective and a conductive contribution as, $\textbf{J}=\rho _e\textbf{u}+\kappa \textbf{E}$, where $\kappa$ is the Ohmic conductivity, $\textbf{E}$ the electric field strength and $\rho _e$ the charge density associated with the free charge carriers in the liquid. The latter is related to electric field by Gauß’s law, $\rho _e= \nabla \cdot \varepsilon \textbf{E}$, where $\varepsilon$ is the dielectric permittivity of the medium. Introducing an electric potential, $\textbf{E}=-\nabla \varphi$, the charge conservation law in the flowing medium reads

$$\begin{aligned} \nabla \cdot \varepsilon \nabla \varphi&= -\rho _e \, , \end{aligned}$$

(3)

$$\begin{aligned} \frac{\partial \rho _e}{\partial t}+\nabla \cdot \left( \rho _e\textbf{u}\right)&= \nabla \cdot \kappa \nabla \varphi \, . \end{aligned}$$

(4)

Note that the charge density can actually be eliminated from the system of equations. However, this results in a third order equation for the electric potential; a form which is less suitable for numerical calculations. In conductive solids ($\textbf{u}=\textbf{0}$), the system of Eqs. (3)–(4) reduces to the conventional EQS formulation for low frequency fields in resting media. In nonconducting dielectrics ($\textbf{u}=\textbf{0}$, $\kappa = 0$), the equations reduce to Laplace’s equation for the electrostatic potential. Thus, the electric field problem is well posed in the fluid as well as in the solid phases provided that appropriate boundary and initial conditions are specified for the electric potential and charge density, respectively.

Finally, given the electric field distribution, the electric force density applied on the fluid can be obtained from the Maxwell stress tensor [18] as

$$\begin{aligned} \textbf{f}_\textrm{e}&= \nabla \cdot \left( \varepsilon \textbf{E}\otimes \textbf{E}-\frac{1}{2}\varepsilon E^2\textbf{I}\right) \, , \end{aligned}$$

(5)

where $\textbf{I}$ represents the third-order unit tensor.

2.2 Contact Line Model

Droplet dynamics is closely related to the properties of the three-phase contact line between the liquid droplet, surrounding air and a solid body. Along this line, the surface tension force is ill-defined. Therefore, the flow properties at the contact line must be imposed by a boundary condition that is typically obtained from experiments. A common boundary condition consists in prescribing the apparent contact angle, $\theta$, between the droplet surface and solid wall [8]. In static equilibrium, the contact angle attains a range of admissible values, $\theta _\textrm{rec}< \theta < \theta _\textrm{adv}$, where $\theta _\textrm{rec}$ denotes the minimal contact angle required for a receding contact line motion and, $\theta _\textrm{adv}$ is the maximal angle required for the contact line to perform an advancing motion. Within this range of contact angels, the droplet exhibits hysteretic motion with the position of the contact line remaining pinned on the solid wall.

In the dynamic case involving a moving contact line, the contact angle will generally depend on the local contact line velocity along the solid wall, $u_\textrm{cl}$. The dynamic contact angle model adopted in this paper is given by Kistler [13]:

$$\begin{aligned} \theta&= f_\textrm{H}\left( C_a+f_\textrm{H}^{-1}\left( \theta _\mathrm {adv/rec}\right) \right) \, , \end{aligned}$$

(6)

$$\begin{aligned} \text{ with } \quad f_\textrm{H}\left( x\right)&= \arccos \left( 1 - 2 \tanh \left[ 5.16\left( \frac{x}{1+1.31x^{0.99}}\right) ^{0.706}\right] \right) \, . \end{aligned}$$

(7)

In (6), the capillary number, $C_a=\mu u_\textrm{cl} / \gamma$, is a signed quantity. It is considered to be positive when the contact line is advancing and is otherwise negative. Similarly, $\theta _\mathrm {adv/rec}$ attains the value $\theta _\textrm{adv}$ or $\theta _\textrm{rec}$ depending on whether the contact line performs a receding or an advancing motion, respectively. Thus, the limiting contact angles, $\theta _\textrm{adv}$ and $\theta _\textrm{rec}$, represent two model parameters that need to be determined experimentally. Note that these limiting angles are directly related to the wettability of the solid surface for the given liquid.

