# Exploring fluorescence lifetime potential for analyzing the composition of vaporizing bicomponent droplets

Minghao Wang, Mehdi Stiti, Hadrien Chaynes, Guillaume Castanet

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Minghao Wang, Mehdi Stiti, Hadrien Chaynes, Guillaume Castanet. Exploring fluorescence lifetime potential for analyzing the composition of vaporizing bicomponent droplets. Experiments in Fluids, 2025, 66 (10), pp.181. 10.1007/s00348-025-04108-8. hal-05288384

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# Exploring fluorescence lifetime potential for analyzing the composition of vaporizing bicomponent droplets

Minghao Wang<fnref n="1" /> · Mehdi Stiti<fnref n="2" /> · Hadrien Chaynes<fnref n="1" /> · Guillaume Castanet<fnref n="1" />

Accepted: 20 August 2025

© The Author(s) 2025

## Abstract

Predicting droplet evaporation is particularly complex when the liquid phase consists of multiple components. To date, only a limited number of physical optical phenomena have been used to non-intrusively measure the composition of droplets. Laser-induced fluorescence is a promising approach, as the emission and absorption of certain fluorescent dyes are known to depend on solvent polarity, viscosity, and, more generally, the chemical environment. However, a challenge is that fluorescence signal intensity is generally sensitive to both temperature and composition. This study investigates fluorescence lifetime measurements as a robust alternative. We demonstrate that, with a well-chosen fluorescent dye, it is possible to measure the composition of bicomponent droplets using a single dye and a single detection band, with minimal constraint on detection band selection, and without ambiguity due to temperature variations. To validate the technique, it is applied to acoustically levitated droplets across several mixtures that exhibit markedly different behaviors.

## 1 Introduction

Droplet evaporation plays a pivotal role in numerous engineering applications, such as evaporators, combustors, gas scrubbers and spray dryers. In these processes, dispersing liquid into fine droplets significantly accelerates heat and mass transfer by greatly increasing the surface area between the liquid and gaseous phases. In most practical cases, droplets consist of a mixture of several components with different volatilities, making the evaporation process increasingly complex (Brenn et al. 2007; Liu et al. 2016; Jiménez-García and Rivera 2023). Over the evaporation process, the composition of these droplets evolves, modifying the physical properties of both the liquid and vapor phases. Concentration gradients develop inside the droplets due to convective transport and species diffusion. Additionally, if the mixture exhibits non-ideal behavior, advanced thermodynamic models may be required to accurately define the parameters governing liquid-vapor equilibrium at the droplet surface, particularly the chemical activity (Poling et al. 2001). Given

the many challenges involved in modeling the evaporation of multicomponent droplets, obtaining some insights through measurements of the droplet’s chemical composition would be highly beneficial. To date, only a limited number of physical phenomena have been employed for non-intrusive measurement of droplet composition in sprays (Lemoine and Castanet 2013). Some of these methods are sensitive to both temperature and composition, which poses challenges and limits their applicability, as both variables change simultaneously during the vaporization of multicomponent droplets.

Raman spectroscopy is a widely used technique for detecting chemical species (Pan et al. 2022). By analyzing the positions and intensities of peaks in a Raman spectrum, quantitative information about chemical composition can be obtained. However, the weak Raman signal presents a challenge when working with small scattering particles such as droplets in sprays. The application of Raman spectroscopy becomes even more difficult due to morphology-dependent resonances (MDR), which can alter the relative intensities of the Raman peaks, making the interpretation of the spectrum quite complex (Owen et al. 1982). Rainbow refractometry has demonstrated significant capability in measuring the temperature and composition of droplets in sprays (Lemoine and Castanet 2013). The angular position of the rainbow produced by individual droplets or groups of droplets shifts as the refractive index changes with composition and temperature. Composition measurements have been

<footnote>Mehdi Stiti
mehdi.stiti@cnrs.fr</footnote>

<footnote n="1">¹ LEMTA, Université de Lorraine, CNRS, 5400 Nancy, France</footnote>

