# Unraveling droplet drying kinetics under electric fields: A study on moisture content and evaporative flux profiles

A. Dokmak, T. Deleau, A. Maudhuit, P. Arlabosse, Maria-Inês Ré

► To cite this version:

A. Dokmak, T. Deleau, A. Maudhuit, P. Arlabosse, Maria-Inês Ré. Unraveling droplet drying kinetics under electric fields: A study on moisture content and evaporative flux profiles. Chemical Engineering Research and Design, 2025, 221, pp.176-186. 10.1016/j.cherd.2025.08.004. hal-05248454

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https://imt-mines-albi.hal.science/hal-05248454v1

Submitted on 10 Sep 2025

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# Unraveling droplet drying kinetics under electric fields: A study on moisture content and evaporative flux profiles

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

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

### Keywords:

Charged droplet

Drying kinetic

Evaporative flux

Moisture profile

Electric field effect

## ABSTRACT

This study investigates the influence of electric fields (EF) on the drying kinetics and evaporative flux of CuSO₄.5H₂O aqueous droplets at different solid contents. The drying behavior of pendant droplets was examined under both convective and diffusive conditions with varying EF intensities. The results indicate that the EF increases the surface area of the pendant droplet during both drying scenarios. In diffusive drying, the drying kinetic is different with and without EF. Indeed, the EF reduces moisture removal and evaporative flux in the early stages for highly concentrated droplets having high response to the electric field. However, as the droplet dries, the moisture removal rate and evaporative flux increase and converges to constant values, a trend observed under no-field conditions. In convective drying, the application of an electric field maintained a constant moisture removal rate until the droplet had lost 30 % of its water content, while in the no-field conditions this rate decreased up to 25 % as the drying progresses until the same water loss. Furthermore, when the electric field is applied, the evaporative flux increases with the drying time to reach constant values with the drying time. As a conclusion, this work uniquely combines pendant droplet drying under controlled drying conditions and high electric fields with a real time mass measurement and image-based surface area and deformation tracking, providing experimental access to electrohydrodynamic evaporation fluxes not previously reported in literature.

### Abbreviation

CuSO₄.5H₂O

EF

EHD

FFT

HVG

### Nomenclature

De

E

m

N

RH

S

T

t

v

X

Y

### Greek letters

γ

ε

Copper sulfate pentahydrate

Electric field

Electrohydrodynamic

Fast Fourier Transform

High voltage generator

Deformation

Electric field intensity

Mass

Evaporative flux

Relative humidity

Surface

Temperature

Time

Airflow

Moisture content (dry basis)

Absolute humidity

Interfacial tension

Permittivity

-

V/m

kg

g·s⁻¹·m⁻²

%

m²

(°C)

s

m/s

kg<sub>water</sub>/kg<sub>dry solid</sub>

g/kg dry air

N/m

F·m⁻¹

(continued on next column)

(continued)

<table><tr><td>ρ</td><td>Density</td><td>kg·m⁻³</td></tr><tr><td>μ</td><td>Viscosity</td><td>Pa·s</td></tr><tr><td>σ</td><td>Electrical conductivity</td><td>S·m⁻¹</td></tr><tr><td colspan="3"><strong>Subscripts</strong></td></tr><tr><td>0</td><td colspan="2">At the beginning of the experiment</td></tr><tr><td>crust</td><td colspan="2">Crust formation</td></tr><tr><td>exp</td><td colspan="2">Experimental balance time frame</td></tr><tr><td>fp</td><td colspan="2">Final particle</td></tr><tr><td>solid</td><td colspan="2">Appearance of the first solid</td></tr><tr><td>video</td><td colspan="2">Processed frames</td></tr></table>

## 1. Introduction

Advancements in drying processes have increasingly focused on incorporating electric fields (EF) into drying processes to enhance their efficiency. Among these innovations, electrostatic assisted spray drying has emerged as a promising technique (Jayaprakash et al., 2023). In this process, the solution to be dried is atomized using an

<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, Treillières 44119, France</footnote>

<footnote>* Corresponding author at: Université de Toulouse, Mines Albi, Centre RAPSODEE UMR CNRS 5302, Campus Jarlard, Albi, France
E-mail address: akaber.dokmak@spray.com (A. Dokmak).</footnote>
electrostatic-assisted bi-fluid nozzle, generating charged droplets entering in contact with hot and inert drying gas, usually nitrogen, within a drying chamber. Through heat and mass transfers, the droplets dry to form solid particles.

Experimental studies on the scale of the electrostatically assisted spray drying process have demonstrated the potential of this process to achieve drying at lower gas temperatures compared to conventional spray dryers (Beaumeux et al., 2024, Jayaprakash et al., 2023, Jayaprakash et al., 2023, Mutukuri et al., 2021, Wang et al., 2022). For example, authors in Mutukuri et al. (2021) showed that drying and encapsulation of monoclonal antibodies were achieved at an outlet temperature of 70 °C with the application of 5 kV using electrostatic assisted spray drying, whereas a higher temperature of 130 °C was required to obtain a similar product under no-field. Another example concerns the encapsulation efficiency of β-galactosidase enzyme using various carrier materials, comparing conventional spray drying with electrostatic assisted spray drying (Jayaprakash et al., 2023). The authors achieved drying at an outlet temperature of 44 °C with the application of 1-3 kV using electrostatic assisted spray drying, whereas conventional spray drying required a higher outlet temperature of 110 °C. However, the effect of the electric field on the drying mechanisms that leads to reducing the drying temperatures remains insufficiently understood. To understand these mechanisms, it is essential to establish and study the influence of the electric field on drying kinetics. These drying kinetics have been extensively investigated for conventional drying techniques by monitoring the evolution of the properties of drying droplets (Al Zaitone et al., 2020, de Souza Lima et al., 2020, Sadek et al., 2015). To establish such insights, it is common to scale down to the single droplet level, since tracking droplets drying in large scale spray dryers is challenging. By analyzing changes in droplet mass and morphology during drying, moisture removal rate (defined as the rate at which the droplet moisture content decreases over time) and water evaporative flux (reflecting the evaporation at the droplet interface) can be determined.

In conventional drying processes, drying can be divided in different periods (Boel et al., 2020, Tsotsas et al., 2000). The starting period is a warm or cooling period, where the droplet temperature evolves toward wet bulb temperatures. Once this equilibrium has been established, the process moves on to the first drying period, where the droplet surface remains entirely covered with solvent and surface evaporation dominates as the main drying mechanism. This phase is characterized by a constant evaporation flux.