3 Numerical Implementation

Equations (1)–(2), (3)–(5) and (6), (7) provide a complete description of the EHD problem. We use the classical Volume of Fluid (VoF) method [12] for discretizing the equations, an iso-surface interface reconstruction approach for curvature and contact line dynamics and a dedicated mesh coupling procedure for transferring data between the fluid flow and electric field problems. These approaches are implemented in the computational framework OpenFOAM [21] as described in the following.

3.1 The Fluid Flow Problem

For the solution of the Navier-Stokes equations, (1)–(2), a diffuse-interface approach is adopted. The density of the liquid phase is represented on the mesh by the volume fraction, $\alpha$, obeying the transport equation,

$$\begin{aligned} \frac{\partial \alpha }{\partial t} + \nabla \cdot \left( \alpha [\textbf{u}+\textbf{u}_c]\right) = 0 \, , \end{aligned}$$

(8)

with $\alpha \in \left[ 0,1\right]$, where $\textbf{u}_c$ is an artificial compression velocity pointing in the direction normal to the phase boundary. The effective fluid properties used in the discretization of (1)–(2) are determined as weighted averages with the volume fraction field, $\alpha$, on the mesh [12]. The diffuse-interface representation of the phase boundary allows to compute the surface tension force as [3],

$$\begin{aligned} \textbf{f}_\textrm{s} = -\gamma \nabla \cdot \left( \frac{\nabla \alpha }{||\nabla \alpha ||}\right) \nabla \alpha \, . \end{aligned}$$

(9)

In order to apply the contact line model given by (6), (7) the velocity of the contact line along the solid wall must be evaluated numerically. In this work, we have adopted the scheme proposed in [15] where the contact line velocity is given by

$$\begin{aligned} u_\textrm{cl}=\frac{\textbf{u}_\textrm{w}\cdot \textbf{n}_\textrm{i}}{\sqrt{1-(\textbf{n}_\textrm{w}\cdot \textbf{n}_\textrm{i})^2}} \, , \end{aligned}$$

(10)

where $\textbf{u}_\textrm{w}$ is the fluid velocity near the solid wall, $\textbf{n}_\textrm{w}$ is the normal vector to the wall and $\textbf{n}_\textrm{i} = \nabla \alpha / ||\nabla \alpha ||$ is the normal to the diffuse interface. In order to account for contact line pinning, the volume fraction in the boundary cells is kept constant as long as the contact angle lies between its static limits $\theta _\textrm{adv}$ and $\theta _\textrm{rec}$, respectively.

Given the above model for the phase boundary, the pressure-velocity Eqs. (1), (2) are solved numerically using the piso algorithm implemented in the interFoam solver [21]. The handling of the contact line dynamics has been adopted from a specialized implementation of the code; the interFoamExtended solver developed at the Technical University of Darmstadt [25].

3.2 The Electric Field Problem

The solution of the electric field Eqs. (3) and (4) can be performed on the same footing with the flow problem using a VoF-scheme with the volume fraction (8). The electric potential and charge density are allocated at cell centres on the same computational mesh as the fluid flow quantities. At the diffuse electric interface between fluid phases $\textrm{1}$ and $\textrm{2}$, the effective dielectric permittivity and electric conductivity are computed as weighted harmonic averages,

$$\begin{aligned} \frac{1}{\varepsilon } =\frac{\alpha }{\varepsilon _\textrm{1}}+\frac{1-\alpha }{\varepsilon _\textrm{2}} \quad \textrm{and}\quad \frac{1}{\kappa } =\frac{\alpha }{\kappa _\textrm{1}}+\frac{1-\alpha }{\kappa _\textrm{2}} \, , \end{aligned}$$

(11)

respectively. The choice of harmonic averaging in (11) has been shown to be numerically more accurate for strongly contrasting material properties [31] as is the case for a conductive droplet in an essentially nonconducting gaseous environment.

