Evaporation of a Sessile Water Drop Subjected to Vertical Vibration: The Drying Kinetics near the Resonance Frequency

Abstract

The evaporation of sessile water drops involves coupled heat and mass transfer and is influenced by temperature, relative humidity, and the nature of the surface on which the drop rests. This work investigates the possibility of using vibration to enhance evaporation kinetics. For this purpose, experiments were conducted with vertical vibration near the resonant frequency. An original experimental device was designed, including a shaker controlled by a signal generator and an amplifier, a high-speed camera, and an adapted lighting system. The amplitude–frequency relationship was first examined to select the resonance frequency. As expected, the evaporation kinetics of two drops—one with vibration at the resonance frequency and the other without vibration—demonstrate that vibration accelerates evaporation and reduces drying time by 20.6% on PTFE substrate and by 23.5% on glass substrate.

1. Introduction

The evaporation of sessile drops, a drop resting on a solid or liquid surface, is common in many situations, such as car windshields or plant leaves after morning dew, rain, irrigation, or cultural treatment. Such evaporation is also frequent in industrial processes: biochemical and pharmaceutical processes, paint [1], printing techniques [2], internal combustion engine performance [3], etc.

The physical phenomenon of evaporation of a drop placed on a solid surface was investigated by several researchers. A numerical study on the influence of the thermal diffusivity of the substrate and the volatility of the liquid on the evaporation phenomenon under unheated and heated conditions was carried out by Maatar et al. [4]. The results demonstrated that for an unheated substrate, volatility reduces droplet lifetime. Another numerical and experimental study by Wakata et al. [5] investigated the evaporation of ethanol–water droplets. They showed that the addition of a volatile component like ethanol decreases the droplet lifetime compared to pure water, accelerating the evaporation process even without external heating. A theoretical analysis was developed by Picknett et al. [6] to study the evaporation rate and residual mass at any time during the life of the drop under two evaporation modes: at constant contact angle and at constant contact area. These predictions show that the theory is reasonably accurate for drops of mass ranging between 1 pg and 40 mg. A study was conducted by Yu et al. [7] on the evaporation of droplets on PDMS and PTFE surfaces under different modes: constant contact area, constant contact angle, and mixed mode. The results demonstrate the validity of the theoretical solution, which would assist in a better understanding and application of water droplets on solid surfaces. A theoretical and experimental study of the evaporation of water droplets on silicon and Teflon substrates was conducted by Nguyen et al. [8]. The study demonstrated the significant influence of surface roughness, hydrophobicity, and contact angle hysteresis on the evaporation of sessile droplets. Mollaret et al. [9] report an experimental and numerical study that describes the hydrodynamics inside the evaporating drop and the effect of the humidity on the evaporation process. Major research on sessile drop evaporation revealed three behavioral evaporation situations: the pinned mode, the unpinned modes, or the stick–slip mode [10,11]. The stick–slip behavior of the droplet triple point has been investigated using Lattice Boltzmann simulations [12,13]. Ge et al. [12] studied droplet spreading and permeation on porous substrates, showing that surface wettability and pore properties strongly influence the process. Frank et al. [13] examined low-viscosity droplets on various porous surfaces and introduced an effective equilibrium contact angle to describe the overall spreading dynamics accurately. An experimental study by Zhao et al. [14] investigated the influence of surface roughness on the evaporation dynamics of sessile droplets. They showed that increasing the roughness of a copper substrate accelerates droplet evaporation by enhancing spreading and reducing the apparent contact angle, highlighting the importance of selecting an appropriate substrate to control evaporation efficiency. The experimental study of Khilifi et al. [15] and the numerical study of Foudhil et al. [16] investigated several drops deposited alongside each other on the same substrate under the same conditions. The comparison explained the effect of the interaction between neighboring drops, which delayed the evaporation of these drops, particularly the central drop. The results showed that the interaction phenomenon becomes less important as the distance between the drops increases. Another study conducted a numerical simulation of droplet evaporation on heated flat and microstructured hydrophobic surfaces [17]. The results showed that complex phenomena, such as the influence of substrate roughness and hydrophobicity on contact-line dynamics, can occur during the evaporation process.