As drying progresses, the droplet surface is no longer saturated with solvent, leading to the formation of a solid crust at the interface, which marks the onset of the second drying period also known as falling drying rate period. In this phase, drying occurs through water evaporation inside the droplet and diffusion toward the surface, resulting in a reducing evaporative flux (Fu et al., 2012, Porowska et al., 2016). Extending this understanding to electrically charged droplets allows to understand how the application of an electric field influences the drying process.

It is well established that previous studies have examined the behavior of electrically charged droplets during evaporation, either under ambient drying conditions or on heated substrates. These investigations have primarily focused on single droplet drying using sessile (Chen and Gan, 2024, Wang et al., 2024) or suspended configurations (Dhar et al., 2018, Jaiswal et al., 2020, Jaiswal, 2019), often incorporating thermal imaging, high-speed microscopy, or numerical modeling phenomena (Xu et al., 2021, Xu et al., 2024). However, much of this work has been limited to low electric field intensities, which are not representative of the field strengths used in industrial electrostatic-assisted drying systems. Furthermore, most studies involve sessile droplets placed directly on charged electrodes, where wetting dynamics and contact line behavior are the dominant physical (Vancauwenbergh et al., 2013). For instance, authors in Wang et al. (2024) observed a 32.9% reduction in evaporation time for a sessile water droplet under a 280 kV/m field due to droplet deformation and

increased contact with the heated substrate. Authors in Jaiswal (2019) experimentally demonstrated that applying a vertical EF (up to 62 kV/m) to pendant NaI and CuSO₄·5H₂O droplets, slows the diameter reduction during evaporation at ambient conditions. Same authors in Jaiswal et al. (2020) demonstrated that applying the field horizontally enhances the droplet diameter reduction.

However, to our knowledge, experimental work on pendant droplets suspended on grounded filaments under high voltage electric fields in air and under controlled drying conditions remains very limited. Moreover, existing studies focused on tracking morphological evolution and visually observing the lifetime of a naturally evaporating charged droplets, without a real-time mass tracking for such configurations. Consequently, none of the existing experimental setup focused on monitoring drying kinetics and establishing the evaporative fluxes of electrically charged droplets under controlled airflow conditions. These shortcomings underlie the need to carry out studies that evaluate simultaneously and directly moisture content, surface area, deformation and the and evaporative flux profiles of charged droplets drying under intense electric field intensities and controlled drying environment. The aim of our study is to make a scientific contribution to meeting this need by investigating the effects of the electric field on the drying kinetics of pendant droplets having different electrical and physical properties.

## 2. Materials and methods

### 2.1. Materials

Copper sulfate pentahydrate (CuSO₄·5H₂O) aqueous solutions at 6%, 19% and 25% wt/wt solid content were prepared by dissolving appropriate amounts of CuSO₄·5H₂O (purity > 98% from Sigma Aldrich) in ultrapure water (Purelab purification system) at room temperature, with stirring until complete dissolution. The properties of the solutions, including electrical conductivity, viscosity, dielectric permittivity, and density, were characterized as described in our previous work (Dokmak et al., 2025). Some of these properties are presented in Table 1.

### 2.2. Experimental setup

The used experimental setup is a homemade system designed to track the morphological evolution of a drying charged pendant droplet using a high speed camera (Dokmak et al., 2025). In the actual study, in order to track in situ the droplet's mass evolution, a precision microbalance is integrated to the system, as presented in Fig. 1 and Fig. 2 and described hereafter.

The droplet is suspended inside the suspension zone (Fig. 1(e)). It is suspended on a metallic needle, as presented in Fig. 2(a), positioned 30 cm away from the drying air inlet to minimize flow disturbances. The needle is grounded and Teflon coated to minimize its influence on the droplet's charge distribution. The suspension zone is housed in a L-shaped drying chamber made of plexiglass (Fig. 2(f)), which ensures electrical insulation. The metallic needle is hung from the bottom of a precision microbalance (Ohaus Pioneer PX series PX125D, Switzerland) (Fig. 1(c) and Fig. 2(e)), to provide real time measurements of the droplet's mass during drying. The precision microbalance has a

<table><caption>**Table 1**

Electrical and physical properties of the tested solutions CuSO₄·5H₂O aqueous solutions.</caption><thead><tr><td>Solid content (% wt.)</td><td>Electrical conductivity (mS/cm)</td><td>Relative permittivity ε<sub>r</sub> (Akilan et al., 2006)</td><td>Interfacial tension (mN/m) (Dokmak et al., 2025)</td></tr></thead><tbody><tr><td>6</td><td>1.79</td><td>88.0</td><td>72.32</td></tr><tr><td>19</td><td>4.29</td><td>81.0</td><td>84.51</td></tr><tr><td>25</td><td>5.11</td><td>78.7</td><td>91.73</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Fig. 1. The main sections of the homemade electrically charged single pendant droplet drying setup: (a) L shaped drying chamber, (b) high speed camera, (c) precision microbalance, (d) heating zone and (e) is the droplet suspension zone formed of a grounded metallic needle, cylindrical charged electrode and a thermocouple.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Drying setup of the pendant droplet in the presence of an electric field, composed of: (a) grounded metallic needle, (b) charged cylindrical electrode connected to an HVG, (c) high speed camera, (d) LED light, (e) precision microbalance, (f) plexiglass drying chamber, (g) dehumidification column, (h) air preparation zone, (i) air mixing zone and (j) heating zone.</figcaption></figure>

readability of 0.01 mg, repeatability of ± 0.02 mg, and linearity of ±0.1 mg.

The electric field used to charge the droplet is generated using a cylindrical electrode (diameter: 0.3 cm, height: 1 cm), placed 1 cm below the needle (Fig. 2(b)), and connected to a high voltage generator (from Fluid Air, USA) (0 to ± 35 kV). In this study, the electric field intensity (E) is designated as the ratio of the applied voltage to the

distance between electrodes.

The system also includes a high-speed monochrome camera (MB133USB by Sentech America) for capturing droplet morphology (Fig. 1(b) and Fig. 2(c)). A white painted background and LED light are placed facing the camera to optimize image contrast (Fig. 2(d)).

The simultaneous use of the precision microbalance and the camera allows to track both the mass and morphological evolution, which will
be used to measure the drying kinetics and consequently calculate the evaporative flux.

The setup enables to dry the suspended charged droplet using controlled airflow conditions (drying airflow: 0.01–0.47 m.s⁻¹, drying air temperature: 20–140 °C, and absolute humidity: 0.01–10 g.kg⁻¹). The air is dehumidified using a silica gel column (Fig. 2(g)), then conditioned and mixed with saturated air to reach the required absolute humidity and airflow (Fig. 2(h) and (i)). The air is then heated before entering the drying chamber (Fig. 1(d) and Fig. 2(j)).