<footnote n="2">² IMFT, Université de Toulouse, CNRS, 3100 Toulouse, France</footnote>
successfully achieved in n-alkanes, based on the variation of the refractive index with the number of carbon atoms. More recently, Promvongsa et al. (2017) characterized the composition of ethanol/water droplets. A key advantage of rainbow refractometry is its strong sensitivity to refractive index gradients near the droplet surface. However, this sensitivity often requires assumptions regarding the spatial distribution of the refractive index inside the droplet when composition gradients are suspected. Furthermore, in some cases, the relationship between refractive index and component volume fraction is not strictly monotonic, leading to reduced accuracy within certain composition ranges. Laser-induced fluorescence (LIF) can be used to characterize the chemical composition of droplets (Castanet et al. 2007), with approaches primarily based on the fluorescence signal intensity. Maqua et al. (2007) studied the vaporization of acetone droplets, a mixture in which acetone fluoresces in the UV range, unlike ethanol. Due to acetone's higher volatility compared to ethanol, the fluorescence intensity gradually decreases as the droplets vaporize. To accurately interpret this reduction, a fluorescence model is needed to account for variations in droplet size and laser absorption by acetone, both of which contribute to the decrease in fluorescence, along with the compositional changes.

Other types of LIF measurements have been made possible by identifying fluorescent dyes whose emission and absorption spectra are sensitive to the liquid solvent (polarity, viscosity, pH, etc.). Koegl et al. (2020a, b) characterized the composition of *n*-decane/butanol blends using Nile Red after observing significant shifts between absorption and emission spectra when varying butanol volume fractions in the mixture. Similarly to droplet temperature measurements in sprays (Mishra et al. 2016), a ratiometric approach was used to eliminate factors that affect the fluorescence signal, such as droplet size and position relative to the optics. However, Nile Red is highly sensitive to temperature, which rendered these measurements ambiguous, as it was impossible to control the temperature of the multicomponent droplets during evaporation. Maqua et al. (2008) developed a ratiometric method based on three spectral detection bands. Using two intensity ratios, rather than just one, can theoretically allow for determining both temperature and composition. However, accurately measuring composition with this approach proves challenging due to the difficulty of identifying detection bands with clearly separated sensitivities to temperature and composition.

In this study, the chosen approach is based not on fluorescence intensity, but on fluorescence lifetime, which refers to the average time that fluorescent molecules remain in the excited state. Fluorescence lifetime measurements have recently emerged as an attractive solution for measuring droplet temperature in sprays (Mehdi et al. 2021; Wang et al. 2022). Unlike intensity, lifetime is an absolute quantity.

Measurements in droplet flows can be significantly improved and simplified, offering an excellent signal-to-noise ratio through single-photon counting. They also require only a single detection band and are minimally dependent on the selected detection wavelengths (Mehdi et al. 2021). Using an ultrashort pulse laser, it is relatively straightforward to achieve two-photon excitation of the fluorescence, which helps eliminate the effects of multiple light scattering in dense sprays (Wang et al. 2022). Although a few studies in fluid mechanics, such as (Mendels et al. 2008; Stiti et al. 2024), have used fluorescence lifetime to measure composition in fluid flows, their work did not fully address the specific challenges posed by simultaneous changes in temperature and composition, as well as the enrichment of fluorescent dye when evaporation of the liquid solvent takes place.

This study will examine the fluorescence lifetime of various dyes as a function of key parameters, including temperature, dye concentration and liquid solvent, to identify the most suitable candidates for measuring chemical composition in complex mixtures. Subsequently, the focus will transition to characterizing the composition of droplets formed from different binary mixtures undergoing evaporation under acoustic levitation.

## 2 Fluorescence lifetime in different liquid solvents

In this section, we examine the effect of solvent and temperature on the fluorescence lifetime of various dyes. Although not exhaustive, the findings highlight distinctive behaviors, which are essential to explain how to prepare the droplet evaporation experiments.

### 2.1 Fluorescence dyes and liquid solvents

Widely used in dye lasers, the studied dyes exhibit relatively high quantum efficiency and strong resistance to photodegradation. They include Eosin Y (EY), Oxazine 4 (Ox4), Pyromethene 556 (PM556), Rhodamine 640 (Rh640), Rhodamine 6G (Rh6G) and Rhodamine B (RhB). For reference, Fig. 1 shows the emission and absorption spectra of these dyes in water at 293 K and pH=6.8 (measured prior to their dilution into water). Spectra were recorded at low dye concentrations (10⁻⁷ mol L⁻¹) and with an optical path length around 1 mm, ensuring a negligible effect of fluorescence reabsorption.