A fundamental study by Colin et al. [18] focused on the behavior of a water column, referred to as the “index,” inside a capillary tube, which simulates a single pore. The position and movement of this water index were tracked using image analysis, allowing a detailed assessment of liquid displacement under sinusoidal vibrations. Interestingly, they demonstrated that the movement of the index can be directed in a chosen orientation by applying an asymmetrical signal. Additionally, Vorobev et al. [19] studied immiscible liquid displacement in a single vibrating capillary. They showed that strong high-frequency vibrations can alter meniscus shapes and slow down or even stop liquid flows, contrary to the expectation that vibrations facilitate fluid movement. This provides key insights into droplet dynamics in vibrating capillaries. Similar results were reported by Kim [20] for a pendant drop (with a free contact line) on a smooth solid vibrating surface. Another study by Ercolin et al. [21] investigated droplet formation in vibrating mesh atomizers. The results showed that the volume flow rate increases near the resonance frequency of the atomizer, while the droplet size distribution remains nearly constant over varying flow rates. The formation mechanism was identified as Rayleigh jet break-up, and the atomization efficiency was found to be comparable to that of pneumatic atomizers. These findings highlight the strong influence of vibration frequency on the droplet generation process. Studies by DePaoli et al. [22] and Wilkes et al. [23] focused on drops subjected to large-amplitude vibrations. They showed that bound drops behave like flexible non-linear oscillators and can exhibit hysteretic responses. A different study by Adachi et al. [24] experimentally studied a drop of liquified gas placed on the floor at room temperature. The drop showed a characteristic polygonal vibration under the actions of gravity and surface tension, where the mode number changed suddenly as the radius of the drop decreased. The measured vibrational frequency was proportional to (wave number)^3/2 and depended simply on the surface tension and the density of the liquid. In another work by Okada et al. [25] presented an experimental study of the dynamics of water drops on a horizontal oscillating plate. They investigated various drop behaviors, such as axisymmetric polygonal vibration and non-axisymmetric polygonal vibration. They numerically described certain forms of non-axisymmetric polygonal vibrations according to Rayleigh’s theory. Further research by Ilyukhina et al. [26] calculated the perturbed shape of a drop of an ideal incompressible liquid on a vertically vibrating substrate. They determined the vibration effect on the self-assembly of micro- and nanostructures in the disperse system represented by the drop on the substrate. A numerical study proposed by Dong et al. [27] identified the resonance modes of a drop deposited on a hydrophobic surface by calculating either the extreme values of the contact angles, the velocity gradient, or the net hydrostatic force. The study by Sanyal et al. [28] looked at the behavior of droplets that oscillate and evaporate. The authors notice things like the droplet rising, vortices forming at the surface, and horizontal layers developing, all caused by the interaction between the flow of the surface and the droplet’s oscillation. They also show that these behaviors change as the droplet evaporates. The experimental study by Rahimzadeh et al. [29] showed that a vibrating support leads to a significant decrease in the lifetime of the drops and affects the behavior of the stick–slip mechanism. They studied the effect of imposing vertical and horizontal ultrasonic vibrations (40 kHz) on the dynamics and evaporation of sessile droplets of dimethylformamide (DMF), isopropyl alcohol (IPA), and water on Teflon and glass substrates. They demonstrated experimentally that in the case of vertical vibration, the left and right contact angles oscillate in-phase, whereas, in the case of horizontal vibration, there is a 180-degree phase difference between left and right contact angles.

Despite the extensive literature on droplet evaporation and droplet dynamics under external excitation, there is still a lack of studies that simultaneously address both aspects. In particular, the combined effect of vibration and evaporation on drying kinetics has not been systematically investigated. The novelty of the present work lies in filling this gap by coupling these two phenomena. Specifically, we compare the evaporation of two droplets—one subjected to vibration and one without vibration—while analyzing the role of substrate nature, surface roughness, and relative humidity. By performing most experiments near the resonance frequency, where oscillation amplitude is maximal, this study provides new insights into how vibration can influence the drying process of droplets.