The operating conditions tested in this study are presented in Table 2. Two drying modes are tested: diffusive and convective drying modes. In diffusive drying, a minimal airflow (v=0.01 m/s) is used, whereas in convective drying, a higher airflow (v=0.47 m/s) is used to enhance heat and mass transfer. To isolate the effects of airflow and the applied electric field intensity, the drying temperature was fixed at 25 °C in both cases, ensuring that any observed differences in drying behavior were not influenced by thermal variations. Additionally, the absolute humidity was maintained at 0 g/kg.

The tested electric field intensities were chosen based on previous study (Dokmak et al., 2025), where it was demonstrated that in the presence of an electric field, CuSO₄·5H₂O droplet deforms into a prolate shape (elongates in the direction of the electric field), as shown in Fig. 3. The deformation factor (De), calculated using Eq. (1), was used to classify the magnitude of deformation: small for De ≤ 0.02 and large for De > 0.02.

$$De = \frac{a - b}{a + b} \qquad (1)$$

Where a is the droplet diameter in the direction of the electric field and b is the droplet diameter perpendicular to the electric field.

The results of the previous study demonstrated that applying –400 kV/m for solutions at 6, 19 and 25 % wt/wt solution and –800 kV/m for solution at 6 % induces small deformations while applying –800 kV/m for droplets at 19 % and 25 % wt/wt induces high deformations.

Under no-field conditions, the droplet diameter remains nearly spherical, with both vertical (a) and horizontal (b) diameters close to 1.5 × 10⁻³ m. At –800 kV/m and 25 % wt/wt, deformation becomes more pronounced, with a reaching 1.9 × 10⁻³ m and b decreasing to 1.4 × 10⁻³ m. These values were extracted from high-resolution contour images using the image processing methodology described in Dokmak et al. (2025).

Therefore –400 and –800 kV/m were selected for the electric field intensity in this study to explore the drying kinetics.

To perform the experiments, first the setup is preheated and then a 5 µL solution droplet is generated using a micropipette (model 0.5 ± 0.02 µL/10 ± 0.05 µL, Ergonomic High-Performance Pipettor, VWR International, USA) and hung on the metallic needle. The EF is applied, and the drying experiment starts. The balance records the droplet mass every 1 s and the camera captures images every 5 s. At the end of the drying experiment, the dried particle is collected from the metallic needle and its mass m<sub>fp</sub> is weighed using a precision microbalance (Sartorius Ultra-micro balance SE2, Goettingen, Germany).

For each set of the investigated operating conditions, experiments were carried out in triplicate and average values are presented in the following sections.

<table><caption>**Table 2**
Setpoints of the tested drying operating conditions.</caption><thead><tr><th>Drying mode</th><th>Airflow v (m/s)</th><th>Temperature T (°C)</th><th>Absolute humidity Y (g/kg)</th></tr></thead><tbody><tr><td>Diffusive drying</td><td>0.01</td><td>25</td><td>0</td></tr><tr><td>Convective drying</td><td>0.47</td><td>25</td><td>0</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Deformation of a droplet when subjected to an electric field, acting on the vertical axis of the droplet. The spherical shape represents the non-deformed droplet before application of the EF while (–) is the prolate shape deformation and (–) is the oblate shape deformation.8</figcaption></figure>

## 2.3. Data processing

The collected balance and camera data are used to calculate the moisture content on a dry basis and the evaporative flux of the droplet.

The moisture content on a dry basis X can be calculated using Eq. (2):

$$X = \frac{m - m_{fp}}{m_{fp}} \qquad (2)$$

Where m is the measured mass of a droplet (g) and m<sub>fp</sub> is the weight of the final dry solid particle (g).

The evaporative flux calculation represents the mass loss per unit droplet surface area over time, calculated using Eq. (3):

$$N = \frac{-m_{fp}}{S} \times \frac{dX}{dt} \qquad (3)$$

Where N is the evaporative flux (g of evaporated water·m⁻²·s⁻¹), S is the surface area of the droplet (m²), dX/dt is the derivative of the moisture content in (g<sub>water</sub>·g<sub>solid</sub>·s⁻¹).

### 2.3.1. Captured image processing

For image processing, a MATLAB R2024a code was developed to calculate the droplet surface area using Guldin's theorem, (also known as Pappus's centroid theorem), which provides a geometrical method for determining the surface area of a solid of revolution. According to this theorem, the surface area generated by rotating a planar curve around an external axis is equal to the length of the curve multiplied by the distance traveled by its centroid. This approach assumes that the droplet remains axisymmetric (Goodman and Goodman, 1969, Schmid, 2025).

The droplet contour was extracted from the droplet images and smoothed using Fast Fourier Transform (FFT) to reduce noise. The surface area was then computed by revolving the smoothed contour around the vertical axis using Guldin's theorem. Calculations were initially performed in pixels and later converted to square meters using a calibrated spatial resolution factor of 2.75 × 10⁻⁶m/pixel. This method, as detailed in an earlier publication (Dokmak et al., 2025), allows for accurate tracking of surface area evolution during drying, even under high electric field intensities. At high intensities, deformation of electrically charged copper sulfate droplets, previously analyzed using the same experimental setup (Dokmak et al., 2025), confirmed that while the
droplets deform and elongate vertically under the electric field, they retain axis-symmetry. This validated the continued application of Guldin's theorem even in the presence of deformation.

However, as the drying progresses, variations in surface area and the formation of dry particles may disrupt the symmetry, leading to deviations from axisymmetric conditions. Consequently, the number of images that are processed and the corresponding surface area calculations are limited to a specific timeframe period $t_{\text{video}}$ during which the droplet is axisymmetric.

Furthermore, the captured images serve to visually determine two characteristic drying times, i.e. the time of appearance of the first solid on the droplet $t_{\text{solid}}$ and the time at which the surface of the droplet is fully covered by a solid crust $t_{\text{crust}}$.

### 2.3.2. Balance measurements processing

Several limitations in the acquired mass data need to be addressed before the computation of $X$ and $N$:

* The balance recorded signal is noisy making it challenging to calculate the moisture content derivative.

* Mass data is recorded every second generating a temporal vector $t_{\text{exp}}$, while surface area data is extracted from video frames and calculated while the droplet is axisymmetric, generating a temporal vector $t_{\text{video}}$. To ensure accurate evaporative flux calculations, both datasets must be synchronized to a common time frame ($t$).

