# A new device for monitoring deformation dynamics and droplet shrinkage kinetics (during drying) of a charged droplet suspended in air

A. Dokmak, Thomas Deleau, Jean-jacques Letourneau, A. Maudhuit, Patricia Arlabosse, Maria-Inês Ré

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## ► To cite this version:

A. Dokmak, Thomas Deleau, Jean-jacques Letourneau, A. Maudhuit, Patricia Arlabosse, et al. A new device for monitoring deformation dynamics and droplet shrinkage kinetics (during drying) of a charged droplet suspended in air. Journal of Electrostatics, 2025, 134, pp.104033. 10.1016/j.elstat.2025.104033. hal-04926769

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https://imt-mines-albi.hal.science/hal-04926769v1

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# A new device for monitoring deformation dynamics and droplet shrinkage kinetics (during drying) of a charged droplet suspended in air

A. Dokmak<sup>a,b,*</sup>, T. Deleau<sup>a</sup>, J.J. Letourneau<sup>a</sup>, A. Maudhuit<sup>b</sup>, P. Arlabosse<sup>a</sup>, M.I. Ré<sup>a</sup>

## ARTICLE INFO

Keywords:

Deformation kinetics

Electric capillary number

Pendant droplet

Interfacial tension

Electric field

## ABSTRACT

In this study, a new device was developed to study the deformation kinetics of a pendant droplet under controlled drying conditions and in the presence of an electric field (EF). It has made it possible to investigate the deformation and air-drying kinetics of conductive and electrically charged pendant CuSO₄·5H₂O droplets. Various EF intensities (0–800 kV/m) and solid content concentrations (0–25 % wt/wt) are applied. Results show significant reduction (up to 42 %) in interfacial tension with increasing EF, along with a quadratic relationship between EF intensity and interfacial tension. Droplet surface area expansion and deformation are influenced by the EF intensity and the solution solids content. It has also been shown that, during drying, EF application can influence the deformation kinetics and shrinkage rate during the early drying stages, particularly at higher solid concentrations and high EF intensity, revealing new insights into droplet dynamics under EF.

## 1. Introduction

Applying electric fields to enhance the drying processes techniques have been widely investigated in industries like pharmaceuticals, food processing, ink printing applications among others where controlling the droplet properties is important for optimizing the process efficiency. Techniques like electrospray drying [1,2], electrohydrodynamic air drying [3,4], and electrostatic assisted spray drying [5,6] have been the focus of research and development to improve drying and enhance the product characteristics. While many experimental studies have highlighted the benefits of these techniques, it is crucial to understand the effect of electric fields at the scale of drying droplets.

Existing studies have focused on droplet-scale behavior in the presence of an electric field in controlled environments. However, most of this research has focused on conductive droplets suspended in immiscible continuous viscous fluids [7,8], such as water droplet in oil, or sessile droplets placed on a hydrophobic substrate, where stability of the droplet is easier to control [9,10].

For instance, studies have demonstrated that the application of an electric field to the droplet suspended in an immiscible fluid induces electrical forces that act on the droplet, thus modifying the balance of forces due to the additional Maxwell stress [11,12]. This gives rise to the following expression of the droplet equilibrium presented in Equation

(1) in which Young-Laplace equation is used to describe the capillary pressure difference sustained across the interface.

$$ \gamma \left( \frac{1}{r_1} + \frac{1}{r_2} \right) = \Delta P_0 - \Delta \rho gz + \frac{1}{2} \varepsilon_c E_n^2 \quad (1) $$

Where $\gamma$ is the interfacial tension (N/m), $\varepsilon_c$ is the permittivity of the continuous medium, such as air (F·m⁻¹), $r_1$ and $r_2$ are the principal radii of curvature of the drop (m), $\Delta P_0$ is the pressure difference across the interface at the apex (Pa), $\Delta \rho$ is the density difference between the droplet and its surrounding medium (kg·m⁻³), $g$ is the acceleration due to gravity (m/s²) and $z$ is the vertical coordinates (m) where the origin is assumed to be at the apex.

$E_n$ is the normal component of the electric field acting on the interface of the droplet. The term $\frac{1}{2}\varepsilon_c E_n^2$ denotes the normal component of the Maxwell stress. This term is particularly significant for perfectly conducting liquid droplet in a dielectric continuous medium [12].

Authors in Ref. [13] explored the impact of the external static electric field on the interfacial tension of pendant droplets (water, whey protein isolates solutions and chickpea liquor), using Rotenberg method to resolve the classical Young Laplace equation: from an approximated initial value of the interfacial tension, they numerically integrated the Laplace's equation by successive iterations using the Runge-Kutta method until the computed and the digitized profiles of the droplet

<footnote><sup>a</sup> Université de Toulouse, Mines Albi, Centre RAPSODEE UMR CNRS 5302, Campus Jarlard, Albi, France</footnote>

<footnote><sup>b</sup> Fluid Air, Division de Spraying Systems, 28 rue Louis Pasteur, 44119, Treillières, France</footnote>

<footnote>* Corresponding author. Université de Toulouse, Mines Albi, Centre RAPSODEE UMR CNRS 5302, Campus Jarlard, Albi, France.</footnote>

<footnote>E-mail address: Akaber.dokmak@spray.com (A. Dokmak).</footnote>
coincide. Their findings stated that interfacial tension decreases pro-
portionally with increasing the applied voltage. They also noted that in
the presence of an EF the Young-Laplace equation needs to be modified
to introduce the electrical Maxwell stress.

Additionally, these electrical stresses acting on the droplet leads the droplet to deform. Authors in Ref. [14] studied the morphological deformation dynamics of a sessile drying droplet subjected to a uniform electric field. They found that the interfacial tension of the droplet decreases, and the droplet deforms due to the accumulation of charges, leading to an enhanced droplet evaporation. However, at large deformations, the heat transfer was weakened due to the reduction of the contact diameter with the substrate. Furthermore, at larger EF, the droplet broke due to the competition between electrical and capillary forces.

Additionally, numerical studies and theoretical models aimed to predict the theoretical deformation of a droplet pendant in controlled viscous fluids under electric fields. They are based on the properties of both fluids' conductivity, permittivity, viscosity and interfacial tension. Taylor and Melcher [15,16] developed a theoretical model to estimate the small deformation for leaky dielectric systems (conductive or dielectric droplet in an insulating continuous medium). Leaky dielectric solutions, such as oils, polymers, electrolytes saline solutions and water are solutions that possess both dielectric properties and conductive properties [17]. Droplets were found to be more deformed than would be expected from the theoretical prediction, in many cases by as much as a factor of two to four. Ajayi improved the Taylor and Melcher model (presented later in section 2.2.3) to estimate higher deformations by extending the linearized model of Taylor to second degree polynomial [18]. Although Ajayi's more accurate theory indicates an increase in drop deformations, the higher-order correction was found to be insufficient to fully remove the discrepancy between experimental and theoretical deformations.

However, despite all the progress made in this field, little attention
has been given to studying the behavior of a conductive stable pendant
droplet in air medium under electric field, especially in drying scenarios
where the droplet is pendant in controlled air environment (tempera-
ture, humidity, airflow) rather than in another immiscible fluid such as
oil. This scenario helps to understand and optimize the electrostatic
assisted drying process, where charged droplets are in contact with
drying air. In this scenario, droplets may undergo different deformation
dynamics, charge distribution or drying patterns due to the absence of
the viscous host medium or the hydrophobic substrate. Authors in
Ref. [19] studied the effect of the electric field on the natural evapora-
tion of static pendant saline droplets as well as the effect on internal
advection but the droplet deformation aspect was not presented. Addi-
tionally, their system was limited to evaporation without the possibility
to control the drying air medium and the application of the electric field
intensities was limited to around 67 kV/m.

