This study quantifies the evaporation rate of sessile droplets using a quartz crystal microbalance (QCM). Specifically, we analyze the evaporation of water droplets on a gold-coated flat surface exposed to dry nitrogen at different temperatures. In this approach, we use the QCM as a radius sensor and determine the contact angle by droplet imaging, which allows calculating the instantaneous volume and the evaporation rate. For comparison, we quantify evaporation using computational modeling and an experimental technique based on droplet imaging alone. In general, the QCM-based approach was found to provide higher accuracy and a better agreement with the model predictions compared to the approach using imaging only. With modeling and experiments, we also elucidate the role of droplet self-cooling, vapor advection, and diffusion on the net rate of evaporation of sessile droplets. For all the conditions analyzed in this study, the evaporation rate was found to decrease monotonically. We found this reduction to take place even in the presence of a steadily increasing droplet temperature due to a shrinking evaporation area. Considering the vapor transport mechanisms occurring in the ambient, we find diffusion to be the rate-limiting process controlling the net evaporation rate of the droplet.

## I. INTRODUCTION

Droplet evaporation has been studied for over a century, starting with the works of Maxwell and Niven^{1} and Langmuir.^{2} The continued interest in this field is due to the role of droplet evaporation in applications such as evaporative cooling,^{3–7} fuel combustion,^{8,9} inkjet printing,^{8} microfluidics,^{10,11} desalination,^{12} and natural phenomena like dew formation and precipitation.^{9} Sessile droplet evaporation is a highly coupled heat and mass transfer phenomenon, often involving convection and diffusion of heat and vapor into the ambient atmosphere and heat conduction through the substrate. Besides, the self-cooling of the liquid droplet can result in a temperature gradient, which can lead to thermo-capillary flow.^{9} This process also depends on other parameters such as the surface topology and chemistry, ambient vapor composition, and temperature.^{9,13}

A quartz crystal microbalance (QCM) can be used to investigate the evaporation of liquids. QCMs are resonators consisting of a quartz substrate with metal electrodes. They are used extensively in deposition processes as thickness monitors with resolutions better than 1 nm. The high sensitivity to surface phenomena allows the use of QCMs to study evaporation^{14–18} and wetting characteristics of liquids.^{19–35} The use of a QCM to study droplet evaporation was first carried out by Joyce *et al*., who compared the evaporation of various alcohols qualitatively.^{17} Joyce *et al*. saw unique frequency signatures for each alcohol. They also noted the changes in the QCM response when the droplets were evaporating in the constant contact angle and the constant contact radius modes, which was also described by Picknett and Bexon.^{17,36} While the study by Joyce *et al*. compares the frequency response and the droplet retreating speed of several alcohols, the evaporation rates of the droplets were not quantified. Similarly, Pham *et al*. used a QCM to study evaporating droplets loaded with latex microparticles. Although the study compares the frequency responses of different evaporating droplets, it does not quantify the evaporation rates. Zhuang and co-authors have also used the QCMs to study droplets. Their studies cover different aspects of the interfacial phenomenon, including droplet spreading,^{37} colloidal suspensions,^{14} and rheological behavior of complex fluids like silicone oils.^{26} However, these studies do not consider droplet evaporation. Lee *et al*. investigated evaporating droplets on nanoporous superhydrophobic surfaces.^{18} This study found that droplet evaporation from a superhydrophobic surface followed a constant contact radius mode in the early stage of evaporation and a combination of a constant contact radius and constant contact angle modes without a Cassie–Wenzel transition in the final stage. Again, the evaporation rate was not quantified. Couturier *et al*. studied the evaporation of water droplets on gold electrodes by obtaining the frequency responses at multiple overtones or resonances. Similar to the study conducted by Joyce *et al*., Couturier *et al*. also note the variation in the frequency when the droplet is pinned vs when it is receding.^{15} However, they observed a weak increase in the frequency, although the droplet appeared pinned. They concluded that this observation is “related to the edge of the droplet,” and there was no quantification of the evaporation rate. Indeed, the QCM response is related to the contact line dynamics. However, this aspect was clarified much later in a study conducted by Murray and Narayanan, wherein the frequency response of a QCM loaded with a droplet was predicted using detailed computational modeling and experiments.^{21} They concluded that under most situations involving large droplets, where the droplet height is much larger than the viscous decay length, changes in the contact angle alone would be imperceptible in the frequency response of the QCM. On the other hand, the QCM is indeed quite sensitive to microscopic changes in the droplet radius.