The solvents considered in this study are listed in Table 1; some are protic, including water, methanol, ethanol, *n*-propanol, isopropanol, *n*-pentanol, ethylene glycol and glycerol, while others are aprotic, such as acetone and 3-pentanone. Due to their ionic nature, several of the studied dyes are

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Absorption and emission spectra of EY, OX4, PM556, Rh640, Rh6G and RhB in water (T=293 K, pH=6.8) Emission spectra were acquired using a laser at 490 nm</figcaption></figure>

<table><caption>**Table 1** Properties of the tested liquids. The values of ε<sub>r</sub>, n, δ, η and ET(30) were obtained from the Stenutz database for Organic Chemistry (https://www.stenutz.eu/chem/, 2024)</caption><thead><tr><th></th><th>Relative permittivity ε<sub>r</sub> [-]</th><th>Refractive index n [-]</th><th>Dipole moment δ [D]</th><th>Orientation Polarizability Δν [-]</th><th>ET(30) [kcal/mol]</th><th>Viscosity η [Pa s] × 10<sup>-3</sup></th></tr></thead><tbody><tr><td><strong>1-Hexanol</strong></td><td>13.3</td><td>1.418</td><td>1.60</td><td>0.244</td><td>48.8</td><td>4.59</td></tr><tr><td>Pentanol</td><td>13.9</td><td>1.409</td><td>1.70</td><td>0.249</td><td>49.1</td><td>4.00</td></tr><tr><td><strong>3-Pentanone</strong></td><td>17.3</td><td>1.391</td><td>2.70</td><td>0.266</td><td>39.3</td><td>0.47</td></tr><tr><td>2-Propanol</td><td>18.2</td><td>1.377</td><td>1.59</td><td>0.273</td><td>49.2</td><td>2.40</td></tr><tr><td><strong>1-Propanol</strong></td><td>20.1</td><td>1.383</td><td>1.65</td><td>0.274</td><td>50.7</td><td>2.30</td></tr><tr><td>Acetone</td><td>21.4</td><td>1.359</td><td>2.69</td><td>0.285</td><td>42.2</td><td>0.36</td></tr><tr><td><strong>Ethanol</strong></td><td>24.5</td><td>1.361</td><td>1.66</td><td>0.288</td><td>51.9</td><td>1.10</td></tr><tr><td>Propylene glycol</td><td>32.0</td><td>1.431</td><td>2.27</td><td>0.271</td><td>54.1</td><td>43.4</td></tr><tr><td><strong>Methanol</strong></td><td>32.6</td><td>1.328</td><td>2.87</td><td>0.308</td><td>55.4</td><td>0.59</td></tr><tr><td>Ethylene glycol</td><td>41.4</td><td>1.432</td><td>2.27</td><td>0.276</td><td>56.3</td><td>20</td></tr><tr><td><strong>Glycerol</strong></td><td>42.5</td><td>1.475</td><td>2.56</td><td>0.263</td><td>57</td><td>1412</td></tr><tr><td>Water</td><td>80.1</td><td>1.333</td><td>1.87</td><td>0.320</td><td>63.1</td><td>1</td></tr></tbody></table>

The solvent polarizability Δν is calculated using the formula $\frac{\epsilon_r - 1}{2\epsilon_r + 1} - \frac{n^2 - 1}{2n^2 + 1}$, where n and ε<sub>r</sub> represent the refractive index and relative permittivity of the solvent, respectively.

insoluble in nonpolar solvents, such as *n*-alkanes; Rhodamine B (RhB) being one example.