Equations (3), (4) are advanced in time by the staggered scheme,

$$\begin{aligned} \nabla \cdot \varepsilon ^{n+1/2}\nabla \varphi ^{n+1/2}&= -\rho _{e}^{n+1/2} \, , \end{aligned}$$

(12)

$$\begin{aligned} \frac{\rho _{e}^{n+1}-\rho _{e}^{n}}{\Delta t}+\nabla \cdot (\rho _{e}^{n+1/2}\textbf{u}^{n+1/2})&=\nabla \cdot \kappa ^{n+1/2}\nabla \varphi ^{n+1/2}\, , \end{aligned}$$

(13)

where $\Delta t$ is the time step and n the time level. In addition, a fixed-point iteration is applied in every time step (cf. Fig. 1) in order to ensure numerical consistency between the charge density and electric potential solutions.

3.3 Coupling Procedure

Most EHD problems include the presence of solid bodies such as dielectric substrates or electrode coatings. While these solids are irrelevant for the fluid flow problem, the solution of the EQS problem is often determined by their geometry and material properties. Since the solution domains for the fluid flow and EQS problems are generally not identical, a two-mesh coupling procedure is applied. The computational mesh for the EQS problem includes the fluid domains as well as eventually available dielectric solids. A second computational mesh for the flow problem is constructed as a subset of the EQS mesh by excluding solid parts. Index maps are used to transfer discrete fields from one mesh to the other. Hereby, no explicit interpolation is required since the field data reside on the same locations in the mesh.

The complete flow chart of the coupling scheme including the inner and outer fixed-point iteration loops is shown in Fig. 1. In addition to the depicted steps, a parallel adaptive mesh refinement procedure with dynamic load balancing is applied in every time step of the simulation. The reader is referred to [26] for a detailed description of the parallel mesh refinement procedure that we have omitted in this article for the sake of compactness.

Fig. 1

Adapted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Flow chart of the time stepping scheme for the coupled fluid dynamics and electric field problems including the inner and outer iteration loops that ensure solution consistency in every time step of the simulation. For the notation used in the momentum calculation step, the user is referred to  [21].

4 Applications

The simulation model is applied in the study of two droplet problems of practical relevance. The first consists in the investigation of forced droplet oscillations on the surface of outdoor HV insulators for power line applications. The assessment of the dynamic droplet contact angle provides the starting point for an estimation of inception fields for the partial discharges caused by these droplets. The second application is the characterization of electrosprays in the cone-jet mode in terms of the relationship between droplet charge and its radius.

4.1 Sessile Droplets in HV Fields

Water droplets caused by rain or dew on the surface of HV outdoor equipment undergo a continuous oscillatory motion under the influence of AC electric fields. The sharp contact angle between droplet surface and insulating substrate results in a field singularity along the contact line that is responsible for electric field enhancement on the insulator surface. As a result, electrical discharge events occur that gradually damage the insulator surface [11].

We consider the experimental setup used in [19] and depicted in Fig. 2. It consists of a pair of cylindrical electrodes that are embedded in an silicone rubber substrate. A water droplet placed between the electrodes experiences a nearly homogeneous AC electric field that is oriented along the substrate surface. As seen in the Figure, the presence of a (static) droplet causes field enhancement at the droplet’s contact line from which the partial discharge events are initiated.

Fig. 2

Adapted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Left: simulation setup including HV electrodes and water droplet on the substrate. Right: electric field strength along the substrate surface parallel to the applied field. The field is singular at the contact line between droplet and substrate. For simplicity, the case of a static droplet is shown.

4.1.1 Droplet Oscillation Modes

In the simulations, a $20\,\upmu \textrm{L}$ water droplet is considered with a peak voltage of $7\,\textrm{kV}$ at frequency $27\,\textrm{Hz}$ applied between the electrodes. The initial droplet shape is calculated semi-analytically by numerically integrating the Young-Laplace equilibrium equation according to the procedure described in [27]. The hysteresis range for water droplets on silicone rubber is obtained experimentally resulting in the limiting receding and advancing angles, $\theta _\textrm{rec} = 60^\circ$ and $\theta _\textrm{adv}=140^\circ$, respectively [19].