The prior efforts indicate the potential to use QCMs to study evaporation of various liquids,^{14–18} wherein using a QCM in place of a traditional approach may allow for higher sensitivity and better resolution. This study uses a QCM, for the first time, to quantify the evaporation rate of a droplet. Unlike the previous approaches to study evaporation, this study leads to a quantitative relationship between evaporation flux and the frequency response of the QCM. Additionally, this method allows quantifying the evaporation rate with higher accuracy compared to techniques based on imaging alone. This technique is especially useful when a gravimetric approach and overhead imaging are infeasible.

## II. BACKGROUND THEORY

The Sauerbrey equation [Eq. (1)] relates the change in resonance frequency, $\Delta f$, to the apparent mass of a thin, rigid film, Δ*m*,^{38} where *ρ _{q}* and

*μ*are the density and shear modulus of the quartz substrate, respectively, and

_{q}*A*is the active electrode area of the QCM. Here, the unloaded frequency is $f0$ and the loaded frequency is

_{q}*f*,

For an unloaded QCM, the surface can be considered stress free. However, when submerged in a semi-infinite liquid with viscosity *μ* and density *ρ,* the viscoelastic coupling between the QCM and liquid will cause a decrease in resonant frequency. This decrease can be derived from a force-balance, as carried out by Kanazawa and Gordon^{39,40} [Eq. (2)]. For example, for a 10 MHz quartz crystal, the frequency shift when one side of the QCM is submerged in water is about −2000 Hz,

The oscillating crystal surface drags liquid close to it in phase with it. The velocity in the liquid quickly decays away from the surface, and the displacement is no longer in phase with the crystal surface. Mathematically, this is similar to the Stokes problem.^{21} The decay length *δ* of the velocity in the liquid is related to the fluid properties and the frequency of the QCM [Eq. (3)].^{28} For a 10 MHz QCM submerged in water, *δ* is close to 170 nm,

Equations (1) and (2) can be made analogous by relating the mass density, Δ*m/A _{q}* to

*δρ/*2,

^{28}which indicates that the mass layer is effectively

*δ/*2 thick. Consequently, most surface phenomena occurring farther than

*δ/*2 from the QCM are typically not sensed.

Surface roughness,^{16,41} hydrostatic pressure,^{42} the conductivity of the liquid,^{43,44} and compressional waves^{45–47} have known to affect the frequency response. Under certain conditions, these effects can be negligible. For example, for smooth polished QCM electrodes loaded with small droplets, the roughness and gravitational effects can be neglected. In the case of de-ionized water, the electrical conductivity is low (∼5*.*5 × 10^{−6} S/m), which will not influence the frequency of the QCM.^{44} For sessile droplets, while the effect of compressional waves is possible, it only occurs at specific radii and heights. In this case, the frequency response gets magnified when the droplets are placed off-center relative to the center of the QCM electrode.^{48,49} Since compression waves are minimal at the center of the electrode,^{45,46} we can also neglect their effect in small $(rd\u226are)$ centered droplets. Besides, the compression waves also become less significant for crystals of high natural frequency (*f*_{0 }*> *4 MHz) and planar surface.^{49}

The QCM response for sessile droplets is different compared to a semi-infinite liquid layer. In this regard, knowing the radial sensitivity of the QCM is essential, which depends on the velocity profile at the surface of the QCM.^{19,50} Prior studies assume the velocity distribution on the QCM surface to be a Gaussian function centered at the QCM electrode,^{19,50,51} which is supported by experiments^{46} as well as computational modeling.^{21} Using the radial sensitivity of the QCM across the droplet contact area, the frequency response of the QCM can be derived, as shown in Eq. (4).^{19} Here, we neglect the angular dependence of the QCM sensitivity, as done in prior studies.^{46,50}

Equation (4) describes the frequency shift based on the finite contact radius, *r*, of a fluid on a QCM with electrode radius *r _{e}*. For a large contact radius, Eq. (4) becomes Kanazawa's equation, Eq. (2). Here, the constant

*a*is intrinsic to the QCM. Although its value is not precisely determined, multiple efforts have reported values of unity for the type of QCM used in this work [Fig. 1(a)], which is a planar AT-cut crystal with concentric keyhole-shaped electrodes.

^{19,21,45,52}Consequently, $a=1$ in this work as well. However,

*a*could have other values since it depends on several factors, including the electrode patterns on the QCM and the crystal concavity.

^{19,45,52}This constant could be calibrated by placing a series of varying radius droplets at the center of the QCM and measuring their radius by other methods, such as overhead imaging and performing a least-squares fit of Eq. (4) to obtain

*a*for a particular QCM.