This paper aims to investigate the electrohydrodynamic behavior of
a static conductive pendant droplet suspended in air under the influence
of an electric field. A new setup is developed, enabling the application of
an electric field to a pendant droplet throughout the drying while
simultaneously tracking its deformation and evaporation using image
processing techniques. This work focuses on two main areas: deforma-
tion dynamics (without drying) with the application of the electric field
and the deformation kinetics of the charged droplet during drying.

## 2. Materials and methods

### 2.1. CuSO₄·5H₂O aqueous solutions preparation

The composition and properties of the solution determines the droplet's response to the applied electric field. Ionic species align and migrate under the influence of the electric field [20], while dipoles polarize in response to the field's application [21,22]. Furthermore, droplet behavior is linked to the properties of the ionic species, including

their molecular weight, valence, and solvation characteristics [20]. Saline solutions are interesting model systems due to their dual nature: they contain both free ionic species and polar molecules. This combination allows for a complex response to electric fields, where ionic species migrate and align according to the field, and dipoles undergo polarization. These solutions allow investigating the complex relationship between electric field properties, solution properties, and environmental conditions. Through the variation of solid content concentration in these solutions, it is possible to modulate the electro-physical properties of the solution, specifically its conductivity and permittivity.

In this study the CuSO₄·5H₂O solution was selected based on previous findings of authors in Ref. [19] on the influence of the EF on the interfacial tension and evaporation kinetics of pendant saline droplets. Copper sulfate pentahydrate CuSO₄·5H₂O salt (purity higher than 98 % in mass) was purchased from Sigma Aldrich and ultrapure water was collected using Purelab purification system. A broad range of electro physical chemical properties was investigated through the manipulation of the solid content in the solution. The systems tested have a saturation index (defined as the ratio of the solid content to the saturation solid content) ranging between 20 and 95 % which is equivalent to a solid content ranging between 6 and 25 % wt/wt. The aqueous solutions were prepared by dissolving weighted amounts of CuSO₄·5H₂O salt into ultrapure water at room temperature and stirring the solution until complete dissolution. The properties of the solutions were characterized as follows:

The electrical conductivity (σ) was measured using a conductivity
meter (CDM210, MeterLab™, from Radiometer Analytical). The vis-
cosity (μ) was measured using a viscometer (DV- E viscometer). The
dielectric permittivity values ε<sub>r</sub> of CuSO<sub>4</sub>·5H<sub>2</sub>O aqueous solutions at
25 °C were obtained from literature [23]. Density (ρ) of the different
solutions were obtained by generating a droplet of 5 μL using a micro-
pipette (model 0.5 ± 0.02 μL/10 ± 0.05 μL, Ergonomic
High-Performance Pipettor, VWR International, USA) and positioned on
a precision microbalance (Ohaus Pioneer PX series PX125D,
Switzerland). Properties of the tested solutions are presented in Table 1.

## 2.2. Experimental setup and procedure

### 2.2.1. Pendant droplet setup

The experimental setup was developed in-house and designed to track morphological evolution of a pendant droplet drying in controlled conditions. The setup, presented in Fig. 1, allows controlling the drying operating conditions, such as the drying air temperature, humidity, flow and drying experiment time, by the means of a LabVIEW software, where setpoints can be specified: Airflow (0.01 m s⁻¹ to 0.47 m s⁻¹), drying air temperature (20 °C–140 °C), absolute humidity of drying air (0.01 g kg⁻¹ to 10 g kg⁻¹) and the desired experiment time. Once these setpoints are specified, the system starts preheating. The airflow is conditioned by dehumidifying it using a silica gel column then mixing dry and water saturated air in a mixing cylinder to achieve a specific absolute humidity level. A hygrometer (Model HC2-SM stainless steel, Michell Instruments, France) is placed at the end of the mixing cylinder to measure the air temperature and relative humidity. The air is then heated to the desired temperature using a heating element (Model ASI, Chromalox, Pennsylvania, U.S.A.). A thermocouple (Type K) is placed at the exit of this zone to control the drying air temperature.

To generate the electric field around the pendant droplet, a grounded metallic needle is used to suspend the droplet. A cylindrical electrode (diameter d = 0.3 cm, height L = 1 cm) is placed at 1 cm below the needle, as presented in Fig. 1. The cylindrical electrode is connected to a high voltage generator (HVG-Dual Polarity-870,513B from Fluid Air), allowing to control the applied high voltage (0 to ±35 kV). The needle is grounded and Teflon coated to minimize its influence on the droplet's charge distribution.

The whole system is housed in an L shaped drying chamber, which is
<table><caption>**Table 1**

Physical-chemical properties of the CuSO₄·5H₂O solutions at 25 °C used in the study.</caption><thead><tr><th>CuSO<sub>4</sub>·5H<sub>2</sub>O solid content (%wt/wt)</th><th>Saturation index (%)</th><th>Conductivity σ (S·m<sup>-1</sup>)</th><th>Relative permittivity ε<sub>r</sub> [23]</th><th>Density ρ (kg·m<sup>-3</sup>)</th><th>Viscosity μ (x 10<sup>-3</sup> Pa s)</th></tr></thead><tbody><tr><td>0</td><td>0</td><td>0.00077</td><td>77.5</td><td>970</td><td>0.9</td></tr><tr><td>6</td><td>23</td><td>1.79</td><td>88</td><td>1030</td><td>1.0</td></tr><tr><td>15</td><td>55</td><td>3.53</td><td>86.4</td><td>1050</td><td>1.2</td></tr><tr><td>19</td><td>73</td><td>4.29</td><td>81</td><td>1066</td><td>1.5</td></tr><tr><td>25</td><td>95</td><td>5.11</td><td>78.7</td><td>1086</td><td>2.0</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Setup of the pendant droplet drying in the presence of an electric field, composed of the following components: A LED light, the isolated drying chamber, a high-speed camera, a heater for drying air and a suspension zone. The suspension zone of the droplet inside the drying chamber is formed of a cylindrical electrode, connected to an HVG and a metallic grounded needle used to generate the EF.</figcaption></figure>

made of plexiglass for insulation purposes and to prevent the electrification of the system, as presented in Fig. 1. The suspension zone, formed of the metallic needle, is placed 30 cm away from the air inlet to minimize airflow disturbances. Two glass windows facing the needle are placed on both sides of the drying chamber, one providing access to manipulate the droplet, and another allowing image acquisition with the camera during drying.

The monochrome high-speed camera (MB133USB by Sentech America) is used to capture high resolution images of the droplet, enabling the analysis of morphological changes such as surface area, volume, interfacial tension, and deformation using a specified MATLAB code.

To ensure optimal image quality, a 3D mobile system is attached to the camera to allow precise positioning. Additionally, the glass window providing access to the suspension zone is covered with white paint for a clear background of the images. A LED light source is positioned behind the white background to ensure proper illumination for better contrast.