Experimental evidence shows that droplets on insulator surfaces under outdoor conditions may contain a certain amount of net charge that is due to surface contamination or ionization events in air [5]. A droplet charge in the order of $100\,\textrm{pC}$, as suggested in [17], is quite significant compared to the displacement charge induced by the external field. This charge will have a strong impact on droplet’s motion. Therefore, in the following, two droplet scenarios are considered. In the first scenario, an uncharged water droplet is considered. In the second one, a net charge of $100\,\textrm{pC}$ is given to the droplet before the electric field is switched on. The initial charge distribution within the droplet is found from the steady state solution of the EHD problem in absence of external fields.

Figure 3 shows simulation results for the uncharged droplet. The oscillation pattern shown in the Figure is settled after a few periods of the applied voltage. The motion consists essentially in a forced vertical oscillation at twice the applied frequency. This is the expected behaviour, since the dominant driving force on the droplet is due to dielectric polarization. Also in the Figure, the simulation results are compared with droplet images obtained experimentally using a high-speed camera in the High Voltage Lab at the TU Darmstadt. The striking agreement between simulation and measurement demonstrates the reliability of the simulation model.

Fig. 3

Reprinted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Motion of a $20\,\upmu \textrm{L}$ droplet when a $7\,\textrm{kV}$ peak voltage at $27\,\textrm{Hz}$ is applied. The dynamics over half a period of the AC voltage is shown. First and third lines: simulation results. The colour map depicts electric field strength. Second and fourth lines: high-speed camera images obtained experimentally.

Figure 4 shows an analogous simulation for a charged droplet. The difference to the uncharged case is obvious. A horizontal mode in the direction of the applied field is excited with a considerably larger amplitude than in the charged case. In addition, the frequency of oscillation is nearly the same as that of the applied field. This indicates that the dominant driving force in the charged case is given by a monopole-type interaction of the external field with the net charge of the droplet in its centre of mass.

Fig. 4

Reprinted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Oscillations of a charged droplet at $27\,\textrm{Hz}$. Droplet profiles at three time instants over half a period are shown. The colour map depicts electric field strength.

The above results can be quantified in terms of harmonic oscillation modes. For this purpose the motion of the droplet centre of mass in all three directions is considered. The spectral content of this motion is shown in Fig. 5. As observed, the dominant mode for the uncharged droplet is a vertical oscillation at $\sim 58\,\textrm{Hz}$, that is slightly larger than double the frequency of the applied voltage. However, a horizontal oscillation of smaller amplitude is observed as well. The latter occurs at around the same frequency as the applied voltage. This is consistent with the finding in [4, 28] for uncharged droplets, where azimuthally degenerate droplet modes were found to oscillate at half the frequency of the driving force.

The case of the charged droplet is shown in Fig. 5 (right). The modal patterns and frequencies are essentially the same as in the uncharged case. However, the horizontal mode at the lower frequency, $27\,\textrm{Hz}$, is clearly the dominant one. Thus, the dynamics of sessile droplets on insulator surfaces depends strongly on the net charge carried by the droplet. This behaviour has immediate consequences on the electric field threshold for partial discharge inception as discussed in the following.

Fig. 5

Adapted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Oscillating modes of a $20\,\upmu \textrm{L}$ sessile droplet for an applied voltage at $27\,\textrm{Hz}$. The coordinate system is as in Fig. 2. Left: uncharged droplet. Right: droplet with a net charge of $100\,\textrm{pC}$.

4.1.2 Inception Field Estimation

The basic assumption in the analysis is that partial discharges originate from the electric field enhancement at the droplet’s contact line only. The field singularity at this location is essentially determined by the contact angle between droplet surface and dielectric substrate, where the strength of singularity is larger for smaller contact angles [22]. Thus, the discharge inception field may be estimated from the worst case scenario consisting in the smallest value of contact angle attained by the droplet at the peak applied voltage. This approach simplifies the analysis substantially by allowing to consider the droplet dynamics and the discharge inception problems one at a time.

The transient behaviour of the contact angle is shown in Fig. 6 for the two considered cases of an uncharged and a charged droplet, respectively. The thin grey lines in each of the graphs depict instantaneous contact angle values measured along the the y-direction over half a period of the applied field. Since droplet motion is quasiperiodic due to modal superposition, the contact angles over several periods will span a whole range of values. This range is described in the figure by the blue-shaded areas corresponding to the envelope of the contact angle curves recorded for different periods at different phases of the applied field.