2. Materials and Methods

Our experimental setup was designed to carefully measure the evaporation kinetics of a sessile drop. The kinetics of the non-vibration configuration are assessed by an electronic balance. This information also serves as a reference value to validate volume determination by image processing. For all cases, the drop shape is assessed by a high-speed camera. The images are subsequently used to follow the oscillations, but also to calculate the volume by image processing. Several experiments were carried out with 10 µL drops of pure water on Teflon (PTFE) and glass substrates. During all the experiments, the temperature was approximately 22 °C and the relative humidity was 40% measured by a temperature and humidity sensor (Sensirion SHT75; RH from 0 to 100 ± 1.8%; temperature from −40 to 123.8 ± 0.3 °C).

2.1. Experimental Setup

The experimental setup consisted of a vibration chain, an optical bench, a balance, and a computer (Figure 1). In the case of vibration, the substrate was placed on the shaker (Vibration Exciter Brüel & Kjaer type 4809; frequency range from 10 Hz to 20 kHz; maximum displacement 8 mm peak-to-peak) set on an anti-vibration table and driven by the amplifier (Power Amplifier Brüel & Kjaer type 2718; continuously variable current limit from 1A to 5A (RMS); 40 dB voltage gain) and a function generator (Agilent 33120A; 50 Ω impedance, amplitude from 50 mVpp to 10 Vpp; wave output from 1 Hz to 15 MHz). Here, mVpp (millivolts peak-to-peak) refers to the voltage difference between the maximum and minimum of the sinusoidal input signal. This electrical input is converted into the mechanical displacement of the shaker, which was directly measured from the droplet motion using the high-speed camera. For instance, an input of 100 mVpp typically corresponds to a substrate vibration amplitude of about ±1 mm. The behavior of the drop was observed using a high-speed camera (Photron, AX100, up to 4000 frames per second at the max. resolution of 1024 × 1024) with a macro lens (Canon, MP-E, 65 mm-f/2.8, 1 to 5×). In the absence of vibration, the balance (Mettler Toledo, AG245; readability of 0.1 mg to 0.01 mg; taring range of 0–210 g) was used to follow the mass of the drop. This setup ensures the relevance and ease of subsequent image processing.

2.2. Preparation of the Experiments

The surface state of a substrate influences the contact angle and the evaporation kinetics of a drop, as the contact angle of a liquid and the mode (pinned, sliding) depend on the roughness. To ensure a reproducible wetting angle, the surface of the Teflon (PTFE) substrate was polished with a Struers polisher (POTOPOL 22, rotation speed 150 rpm), with sandpaper grit size 4000. After polishing, the PTFE surface exhibited a roughness on the order of ~2 µm. The quality of the polished surface was confirmed by observations with a confocal laser scanning microscope (Carl Zeiss, LSM 700, laser line—405 nm) (Figure 2).

Impurities, such as dust and small drops of oil, can spread over the polished substrate. Their existence can seriously affect the experimental results. To avoid this problem, the substrate was cleaned just before each experiment with a surfactant solution (3-vol%, Mucasol, Merz) and thoroughly rinsed with deionized water (0.054 mS/cm of electrical conductivity and 3 ppb of TOC).

2.3. Measurement Protocol

Two different measurement protocols were applied for non-vibrating and vibrating droplets. For droplets at rest, the analytical balance was first cleaned, and a 10 µL water droplet was deposited using a syringe. The balance was closed, and the droplet weight was recorded every 30 s until complete evaporation, while photographs of the droplet were simultaneously taken. For vibrating droplets on PTFE, a 10 µL droplet was deposited using a micropipette (Eppendorf Research, 0.5–10 µL), and room temperature and humidity were monitored with a Sensirion SHT75 sensor. Vibration was applied by a function generator (33120A, Agilent, Santa Clara, CA, USA) amplified through a Brüel & Kjaer Type 2718 amplifier (0.1 A, 0.3 V, 20 dB gain), with the frequency adjusted to the desired resonant value. The droplet shape was observed using a high-speed camera (FASTCAM Mini AX), and the evaporation process was recorded photographically in a time-lapse sequence. Each acquisition sequence consisted of 50 images, with an interval of 30 s between sequences, and a total of 180 sequences were recorded.