^{21}

Note that Eq. (4) does not indicate any contact angle dependence. Our previous study found the contact angle to be relevant only for microscopically tiny droplets.^{21} For example, for a water droplet of diameter 1 mm, using a 10 MHz QCM, changes in the contact angle can affect the frequency response only when the contact angle is less than 0.02°. For such low contact angles, the droplet height approaches the decay length, *δ,* which can affect the frequency response. This dependence is negligible for large contact angles, and the frequency response is related only to the droplet radius. We leverage this knowledge to quantify evaporation using the QCM.

Moreover, a thin film surrounding a macroscopic droplet can also affect the QCM's frequency response. However, we expect its contribution to the overall frequency response to be negligible. For example, for the QCM used in this study, Eq. (4) will yield a frequency response of −136 Hz for a 1 mm diameter water droplet. For a thin film surrounding the droplet, with the shape of an annulus of inner radius 0.5 mm, outer radius 0.6 mm, and a thickness of 10 nm, the frequency response would be −6.8 Hz, using Eq. (6) of Ref. 21 integrated from 0.5 to 0.6 mm with an areal mass density of 9.98 mg/m^{2}. In this extreme example involving a 100 *μ*m-wide thin film, the total frequency shift would be −142.8 Hz, wherein the contribution from the thin film is less than 5%. In reality, the thin film region can be significantly narrower and thinner, having a relatively smaller effect on the overall frequency response. Similarly, its contribution is negligible when analyzing large droplets. For example, a 100 *μ*m-wide and 10 nm-thick film surrounding a 2 mm diameter droplet will only contribute 2% toward the total frequency response.

## III. EXPERIMENTAL METHODS

We use two approaches to calculate the instantaneous droplet volume. The first approach is based entirely on droplet visualization (method 1), and the second approach uses a combination of imaging and QCM measurements (method 2), as listed in Table I.

. | Method 1 (M1) . | Method 2 (M2) . |
---|---|---|

Height | Imaging | Imaging + QCM |

Radius | Imaging | QCM |

. | Method 1 (M1) . | Method 2 (M2) . |
---|---|---|

Height | Imaging | Imaging + QCM |

Radius | Imaging | QCM |

Method 1 or M1 is image-based, which uses a single side-view image of a droplet to obtain the height as well as the radius. Method 2 or M2 is a QCM-based approach, which uses the frequency response to determine the radius, and the side-view image to determine the height. Both methods are performed simultaneously for comparison and use the same images for height determination.

For both methods, the calculation of height, $h=rtan(\theta /2)$, assumes a spherical cap, where the contact angle, $\theta $ was measured using a goniometer (Ramé-Hart), and droplet radius,$r$, was obtained from the respective radius measurements. In general, the contact angle approach is not strictly necessary, and height, *h* could be measured directly from images using a calibrated camera or a cathetometer. In this study, we use a goniometer to measure contact angles due to its high accuracy and sampling rate.

The goniometer, consisting of a light source and a camera, finds the contact angle and droplet radius by finding the liquid–vapor interface by locating the regions that correspond to the maximum change in the intensity of light or image contrast. The interface is then fit to a circular profile by a least-squares curve fit. The contact angle is found by numerical differentiation at the three-phase contact point. The width of the profile at this location determines the radius for M1. For accuracy, the droplet dimensions were obtained by scaling the images using a precision calibration target.

The radius measured by the QCM for M2 does not depend on the spherical cap assumption. Here, Eq. (4) is employed to find the radius. In order to calibrate the QCM, it is submerged in de-ionized water multiple times to obtain the average $\Delta f$, which corresponds to the $\Delta fK$ in Eq. (2) and the leading terms in Eq. (4). In this case, the error in $\Delta fK$ is the standard deviation of $\Delta f$ measured in the submersion trials. Equation (4) is then solved to determine *r*, the radius of the droplet. In each experiment, the unloaded frequency corresponds to the average frequency recorded before the deposition of the droplet.

The spherical cap assumption is a good approximation for droplets with dimensions smaller than the capillary length, $\kappa \u22121=\gamma /\rho g$, where $\gamma $ is the surface tension, and *g* is the acceleration due to gravity. All droplets examined in this work have dimensions smaller than the capillary length for water, which is approximately 2.7 mm. Larger droplets can be considered by replacing the spherical cap assumption with an axisymmetric assumption and using the droplet profile to determine the height, which along with radius from the QCM, can determine the volume.