The solvent is well known to significantly influence the fluorescence of dye molecules (Lakowicz 2006). One of the most documented effects is solvatochromism, where the Stokes shift, the wavelength difference between the absorption and emission maxima of the dye varies with the solvent polarity. Fluorescent molecules generally acquire a larger dipole moment upon excitation. Consequently, in polar solvents, surrounding solvent molecules

tend to reorient themselves around the excited fluorescent molecules. This effect intensifies with increasing solvent polarity, leading to a lower energy in the excited state and, therefore, emission at longer wavelengths. The dielectric constant values shown in Table 1 offer an estimation of solvent polarity. Generally, solvents with a dielectric constant (ε<sub>r</sub>) below 15 are classified as nonpolar. The orientation polarizability has also been proposed as a relevant parameter for the solvent relaxation mechanism outlined above (reference). However, neither the dielectric constant
($\epsilon_r$) nor the orientation polarizability ($\Delta\nu$) can fully account for all the effects of solvent polarity. For instance, intramolecular charge transfer (ICT), which involves the redistribution of charge between different regions of the molecule from an electron-donating group to an electron-accepting group can be significantly affected by the solvent polarity in certain fluorescent molecules. This process leads to structural changes that stabilize the molecule in the excited state. In polar solvents, the energy of the ICT state is typically lower due to solvent stabilization of the charge-separated state, which reduces the energy gap between absorption and emission, causing a redshift or reduced frequency shift in the emitted fluorescence (Lakowicz 2006). Due to the intricate and complex nature of the phenomena associated with solvatochromism, empirical scales based on reference dyes are commonly employed as indicators (Reichardt 1994). One of the most widely used empirical scales is the ET(30) scale, which represents the transition energy in kcal/mol between the ground state and the lowest excited state of betaine. Table 1 displays the ET(30) values corresponding to the solvents selected for this study. It is noteworthy that ET(30) values are relatively well correlated with the solvent polarity indicated by the dielectric constant $\epsilon_r$.

It is worth mentioning that solvatochromism does not exclusively stem from the polarity of the solvent. For example, in certain fluorescent species known as molecular rotors (Jiao 2019), the fluorescence quantum yield increases in highly viscous solvents. This phenomenon occurs because the reduced molecular mobility in viscous environments limits collisional quenching, thereby enhancing fluorescence efficiency.

Solvatochromism has been extensively studied through studies of absorption and emission spectra. In contrast, data on fluorescence lifetime are less accessible, and correlating fluorescence lifetime with solvatochromism is not straightforward. This makes it challenging to identify dyes suitable for measuring chemical composition based on fluorescence lifetime and motivates this study.

## 2.2 Measurement of the fluorescence lifetime

Fluorescence lifetime measurement is conducted using the time-correlated single-photon counting (TCSPC) method. Our experimental setup, shown in Fig. 2, is comparable to that used in a previous study by Mehdi et al. (2021) for measuring droplet temperature in a spray. A femtosecond Ti/Sapphire laser (Coherent Chameleon Ultra II) emits short pulses with a width of 140 fs and a wavelength tunable between 690 and 1080 nm. To achieve photoexcitation of the dyes in the visible range, the laser frequency is doubled using a second harmonic generator (SHG, HarmoniXX from APE GmbH). A hybrid photomultiplier tube (PMT, R10467, Hamamatsu), cooled by a Peltier cell, is used to detect and count single photons. A bandpass filter is mounted in front of the PMT to limit perturbations due to the ambient light. The filter's bandwidth and central wavelength are selected based on the emission spectrum of the dye. For EY, RhB, Rh6G and PM556, the detection band is [570–580 nm]. For Rh640 and Ox4, the detection is shifted to the spectral band [610–630 nm] more suitable for obtaining signal. The output current from the photomultiplier tube (PMT) is sent to a time-correlating counter (HydraHarp 400 controller, PicoQuant GmbH) to record photon arrival times. A fast photodiode (TDA 200 from PicoQuant GmbH), connected to the same timing electronics, receives laser light reflected by a beam splitter (99% transmission, 1% reflection) positioned at the laser output and serves as the time reference. The time difference between the laser pulse emission, received by the fast photodiode, and the detection of a photon by the PMT is measured repeatedly for a large number of laser pulses. These measurements are recorded in a histogram showing the number of photons as a function of time. Examples of resulting histograms are shown in Fig. 3A, illustrating the fluorescence decay of EY over several nanoseconds with a logarithmic scale on the y-axis for different solvents. Fluorescence is commonly assumed to follow an exponential decay:

<figure><img src="image_4.png" /><figcaption>**Fig. 2** Experimental setup for fluorescence lifetime measurements</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3 A Fluorescence decay of EY in various solvents. These measurements were obtained in a cuvette at room temperature ($C_{EY}$ = $10^{-6}$ mol/L, $T$ = 293 K). B Fluorescence lifetime of EY as a function of the polarity scale ET(30)</figcaption></figure>

$$F(t)/F(t = 0) = e^{-t/\tau}, \qquad (1)$$

Here, $\tau$, the fluorescence lifetime, represents a characteristic time for the decay of the dye's fluorescence. However, in practice, rather than a mono-exponential law, the fluorescence response $F(t)$ is often better represented by a combination of two exponential functions (Boens et al. 2007; Casadevall et al. 2010; Kristoffersen et al. 2014; Mehdi et al. 2021):

$$F(t)/F(t = 0) = ae^{-t/\tau_1} + (1-a)e^{-t/\tau_2}, \qquad (2)$$

where $a$ is an average parameter between 0 and 1. The parameters $a$, $\tau_1$ and $\tau_2$ are determined experimentally by fitting the decay curve using a nonlinear least squares method, which aims to minimize the difference between the theoretical decay described by Eq. (2) and the experimentally measured decay. From Eq. (2), an equivalent fluorescence lifetime, named $\tau$ hereafter, can be computed as follows:

$$\tau = a\tau_1 + (1-a)\tau_2. \qquad (3)$$

The timing jitter of the various components in the TCSPC chain is a crucial factor in measurement precision. The instrument response function (IRF), which represents the time spread introduced by all components, was evaluated at a width of 90 ps at half maximum (FWHM). This is small compared to the fluorescence lifetime, which is on the order of nanoseconds for the fluorescence species considered.

Nevertheless, the theoretical model in Eq. (2) must be corrected by convoluting the IRF with the decay model to obtain a more accurate assessment of $a$, $\tau_1$ and $\tau_2$.

The TCSPC technique is notable for its high signal-to-noise ratio. The noise level (count rate in the dark) is approximately 50 counts per second (Cps), while the count rate during measurements typically ranges from $10^5$ to $10^6$ Cps for both the calibrations in a cell and the results obtained on the acoustically levitated droplet. This high SNR allows for the detection of variations of the fluorescence signal over more than three decades, as demonstrated in the example of fluorescence decay presented in Fig. 3B. This capability is expected to facilitate future applications of the measurement technique in dense sprays, where light is significantly attenuated by scattering (Wang et al. 2022). The uncertainty in the measurement of the lifetime $\tau$ was estimated to be approximately 0.03 ns. This estimation is discussed in Appendix 1, based on the scattering of the measurement points.

## 2.3 Effect of the solvent on the fluorescence lifetime

Fluorescence times were measured in various solvents, with the results summarized in Table 2. All dyes exhibit sensitivity to their chemical environment to varying degrees. Except for RhB, the longest lifetimes are observed in acetone, a solvent that stands out particularly for Rh640 and Rh6G. Among the dyes, EY shows a marked solvent dependency, with its lifetime increasing by a factor of 3.7 between water
<table><caption>**Table 2** Fluorescence lifetime of the dyes in various solvents at room temperature (293.15 K)</caption><thead><tr><th></th><th>EY (ns)</th><th>OX4 (ns)</th><th>Rh640 (ns)</th><th>RhB (ns)</th><th>Rh6G (ns)</th><th>PM556 (ns)</th></tr></thead><tbody><tr><td>1-Hexanol</td><td>2.94</td><td>3.21</td><td>3.99</td><td>2.63</td><td>3.26</td><td>3.51</td></tr><tr><td>Pentanol</td><td>2.99</td><td>3.18</td><td>4.05</td><td>2.74</td><td>3.56</td><td>4.33</td></tr><tr><td>3-Pentanone</td><td>3.8</td><td>3.86</td><td>4.46</td><td>2.43</td><td>4.38</td><td>5.06</td></tr><tr><td>2-Propanol</td><td>2.83</td><td>3.42</td><td>4.22</td><td>2.72</td><td>3.67</td><td>4.34</td></tr><tr><td>1-Propanol</td><td>2.82</td><td>3.35</td><td>4.55</td><td>2.88</td><td>3.73</td><td>4.37</td></tr><tr><td>Acetone</td><td>4.03</td><td>5.02</td><td>6.13</td><td>1.9</td><td>4.89</td><td>5.08</td></tr><tr><td>Ethanol</td><td>2.92</td><td>3.45</td><td>4.67</td><td>2.49</td><td>3.94</td><td>4.61</td></tr><tr><td>Propylene glycol</td><td>2.41</td><td>2.91</td><td>4.03</td><td>2.99</td><td>3.57</td><td>4.4</td></tr><tr><td>Methanol</td><td>2.77</td><td>3.65</td><td>4.61</td><td>2.08</td><td>4.11</td><td>4.85</td></tr><tr><td>Ethylene glycol</td><td>2.39</td><td>2.91</td><td>3.92</td><td>2.51</td><td>3.41</td><td>4.34</td></tr><tr><td>Glycerol</td><td>2.11</td><td>2.74</td><td>3.94</td><td>3.33</td><td>3.78</td><td>3.96</td></tr><tr><td>Water</td><td>1.20</td><td>2.5</td><td>4.22</td><td>1.66</td><td>3.8</td><td>4.71</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 4** Fluorescence lifetime of RhB as a function of the solvent viscosity at 293 K. The dashed lines are only included to highlight the trend</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5** Influence of temperature on the fluorescence lifetime of the dyes in water. The dye concentration was set at $10^{-5}$ mol/L. Measurements were performed using deionized water at initial pH=6.8 at 293 K</figcaption></figure>