2.4. Data Analysis

The first script begins by reading successive images of a vibrating drop from TIFF files. Each image is converted to greyscale to identify the relevant areas using Otsu thresholding, a method that automatically determines an optimal threshold for separating the object of interest (the drop) from the background. Once the image has been binarized, a morphological closure operation is applied to fill in any gaps and improve detection of the drop’s contours. The script then focuses on quantifying the position of the drop and the support. It identifies the minimum position of the blob and the support in the image and then calculates intensity averages in specific regions representing the blob and the support. These averages are then used to track the evolution of the drop over time. For each processed image, the positions of the drop and the support are recorded, as well as the difference between these two values. Curves representing these positions are generated and displayed. The vibration amplitude is then calculated by measuring the difference between the maximum and minimum positions of the drop relative to the support. The second Python (version 3.8.0) script is designed to estimate the volume, diameter, and height of a drop using image processing. It begins by loading two images: one containing the drop and a reference image used to correct the background. After conversion to greyscale, thresholding is performed using the triangle method, a thresholding technique used in image processing to segment an object of interest by determining an optimum binarization threshold. It is applied to generate a binary image, enabling the drop to be isolated from the background. A morphological opening operation is then performed to eliminate noise and improve segmentation. The center of mass of the drop is determined by averaging the coordinates of the white pixels. The volume is estimated by integrating the distances of the drop’s pixels from its center. The diameter is obtained from the extreme left and right positions of the drop, while the height is calculated by measuring the distance between the highest and lowest pixels. This script uses an approximation based on the pixels detected to reconstruct the volume of the drop.

3. Results and Discussion

3.1. Study of Vibration

When subjected to vibration, a liquid droplet can resonate at specific frequencies, which enables it to adopt characteristic shape modes. Figure 3 illustrates the impact of vertical vibrations on a 10 μL droplet at varying frequency values.

These observations indicate that the dynamics of the drop depend closely on the frequency of vibration. At low frequencies, surface tension exerts a stabilizing effect, maintaining regular oscillations. On the other hand, as the frequency increases, the drop undergoes non-linear deformations that can lead to capillary instabilities or even bursting if the excitement becomes too intense. So, the transition from a stable to an unstable regime occurs progressively as the excitation frequency increases. Daniel et al. [30] studied the oscillation modes of sessile droplets subjected to asymmetric horizontal vibrations. They observed that, in this configuration, the first mode corresponds to a horizontal oscillation around the droplet’s center of mass, while the second mode is associated with a vertical motion. These results agree with the theoretical predictions of Barwari et al. [31], who demonstrated that the vibration frequency influences the shape of the droplet and the direction of oscillation. They identified the first mode as a forward-backward motion and the second mode as an up-down oscillation.

However, in our study, where the droplet is excited by vertical vibrations, we observe that the first resonance mode corresponds directly to a vertical oscillation (up-down motion), unlike the horizontal oscillation reported by Daniel et al. [30]. This suggests that the nature of excitation plays a crucial role in determining the dominant oscillation mode.

3.2. Determination of the Resonance Frequency

Figure 4 presents a temporal evolution of the height of the water droplet and the PTFE substrate, as computed automatically by the first Python script. The figure depicts the absolute position of the top of the drop and of the support versus time, as well as the relative position of the drop (distance between the support and the top of the drop). The vibration amplitude is automatically computed by the script (orange dot-dash lines). In this case, one can see that the drop amplitude is much larger than the support amplitude, which is characteristic of a resonance.

In this study, we concentrated on the second resonance mode, which is characterized by vertical deformation of the drop. This mode occurs visually in a frequency range around 70 Hz. In order to determine more precisely the frequency mode, we plotted the resonance curve, as illustrated in Figure 5. The maximum amplitude of the oscillations is observed at 70 Hz, which induces an intense vibratory movement of the drop, with a potential to accelerate evaporation and contribute to faster drying. The drop can vibrate in different modes, which correspond to different resonant frequencies. The increase at 67 Hz likely corresponds to another mode of vibration of the water droplet, distinct from the dominant mode at 70 Hz.

It is well established that, under ambient conditions, the evaporation of millimetric sessile droplets is predominantly governed by vapor diffusion into the surrounding gas phase, while the contribution of internal convective heat transfer is secondary (Picknett & Bexon [6]; Hu & Larson [32]; Hu & Larson [33]; Barash et al. [34]). In contrast, substrate vibration significantly accelerates evaporation by inducing depinning of the contact line, modifying droplet geometry, and enhancing renewal of the liquid–air interface (Rahimzadeh & Eslamian [35]). Therefore, the observed differences in evaporation behavior between the two substrates can be primarily attributed to surface wettability and liquid–solid interactions, whereas the dominant enhancement mechanism arises from the applied vibration rather than from internal convection.