A Fast Fourier Transform (FFT) method (Deleau et al., 2020) is performed to filter the noisy mass signal and to calculate the moisture content on a dry basis. FFT decomposes the signal into its frequency components, making it possible to filter out unwanted fluctuations while retaining meaningful trends. However, this process can introduce artificial oscillations in the time derivative of moisture content $\frac{dX}{dt}$. Consequently, to further refine the data, the smooth function with "Gaussian" filtering and a window size of 100 is applied, reducing oscillations caused by the FFT on $\frac{dX}{dt}$. Then using Eq. (3) the evaporative flux can be calculated. The data processing steps are presented in Fig. 4

## 3. Results

### 3.1. Moisture content and evaporative flux in diffusive drying

In this section the results of convective drying ($T = 25$ °C, $Y=0$ g/kg dry air and $v=0.01$ m/s) are discussed.

Using captured images, the appearance of the first solid $t_{\text{solid}}$ and

the solid crust $t_{\text{crust}}$ were visually identified from the droplet images presented in Fig. 5. These critical times are then coupled with the balance measurement and moisture content and helps in identifying the different drying periods.

The normalized moisture content $\frac{X}{X_0}$ curves under various applied electric field intensities, where $X_0$ is the moisture content of the droplet at the beginning of the drying, are presented in Fig. 6.

For droplets experiencing small deformations, i.e., for solutions at 6%, 19% and 25% wt/wt dried in the absence of an EF and at -400 kV/m as well as the solution at 6% wt/wt at -800 kV/m, the drying process revealed two distinct phases. During the initial stage, moisture content decreased rapidly and linearly, driven primarily by the surface evaporation, until the first solid appeared at the droplet interface. As drying continued and a solid crust formed, the rate of moisture loss diminished, signifying a shift from surface dominated evaporation to a slower diffusion-controlled process.

In contrast, droplets at 19% and 25% wt/wt subjected to -800 kV/m, where significant deformations ($De > 0.1$) were estimated, demonstrate a different drying behavior during the early stages. During the early drying period, the rate of moisture loss was slower compared with cases without an EF. However, as the drying time progresses, the drying rate is increased, eventually aligning with the no-field profiles. Following the crust formation, when the drying entered the diffusion-limited phase, the moisture content values converges to values similar for the no field conditions.

The altered moisture loss rate in the early drying period highlights the impact of the EF, particularly at high concentrations and field intensities.

Furthermore, the influence of initial solid content on the moisture content profiles of droplets drying under fixed electric field intensities is presented in Fig. 7. Under no field conditions (0 kV/m), it is noteworthy that during the initial stage of drying, when the droplet surface remains saturated with water, the moisture removal rate appears similar across all solid content concentrations. However, as drying progresses (around 500 s), the normalized moisture content of the 25% wt/wt solution becomes lower than that of the 19% wt/wt and 6% wt/wt solutions. Additionally, higher initial solid content is associated with earlier onset of solid formation and crust development. For example, solid crust formation was observed at approximately 1404 s for the 25% wt/wt solution, compared to 1995 s for the 6% wt/wt solution.

Under an applied electric field of -800 kV/m, the moisture removal rate during the initial drying period also varies with solid content. In this case, droplets with 25% wt/wt and 19% wt/wt concentrations exhibit a slower moisture loss than those with 6% wt/wt, suggesting that the

<figure><img src="image_6.png" /><figcaption>Fig. 4. Data processing of the drying experiment (balance mass data and captured images) to calculate the moisture content, the surface and the evaporative flux over time.11</figcaption></figure>
<table><thead><tr><th>Solution</th><th>Initial droplet</th><th>First solid</th><th>Crust (lock point)</th></tr></thead><tbody><tr><td>25 % wt/wt,<br>0 kV/m<br>De = 0</td><td></td><td></td><td></td></tr><tr><td>25 % wt/wt,<br>-400 kV/m<br>De = 0.0252</td><td></td><td></td><td></td></tr><tr><td>25 % wt/wt,<br>-800 kV/m<br>De > 0.1260</td><td></td><td></td><td></td></tr><tr><td>19 % wt/wt,<br>0 kV/m<br>De = 0</td><td></td><td></td><td></td></tr><tr><td>19 % wt/wt,<br>-400 kV/m<br>De = 0.0241</td><td></td><td></td><td></td></tr><tr><td>19 % wt/wt,<br>-800 kV/m<br>De > 0.1100</td><td></td><td></td><td></td></tr><tr><td>6 % wt/wt,<br>0 kV/m<br>De = 0</td><td></td><td></td><td></td></tr><tr><td>6 % wt/wt,<br>-400 kV/m<br>De = 0.0006</td><td></td><td></td><td></td></tr><tr><td>6 % wt/wt,<br>-800 kV/m<br>De > 0.0220</td><td></td><td></td><td></td></tr></tbody></table>

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<figure><img src="image_33.png" /><figcaption>Fig. 5. Images of CuSO₄·5H₂O (25, 19 and 6 % wt/wt) droplets drying in diffusive condition and at different EF intensities (0, -400 and -800 kV/m). Images correspond to three critical times: t = 0 s, time of apparition of the first solid (t solid) and time of formation of the solid crust (t crust).13.</figcaption></figure>
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<figure><img src="image_36.png" /><figcaption>**Fig. 6.** Normalized moisture content profile for diffusive drying of CuSO₄·5H₂O droplet at different EF intensities, 0, -400 and -800 kV/m: (a) 6 % wt/wt and (b) 19 % and (c) 25 % wt/wt. Diamonds and circles mark the critical points of first solid formation and solid crust fully formation, respectively. The shaded areas indicate plus or minus one the standard deviation around the mean.15</figcaption></figure>

electric field affects early drying dynamics more strongly at higher concentrations. A similar trend is observed regarding the timing of first solid appearance and crust formation as in the no-field condition.

The surface area evolution of CuSO₄·5H₂O droplets at 6 %, 19 % and 25 % wt/wt during drying under electric field intensities of 0, -400, and -800 kV/m is presented in Fig. 8(a), (b) and (c) respectively.