The first objective of this work is to study the behavior of a static pendant droplet in air medium. For these experiments, the pendant droplet system is used without starting the drying experiment nor controlling the droplet environment. For this purpose, the position of the camera is adjusted using the 3D mobile system as well as the LED lighting to obtain a net and clear image of the droplet. The high voltage generator is set at 0 kV and a white reference image of the empty metallic needle is captured. Next, a droplet of an initially theoretical volume of 5 µL is generated using the micropipette and deposited on the tip of the metallic needle. A photo of the droplet is captured using the high-speed camera then the electric field is applied. The ratio of applied voltage to the distance between the needle and the cylindrical electrode is used to designate the electric field intensity (E).

The electric field intensity is increased by 50 kV/m. The EF intensity value is kept constant for 15 s to allow the droplet to equilibrate. An image of the droplet is captured, and after 15 s the electric field is increased again. This cycle is repeated until the detachment of the droplet which indicates the maximum voltage that can be supported by the droplet.

In this study, both positive and negative polarity voltages are tested. To simplify the designation of the EF, the polarity of the EF will be denoted by ± followed by the applied intensity E. Three repetitions are performed for each solution and the average values are calculated. Average results are presented and discussed in the results section.

For drying experiments, the system is preheated, and the EF is applied. Diffusive drying (Peclet number inferior to 1) in a controlled environment is performed: airflow is fixed at 0.01 m/s (the minimum airflow that allows to control the air humidity in the system), temperature at 25 °C and absolute humidity at 0.04 ± 0.01 g water/kg dry air. For the EF generation, the distance between the electrodes is fixed at 1 cm and the voltage setpoints on the HVG were fixed at 0, -4 kV and -8 kV corresponding to EF of 0, -400 and -800 kV/m (the choice of these electric field intensities is explained in the results section).

Once preheating is completed, a 5 µL droplet is suspended on the needle by using the micropipette.

Air is directed into the drying chamber through a flow straightener to ensure uniformity. The airflow initiates the drying process, and the camera captures images of the droplet every 5 s until the end of the experiment. Each drying experiment is performed in triplicates and average values are presented.

### 2.2.2. Physical properties calculation

The different steps used to calculate the physical properties of the droplet, such as the interfacial tension, volume, surface and diameter, are described in this section. The calculation of these properties requires analyzing the images of the pendant droplet. It also requires a model describing the contour of the droplet.

#### 2.2.2.1. Model describing the droplet profile at equilibrium.
The properties are determined by using a pendant droplet technique, which involves obtaining the contour of a pendant droplet (presented in Fig. 2) through image analysis and by applying numerical equations, described as follows:

First, the pendant droplet is supposed to be in thermodynamical
<figure><img src="image_4.png" /><figcaption>Fig. 2. Geometry of a pendant droplet representing the different parameters used to calculate the interfacial tension: example for a Bond number equal to 0.25.</figcaption></figure>

equilibrium with the surrounding medium (air). The droplet is therefore both static and axisymmetric. Additionally, it is assumed that the droplet is sufficiently large so that the thickness of the interface can be neglected.

Under these two assumptions, the mechanical equilibrium of the droplet is governed by the Laplace equation: While crossing the droplet interface from the outside, the pressure increases by $2\gamma H$ and thus at a point $M(r, z)$, the pressure difference between the internal and external environments is expressed in Equation (2):

$$P_{\text{int}} - P_{\text{ext}} = 2\gamma H_M \quad (2)$$

With $P_{\text{int}}$ and $P_{\text{ext}}$ the internal and external pressure at a point $M$ respectively, $\gamma$ is the interfacial tension.

$H_M$, presented in Equation (3), is the average curvature at $M$, defined as the average value of the two main curvatures $\frac{1}{r_1}$ and $\frac{1}{r_2}$, corresponding to the smallest and biggest oscillating circles at point $M$, respectively $C_{r_1}$ and $C_{r_2}$ as shown in Fig. 2.

$$H_M = \frac{1}{2} \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \quad (3)$$

Within the static droplet, since it is in equilibrium and only subjected to gravitational forces, the internal pressure at any point $M(r, z)$ depends on the pressure at the apex $A(0, 0)$ of the droplet, following Equation (4):

$$P(z) = P_A - \rho gz \quad (4)$$

$P_A$ is the pressure at the apex, $\rho$ is the droplet density, $g$ is the gravitational acceleration and $z$ is the vertical position of point $M$.

The external pressure at this point is equal to the atmospheric pressure. This leads to the following Equation (5):

$$\gamma \left( \frac{1}{r_1} + \frac{1}{r_2} \right) = \Delta P_0 - \Delta \rho gz \quad (5)$$

Using curvilinear abscissa and the variables $(r, z, \Phi)$, defined in Fig. 2, Equation (5) is transformed into a system of ordinary differential equations (ODEs), presented in Equations (6)–(8).

$$\frac{dr}{ds} = \cos \Phi \quad (6)$$

$$\frac{dz}{ds} = \sin \Phi \quad (7)$$

$$\frac{d\Phi}{ds} = \frac{2}{r_A} - \frac{\sin \Phi}{r} - \frac{\Delta \rho gz}{\gamma} \quad (8)$$

With origin set at the apex ($s=0$) where $(r, z, \Phi) = (0,0,0)$.

For numerical purposes, it is of interest to nondimensionalize the system using the curvature radius at the apex $r_A$, as the characteristic length. The resulting nondimensionalized variables are noted $r^*$, $s^*$ and $z^*$. The nondimensionalized system is then represented using Equations (9)–(11):

$$\frac{dr^*}{ds^*} = \cos \Phi \quad (9)$$

$$\frac{dz^*}{ds^*} = \sin \Phi \quad (10)$$

$$\frac{d\Phi}{ds^*} = 2 - \frac{\sin \Phi}{r^*} - B_{O_A} z^* \quad (11)$$

With $B_{O_A}$ is the Bond number at the apex, is the ratio between the gravitational forces and capillary forces and is defined in Equation (12):

$$B_{O_A} = \frac{\Delta \rho gr_A^2}{\gamma} \quad (12)$$

The resolution of the system presented in Equations (9)–(11) gives the geometrical contour of the pendant droplet for a given Bond number. The obtained value from resolving the system is fitted to the contour obtained with the experimental one obtained by image processing. The procedure will be described hereafter:

A MATLAB code was developed to analyze the morphological changes of pendant droplets during drying. The process involves several essential steps of image and numerical processing: 1- Image processing; 2- Droplet isolation and processing; - Binarization and contour identification; 4- FFT and noise suppression; 5- Apex, axis and major curvature radius determination, 6- Determination of surface and volume of the drop using Guldin theorem, 7- Interfacial calculation coupling ODEs solving and non-linear least square minimization based on the experimental drop contour.

Firstly, the external droplet contour is extracted by a segmentation binarization transform of the experimental droplet image presented in Fig. 3(a) and (b). This contour is smoothed using a Fast Fourier transform (FFT) which leads to a reduction of the "noise" mostly inherent to the camera resolution. This step is critical to ensure a good convergence of subsequent optimization and avoid the problem due to the presence of multiple local minima when minimizing the distance between models and contour coordinates. The second step concerns the detection of the apex position and the axial revolution of the drop. Usually, the camera is assumed to be perfectly aligned on a vertical axis. Unfortunately, even a faint inclination of the vertical axis leads to important deviation value of the tension. A good and simple technique of axis and apex detection is given by using a numerical folding of the contour around the axis and minimizing the distance between the two semi-contours by acting on the two parameters of the axis. The droplet mechanical equilibrium leads to a perfect overlap and the apex and axes positions obtained permits to straighten the contour (Fig. 3(c)). The last image processing step before calculation is the determination of the curvature radius at the apex which is easier with the FFT noise suppression. It is to be noticed that most of this operation is numerically simplified by working with polar coordinates to obtain monotonical behavior of one of the new coordinates - the angle.