Thanks to its hydrophobic properties, PTFE influences the dynamics of water droplets, particularly their resonance behavior. Several studies have highlighted its significant impact on the frequency and amplitude of water droplet resonance. In particular, water droplets on PTFE exhibit lower resonance frequencies than those reported on other hydrophobic materials, such as silicone rubber (SR) [36]. Furthermore, the resonant frequencies of the droplets vary depending on the substrate and can deviate from theoretical predictions of free oscillating droplets, highlighting the importance of substrate properties in determining these frequencies [37]. The impact of the substrate is not limited to the resonant frequency; it also plays a determining role in the amplitude of droplet vibrations. Hydrophobic surfaces, such as PTFE, modify impact dynamics by limiting spreading and favoring accumulated rebound, which can amplify resonance amplitudes [38]. In addition, the interaction between the droplet and the PTFE substrate can generate complex resonance modes, where several modes coexist, thus offering the overall amplitude of the vibrations [37].

For droplets on a glass substrate, the resonance frequency was determined visually. We observed the droplet shape and identified the frequency at which it exhibited the largest amplitude (triangular shape). This frequency was found to be close to the frequency range observed for PTFE, around 70 Hz.

3.3. Effect of Vibration

At first, it is interesting to observe the evolution of the drop size and shape during drying. Figure 6 presents the temporal evolution of a droplet deposited on a PTFE substrate, with and without vibration. One can observe that the vibration amplitude decreases as the drop size shrinks. This behavior can be explained by two effects:

• The frequency was chosen to obtain resonance with the initial 10 μL drop;
• The potential vibration amplitude necessarily decreases with drop size.

It should be noted that the resonance frequency of a droplet depends on its size, and as evaporation progresses, the actual resonance frequency gradually increases. In the present study, a fixed frequency of 70 Hz corresponding to the resonance of the initial droplet was employed for all vibration experiments. While this approach does not perfectly track the resonance throughout the entire evaporation process, it allows a clear comparison between with and without vibration droplets. The potential decrease in vibration amplitude as the droplet shrinks is acknowledged, and the limitations of using a fixed frequency approach versus full process resonance tracking are discussed. Despite this, the observed trend of shorter drying times for droplets with vibration compared with static droplets remains valid, providing meaningful insight into the impact of vibration on evaporation. For the drop at rest, one can observe a gradual decrease in the contact angle, with a decrease in the contact area only at the end of drying. This reveals a constant contact area mode during most of the drying, followed by a constant contact angle mode (or possibly a mixed mode).

This figure shows the evolution of the size of a drop of water (10 µL) deposited on a PTFE substrate, in the presence (at 70 Hz) and absence of vibrations, at different evaporation times. It was observed that, under both conditions, the size of the drop decreased as evaporation progressed.

In the absence of vibration, the drop maintains a relatively stable shape and evaporation is more regular and less rapid, with a gradual reduction in its height and diameter.

On the other hand, under the effect of vibrations, the drop undergoes periodic oscillations, with minimum and maximum height positions. Figure 7 shows the dynamic evolution of the shape of the drop under a vibration of 70 Hz for 30 s, clearly illustrating the oscillation between elongated and spherical shapes. These oscillations are due to the mechanical forces applied by the vibrations, which disturb the boundary layer of the drop and thus increase the exchange surface.

At 3600 s (Figure 6), the three configurations (without vibration, with vibration at the minimum and maximum height) show a significantly reduced drop. However, the vibrated drop has a smaller volume than the non-vibrated drop. This phenomenon can be attributed to the altered distribution of capillary and surface tension forces under the effect of vibration.

This section presents a comparative analysis of the drying kinetics of two drops, one subjected to vibration and the other not, under identical environmental conditions (temperature and relative humidity) on glass and PTFE substrates. Additionally, the relationship between vibration, substrate type, and evaporation rate is examined.

Contact angle and diameter comparison:

The dynamic contact angles were measured using a drop tensiometer equipped with an automatic actuator and dedicated software. A 5 µL droplet, contained in a syringe attached to the actuator, was dispensed on the substrate. The actuator automatically advanced and retracted the droplet, and the software calculated the advancing and receding contact angles based on the droplet shape during this process.