Results show a decrease in surface area with time across all conditions, indicating reduced mass transfer area over drying time. At 0 and -400 kV/m, the evolution of the surface area shows minimal

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /><figcaption>**Fig. 7.** Normalized moisture content profile for diffusive drying of CuSO₄·5H₂O droplet at 6, 19 and 25 % wt/wt solid content: (a) 0 kV/m (b) -800 kV/m. Diamonds and circles mark the critical points of first solid formation and solid crust fully formation, respectively.16</figcaption></figure>

differences, while at -800 kV/m, droplets exhibit a larger initial surface area due to the deformation induced by the electric field, whatever the solution concentration. Indeed, the electric field generates electrical forces acting on the droplet and elongating the droplet toward the cylindrical electrode (Dokmak et al., 2025).

As the droplet dries over time, the surface area at -800 kV/m gradually converges to values like those observed under no-field conditions. This phenomenon is likely due to the increasing salt concentration within the droplet as it dries. It has been previously demonstrated that at a given solid content concentration, the increase of the EF intensity lead to a decrease in the interfacial tensions of the droplet. Furthermore, it has been demonstrated that at a given EF intensity, as the solid content of the solutions increases, its interfacial tension increases (Dokmak et al., 2025). Thus, a higher salt concentration increases the interfacial tension as summarized in Table 1. Consequently, it can be hypothesized that at a given EF intensity, as the drying progresses, the concentration of salt in the droplet increases and consequently the interfacial tension increases. This increase in the interfacial tension may consequently hinder the effect of the field on the surface area deformation.

Finally, the evaporative flux of drying droplets, calculated using moisture content and surface area data, is shown in Fig. 9. Since axisymmetric was not maintained throughout the entire drying process, particularly for highly concentrated droplets, the surface area calculation was limited to the time interval during which the droplet remained sufficiently axisymmetric. This, in turn, constrained the range of
<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /><figcaption>**Fig. 8.** Surface area as a function of the time of acquisition of axis-symmetrical images during diffusive drying for the CuSO₄·5H₂O droplet at different EF intensities, 0, –400 and –800 kV/m: (a) 6 % wt/wt and (b) 19 % wt/wt and (c) 25 % wt/wt. The shaded areas indicate plus or minus one the standard deviation around the mean.18</figcaption></figure>

moisture content (X) over which the flux could be reliably calculated. To ensure consistency and comparability across all experimental conditions, the flux calculation was restricted to the range 0.7 < X/X₀ < 1, corresponding to the period in which the axisymmetric assumption held for all tested droplets.

During this investigated period, droplets with small deformation (i.e.

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /><figcaption>**Fig. 9.** Initial drying flux during for diffusive drying of CuSO₄·5H₂O droplet at different EF intensities, 0, –400 and –800 kV/m: (a) 6 % wt/wt, (b) 19 % wt/wt and (c) 25 % wt/wt.20</figcaption></figure>

for solutions at 6, 19 and 25 % wt/wt at no field and –400 kV/m and the 6 % wt/wt solution at –800 kV/m) exhibited a nearly constant evaporative flux (N around 0.2 g·s⁻¹·m⁻²), with a slight increase observed at the beginning of the drying process. This initial increase is likely due to transient thermal effects where the droplet undergoes temperature equilibration with the surrounding environment. Additionally, uncertainties in the FFT signal during this early stage of evaporation may contribute to this initial fluctuation. During the initial drying phase, the droplet surface remains fully covered with solvent, and moisture loss is primarily controlled by external heat and mass transfer.
However, droplets at 19 % and 25 % wt/wt subjected to a -800 kV/m field, showed notable and more pronounced lower initial flux compared to no-field conditions. This decrease in flux, particularly for the range of 0.85 < $\bar{X}/X_0$ < 1, is explained by the slower moisture reduction and the increased surface area. As the drying process continues, the evaporative flux increases and converges toward a constant value, a similar trend to that at no field condition.

## 3.2. Moisture content and evaporative flux in convective drying

In this section the results of convective drying (T = 25 °C, Y= 0 g/kg dry air and v =0.47 m/s) are discussed. Fig. 10 presents the normalized moisture content profiles during convective drying. Here, only the results for the solutions at 19 and 25 % wt/wt are presented, as the effect of the electric field on moisture content removal during diffusive drying was particularly evident for these concentrations.

As for diffusive drying, two distinct phases are observed: an initial steep reduction in moisture content, followed by a slower rate of reduction, separated by the formation of a solid crust on the droplet surface. The transition between these phases occurs more quickly in convective drying compared to diffusive drying. For instance, Fig. 11 represents a comparison between the effect of diffusive and convective drying conditions on the normalized moisture profiles for the solution at 25 % wt/wt and drying under different field intensities.

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /><figcaption>**Fig. 10.** Normalized moisture content (dry basis) evolution for the CuSO₄.5H₂O droplet at (a) 19 % wt/wt and (b) 25 % wt/wt, with an airflow of 0.47 m/s and at different EF intensities, 0, -400 and -800 kV/m. The diamonds represent the apparition of the first solid and the circles represent the fully formed solid crust.22.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>**Fig. 11.** Comparison between the Normalized moisture content (dry basis) evolution in diffusive and convective drying conditions for the CuSO₄.5H₂O droplet at 25 % wt/wt and at different EF intensities, 0, -400 and -800 kV/m. The diamonds represent the apparition of the first solid and the circles represent the fully formed solid crust.22.</figcaption></figure>

For droplets with 25 % wt/wt solid content drying at no-field (0 kV/m) with an airflow speed of 0.47 m/s, the solid crust forms at approximately 500 s, whereas in diffusive drying, the crust forms at around 1500 s. This faster drying rate is due to the combined convective and diffusive transport of water vapor from the droplet to the surrounding air.

Moreover, it can be remarked from Fig. 10 that the presence of an electric field does not significantly affect the rate of moisture reduction during convective drying. Notably, the slowdown in moisture reduction during the initial stage, observed for solutions at 19 % and 25 % wt/wt at -800 kV/m in diffusive drying conditions, is not evident with the presence of the airflow, suggesting that the drying airflow hinders the EF effect on moisture content.

Furthermore, when comparing different EF intensities during convective drying, the time at which the solid crust forms is similar across all EF conditions.

To better understand the effect of the electric field on the drying process, in both diffusive and convective conditions, despite the effect of the EF on deforming the droplet surface area, the moisture removal rate ($-m_{fp} \times \frac{dX}{dt}$) is calculated and presented in Table 3: Moisture removal rate ($-m_{fp} \times \frac{dX}{dt}$) for the CuSO₄.5H₂O droplet at (a) 19 % wt/wt and (b) 25 % wt/wt drying at 0 and -800 kV/m in both diffusive ($v = 0.01$ m/s) and convective ($v = 0.47$ m/s) conditions: for $\bar{X}/X_0 = 1$ and $\bar{X}/X_0 = 0.7$.