#### 2.2.2.2. Properties calculations.
Now a minimization of the distance between the drop contour and the modeled contour is done by adjusting the Bond value (hence the interfacial tension). In each iteration of this optimization the ODE model presented in Equations (9)–(11) must be solved: The droplet profile is obtained by solving the system of non-dimensionalized differential equations using the "ode45" MATLAB
<figure><img src="image_5.png" /><figcaption>Fig. 3. Output of each step of the MATLAB code for image processing: (a) Reading of the droplet image, (b) Droplet isolation from the needle, (c) Identification of the contour, (d) Morphological properties calculation and (e) Interfacial tension calculation.</figcaption></figure>

function. Once the initial solution is computed, the next step involves refining the parameters to improve the agreement between the theoretical and experimental droplet contours. To achieve this, the optimization process utilizes the “fminsearch” MATLAB function. This function is employed to minimize the discrepancies between the theoretical predictions and the experimentally observed droplet contour by adjusting the Bond number.

Once the optimization converges, the final Bond number is obtained, and the interfacial tension can be calculated using Equation (12). This approach ensures that the droplet’s contour is captured with high precision, accounting for the influence of gravity and surface tension forces, while also considering the accuracy of the initial assumptions regarding thermodynamic equilibrium and axis symmetry. Finally, the volume and the surface area of the droplet are calculated, in pixels, using the drop axial symmetric property and the Guldin theorem. Additionally, by identifying the maximum and minimum ‘x’ and ‘y’ coordinates on the contour data, the horizontal and vertical diameters, in pixels, of the droplet can be calculated (Fig. 3(d)). All these physical properties are then converted from pixels to cubic and square meters units using the resolution factor calculated previously.

For image processing to be effective, it is critical that the droplet remains axisymmetric. This restricts the number of frames that can be processed for a drying experiment. As a result, only the images where the droplet remains axis symmetrical, particularly during the initial stages of drying are analyzed and discussed hereafter. Consequently, for drying experiments, the results are reported up to the point where the droplet maintains its axis symmetry, referred to as $t_{video}$ (the duration of the experiment during which the camera captures images of an axisymmetric droplet).

Furthermore, it should be noted that the first image is captured 5 s before the start of the drying experiment. Using the Fick’s law for diffusion, to calculate the evolution of a droplet during diffusive drying, it could be remarked that during the first 5 s before the capture of the first image, the droplet loses less than 1.6 % of its volume due to evaporation. This loss is minimal and does not significantly impact the overall drying kinetics analysis. This estimation has been calculated under the specific experimental conditions of diffusive drying (airflow

around 0.01 m/s, setpoint drying temperature of 25 °C, and absolute humidity set to 0 g/kg dry air).

### 2.2.3. Non dimensional analysis

Once images of the droplet are processed, the deformation factor of the droplet can be calculated using Equation (13):

$$De = \frac{a - b}{a + b} \qquad (13)$$

where a and b are respectively the vertical (same direction of the applied electric field) and the horizontal diameters of the droplet, as presented in Fig. 3(d).

To quantify the deformation dynamics of the pendant droplet in air as a function of the forces acting on it, the non-dimensional bond number $Bo$ and the electrical capillary number $Ca_E$ are calculated. The Bond number $Bo$, defined in Equation (14), is the ratio of gravitational force to capillary force:

$$Bo = \frac{\Delta \rho gr^2}{\gamma} \qquad (14)$$

With $\Delta \rho$ (kg.m$^{-3}$) is the difference of density between the droplet and the continuous medium, g is gravitational acceleration (m/s$^2$), r is the horizontal radius of the droplet (m) and $\gamma$ is the interfacial tension (N/m). A $Bo$ inferior to 1 suggests that the interfacial tension dominates over gravitational forces and consequently a more spherical droplet is obtained.

Electric capillary number $Ca_E$, defined in Equation (15), is the ratio of the electrical force to the capillary force acting on the droplet:

$$Ca_E = \frac{\epsilon_c \epsilon_0 E^2 r}{\gamma} \qquad (15)$$

Where $\epsilon_c$ is relative permittivity of the surrounding medium (air in the case of this study), $\epsilon_0$ is the dielectric permittivity of vacuum ($\epsilon_0 = 8.854 \times 10^{-12} F m^{-1}$), E is the intensity of the applied electric field, r is the horizontal radius of the droplet (m) and $\gamma$ is the interfacial tension of the droplet (N/m).
**Electrohydrodynamic models.** Electrohydrodynamic models used to simulate the influence of the applied electric field on the deformation of a pendant droplet in an immiscible viscous fluid exists in literature.

Taylor and Melcher model, described in Equation (16) was initially established for leaky dielectric droplet; However, it can also be applied for conductive droplets suspended in a perfectly insulating medium when the deformation is smaller than 0.02 [24,25]. It is noteworthy that this model has been mainly developed for scenarios where the droplet and the continuous medium have comparable viscosities.

$$De = KCa_E \text{ With } K = \frac{9}{16(2R' + 1)^2} \left[ (1 + R'^2 - 2QR'^2) + 3R'(1 - QR') \frac{2 + 3\lambda}{5 + 5\lambda} \right] \quad (16)$$

Where $R' = \frac{\sigma_c}{\sigma_d}$, $Q = \frac{\epsilon_d}{\epsilon_c}$, and $\lambda = \frac{\mu_d}{\mu_c}$ denote respectively the ratios of resistivity, permittivity and viscosity of the drop to those of the continuous phase (air in the actual study).

However, to apply the model for conductive droplets of higher deformation ($De < 0.1$), Ajayi improved Taylor and Melcher model to a second-degree polynomial model. The Ajayi model is presented in Equation (17):

$$De = K_1 Ca_E^2 + K_2 Ca_E \quad (17)$$

$$\text{With } K_1 = \frac{9}{16} \frac{F_d(R', Q, \lambda)}{(1 + 2R')^2} \\ \text{and } K_2 = \frac{K_1}{(1 + 2R')^2} \left[ \left( \frac{9}{5} \frac{1 - R'}{1 + 2R'} - \frac{1}{16} \right) F_d + R'(1 - R'Q)\beta(\lambda) \right] \\ F_d(R', Q, \lambda) = (1 - R')^2 + R'(1 - R'Q) \left( 2 + \frac{3}{5} \frac{2 + 3\lambda}{1 + \lambda} \right) \\ \beta(\lambda) = \frac{23}{20} - \frac{139}{210} \frac{(1 - \lambda)}{(1 + \lambda)} - \frac{27}{700} \frac{(1 - \lambda)}{\lambda} \quad (2)$$

Where $R' = \frac{\sigma_c}{\sigma_d}$, $Q = \frac{\epsilon_d}{\epsilon_c}$, and $\lambda = \frac{\mu_d}{\mu_c}$ denote respectively the ratios of resistivity, permittivity and viscosity of the drop to those of the continuous phase.