Evaporation of sessile droplets was studied using a planar 10 MHz AT-cut and polished QCM with keyhole-shaped gold electrodes, as shown in Fig. 1(a). All experiments use de-ionized (DI) water. The frequency of the QCM was monitored and recorded by a commercially available system (eQCM, Gamry Instruments). The experiments were conducted by placing the QCM inside an enclosure maintained at the desired temperature and moisture, as illustrated in Fig. 1(b). The enclosure was supplied with nitrogen gas at low speeds to displace moist air. The humidity in the enclosure was held constant at 0%–2%. Temperature measurements indicate a constant temperature during the experiment in the enclosure. Measurements were acquired using humidity (Honeywell HIH-4000) and temperature sensors (J-Type thermocouples). A manual syringe with a 22-gauge needle was used to deposit droplets on the QCM surface. The needle entered the enclosure from the top through a 1/8 in. diameter hole. In this study, water droplets of approximately 2.5 *μ*l in volume were deposited onto a QCM. An IR camera (FLIR A655sc) monitored the temperature of the QCM during evaporation. Three environmental temperatures (26 °C, 35 °C, and 40 °C) were investigated to demonstrate the feasibility of using a QCM for measuring the evaporation rate of the droplets at different operating conditions. These enclosure temperatures, which were also the initial states of the QCM, were maintained by heating the enclosure walls and the nitrogen gas supply line.

The frequency response and the visual and infrared images were recorded as the droplet evaporated. The volume of the sessile droplet is calculated as $V=\pi (3r2h+h3)/6$ assuming a spherical cap, where *V*, *r*, and *h* are the volume, radius, and height of the droplet, respectively. The evaporation rate is determined by taking the time derivative of the volume. In order to minimize the noise in the calculated derivative of the droplet volume, a piecewise linear fit was applied to the raw volume data to determine the evaporation rate.

## IV. MODEL OVERVIEW

A computational model was used to compare with the evaporation rates determined by the QCM and visualization approach. Assuming evaporation to be quasi-steady, the model determines the evaporation rate for a given radius and height. The model, which is symmetric about the xz-plane, consists of a water droplet on a flat quartz surface surrounded by vapor, as shown in Fig. 2. The coupling of the water and vapor domains involves conduction, convection, radiation, and evaporation mechanisms, as described below. A commercial finite element software (COMSOL Multiphysics®^{53}) was used to solve this multiphysical system.

Since the characteristic velocity and temperature difference is relatively small in the liquid domain, heat transfer by convection and radiation are neglected compared to heat conduction, as shown in Eq. (5). Moreover, thermocapillary flow in small water droplets was found to be negligible by Girard *et al*.^{54} Therefore, Eqs. (5) and (6) govern the droplet and QCM domains with a velocity of $u=0$. Heat transfer in the vapor domain occurs by convection and conduction, as shown in Eqs. (5) and (6). For the 35 °C and 40 °C trials, the radiation from the enclosure walls on the droplet and QCM is also included, as given in Eq. (7),

Here ** q** is the heat flux vector,

*k*is the thermal conductivity,

*T*is temperature, $\rho $ is the fluid density, $Cp$ is the specific heat capacity, and $u$ is the velocity vector in the vapor domain. For radiation, $n$ is the vector normal to the surface, $\u03f5$ is the emissivity of the surfaces, taken to be 0.95 in the IR wavelength range, and $\sigma $ is the Stefan–Boltzmann constant. The density, specific heat capacity, and thermal conductivity above correspond to the respective domains on which Eqs. (5) and (6) are applied.

In the vapor domain, the temperature and moisture concentration gradients can affect the density causing a buoyancy-affected flow. The equations governing this flow are

Here, $\mu v$ is the dynamic viscosity of pure nitrogen gas, *g* is the acceleration due to gravity, $I$ is the identity matrix, and superscript *t* denotes transpose. The transport of water vapor through nitrogen gas occurs via diffusion and convection, which depends on the nonisothermal flow parameters, as shown in Eq. (10),

Here, *D* is the diffusion coefficient, and *c* is the concentration of water vapor in the vapor domain.

The study models evaporation at the liquid–vapor interface as a boundary heat sink to account for evaporative self-cooling, as shown in Eq. (11), where $Lv$ is the latent heat of evaporation,

The evaporation flux, $Jevap$, is determined from the vapor flux at the liquid–gas interface. For this calculation, the vapor concentration at the liquid–vapor interface is the saturation concentration corresponding to the interface temperature.

We solved the governing equations (5)–(11) with boundary conditions consisting of a flow inlet on the left side of the domain with 0% humidity and a velocity (∼10 cm/s) parallel to the QCM surface (see Fig. 2). The top, bottom, and right sides of the computational domain are flow outlets representing atmospheric pressure. The xz-plane represents the plane of symmetry. The QCM and droplet surface represent no-slip boundaries. All the required temperature-dependent material properties were obtained from Ref. 55, except for the specific heat capacity of quartz, which was obtained from Ref. 56. The density of dry nitrogen, $\rho N2$, was modified to include moisture, as shown in Eq. (12), where $MWandMN2$ are the molar masses of the water vapor and nitrogen gas, respectively. The thermal conductivity, specific heat capacity, and viscosity of the vapor domain were assumed to be the same as pure nitrogen gas since the effect of moisture on these properties will be less than 2%.^{57} All properties used in the computational model were temperature-dependent except for the thermal conductivity and density of quartz, which was assumed constant,