Results show that in diffusive drying conditions, the moisture removal rates of droplets at 25 and 19 % wt/wt, in the absence of the electric field, decreases by 25 % when the droplet loses 30 % of its moisture content. However, when applying -800 kV/m, both solutions exhibited an increase in the moisture removal rate by 3 times when the droplet dries and loses 30 % of its moisture content.

Differently, at no field conditions and under convective drying conditions, when the droplet loses 30 % of its moisture content, its moisture removal rate decreases by around 30-35 % while this rate remains constant when applying -800 kV/m.

<caption>**Table 3**</caption>

Moisture removal rate ($-m_{fp} \times \frac{dX}{dt}$) for the CuSO₄.5H₂O droplet at (a) 19 % wt/wt and (b) 25 % wt/wt drying at 0 and -800 kV/m in both diffusive ($v = 0.01$ m/s) and convective ($v = 0.47$ m/s) conditions: for $\bar{X}/X_0 = 1$ and $\bar{X}/X_0 = 0.7$.

<table><thead><tr><th></th><th></th><th colspan="2">No field (0 kV/m)</th><th colspan="2">-800 kV/m</th></tr><tr><th>$-m_{fp} \times \frac{dX}{dt}$ (g.s<sup>-1</sup>)</th><th></th><th>$\bar{X}/X_0 =$</th><th>$\bar{X}/X_0 =$</th><th>$\bar{X}/X_0 =$</th><th>$\bar{X}/X_0 =$</th></tr></thead><tbody><tr><td>(a) 19 % wt/ wt</td><td>Diffusive</td><td>0.36</td><td>0.7</td><td>1</td><td>0.7</td></tr><tr><td></td><td>Convective</td><td>1.16</td><td>0.74</td><td>0.16</td><td>0.34</td></tr><tr><td>(b) 25 % wt/ wt</td><td>Diffusive</td><td>0.34</td><td>0.25</td><td>0.14</td><td>0.34</td></tr><tr><td></td><td>Convective</td><td>1.07</td><td>0.74</td><td>0.87</td><td>0.81</td></tr></tbody></table>
These observations demonstrate the ability of the electric field to enhance or sustain a high moisture removal rate during the drying process, despite the variations in the droplet surface area.

Surface area and evaporative flux N values, calculated at $\frac{X}{X_0} = 1$ and 0.7 are presented in Table 4.

Table 4 shows that the evaporative flux for droplets at 19% and 25% wt/wt drying at no field conditions is approximately 0.6 g.s⁻¹·m⁻², which is 3 times higher than the flux observed in diffusive drying in Fig. 9.

Furthermore, CuSO₄·5H₂O droplet at 19% and 25% wt/wt subjected to -800 kV/m has a higher initial surface area compared to the no field conditions. This highlights that the effect of the EF in deforming and elongating the droplet remains even in the presence of the convective airflow. As the droplet dries, its surface area decreases, and this reduction is more pronounced at -800 kV/m conditions. For instance, for the solution at 25% wt/wt, the droplet's surface area with no field conditions decreases by around 18% when the droplet loses 30% of its moisture compared to 30% loss of surface area for the droplet at -800 kV/m.

Moreover, when subjected to -800 kV/m, the initial evaporative flux at $\frac{X}{X_0} = 1$ is around 0.4 g.s⁻¹·m⁻², approximately 33% lower than the values observed under no-field conditions. This reduction is attributed to the increased surface area available for mass transfer, while moisture content variation remains unchanged. As the droplet dries and reaches $\frac{X}{X_0} = 0.7$, the surface area converges to values like those observed at 0 kV/m, and consequently the evaporative flux returns to values between 0.5 g.s⁻¹·m⁻², like no-field conditions at $\frac{X}{X_0} = 0.7$.

## 4. Discussion and hypotheses at the molecular scale

The observed impact of the electric field on drying behavior can be hypothesized by examining molecular scale mechanisms as suggested in the literature.

The EF reorganizes dipoles and ions within the droplet through polarization and electromigration (Kerisit et al., 2023, Rolle et al., 2022). This reorganization, coupled with electrical forces acting on the droplet, deforms the droplet and reduces its interfacial tension. However, it is known in literature that when the concentration of a saline solution increases, its interfacial tension increases as well (Dokmak et al., 2025, Widodo et al., 2021). Thus, one proposed hypothesis can be that, at a constant EF intensity, as drying progresses and salt concentration in the droplet increases, the interfacial tension rises, gradually reducing the influence of the EF on the droplet morphology. This may explain the observed convergence of the surface area and the evaporative flux to no field values during the later stages of drying under both diffusive and convective conditions.

During diffusive drying, the decreased moisture reduction rate and

<table><caption>**Table 4**

Surface area and drying flux for the CuSO₄·5H₂O droplet at (a) 19% wt/wt and (b) 25% wt/wt drying at 0.47 m/s at 0 and -800 kV/m for $\frac{X}{X_0} = 1$ and $\frac{X}{X_0} = 0.7$.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="2">No field (0 kV/m)</th><th colspan="2">-800 kV/m</th></tr><tr><th><sup>X</sup>/<sub>X₀</sub> = 1</th><th><sup>X</sup>/<sub>X₀</sub> = 0.7</th><th><sup>X</sup>/<sub>X₀</sub> = 1</th><th><sup>X</sup>/<sub>X₀</sub> = 0.7</th></tr></thead><tbody><tr><td rowspan="2">(a) 19% wt/wt solid content</td><td>Surface area (x10<sup>-5</sup> m<sup>2</sup>)</td><td>1.91</td><td>1.52</td><td>2.13</td><td>1.62</td></tr><tr><td>Evaporative flux (g.s<sup>-1</sup>·m<sup>-2</sup>)</td><td>0.61</td><td>0.49</td><td>0.40</td><td>0.52</td></tr><tr><td rowspan="2">(b) 25% wt/wt solid content</td><td>Surface area (x10<sup>-5</sup> m<sup>2</sup>)</td><td>1.75</td><td>1.42</td><td>2.14</td><td>1.58</td></tr><tr><td>Evaporative flux (g.s<sup>-1</sup>·m<sup>-2</sup>)</td><td>0.61</td><td>0.52</td><td>0.41</td><td>0.51</td></tr></tbody></table>

evaporative flux in the early stages for highly deformed droplets could be attributed to the interaction between electric forces and internal fluid dynamics. According to molecular dynamics simulations reported in the literature, the EF generates electrohydrodynamic (EHD) flows within the droplet (Ko et al., 2008, Pradhan et al., 2023)). The nature of these flows is dependent on the EF direction. For example, Dhar et al. (2018) demonstrated that applying a vertical EF to a pendant saline droplet suppresses internal circulation, retarding evaporation, whereas a horizontal EF enhances circulation, accelerating evaporation. This mechanism provides a plausible hypothesis for the initial retardation in moisture loss observed in the actual study. Another key hypothesis supported by molecular simulations is that the EF direction significantly influences dipole organization altering consequently the evaporation (Okuno et al., 2009). These effects are proposed to play a critical role in the variations observed in evaporative flux and moisture removal during drying.