The deformation is usually categorized in small deformation ($De < 0.02$) and large deformation ($0.02 < De < 0.1$) [24,25]. Small deformations can be estimated using the first order Taylor and Melcher model while the improved second order Ajayi model is usually used to estimate large deformation.

Both theoretical models are tested and compared to experimental results in the following section. However, it is important to note that in the actual setup, the droplet is attached to a needle, which can introduce a mechanical force exerted by the needle that can deform the droplet. The influence of this force is not accounted in the Taylor and Melcher and Ajayi models. Thus, these models can be applied after validating the assumption that, in the absence of the field, the metallic needle has no influence on the droplet shape.

### 3. Results

#### 3.1. Electric field effect on droplet properties

##### 3.1.1. Interfacial tension

The effect of the electric field intensity on the interfacial tension of ultrapure water and saline solutions is presented in Fig. 4. At 0 kV/m, the measured interfacial tension of ultrapure water is in accordance to literature values (72.28 mN/m) [26]. Still at 0 kV/m, the introduction of the CuSO₄·5H₂O salt into the solution increases the interfacial tension, with solutions containing 6 % and 2 5 % solids by weight exhibiting interfacial tension values of 72 mN/m and 91 mN/m respectively. This trend is consistent with the established effect of inorganic salts, which

<figure><img src="image_6.png" /><figcaption>Fig. 4. Interfacial tension of the pendant droplets as a function of the electric field intensity, for ultrapure water and CuSO₄·5H₂O solutions at 6, 15, 19 and 25 % wt/wt. (o) are the average value of the experimental data obtained at both positive and negative EF polarities and (■) is the quadratic fit of the equation $\gamma = \gamma_0 - \delta E^2$. The error bars are plus or minus one the standard deviation.</figcaption></figure>

increases the interfacial tension values by driving solvated ions away from the interface to the droplet’s bulk [27].

As shown in Fig. 4, applying an EF reduces the interfacial tension for all solutions. Similar decreasing trend of the interfacial tension with the application of the electric field for saline solutions and especially the CuSO₄·5H₂O was reported in literature [19]. Additionally, Fig. 4 shows that as the absolute value of the EF intensity increases, the interfacial tension slowly decreases, then the reduction rate increases and the interfacial tension drops to 40 mN/m for ultrapure water at 700 kV/m which corresponds to a reduction of 42 % compared with 0 kV/m. For the different CuSO₄·5H₂O solutions, the interfacial tension values converge to approximately 50 mN/m at high EF intensities suggesting that at these conditions, the impact of varying salt concentrations on interfacial tension is erased compared to the effect of the EF. Furthermore, the standard deviation between the positive and negative polarities remains below 3 %, indicating consistent interfacial tension results under both positive and negative EF polarities. This suggests that the electrical forces acting on the droplet are of similar magnitude regardless of polarity.

By fitting the experimental results in Fig. 4, a quadratic fit relationship between the interfacial tension and the applied electric field intensity (E) is found to represent the reduction of interfacial tension with the intensity (E) of the applied field, as described in Equation (18):

$$\gamma = \gamma_0 - \delta E^2 \quad (18)$$

Where $\gamma_0$ is the interfacial tension of the droplet at no-field conditions. The parameters obtained from this quadratic fit are presented in Table 2. For all the tested solutions, the R² are higher than 0.93, indicating a strong correlation between the experimental results and the quadratic

<caption>Table 2</caption>

Quadratic fit parameters for the interfacial tension versus the absolute value of the intensity of the EF (E) for different solid content concentrations.

<table><thead><tr><th>CuSO<sub>4</sub>·5H<sub>2</sub>O solid content (%wt/wt)</th><th>γ<sub>0</sub> (10<sup>-3</sup> N·m<sup>-1</sup>)</th><th>δ (10<sup>-14</sup> N m·V<sup>-2</sup>)</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>0</td><td>67.94</td><td>5.54</td><td>0.935</td></tr><tr><td>6</td><td>72.32</td><td>4.34</td><td>0.998</td></tr><tr><td>15</td><td>80.81</td><td>6.62</td><td>0.991</td></tr><tr><td>19</td><td>84.51</td><td>6.86</td><td>0.995</td></tr><tr><td>25</td><td>91.73</td><td>8.63</td><td>0.969</td></tr></tbody></table>
model. Similarly, authors in Ref. [28] proposed a quadratic model to estimate the reduction in interfacial tension, where a reduction factor $γ_E$ was found to be proportional to the square of the applied EF intensity.

The parameter $δ$, which represents the reduction in interfacial tension due to the applied EF, varies depending on the ion concentration in the solution. $δ$ is $5.54 \times 10^{-14}$ N m V$^{-2}$ for ultrapure water, and this value increases at higher solid content, reaching $8.63 \times 10^{-14}$ N m V$^{-2}$ for solution at 25 % wt/wt. The increase of solid content leads to an increase in ions thus increasing the surface charge and consequently leading to a more pronounced reduction in interfacial tension. However, the solution at 6 % wt/wt solids shows a notable exception to this trend, where $δ$ seems to be smaller than the value obtained for ultrapure water. This special behavior at 6 % wt/wt may be explained by the high permittivity of the solution which leads to an important polarization of water dipoles compared to the other solutions. Polarization tends to reorient and align the water dipoles in the direction of the electric field [22]. On the basis of the trends observed, it can be hypothesized that polarization in this case strengthens the hydrogen bonding, which may consequently explain the smaller reduction in the interfacial tension.

### 3.1.2. Morphological evolution

Experimental captured images show a deformation of the droplet with the application of the electric field. Examples for ultrapure water droplet, CuSO₄.5H₂O aqueous droplet at 6 % and 25 % wt/wt are presented in Fig. 5. It can be visually observed that as the electric field increases, the droplet is elongated in the direction of the electric field and toward the charged electrode. To quantify the effect of this deformation, the results of the surface variation and deformation factor will be discussed in this section.

Fig. 6 shows the variation in the relative surface area $S/S_0$ of a pendant CuSO₄.5H₂O aqueous droplet under an applied electric field EF of both negative and positive polarities at various solid content concentrations. $S_0$ is the surface area of the droplet without the electric field application (no-field condition).

Two different behaviors of surface evolution can be observed depending on the intensity of the applied EF, thus indicating the presence of an EF intensity threshold delimiting them:

At low absolute values of EF intensities, $S/S_0$ is less than 1, indicating contraction for all concentrations, particularly pronounced at lower solid content (0 and 6 % wt/wt). At small EF intensities the rate of interfacial tension reduction is slow. This observation hypotheses that the electrical forces generated are insufficient to induce significant deformation in the droplet. Thus, the capillary forces are dominant over electrical forces at low EF intensities. However, to achieve a new equilibrium state, the droplet undergoes a slight shrinkage, reducing its surface area. This may result in a surface area ratio ($S/S_0$) of less than 1.

As the EF intensity increases beyond this threshold, $S/S_0$ begins to increase, indicating surface area expansion. This is primarily due to the increase of the electrical forces induced on the droplet interface and counteracting the interfacial tension, leading the droplet to elongate in the direction of the EF. Interestingly, it is remarked that the threshold value decreases with the increase of solid content. For example, for the 6 % wt/wt solution, the surface area reduction persists up to ±600 kV/m. While at higher solid content, the threshold value of the EF is around -350 kV/m for the droplet at 15 % wt/wt and around -250 kV/m for the droplet at 19 and 25 % wt/wt. At positive polarities, the threshold value of EF is around 500 kV/m for the droplet at 15 % wt/wt and around 400 kV/m for the droplet at 19 and 25 % wt/wt.