## V. RESULTS

### A. Frequency response of the QCM loaded with evaporating droplet

Figure 3(a) shows the normalized frequency response, $\Delta f/|\Delta fmax|=(f\u2212fo)/(fo\u2212fmin)$, of evaporating droplets at different temperatures. The normalized frequency was obtained from raw data shown in Fig. 3(b). After deposition, the first stage of evaporation results in a relatively constant frequency response even as the droplet is evaporating until the frequency begins to increase to reach the unloaded resonant frequency $(\Delta f=0)$.

In general, we can classify the frequency response into two regions, namely, to the constant contact radius (CCR) and the variable contact radius (VCR) modes of evaporation. Note that similar behavior was seen in previous studies as well. For example, Picknett and Bexon ^{36} had described the evaporation of droplets in the constant contact angle and constant contact radius modes in their study. The transition from a nearly constant frequency to increasing frequency response is not surprising, since Eq. (4) indicates that only changes in radius affect the frequency response. During evaporation, the CCR mode lasts until the contact angle reaches the receding contact angle. At this point, the droplet begins to recede (VCR mode), and the frequency begins to increase. Furthermore, higher temperatures result in faster evaporation of the water droplet. This variation is evident in Fig. 3(a) based on the total time taken for the normalized frequency to return to zero. We also observe a proportional decrease in the duration of evaporation in the CCR mode with increasing temperature. The VCR mode may involve a constant contact angle; however, this depends on various factors, including surface wettability and roughness. In this case, the VCR mode typically ends with a simultaneous decrease in both the contact radius and the contact angle. This aspect is evident in Fig. 4, which shows the variations in the contact angle, height, and contact radius of the droplets obtained using both methods (M1 and M2).

### B. Comparison of QCM and imaging-based characterization

Figure 4 shows the transition from the CCR to the VCR mode of evaporation with both methods M1 and M2, indicating when the droplet begins to recede. Although the radius and height calculated from M1 and M2 generally agree, minor differences do exist. For example, Fig. 4(a) shows a case where the variations of the radii from M1 and M2 do not agree between 450 and 600 s. The radius seen from the image (M1) drops quickly and then becomes stable, whereas the radius from the QCM (M2) is steadily decreasing. These differences mainly arise from the calculation of the droplet radius based on an image cross section, which may not always coincide accurately with the droplet center. Besides, the shape of the droplet contact area may also deviate from being perfectly circular and symmetric. However, these issues do not affect the QCM response since it responds to any variations in the contact area of the droplet. Thus, the effective radius determined by M2 monotonically decreases and captures the real change in the contact area of the droplet.

Figure 5 shows the precision in the radius measurement using M1, wherein the typical error in the radius measurements is ∼10 *μ*m. The error depends on the transition of the pixel color in each image at the edge of the droplet from black to white. This calculation involves taking a gradient of the grayscale image across the widest part of the droplet, which results in two peak values at the corners of the droplet. The separation of the peaks determines the diameter of the droplet. The half-width of each peak quantifies the error in determining the diameter, and in turn, the radius. This calculation does not account for any deviation of the focal plane from the exact center of the droplet, which would be the case when the droplet is not perfectly circular, central, and symmetric. For M2, the QCM response is directly related to the amount of area occupied by the droplet.^{21} Hence, the QCM does not require the precise measurement of two edges of the droplet. Using the frequency response of the QCM, the radius is measured with an error of ∼2 *μ*m, which is found by the propagation of uncertainty using Eq. (4).

Using the radius and height measured by M1 and M2, Fig. 6 shows the corresponding volumes throughout the evaporation process for each method and the respective temperatures. The volume vs time graphs are nearly linear; however, toward the end of the evaporation, the slope decreases, and the variation is non-linear. Both methods capture the decreased drop lifetime with increased temperature. The difference in the uncertainty of volumes measured using the two methods is directly related to the accuracy of measuring the radius of the droplet since the contact angle is measured using the goniometer in both M1 and M2. Hence, M2 is more accurate than M1 due to higher accuracy in its radius measurements. While Fig. 6 includes error bands, they are not easily noticeable. For comparison, Fig. 7 shows the error magnitudes using the two methods.

Figure 7 shows how the imaging approach of finding the radius in M1 and M2 can affect the error in the volume measurement. While the error in each trial has a consistent pattern, M2 is generally more accurate than M1. In the initial stages of evaporation, the normalized error of the image-based M1 is approximately 0.03 when the droplet is large while the normalized error of the QCM based M2 is around 0.005, six times more accurate even though both methods share the same contact angle information and the radius measurement for M2 is only four times more accurate than M1. This difference is because the propagated error in the volume measurement is more dependent on the measured radial error than the contact angle error.