In convective drying, the application of the EF does not seem to retard the moisture removal however it allowed to maintain a constant moisture removal rate until the droplet has lost 30% of its moisture content contrary to the results obtained at no-field conditions. This may be explained by the dominant effect of the convective airflow which may hinder the internal EHD flow. However, the application of the EF increases the droplet surface area through elongation in the direction of the electric field, which results in a reduced evaporative flux.

It is conceivable that this slower evaporation flow, at the beginning of drying, in the presence of the EF could protect active compounds present in the droplet from rapid thermal stress and degradation, which is important for drying heat-sensitive compounds (to preserve their functionalities). These findings in convective drying are in line with some experimental results obtained in literature for the drying of monoclonal antibodies and enzymes using electrostatic assisted spray drying, where the process proved its capacity in preserving thermosensitive active compounds during drying and enhancing their encapsulation compared to conventional spray dryers (Beaupeux et al., 2024, Jayaprakash et al., 2023). The aim of the experimental study carried out in this work is to gain a better understanding of the drying behavior of multi-constituent droplets and the effect of an electric field on droplet drying and composition. The relationship between droplet behavior at the scale of observation possible in this study and effects on products observed at the process scale is encouraging for further studies at the droplet scale.

## 5. Conclusions

In this research work, the influence of an electric field on droplet drying was examined under both convective and diffusive conditions. Results demonstrate that the application of an EF initially increases the surface area of the droplet. However, as drying progresses, the effect of the electric field on the droplet morphology diminishes, and the surface area of the droplet converges to values like those observed under no-field conditions.

Different behaviors were observed in terms of moisture evaporation depending on the drying conditions, the salt concentration and the electric field intensity.

During diffusive drying, droplets with a strong response to the EF (high deformation) show a slower moisture removal rate in the early stages compared to those at no-field conditions but increasing after a certain drying point. This initial slowdown in evaporation was hypothesized to be due to the internal electrohydrodynamic (EHD) flows generated within the droplet.

For convective drying, the application of the electric field maintained a constant moisture removal rate when the droplet dries and loses its moisture content (ie 30% of moisture loss in the case of the actual study) while this rate decreased in the absence of the field. Furthermore, it was found that the EF reduces the evaporative flux during the early stages of drying due to the increase of the surface area of the droplets
subjected to –800 kV/m.

While the electric field did not significantly appear to accelerate moisture removal, it modified the droplet's morphology and reduced the evaporative flux in the early drying stage. Assuming the droplet surface remains saturated, this lower flux implies that less thermal energy is needed to evaporate the same amount of water, allowing drying to occur at lower air temperatures. This behavior aligns with findings from electrostatically assisted spray drying, where similar moisture contents are achieved at lower temperatures. These results support the idea that electric fields can reduce thermal stress on sensitive compounds, offering a way to better preserve their structure and functionality during drying.

In conclusion, the interaction between the electric field and airflow creates complex dynamics that influence the moisture content removal and the evaporative flux during the drying stages. However, further research involving detailed measurements of internal droplet flow dynamics in the early drying stages would be wise to fully understand these effects and corresponding electrohydrodynamic phenomena.

## 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.

## References

Akilan, C., Hefter, G., Rohman, N., Buchner, R., 2006. Ion association and hydration in aqueous solutions of Copper(II) sulfate from 5 to 65 °C by dielectric spectroscopy. J. Phys. Chem. B 110, 14961-14970. https://doi.org/10.1021/jp0620769.

Al Zaitone, B., Al-Zahrani, A., Al-Shahrani, S., Lamprecht, A., 2020. Drying of a single droplet of dextrin: drying kinetics modeling and particle formation. Int. J. Pharm. 574, 118888. https://doi.org/10.1016/j.ijpharm.2019.118888.

Beaupex, E., Jayaprakash, P., Dokmak, A., Edorh, J.-M., Gaiani, C., Desobry, S., Maudhuit, A., 2024. Electrostatic spray drying: advantages for thermosensitive actives. Chem. Eng. Res. Des. 203, 688-696. https://doi.org/10.1016/j.chedr.2024.02.016.

Boel, E., Koekoeke, R., Dedroog, S., Babkin, I., Vetrano, M.R., Clasen, C., Van den Mooter, G., 2020. Unraveling particle formation: from single droplet drying to spray drying and electrospraying. Pharmaceutics 12, 625. https://doi.org/10.3390/pharmaceutics12070625.

Chen, N., Gan, Y., 2024. Enhanced evaporation and heat transfer of sessile droplets via coupling electric field and temperature field. Chem. Eng. J. 498, 155259. https://doi.org/10.1016/j.cej.2024.155259.

Deleau, T., Fechter, M.H.H., Letourneau, J.-J., Camy, S., Aubin, J., Braeuer, A.S., Espitalier, F., 2020. Determination of mass transfer coefficients in high-pressure two-phase flows in capillaries using Raman spectroscopy. Chem. Eng. Sci. 228, 115960. https://doi.org/10.1016/j.ces.2020.115960.

Dhar, P., Jaiswal, V., Harikrishnan, A.R., 2018. Electromagnetic field orientation and characteristics governed hydrodynamics within pendent droplets. Phys. Rev. E 98. https://doi.org/10.1103/PhysRevE.98.063103.

Dokmak, A., Deleau, T., Letourneau, J.J., Maudhuit, A., Arlabosse, P., Ré, M.I., 2025. A new device for monitoring deformation dynamics and droplet shrinkage kinetics (during drying) of a charged droplet suspended in air. J. Electrost. 134, 104033. https://doi.org/10.1016/j.elstat.2025.104033.

Fu, N., Woo, M.W., Chen, X.D., 2012. Single droplet drying technique to study drying kinetics measurement and particle functionality: a review. Dry. Technol. 30, 1771-1785. https://doi.org/10.1080/07373937.2012.708002.

Goodman, A.W., Goodman, G., 1969. Generalizations of the theorems of pappus. Am. Math. Mon. 76, 355-366. https://doi.org/10.1080/00029890.1969.12000217.