<figure><img src="image_7.png" /><figcaption>Fig. 5. High speed camera images captured for a 5 µL CuSO₄.5H₂O pendant droplet at ambient operating conditions and subjected to increasing high voltages: (a) Ultrapure water droplet, (b) CuSO₄.5H₂O solution at 6 % wt/wt solid, (c) CuSO₄.5H₂O solution at 25 % wt/wt.</figcaption></figure>
(a)

<figure><img src="image_8.png" /></figure>

(b)

<figure><img src="image_9.png" /><figcaption>Fig. 6. Evolution of the relative surface area (ratio of the surface to surface at no-field condition) with applied EF for CuSO₄.5H₂O droplet (5 µL) at various solid content (0, 6, 15, 19 and 25 % wt/wt) (a) for negative polarities, (b) for positive polarities.</figcaption></figure>

The existence of the threshold EF intensity value, differentiating between expansion and contraction of the droplet, was discussed in literature. Numerical studies found that the deformation of the droplet occurs after a certain voltage threshold, where electric forces become strong enough to overcome the capillary forces. For instance, authors in Ref. [29] found that a sessile water droplet significantly deforms only when an electric potential of 812 kV/m or higher is applied.

By comparing positive and negative polarities it can be remarked that the EF threshold values are higher, and expansion is less pronounced for positive polarities. For example, droplets at 19 and 25 % wt/wt exhibit an increase in surface area between 19 and 30 % when applying –800 kV/m compared to an increase of 10–19 % at +800 kV/m. This higher threshold and lower expansion when applying positive EF polarities may be attributed to the diffusion phenomena. When the counter electrode is charged positively, SO₄²⁻ ions are attracted toward the bottom of the droplet. Since the molecular weight of SO₄²⁻ (96.06 g mol⁻¹) is higher than that of Cu²⁺ (63.55 g/mol), this suggests a slower diffusion of these ions and consequently a slower response to the EF.

Results in Fig. 6 also indicate that the expansion of surface area at high electric field is influenced by the solution's solid content. While all CuSO₄.5H₂O solutions exhibit an increase compared to the no-field state, it is noteworthy that the expansion is generally remarked to be more pronounced with the increase of solid content, except for the solution at 6 % wt/wt CuSO₄.5H₂O, where the expansion appears to be less important than that observed for ultrapure water, regardless of EF polarity. For instance, at 800 kV/m, the surface area of a water droplet increases by 10 % for negative polarities and 6 % for positive polarities, while the droplet at 6 % wt/wt expands by only 5 % for both polarities. In contrast, droplets at 25 % wt/wt solids expand by around 30 % (negative polarity) and 18 % (positive polarity). Authors in Ref. [8] suggest that as the contrast in conductivity between the droplet and the environment increases, the accumulation of charge at the upper and lower portion of the droplet is more influenced and consequently the deformation is more pronounced. In the case of the 6 % wt/wt CuSO₄.5H₂O solution, the expansion is less pronounced compared to other concentrations and ultrapure water. At EF intensity (±800 kV/m), a minimal increase of the droplet surface at 6 % wt/wt is observed suggesting that the electrical forces acting on the droplet does not exhibit strong contraction or expansion due to the high permittivity. One possible interpretation of this observation is the enhancement of hydrogen bonding between water dipoles due to the polarization which may make the droplet less prone to deformation. However, to put in evidence the effect of the electric field on the hydrogen bonding numerical simulations are essential.

### 3.1.3. Pendant droplet deformation

Results of the pendant droplet deformation as a function of the

electric capillary number are presented in Fig. 7. Additionally, the Bond numbers value of the droplet at different electric field intensity are represented in Table 3.

As presented in Fig. 7(a) and (b), in the absence of the EF, droplets of all solid content concentration have De around 0 meaning that the droplet is almost spherically shaped. Minor deviations may be remarked due to experimental errors with handling the droplet and hanging it on the metallic needle as can be remarked in the zoomed-in graph in Fig. 7 (a). At no field condition, calculated bond number Bo, presented in Table 3, are found inferior to 1. For instance, for ultrapure water Bo = 0.3 while for solution at 25 % wt/wt Bo = 0.26, which explains the dominance of interfacial tension and the observed near-spherical form.

When applying for an EF, the Bond number remains inferior to unity, indicating that the capillary forces overcome the gravitational forces. Additionally, it can be remarked from Fig. 7 that with the increase of CaE, De increases, meaning that the droplet is elongated vertically, in a prolate shape and in the direction of the electric field. It is interesting to note that for ultrapure water and droplet at 6 % wt/wt, the deformation factor remains constant with the increase of CaE from 0 to 0.005 before starting to deform into prolate shape in the direction of the electric field. These observations assure that for these solutions, when the CaE is below 0.005 the geometrical shape of the droplet does not change even if it has been found that S/S₀ is less than 1, meaning that the droplet contracts while remaining spherical. This contraction may be due to the polarization and reorganization of water dipoles due to the permittivity values of these solutions and their low ion concentrations compared to solutions at higher solid content.

Additionally, as the CuSO₄.5H₂O solid content increases, a general trend of increasing deformation was observed, except for the solution at 6 % wt/wt. This trend is consistent with existing literature, where high concentrations are associated with increased conductivity, leading to greater ions in the solution and thus a greater charge accumulation at the droplet interface resulting in more pronounced droplet deformations at a given intensity of EF [30]. The special observation related to the small deformation of the solution at 6 % wt/wt (only 0.14) may be explained by the high permittivity value of this solution which consequently means a higher polarization of dipoles in the presence of an electric field.

It can be remarked from experimental results that for both polarities, ultrapure water can sustain high CaE before reaching De = 0.02 (categorized as small deformation). For solution at 6 % wt/wt, deformation remains below 0.02 up to high CaE values, where electric forces are important and pronounced compared to capillary number. For solution at higher solid content, De reaches 0.02 at low CaE values.

It can be remarked from Fig. 7(a) and (b), that Taylor and Melcher model, for small deformation (De < 0.02) does not accurately predict the compression of droplets of ultrapure water and the solution at 6 % wt/wt
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Deformation of the conductive CuSO₄·5H₂O pendant droplet at various solid content (0, 6, 15, 19 and 25 % wt/wt) as a function of the electrical capillary number at negative ((a) and (c)) and positive polarities ((b) and (d)). (o) are the experimental results. In (a) and (b): (-) is the theoretical Taylor and Melcher model. In (c) and (d): (.) is the Ajayi model. In both figures, (-.) is a second-degree polynomial regression.</figcaption></figure>

<table><caption>**Table 3**
Bond number (Bo), calculated for the pendant droplets at different solid content and electric field intensity conditions.</caption><thead><tr><th rowspan="2">|E| (kV/m)</th><th colspan="5">CuSO<sub>4</sub>·5H<sub>2</sub>O solid content (% wt/wt)</th></tr><tr><th>0</th><th>6</th><th>15</th><th>19</th><th>25</th></tr></thead><tbody><tr><td>0</td><td>0.32</td><td>0.32</td><td>0.29</td><td>0.28</td><td>0.27</td></tr><tr><td>100</td><td>0.34</td><td>0.33</td><td>0.29</td><td>0.28</td><td>0.27</td></tr><tr><td>200</td><td>0.34</td><td>0.33</td><td>0.30</td><td>0.30</td><td>0.28</td></tr><tr><td>300</td><td>0.36</td><td>0.34</td><td>0.31</td><td>0.31</td><td>0.30</td></tr><tr><td>400</td><td>0.40</td><td>0.36</td><td>0.34</td><td>0.33</td><td>0.33</td></tr><tr><td>500</td><td>0.43</td><td>0.37</td><td>0.37</td><td>0.37</td><td>0.37</td></tr><tr><td>600</td><td>0.45</td><td>0.41</td><td>0.41</td><td>0.42</td><td>0.43</td></tr><tr><td>700</td><td>0.51</td><td>0.46</td><td>0.47</td><td>0.48</td><td>0.49</td></tr></tbody></table>

CuSO₄·5H₂O. However, once the droplet starts elongating in the direction of the electric field, the model shows better agreement with experimental observations. Fig. 7(c) and (d) also shows discrepancies between the Ajayi's theoretical deformation and experimental values (for solution where De < 0.1).