### C. Quantifying evaporation dynamics using QCM and imaging

In order to determine the evaporation rates, linear segments are fit to the volume curves shown in Fig. 6. Figure 8 shows the normalized evaporation rate of the droplets at different temperatures using both M1 and M2. We calculated the evaporation rates using different segments of the volume vs time data and conducting a weighted least-squares linear regression. These segments are small enough to capture the decreasing trend in the evaporation rate. The weights, *w*, for each data point were calculated using the corresponding uncertainty in volume, *δ*, as $w=\delta \u22122$. Hence, the weight for each data point (i.e., volume) is inversely proportional to the uncertainty in the data point. Then an iterative approach described by York^{58} was used to determine the slope and its standard error for each line segment. The error bars in Fig. 8 represent the standard errors of the slope of the best fit line obtained by the weighted least-squares linear regression.

Figure 8 also shows the evaporation rate predicted by the computational model for different temperatures. Initial testing of the computational model showed little difference in results between using M1 or M2 droplet shapes. Hence, the droplet geometry in the quasi-steady model used the higher accuracy height and radius measurements from M2. In general, the evaporation rate decreases as the droplets get smaller, as seen in previous work.^{59} During the initial stages of the CCR mode, the evaporation rate decreases slowly due to the decreasing contact angle with a nearly constant radius. Subsequently, the evaporation rate begins to increase in the VCR mode since the droplet radius is also decreasing. This transition occurs at approximately 385 s, 255 s, and 175 s for the 26 °C, 35 °C, and 40 °C trials, respectively.

The rates calculated by both the experimental techniques show a similar trend to that of the computational model, although in general, M2 follows the model predictions more closely. Early in the CCR stage, when the droplet is large, the evaporation rate calculations by M1 and M2 are similar and match the model predictions. However, as the droplet contracts, both M1 and M2 deviate from the model predictions to different extents, as shown in Fig. 8. We find M2 to be closer to the model predictions than M1 based on the cumulative deviation evaluated as the root mean square error or $(\u222b(m\u02d9M\u2212m\u02d9C)2dt)/\Delta t$, where $m\u02d9M$ is the normalized evaporation rate measured using either M1 or M2, $m\u02d9C$ is the rate predicted by the computational model, and $\Delta t$ is the total duration. Table II compares the root mean square errors for M1 and M2 for different experiments. Here, the measured evaporation rate was linearly interpolated to match the time points of the model predicted evaporation rate so that integration could be performed. Considering all the data points, the closest agreement between M2 and the model prediction is in Fig. 8(b) (35 °C). Over the three trials, the average root mean square error of M1 is approximately 40% greater than the average error of M2.

. | 26 °C . | 35 °C . | 40 °C . | Average . |
---|---|---|---|---|

M1 | 0.051 | 0.101 | 0.209 | 0.121 |

M2 | 0.056 | 0.039 | 0.166 | 0.087 |

. | 26 °C . | 35 °C . | 40 °C . | Average . |
---|---|---|---|---|

M1 | 0.051 | 0.101 | 0.209 | 0.121 |

M2 | 0.056 | 0.039 | 0.166 | 0.087 |

The deviations seen in the measurements from the model could be due to multiple reasons. The deviations in the evaporation rate using M2 could arise from the estimation of the contact angle, which requires imaging. Furthermore, the droplets could be receding unevenly or non-uniformly toward the center of the QCM, as observed in the IR imaging [Fig. 9(a)]. The QCM sensitivity is also known to decrease away from the center, which can affect the analysis of large droplets. In the case of M1, the significant variations in the evaporation rate could arise from an incorrect estimation of the volume using only images. This variation can be significant for M1 as the droplet reduces in size since it fails to capture the geometry accurately; however, we see a better agreement between M2 and the model predictions due to the use of QCM to determine the droplet radius.

We also want to note that the computational model simplifies the complex nature of flow around the QCM, which could fail to capture any deviations in the evaporation rate due to subtle variations in the flow field, sudden pinning and de-pinning effects, causing disturbances in the boundary layers surrounding the droplet. In summary, although we see a general agreement in rates obtained using M1, M2, and model predictions, each procedure has its inherent limitations. Nevertheless, a combination of imaging, the frequency response of the QCM, and computational modeling could lead to a better understanding of factors affecting the evaporation behavior of liquids.