Fig. 6

Adapted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Range of contact angles in the direction parallel to the field vs. phase of the applied field for a $20\,\upmu \textrm{L}$ droplet at $27\,\textrm{Hz}$. Left: uncharged droplet. Right: droplet with a net charge of $100\,\textrm{pC}$.

As seen in Fig. 6, the contact angle range in the charged droplet case is substantially different from that of the uncharged one. In the uncharged case, the contact angles essentially follow the magnitude of the applied field. The worst case of a minimum contact angle is obtained at the zero-crossing of the applied voltage where a partial discharge is unlikely to occur. In all cases, however, the contact angle remains consistently within the hysteresis range given by the static receding and advancing angles, $\theta _\textrm{rec} = 60^\circ$ and $\theta _\textrm{adv}=140^\circ$, respectively. In other words, for the vertical mode excited in the uncharged case, the contact line remains pinned. Therefore, the knowledge of the limiting receding angle for a given substrate wettability ($60^\circ$ in this case) is fully sufficient to identify a worst case scenario for discharge inception from uncharged droplets.

Fig. 7

Adapted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Left: discharge inception field vs. droplet volume for static droplets with different (but fixed) contact angles. Right: effect of substrate permittivity on partial discharge inception for a $20\,\upmu \textrm{L}$ droplet for different values of the contact angle.

The situation is significantly different in the case of a charged droplet featuring the horizontal oscillation mode seen in Fig. 5 (right). The range of attainable angles is nearly independent from the instantaneous applied voltage. The contact angles at the peak voltage span the full hysteresis range. Furthermore, contact angles beyond the limiting static values can be observed indicating that the contact line performs a stick-slip motion as it has been previously noted, e.g. in [9]. Thus, for charged droplets the discharge inception field is expected to be substantially lower than for uncharged ones. Furthermore, the discharge inception depends not only on the wettability properties of the substrate surface, but also on the net charge carried by water droplets under atmospheric conditions.

The discharge inception fields can be quantified in the range of realistic droplet sizes and contact angles. For this purpose, the electric field distribution at power line frequency is computed numerically for a set of static droplets with different (but fixed) sizes and shapes. Hereby, droplet motion may be ignored, since only the worst case of the smallest attainable contact angle is of interest. Given a numerically computed field distribution, the corona discharge model introduced in [14] is applied for different droplet shapes and sizes using the commercial software Spark3D. The detailed procedure for extracting discharge inception fields is described thoroughly in [22]. In the following, we restrict the discussion to the main results of this study providing the link between the contact angles predicted by droplet dynamics simulations and the discharge characteristics on the surface. This is depicted in Fig. 7 (left), where the inception field is shown as a function of droplet volume for different contact angles assuming a fixed, axially symmetric droplet geometry. The inception field decreases, both, with increasing droplet volume and decreasing contact angle. Over the range of considered volumes, the contact angle causes the inception field to vary by more than a factor of two. The higher sensitivity is observed for small droplets whereas for larger ones the variation of the inception field with droplet volume is less pronounced. The complex relationship between discharge characteristics and problem parameters is demonstrated in Fig. 7 (right). Obviously, the inception field can be maximized by using an insulating substrate with small permittivity. However, this is no longer true for droplet oscillations featuring small contact angles as is the case for intrinsically charged droplets and surfaces with low wettability.

4.2 Electrosprays

Various efforts have been made for the characterization of electrosprays in order to predict the size and charge of the microdroplets produced by electrostatic atomization. Most prominently, Collins et al. [6] and later Gañán-Calvo et al. [10] have suggested the existence of universal scaling laws relating the charge to the radius of the very first ejected electrospray droplet. In the following, the simulation model is used for the investigation of the charge-radius characteristics in cone-jet electrosprays. Thereby, not only the fist ejected droplet but rather the complete transient dynamics of electrospray including the steady state limit is considered.

4.2.1 First Electrospray Droplet

For the study of the first ejection, the onset of electrospray from a (large) sessile droplet placed between two plane electrodes is considered. In the simulations, axial symmetry is presumed. A voltage pulse is ramped up causing the droplet to deform to a conical shape, at the tip of which a jet streams, leading eventually to electrospray atomization. Simulation results for two different test liquids in the initial stage of electrospray are shown in Fig. 8. Clearly, the size (and charge) of the ejected droplets and also the time scales characterizing the process depend on liquid properties.