This suggests that, for saline solutions pendant in air medium where (R > 10¹¹) and λ ≥ 50, the droplet's behavior under the EF becomes more complex and the deformation more pronounced than the Ajayi model can capture. These discrepancies were also remarked by Ref. [8]. By performing numerical simulations, they demonstrated that the deformation of a droplet under the conditions of λ = 2 and R = 2.5, Taylor and Melcher model and Ajayi model deviate from the experimental values especially at high magnitude of CaE (CaE higher than 0.1 and deformation higher than 0.04).

This study involves saline solutions pendant in an air medium where λ is significantly higher (λ ≥ 50) compared to previous studies. In earlier

researches, droplet where typically placed in a viscous medium with λ around 1–2 [8]. This suggests that the Ajayi model fails to estimate the deformation of the droplet in the actual scenario where λ is high. Furthermore, the high value of R (R > 10¹¹) may contribute to additional discrepancies.

A second-degree polynomial regression was conducted to identify a suitable empirical model for the experimental results. The fitting results are summarized in Tables 4 and 5. This empirical model captures experimental observations on droplet deformation under various electric field intensities (both positive and negative) and different solid content concentrations. The coefficients a and b indicate the notable influence of both quadratic and linear terms on the deformation, with their effects becoming more pronounced as the solid concentration increases. For instance, at negative polarities, for ultrapure water, a = 4.69 and b = 0.16. When the concentration increases to 25 % wt/wt, the coefficient 'a' significantly increases to 12.49 and b to 1.34. The high R² values and low RMSE values confirm the precision of the proposed fitting model in predicting the deformation behavior of CuSO₄·5H₂O droplets statically pendant in air and under an electric field.

A second order empirical model is applied to estimate the

<table><caption>**Table 4**
Parameters for second degree polynomial De = aCaE²+bCaE at various solid content (0, 6, 15, 19 and 25 % wt/wt) and negative EF polarities.</caption><thead><tr><th>Solid content (%wt/wt)</th><th>a</th><th>b</th><th>R<sup>2</sup></th><th>RMSE</th></tr></thead><tbody><tr><td>0</td><td>4.69</td><td>0.16</td><td>0.89</td><td>0.0055</td></tr><tr><td>6</td><td>5.22</td><td>0.01</td><td>0.87</td><td>0.0032</td></tr><tr><td>15</td><td>5.15</td><td>1.02</td><td>0.98</td><td>0.0029</td></tr><tr><td>19</td><td>6.75</td><td>1.29</td><td>0.99</td><td>0.0032</td></tr><tr><td>25</td><td>12.49</td><td>1.34</td><td>0.99</td><td>0.0022</td></tr></tbody></table>
<table><caption>**Table 5**
Parameters for second degree polynomial $De = aCa_E^2+bCa_E$ at various solid content (0, 6, 15, 19 and 25 % wt/wt) and positive EF polarities.</caption><thead><tr><th>Solid content (%wt/wt)</th><th>a</th><th>b</th><th>R<sup>2</sup></th><th>RMSE</th></tr></thead><tbody><tr><td>0</td><td>2.46</td><td>0.24</td><td>0.96</td><td>0.0019</td></tr><tr><td>6</td><td>4.74</td><td>0.05</td><td>0.92</td><td>0.0019</td></tr><tr><td>15</td><td>2.84</td><td>0.47</td><td>0.93</td><td>0.0033</td></tr><tr><td>19</td><td>3.52</td><td>0.69</td><td>0.97</td><td>0.0026</td></tr><tr><td>25</td><td>5.29</td><td>0.87</td><td>0.99</td><td>0.0023</td></tr></tbody></table>

deformation of the solution models tested in this study. However, further experiments are needed to develop a robust model for predicting droplet deformation in air under the effect of an electric field. It would be beneficial to conduct further tests to construct a deeper model potentially linking the deformation to the droplet properties, such as viscosity, conductivity and permittivity, as well as to the properties of the surrounding ambient air. Additionally, the suspension system properties and gravity must be considered since it plays a significant role in the deformation of pendant droplets in air. Unlike models in literature where droplets are pendant in a surrounding viscous liquid and gravity is negligible, here, it significantly influences the droplet behavior.

## 3.2. Electric field effect on the drying

Based on the deformation and surface results previously discussed, in order to study the effect of the electric field on the drying at different conditions, two EF intensities –400 kV/m (near threshold value of contraction or expansion) and –800 kV/m (where droplets can be highly deformed) in addition to the no-field (0 kV/m) conditions are tested. Results are presented in this section.

### 3.2.1. Deformation kinetics

Results of the deformation kinetics of a pendant CuSO₄.5H₂O droplet at 25 % and 6 % wt/wt solid content, drying in the presence of an electric field, in diffusive condition at 25 °C and 0.04 g/kg absolute humidity are presented in Fig. 8.

A general trend of decreasing deformation toward more negative values during drying can be remarked for all the drying conditions, regardless of its initial deformation state. This decreasing trend can be divided into two distinct phases: In the first phase, a constant deformation rate of the droplet is observed. De decreases to values below zero, indicating that the horizontal diameter of the droplet becomes larger than the vertical diameter. This suggests significant horizontal spreading of the droplet as it dries. While in the second phase, the rate of deformation increases. This behavior is notable in solutions at 6 % w/wt,

where it is possible to track the droplet deformation for a long period of time (droplet remains axis symmetrical) before the apparition of the first solid and the formation of the solid crust.

When subjecting the droplet to an electric field of –400 kV/m, deformation is more pronounced and tends toward more negative for the solution at 6 % wt/wt compared to the drying at 0 kV/m. While for the solution at 25 % wt/wt, applying –400 kV/m does not highly influence the deformation of the droplet especially during the early drying time. These findings are in accordance with the previous results presented in section 3.1 where it has been demonstrated that an electric field of –400 kV/m has a more pronounced effect of contracting droplets at 6 % wt/wt (which have high permittivity) compared to droplet at 25 % wt/wt where this voltage does not seem to influence the droplet morphology.