### D. Effect of self-cooling and evaporation area

In each trial, the droplet temperature decreases due to evaporation induced cooling. As the quartz substrate is insulating and has a small thermal mass (relative to the heat of vaporization), the droplet and the QCM provide the energy required for evaporation. This heat transfer results in a rapid decrease in the internal energy, and consequently, the temperature of the droplet and the quartz substrate, which is illustrated by infrared images of the QCM (Fig. 9). As the droplet decreases in size, the temperature increases gradually back to the initial temperature of the QCM.

From Fig. 8, we note that the evaporation rate decreases as the droplet becomes smaller, causing less evaporative self-cooling. This decrease allows the droplet and the QCM to gain energy from the ambient nitrogen stream, causing an increase in the temperature. The QCM temperature, which can be a proxy for the droplet temperature, is shown in Fig. 9(d). Here, the exposed quartz surfaces consisting of areas not coated with gold electrodes allow estimating the average temperature of the QCM from the IR images. The self-cooling of the droplet, and by extension, the QCM is evident in the temperature variation shown for each trial. As the ambient temperature increases, the evaporation rate and the rate of self-cooling increase, causing a more substantial decrease in the average temperature of the QCM.

It should be noted that the droplets were deposited at 26 °C for each experiment. However, the instantaneous droplet temperature is difficult to determine by IR imaging due to the angular dependence of emissivity,^{60} non-zero transmissivity, and the variable thickness of the water droplet. Considering the thermal time constant of the droplet (∼1 s), we expect it to equilibrate with the quartz substrate relatively quickly, thus enabling the use of quasi-steady modeling of droplet evaporation. In order to confirm this behavior, we conducted a three-dimensional thermal diffusion analysis to predict the temperature of the droplet and the QCM. Figure 10 shows the predicted variation in the temperature of the droplet and the QCM when the ambient temperature is 35 °C. In this case, after depositing a droplet, initially at 26 °C, the system reaches a quasi-equilibrium state in 5 s. Initially, a finite temperature difference of 9 °C exists between the QCM and the evaporating droplet, which subsequently approaches 3 °C after 5 s. As the droplet evaporates, the temperature difference between the QCM and the droplet decreases further, as shown in the IR images [Fig. 9(b)] and the computational results in Fig. 12, where the droplet approaches the QCM temperature.

This analysis shows that after initial equilibration, the droplet temperature follows a similar trend as the quartz substrate temperature, which is important since it affects the evaporation flux at the liquid–vapor interface. Figure 11 superposes the volumetric evaporation flux (mm^{3}/s/mm^{2}) on the average substrate temperature for different experiments, indicating that the evaporation flux follows the temperature variation and shows a similar trend. And as expected, a higher temperature corresponds to a larger evaporation flux and a shorter droplet lifetime. Although the overall trends are similar, unlike the gradual rise seen in the average temperature of the substrate, we can see minor deviations in the evaporation flux. As mentioned earlier, these deviations can be due to the non-idealities, which include pinning–depinning effects, and non-centric droplet evaporation that can affect the imaging accuracy. Such aspects are not captured in the computational model, which assumes ideal evaporation of a centrally located droplet on the quartz substrate. More importantly, we can see that the average evaporation flux (mm^{3}/s/mm^{2}) increases (Fig. 11), but the net evaporation rate (mm^{3}/s) decreases (Fig. 8). These trends indicate that the decrease in the net evaporation rate observed in the experiments is due to the reduction in the liquid–vapor interfacial area available for evaporation.

### E. Role of vapor advection and diffusion

Figure 12 shows the predicted temperature and vapor concentration distributions at three time-points for the 35 °C trial. Initially, the temperature contours on the droplet, and the QCM surface show large gradients [Fig. 12(a)], which decreases with time [Fig. 12(c)]. The minimum temperature of the droplet in Fig. 12(a) is 28.5 °C. The large temperature gradients agree with those viewed with the infrared camera [Figs. 9 and 10(b)], where the minimum droplet temperature is approximately 30 °C, and the QCM temperature is uniform.

The local cooling of the droplet affects the vapor concentration at the surface of the droplet, lowering the concentration gradients in the vapor, a contributing factor to the decreasing net evaporation rate over time [Fig. 8(b)]. Subsequently, the droplet and the QCM do gain energy from the ambient, which results in a steady rise in temperature. As the temperature gradients decrease, the droplet approaches the QCM temperature [Fig. 12(c)]. This steady rise in droplet temperature should enhance the local evaporation flux. Meanwhile, a continuous decrease in the droplet size lowers the net evaporation area. For the experiments conducted in this study, the net result of these competing effects was an evaporation rate that continually decreased, as shown in Fig. 8(b). In other words, the increase in the droplet temperature did not have as much effect as the decreasing surface area of the droplet. Indeed, Figs. 8(a) and 8(c) demonstrate non-monotonicity in the evaporation rate. However, this variation is not due to the droplet temperature. Any change in droplet temperature would be reflected in the QCM temperature because of the low heat capacity of the quartz crystal, ∼48 mJ/K. In Fig. 9(d), the QCM temperature is either constant or monotonically increasing. Therefore, the droplet temperature must also be monotonic. Hence, the non-monotonic variations in the evaporation rate are due to other effects described earlier.