Fig. 8

Adapted figure with permission from [23], Copyright 2020 by the IEEE

Onset of electrospray from a sessile droplet shown at different time instants including the ejection of the first few droplets. The origin of time is at the instant of first droplet ejection. The colour map depicts electric field strength. Left: methanol. Right: a heptane mixture (with $\gamma =21\mathrm {mN/m}$, $\kappa = 1.9\,\upmu \textrm{S}/\textrm{m}$). The conductivity of the heptane mixture is nearly one order of magnitude smaller than that of methanol [22].

Visual inspection of Fig. 8 indicates that the very first ejected droplet is slightly larger than the succeeding ones. This indicates a different dynamical behaviour of this droplet compared to the rest of electrospray droplets. For this particular droplet, Collins et al. [6] postulate a universal scaling law that provides a liquid-independent relationship between the droplet charge, q and its radius, r. Given the characteristic viscous length $l_\mu =\mu ^2/\rho \gamma$ and the dimensionless droplet charge and radius defined as $q^*=q/\sqrt{\varepsilon \gamma l_\mu ^3}$ and $r^*=r/l_\mu$, respectively, the scaling law reads, $q^*\propto {\left( r^{*}\right) }^{\frac{3}{2}}$, where the proportionality constant is independent from liquid properties. Figure 9 depicts the computed correlation between the first droplet charge and radius for 21 different liquid mixtures spanning a large range of electrical and mechanical properties. The simulation results fit quite well to the predicted power law, thus, providing a numerical confirmation for the universal scaling suggested in [6].

Fig. 9

Adapted figure with permission from [23], Copyright 2020 by the IEEE

Charge-radius correlation for the first electrospray droplet for different applied voltages and a number of liquid mixtures with different electrical and mechanical properties. The black line represents the Collins’s scaling law [6].

4.2.2 Transient Electrosprays

As the electrospray develops, the cone-jet dynamics will affect the ongoing atomization process. Thus, the size and charge of subsequent droplets are expected to differ from that of the first ejection (cf. Fig. 8). The question posed is, to what extent the properties of these subsequent droplets in a transient electrospray process comply to the scaling law in Sect. 4.2.1. It is, furthermore, unclear whether the charge-radius correlation for longer time transients can be represented by a power law, as is the case for the first ejected droplet.

Figure 10 (left) shows the computed droplet radii vs. their breakup time for a heptane mixture. The droplets can be clearly classified into two groups referred to as primary and satellite droplets, respectively, where the radius of the latter is consistently smaller than $2\,\upmu \textrm{m}$. The size of the first few primary droplets is smaller than that of the first ejection. However, the radius of subsequent ejections increases as the electrospray develops. This trend is the same for all considered voltages.

Fig. 10

Adapted figure with permission from [23], Copyright 2020 by the IEEE

Left: droplet size vs. ejection time for a heptane mixture electrospray ($\gamma =21\mathrm {mN/m}$, $\kappa = 1.9\,\upmu \textrm{S}/\textrm{m}$). Right: charge-radius correlation for different applied voltages. The red-framed circle depicts the first ejection.

The charge-radius correlation for all produced droplets in the course of the transient process and for all applied voltages is depicted in Fig. 10 (right). For comparison, the first ejected droplet for each voltage is highlighted by a red-framed circle. Two scaling trends can be observed corresponding to primary and satellite droplets, respectively. Since satellites carry little charge compared to primary droplets, we restrict the discussion on the charge-radius correlation for primary droplets. Clearly, this correlation fits well to a power law. However, the scaling exponent (1.59) differs from that of the universal law for the first droplet. Interesting to note is that the charge density of the first ejection is always higher than that of all succeeding droplets. Detailed simulations with other liquids show that the charge-radius scaling is always consistent with a power law similar to the one obtained in [6] for the first ejection. However, the scaling exponents are different and do, in general, depend on the electrical and mechanical liquid properties. Explicit values of these exponents for acetone, methanol and various heptane mixtures with different electric conductivities are reported in [23].