At –800 kV/m, the deformation at the beginning of drying ($t_{video} = 0$ s) is higher than zero, indicating that the droplet is initially elongated toward the counter electrode. However, for droplets at 25 % wt/wt solid content, the application of –800 kV/m appears to retard the deformation variation rate during the early stages of drying, especially the first 200 s of drying, before reaching deformation rates like those at the no-field conditions. This observation suggests that the electric field influences the early deformation dynamics by delaying the droplet’s response to drying. Similarly, studies in Ref. [19] showed that even though the application of the EF for CuSO₄.5H₂O droplets reduces the interfacial tension of the droplets, the vertical direction of the field reduces the internal advection and therefore reduces the rate of diameter reduction during natural evaporation. It is noteworthy that in this previous study authors focused their study on aqueous CuSO₄.5H₂O of a concentration of 0.1 M equivalent to 2.5 % wt/wt and an electric field intensity of maximum 62.5 kV/m which are inferior to the scope of values studied in this article.

### 3.2.2. Droplet shrinkage during drying process

Fig. 9 represents the volume evolution of the droplets drying under different electric field (EF) intensities for both solutions at 25 % and 6 % wt/wt. At no field conditions, the volume of both solutions decreases steadily over drying time, driven by evaporation, with the 6 % solution exhibiting a slower drying rate due to its lower initial solid content. This reduction rate decreases after a certain period, for instance for solution at 6 % wt/wt this reduction occurs at around 750 s.

At –800 kV/m, the initial volume of the droplets at both 25 % and 6 % wt/wt is increased because of the electric field effect on deforming the droplet. At this electric field condition, a more pronounced initial reduction in volume is observed compared to the no field condition. This is clearly observed for solution at 6 % wt/wt. However, as drying progresses and salt concentration increases, the differences between the

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Deformation profile as a function of the drying time for the CuSO₄.5H₂O droplet (25 % and 6 % wt/wt respectively) diffusive drying at controlled conditions with an absolute humidity of 0.04 g/kg and a drying temperature of 25 °C. Deformation kinetics are studied at different EF intensities: 0, –400 and –800 kV/m. The filling around the curves represents plus or minus one the standard deviation.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 9. Volume reduction as a function of the drying time for the CuSO₄·5H₂O droplet (25 % and 6 % wt/wt respectively) diffusive drying at controlled conditions with an absolute humidity of 0.04 g/kg and a drying temperature of 25 °C without electric field and under an EF of - 800 kV/m.</figcaption></figure>

–800 kV/m and no-field conditions diminishes and volume of the droplet converges to values like no-field indicating that as the drying progress the effect of the EF reduces. This convergence may be explained by the fact that as the droplet dries, the salt concentration increases and consequently the interfacial tension of the droplet increases, thus shadowing the effect of the electrical forces on the droplets. These observations highlight the influence of EF intensity on evaporation rates during the early drying stages, which diminishes over time as interfacial properties dominate.

## 4. Conclusions

This study provides a detailed examination of the deformation kinetics of conductive saline pendant droplets subjected to an electric field, which was made possible by a new device developed for this purpose. Key findings include the reduction of interfacial tension with increasing electric field intensity, particularly at high CuSO₄·5H₂O concentrations, where ion mobility and conductivity lead to pronounced deformation. By analyzing the shape of the pendant droplet and comparing the experimental profile to the theoretical model derived from the Young-Laplace equation, interfacial tension of the droplet was calculated. The results demonstrate a quadratic relationship between interfacial tension and applied EF. Additionally, it has been demonstrated that the application of an electric field deforms a saline droplet pendant in air. This deformation depends on both electric field properties and solution properties: It was found that increasing the solid content of the salt increases the ions present in the solution thus leading to a more pronounced deformation. However, an exceptional behavior was found for droplets at 6 % wt/wt and this may be explained by the high permittivity and low conductivity of the solution.

It was demonstrated as well that it exists a threshold EF intensity above which electric field becomes important and deform the droplet into a prolate shape. Non dimensional analysis was performed to study the relation between deformation and electric capillary number. However, discrepancies were observed between experimental results and existing theoretical models, such as Ajayi's, underscoring the complexity of droplet behavior in the presence of an electric field. The findings of this study demonstrate that a pendant saline droplet in an air medium, resulting in a high viscosity ratio between the droplet and its surroundings, fails the Ajayi model in predicting the deformation of the droplet. Using polynomial regression, this study suggests a second order model to estimate the deformation of the solution models tested in this study. However further experiments should be accomplished to delve deeper into developing a model to predict the deformation of a droplet in air.

Moreover, it was found that during the drying, the deformation

kinetics exhibited two distinct phases: an initial phase characterized by a constant deformation rate and a later phase of accelerated deformation, especially right before the first solid appearance. The application of an electric field, particularly at –800 kV/m, significantly influenced the deformation rate during the early stages of drying. These findings are primary results and indicate that the new experimental setup can successfully monitor the drying kinetics of concentrated saline droplets across different electric field intensities. The deformation curves offer important insights into droplet behavior during drying, facilitating further exploration of the electrohydrodynamic phenomena involved.

The volume evolution of droplets demonstrated that high EF intensities alter the evaporation during the initial stages of drying. At high intensities, the droplet volume shrinkage rate is higher compared to no field conditions. However, over time, the droplet volume converges to values like the no-field conditions, indicating a decreasing effect of the electric field on drying.

Finally, this study showcased the CuSO₄·5H₂O as a robust model system for investigating droplet behavior under electric fields. However, the exploration of alternative salts with differing molecular weights, valences, and solvation characteristics presents a valuable opportunity to deepen the understanding of the mechanisms driving droplet deformation and interfacial dynamics. Such studies could further refine the relationship between ionic properties and electric field interactions, advancing the broader knowledge in this domain.

Our future studies will extend to the introduction of a microbalance to the pendant droplet system enabling both mass loss and evaporative flux to be tracked in real-time. This would enable more precise control and prediction of drying kinetics, particularly under varying electric field intensities. Additionally, convective drying will be studied, as this can offer insights into how heat and mass transfer can enhance or limit deformation and evaporation. Such studies could potentially reveal new perspectives that could inform future applications in fields such as drying droplet-based technologies under electric field.

## CRediT authorship contribution statement

A. Dokmak: Writing – original draft, Methodology, Investigation, Conceptualization. T. Deleau: Writing – review & editing, Supervision, Software, Methodology, Investigation, Conceptualization. J.J. Letourneau: Writing – review & editing, Software, Methodology. A. Maudhuit: Writing – review & editing, Funding acquisition. P. Arlabosse: Writing – review & editing, Supervision, Conceptualization. M.I. Ré: Writing – review & editing, Supervision, Conceptualization.
Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

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Abbreviation

CuSO₄·5H₂O: Copper sulfate pentahydrate

EF: Electric field

Nomenclature

A: Apex

Bo: Bond number

Ca: Capillary number

s: Curvilinear abscissa m

De: Deformation

E: Electric field intensity kV/m

g: Gravitational gravity m.s⁻²

m: Mass g

s': Normalized s

Pe: Peclet number

P: Pressure Pa

r: Radius m

S: Surface m²

t: Time s

z: Vertical coordinate m

μ: Viscosity Pa·s

V: Volume m³

Greek letters

Φ: Angle rad

ρ: Density kg.m⁻³

σ: Electrical conductivity S.m⁻¹

γ: Interfacial tension N/m

e: Permittivity F.m⁻¹

ΔP₀: Pressure difference Pa

Subscripts

0: Vacuum or free space

A: Apex point

c: Continuous medium

d: Droplet

E: Electric

ext: Exterior

int: Interior

n: Normal

r: Relative

video: Recorded frames using camera