In Figure 12, the contours of the vapor concentration, as relative humidity (RH), are also shown as projections on the computational boundaries. The disturbance from the inlet velocity skews the concentration profiles to the left. While vapor advection distorts the RH contour at 10%, the inner contour rings (higher RH values) are nearly circular, indicating that vapor transport near the droplet surface is not significantly affected by vapor advection but controlled by vapor diffusion. This behavior can also be explained based on the mass transfer Peclet number, $Pe=ud/D$, where *u*, *d*, and *D* denote free stream velocity, droplet diameter, and vapor diffusivity in air, respectively. In this case, $Pe\u223c10$ for a substantial duration of evaporation. Hence, any disturbance in the low-speed ambient gas flow is small and only affects the evaporation rate slightly.

Finally, although the predicted evaporation rate trend is in agreement with M2, the quasi-steady computational model cannot capture non-ideal environmental factors such as changes in the flow patterns. Additionally, the non-ideal behavior of the droplet, such as contact line de-pinning, is not captured using the computational model. Such non-ideal phenomena can influence the observed rates of evaporation, and the QCM can capture these effects because of its highly accurate determination of radius. This capability makes the QCM a reliable tool for capturing the evaporation rate of sessile droplets.

## VI. CONCLUSIONS

Evaporation of droplets is an essential phenomenon with wide-ranging applications. Therefore, understanding the role of various factors affecting evaporation dynamics is crucial. Although quartz crystal microbalances (QCMs) have been used to study droplets and evaporation, previous studies have not used QCMs to quantify evaporation rates of liquids. In this work, we quantify the evaporation of water droplets by combining QCM and droplet imaging. We make use of the QCM to measure the instantaneous radius of a sessile droplet while the droplet visualization provides the contact angle. These measurements allow the calculation of the droplet volume and the evaporation rate. In order to test the validity of this approach, we compare the evaporation rate obtained using the QCM with model predictions and a technique using imaging only. We also discuss the physics governing the evaporation process for the conditions analyzed in this study.

This study uses a dry nitrogen stream at different temperatures to cause evaporation. In all environments, although the evaporation rate obtained by the image-based and the QCM-based approach generally agree, they start to differ as the droplet volume decreases. We also found that the QCM results follow the monotonic variation in the evaporation rate predicted by the computational model. Compared to the imaging-only approach, the QCM provides a better match with the model predictions due to higher accuracy in determining the droplet radius, and consequently, droplet volume and evaporation rate. This work shows that a QCM is a useful tool for determining the sessile droplet evaporation rate through its application as a high accuracy radius sensor, especially in environments where gravimetric or overhead imaging is not possible.

In this study, the QCM approach captures the enhancement in the evaporation rates when the water droplet is in a nitrogen stream of higher temperatures. With infrared imaging and computational modeling, we study the effects of droplet self-cooling, heat transfer with the ambient and droplet size on the net evaporation rate. Due to the relatively low thermal mass of the QCM, depositing droplets of low temperature and subsequent evaporation can result in significant thermal gradients. After some time, these gradients become less prominent due to heat transfer from the ambient. Although this heat transfer results in a steady increase in the droplet temperature, which supports a rising evaporation flux, it is not sufficient to offset the effect of the shrinking area of evaporation. Hence, we observe a decreasing trend in the overall evaporation rate. With droplets of diameter ∼1 mm, evaporating into dry nitrogen atmospheres at 25–40 °C, we found vapor diffusion to be the rate-limiting mechanism. We find the QCM-imaging combined approach to introduce less error into the quantifying of the evaporation rate than using droplet visualization only. Additionally, this approach provides better accuracy as the droplet shrinks in size. This technique is especially useful for elucidating the evaporation phenomenon when overhead imaging or gravimetric approaches are not viable. Furthermore, while the current study quantifies the evaporation of pure water into a dry nitrogen environment, similar techniques can be used in the presence of active heating, humid environments, and liquid mixtures to elucidate the evaporation process.

## ACKNOWLEDGMENTS

This paper is based upon work supported by the National Science Foundation's Division of Chemical, Bioengineering, Environmental, and Transport Systems in the Directorate for Engineering under Grant No. 1944323.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (105 Pascals) Pressure and at Higher Temperatures*(U.S. Government Printing Office,