Jaiswal, V., 2019. Interplay of electro-thermo-solutal advection and internal electrohydrodynamics governed enhanced evaporation of droplets. https://doi.org/10.1098/rspa.2019.0046.

Jaiswal, V., Singh, S., Harikrishnan, A.R., Dhar, P., 2020. Competitive electrohydrodynamic and electrosolutal advection arrests evaporation kinetics of droplets. Langmuir 36, 8971-8982. https://doi.org/10.1021/acs.langmuir.0c01619.

Jayaprakash, P., Gaiani, C., Edorh, J.-M., Borges, F., Beaupex, E., Maudhuit, A., Desobry, S., 2023. Comparison of electrostatic spray drying, spray drying, and freeze drying for lacticaseibacillus rhamnosus GG dehydration. Foods 12. https://doi.org/10.3390/foods12163117.

Jayaprakash, P., Gaiani, C., Edorh, J.-M., Beaupex, E., Maudhuit, A., Desobry, S., 2023. Impact of matrices composition and processes on β-galactosidase encapsulation. J. Food Eng. 353, 111547. https://doi.org/10.1016/j.jfoodeng.2023.111547.

Jayaprakash, P., Maudhuit, A., Gaiani, C., Desobry, S., 2023. Encapsulation of bioactive compounds using competitive emerging techniques: electrospraying, nano spray drying, and electrostatic spray drying. J. Food Eng. 339, 111260. https://doi.org/10.1016/j.jfoodeng.2022.111260.

Kerisit, S.N., Simonnin, P.G., Sassi, M., Rosso, K.M., 2023. Electric field effects on water and ion structure and diffusion at the orthoclase (001)-Water interface. J. Phys. Chem. C. 127, 7389-7401. https://doi.org/10.1021/acs.jpcc.2c07563.

Ko, S.H., Lee, H., Kang, K.H., 2008. Hydrodynamic flows in electrowetting. Langmuir 24, 1094-1101. https://doi.org/10.1021/la702455t.

Mutukuri, T.T., Maa, Y.-F., Gikanga, B., Sakhovsky, R., Zhou, Q.T., 2021. Electrostatic spray drying for monoclonal antibody formulation. Int. J. Pharm. 607, 120942. https://doi.org/10.1016/j.ijpharm.2021.120942.

Okuno, Y., Minagawa, M., Matsumoto, H., Tanioka, A., 2009. Simulation study on the influence of an electric field on water evaporation. J. MOL. Struct. THEOCHEM 904, 83-90. https://doi.org/10.1016/j.theochem.2009.02.035.

Porowska, A., Dosta, M., Fries, L., Gianfrancesco, A., Heinrich, S., Palzer, S., 2016. Predicting the surface composition of a spray-dried particle by modelling component reorganization in a drying droplet. In: Proceedings of the 7th International Granulation Workshop on Chemical Engineering Research and Design, 2015 110, 131-140. https://doi.org/10.1016/j.chedr.2016.03.007.

Pradhan, T.K., Banuprasad, T.N., Nandagopal, M.S.G., Chakraborty, S., 2023. Induced flow inside a droplet by static electrical charge. J. Micromech. Microeng. 33, 024004. https://doi.org/10.1088/1361-6439/acaff3.

Rolle, M., Albrecht, M., Sprocati, R., 2022. Impact of solute charge and diffusion coefficient on electromigration and mixing in porous media. J. Contam. Hydrol. 244, 103933. https://doi.org/10.1016/j.jconhyd.2021.103933.

Sadek, C., Schuck, P., Fallourd, Y., Pradeau, N., Le Floch-Fouéré, C., Jeantet, R., 2015. Drying of a single droplet to investigate process-structure- function relationships: a review (np). Dairy Sci. Technol. 95. https://doi.org/10.1007/s13594-014-0186-1.

Schmid, H., 2025. A generalization of the second Pappus-Guldin theorem. Results Appl. Math. 25, 100537. https://doi.org/10.1016/j.rinam.2025.100537.

de Souza Lima, R., Ré, M.-I., Arlabosse, P., 2020. Drying droplet as a template for solid formation: a review. Powder Technol. 359, 161-171. https://doi.org/10.1016/j.powtec.2019.09.052.

Tsotsas, E., Gnielinski, V., Schlünder, E.-U., 2000. Drying of solid materials. In: Wiley-VCH Verlag GmbH & Co. KGaA (Ed.), Ullmann's Encyclopedia of Industrial Chemistry. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. https://doi.org/10.1002/14356007.b02_04.

Vancauwenberghie, V., Di Marco, P., Brutin, D., 2013. Wetting and evaporation of a sessile drop under an external electrical field: a review. Colloids Surf. A Physicochem. Eng. Asp. 432, 50-56. https://doi.org/10.1016/j.colsurfa.2013.04.067.

Wang, W., Huang, X., Wang, L., Teng, L., Luo, X., Li, W., Li, J., Yin, P., Luo, Y., Jiang, L., 2024. Study on the mechanism of sessile droplets evaporation enhanced by the electric field. Chem. Eng. Sci. 288, 119768. https://doi.org/10.1016/j.ces.2024.119768.

Wang, C., Li, J., Zhang, Ya, He, Z., Zhang, Yin, Zhang, X., Guo, Z., Huang, J., Liu, Z., 2022. Effects of electrostatic spray drying on the sensory qualities, aroma profile and microstructural features of instant Pu-erh tea. Food Chem. 373, 131546. https://doi.org/10.1016/j.foodchem.2021.131546.

Widodo, C.S., Sela, H., Santosa, D.R., 2021. 2018. The effect of NaCl concentration on the ionic NaCl solutions electrical impedance value using electrochemical impedance spectroscopy methods. AIP Conf. Proc. 050003. https://doi.org/10.1063/1.5062753.

Xu, H., Wang, J., Tian, J., Huo, Y., Li, B., Wang, D., Zhang, W., Yao, J., 2021. Evaporation characteristics and heat transfer enhancement of sessile droplets under non-uniform electric field. Exp. Therm. Fluid Sci. 126, 110378. https://doi.org/10.1016/j.exthermflusci.2021.110378.

Xu, H., Wang, J., Wang, H., Li, B., Yu, K., Yao, J., Zhang, W., Zuo, L., 2024. Numerical investigation of droplet impact and heat transfer on hot substrates under an electric field. Int. J. Heat. Mass Transf. 229, 125721. https://doi.org/10.1016/j.ijheatmasstransfer.2024.125721.