4.2.3 Steady State Electrosprays

In most applications, electrosprays are applied in continuous-flow mode where a constant liquid flow is supplied in order to sustain a steady atomization process. In order to investigate steady state electrosprays, the simulation setup shown in Fig. 11 (left) is considered. The liquid is provided with different flow rates by means of a metallic capillary. The latter serves at the same time as driving electrode where a voltage of a few kilovolts is applied. The test liquid is heptane enriched with 0.3% stadis-450 ($\gamma =18.6\mathrm {mN/m}$, $\kappa = 1.4\,\upmu \textrm{S}/\textrm{m}$).

Fig. 11

Adapted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Left: simulation setup for continuous-flow electrosprays. A voltage of $4\,\textrm{kV}$ is applied between capillary and the ground plane. Right: droplet size vs. ejection time for the heptane mixture electrospray for three different flow rates.

Figure 11 (right) depicts the electrospray dynamics for three different flow rates. A steady state situation with a nearly constant mean droplet size is reached in all three considered cases. However, the transition times as well as the resulting final droplet sizes are quite different, depending on the applied flow rates. In the case with the largest flow rate, steady state is reached after a few milliseconds corresponding to approximately 200 primary ejections. Note also that for each individual electrospray the two groups of primary and satellite droplets, respectively, can be clearly distinguished.

As seen in Fig. 11, droplet sizes and charges at steady state manifest a significant spread due to fluctuations of the jet profile and perturbations originating from prior ejections. The steady state distributions of primary droplet sizes for different flow rates are shown in Fig. 12 (left). In order to discard transient effects, in this representation, only late-time primary droplets are included that are produced after 200 ejections have already taken place. The relationship between primary droplet sizes and charges at steady state is depicted in Fig. 12 (right). Obviously, the scaling of the mean droplet size and charge at steady state seem to follow a power law when the flow rate is changed with a corresponding scaling exponent of 1.81 for the particular liquid used in the simulations.

Fig. 12

Adapted figure with permission from [22] under the Creative Common License (CC-BY-NC-ND 4.0 International)

Left: droplet size distributions at steady state for three different flow rates. Right: scatter plot of droplet charges and radii at steady state for each flow rate. For reference, the two characteristic scaling curves for steady state electrosprays are shown.

The other interesting observation concerns the relationship between droplet charge and size deviations from their respective mean values for any given electrospray at steady state. As seen in Fig. 12 (right), droplet charge deviations correlate proportionally to volume deviations. This is true for all three flow rates considered. This result is consistent with the experimental finding in [7], where it was observed that the droplet charge distribution at steady state exhibits a significantly larger spread than that of the corresponding size distribution. Furthermore, “...the mean charge  q  and diameter  d associated with these sections are such that the charge density  $6q/ (\pi d^3 )$  varies little within the full range of drop volumes and charges produced by a given spray” [7]. A justification for this behaviour was offered in [7], where Rayleigh’s breakup theory for a cylindrical jet was applied assuming that electric charges are frozen on the jet surface during droplet breakup. Simulations do confirm this result while specifying that the correlation $q \sim r^3$ refers to droplet charge and size deviations from their mean values for a given electrospray at steady state.

5 Conclusions

The strength of the modelling approach consists in the detailed description of contact angle dynamics as well as in the incorporation of electric field distributions resulting from complicated electrode and insulator geometries. This allows for accurate simulations of real world droplet dynamics problems. Two such applications are considered. In the case of water droplets on high-voltage insulators, the oscillations of uncharged and charged droplets are considered. A corona discharge model is applied to determine inception fields as a function of contact angle, droplet size and dielectric permittivity of the substrate. The second application concerns electrosprays. Simulations for an electrospray initiated from a sessile droplet as well as continuous flow electrosprays were performed. Despite the well-known charge-radius scaling of the first droplet, simulations show that a similar scaling applies to the subsequently ejected droplets. The scaling exponent, however, depends on liquid properties. As the electrospray evolves to steady state, yet another characteristic scaling can be observed. It indicates that deviations of the droplet charge distribution from its mean value correlate proportionally with volume deviations. Furthermore, also the correlation of the mean droplet charges and radii at steady state for different flow rates fits well to a liquid specific power law. This latter observation, however, requires further investigations that will be presented in a forthcoming work.