Table 1. Side-view goniometric images of a 5 µL drop of water deposited on a substrate.

Substrate	$\mathbf{Advancing}\mathbf{Angle}{\mathit{\theta }}_{\mathit{a}}$	$\mathbf{Receding}\mathbf{Angle}{\mathit{\theta }}_{\mathit{r}}$
Glass
PTFE

shows the goniometric side-view images of a 5 µL drop of water deposited on each substrate. This selection makes it possible to assess the influence of the physico-chemical properties and roughness of the substrate on contact angles and hysteresis. The choice of glass and polished PTFE substrates is based on their marked differences in terms of wettability, enabling a better understanding of the interactions between the surface and the water droplet. Glass, a hydrophilic material, is widely used in wetting studies because of its smooth surface and high surface energy. Polished PTFE, on the other hand, is a hydrophobic polymer whose properties can be modified by surface treatments, thus influencing its wetting behavior.

The results obtained in

Table 2. Advancing, receding contact angles, and contact angle hysteresis for glass and polished PTFE substrates.

Substrate	Advancing Angle (${\mathit{\theta }}_{\mathit{a}}$)	Receding Angle (${\mathit{\theta }}_{\mathit{r}}$)	Hysteresis (CAH)
Glass	51	25	26
PTFE	112	91	21

show an angle of advance of 51° and an angle of retreat of 25° for glass, leading to a hysteresis of 26°. These values are consistent with those reported in the literature for clean, smooth glass surfaces, where the lead angle typically varies between 40° and 50° and the recoil angle between 10° and 20° (Sikalo et al. [39]; Pittoni et al. [40]).

For polished PTFE, the measured feed angle is 112° and the recoil angle is 91°, with a hysteresis of 21°. These values are in line with previous studies that report lead angles between 110° and 120° and retreat angles close to 90° for polished PTFE surfaces ([41,42]). Compared with conventional PTFE, which generally exhibits a lower hysteresis (around 18°) [43], polished PTFE exhibits a slight increase in hysteresis, probably due to polishing effects that modify the topography on a microscopic scale.

The vibration has an impact on the evolution of the contact angle of water droplets on glass and PTFE (Figure 8a). The contact angle on the glass is approximately half that on the PTFE, which serves to highlight the superior hydrophobicity of the latter, which can maintain a higher and more stable contact angle. During the initial 10 s measurement interval, fluctuations in contact angle were observed, which can be explained by some movements needed to pin to reach stabilized variations. The contrast in contact angle between glass and PTFE directly affects the droplet diameter, as can be seen in Figure 8b. Since the resonance frequency depends on droplet size, it may evolve during the evaporation process. However, within the short 70 s observation window of Figure 8b, the droplet volume and diameter remain nearly constant on both substrates. Consequently, the associated frequency shift is minimal and does not affect the comparison of diameter evolution between glass and PTFE.

Table 3. Total drying time as a function of substrate type and condition.

Substrate	With Vibration	Without Vibration	Difference
PTFE	4380 s	5520 s	+20.6%
Glass	1676 s	2191 s	+23.5%

summarizes the results shown in Figure 9 regarding the total drying time for both PTFE and glass substrates, both with and without vibration.

To quantify the rate at which the drop dries, we calculated the average slope of the decrease in volume over two-time intervals: the first 500 s and the first 1000 s. The slope, defined as the rate of change of volume over time, is given by the following equation:

$$slope={\displaystyle \frac{∆V}{∆t}}={\displaystyle \frac{V_{final}-V_{initial}}{t_{final}-t_{initial}}}$$

(1)

where:

$V_{initial}$ and $V_{final}$ are, respectively, the volumes of the drop at the beginning and end of the period under consideration, and $t_{initial}$and $t_{final}$ are the corresponding times.

Applying this equation to the experimental data, we found that the slope for the case with vibration is always more negative than for the case without vibration. This means that, during the first 500 and 1000 s, the volume of the drop decreases more rapidly when vibration is applied. In other words, the vibration accelerates the drying of the drop.

Table 4. Calculation of the slopes of the droplet evaporation rate after 500 s and 1000 s (PTFE substrate, with and without vibration).

Time Interval (s)	Slope (Without Vibration) (µL/s)	Slope (With Vibration) (µL/s)
0–500	−0.0059	−0.0071
500–1000	−0.0003101	−0.0004427

and

Table 5. Calculation of the slopes of the droplet evaporation rate after 500 s and 1000 s (Glass substrate, with and without vibration).

Time Interval (s)	Slope (Without Vibration) (µL/s)	Slope (With Vibration) (µL/s)
0–500	−0.00551	−0.00814
500–1000	−0.00563	−0.00736

present the slopes corresponding to the curves in Figure 9 for the cases with and without vibration and for the two different substrates, PTFE and glass.

For the first 500 s, the average slope for the case without vibration is [−0.0059 µL/s], while for the case with vibration it is [−0.0071 µL/s]. This difference indicates that vibration causes a more rapid decrease in volume at the start of the drying process. This phenomenon is observed consistently over 1000 s, where the slopes also remain more negative with vibration.

For the first 500 s, the average slope for the case without vibration is [−0.00551 µL/s], while for the case with vibration it is [−0.00814 µL/s]. This difference indicates that vibration causes a more rapid decrease in volume at the start of the drying process. This phenomenon continues over the first 1000 s, where the slope with vibration [−0.00736 µL/s] remains more negative than that without vibration [−0.00563 µL/s], confirming that vibration accelerates evaporation.

This paragraph summarizes all the experiments performed in our study. For the PTFE substrate, six replicates without vibration and three with vibration were conducted, with the droplet volume (10 µL) determined from image processing of a camera-recorded droplet (standard deviations: ±0.00046 µL/s and ±0.00023 µL/s, respectively). For the glass substrate, six replicates were performed for both conditions, with the droplet mass (10 µL) measured using a precision balance (standard deviations: ±0.00117 mg/s and ±0.00116 mg/s). These values reflect the variability throughout the entire evaporation process, from initial deposition to complete evaporation, confirming the reproducibility of the experiments. The evaporation rate of water droplets on a substrate is strongly influenced by the thermal conductivity of the material as well as the contact angle. Glass, with a thermal conductivity of around 0.9 W/m-K, facilitates faster heat transfer to the droplet, which in turn increases the droplet’s surface temperature and accelerates evaporation. Studies have shown that droplets on glass substrates evaporate faster than those on PTFE substrates, which have a much lower thermal conductivity (~0.25 W/m-K) (Sterlyagov et al. [44]; Borodulin et al. [45]). This phenomenon is due to the reduced ability of PTFE to transfer heat, thus slowing down the evaporation process. In addition, the contact angle also plays a key role: on glass, the angle is smaller (~20–40°), favoring a larger contact surface with the air, which improves evaporation (Schofield et al. [46]). On the other hand, on PTFE, where the contact angle can reach high values (~100–160°), the droplet remains more spherical, which limits the contact surface and slows down the process (Schofield et al. [46]). These observations are corroborated by numerical simulations, which have shown that the thermal conductivity of the substrate directly influences the heat and mass transfer mechanisms, thus contributing to faster evaporation on substrates with high thermal conductivity, such as glass [47].

4. Conclusions

The objective of this study was to examine the evaporation of 10 µL water droplets deposited on PTFE (polytetrafluoroethylene) and glass substrates and subjected to vertical vibration. This study provides a better understanding of the mechanisms underlying the influence of the near-resonance vibration frequency on the drying kinetics of water droplets. As part of this study, a meticulously designed experimental setup was developed to visualize the behavior of the drops using an ultra-fast speed camera. This methodical approach aims to accurately determine the physical characteristics of the drops, such as their angle, diameter, amplitude, and volume, under rigorously controlled environmental conditions. The results of the experiment carried out indicate that the drop subjected to vibration at a resonance frequency of 70 Hz has a shorter drying time compared with a fixed drop. In addition, the analysis highlighted an improvement in drying kinetics, with a rate of 20.6% for the PTFE substrate and 23.5% for the glass substrate, under the effect of vibration. These results pave the way for future investigations, particularly into the effect of other parameters such as the amplitude of the vibration, the nature of the substrate, and the viscosity of the liquid. A better understanding of these interactions could enable applications in various fields, such as sputtering control, droplet deposition in microfluidics, or controlled drying of liquids.