and acetone. Figure 3A presents the fluorescence decays of EY dissolved in four solvents: acetone, ethylene glycol, ethanol and water, showing a distinct response in each, with a faster decay in water and a longer one in acetone.

As previously discussed, identifying the mechanisms underlying the solvent's effects can be challenging, as often more than one phenomenon can be involved (Lakowicz 2006). For EY and, to a certain extent, OX4, it is noteworthy that the fluorescence lifetime is longer in aprotic solvents like acetone and 3-pentanone, while in protic solvents, particularly water and bi- or tri-alcohols, the lifetime is shorter, decreasing with increasing solvent polarity. This behavior is illustrated in Fig. 3B, where the fluorescence lifetime of EY decreases approximately linearly with the ET(30) scale. In the case of RhB, viscosity appears to have a significant impact on fluorescence lifetime, as already observed in several studies (Moog et al. 1982). Figure 4 reveals a strong

correlation between the measured fluorescence lifetimes and dynamic viscosity. However, the data indicate lower fluorescence lifetime values for water and polyalcohols (represented in orange) compared to other solvents (represented in blue).

## 2.4 Effect of the temperature

To characterize the effect of temperature, lifetime measurements were performed in a 50 mL glass cuvette equipped with an immersed heating resistor. Figure 5 illustrates the effect of temperature on various fluorescent dyes dissolved in water. Similar trends were observed in other solvents, but for clarity, only the subset of results in water is shown
here. Among the dyes tested, RhB and PM556 exhibit the highest temperature sensitivity. For RhB, this temperature dependency is commonly attributed to the flexibility of the diethylamino groups on its xanthene core, which accelerates non-radiative internal conversion and is further influenced by the temperature-dependent viscosity of water (Casey and Quitevis 1988). To the best of our knowledge, temperature effects on PM556 have not been previously reported. For other dyes, the temperature effect is relatively minor, with only a slight decrease for Ox4 observed over a temperature range of several tens of K.

For EY and RhB, Fig. 6 highlights the effect of temperature across various solvents. In Fig. 6A, the fluorescence lifetime of RhB is observed to vary with temperature in a similar manner across most solvents. However, in the case of ethylene glycol, the decay occurs at a noticeably faster rate. Temperature variations of RhB fluorescence lifetime can be described quite well by an Arrhenius equation. Further details on the coefficients used to obtain the dotted-line curves plotted in Fig. 6A are provided in Appendix 2. The variations of RhB's lifetime make it attractive for temperature measurements. However, it is important to note that, in mixtures, the composition should remain relatively stable, as both temperature and composition significantly influence the fluorescence lifetime. Figure 6B demonstrates that the effect of temperature on EY's fluorescence lifetime remains weak across different solvents. To confirm this observation, additional measurements were conducted in various mixtures, taking care to minimize evaporation in the test cell. It is intriguing to note that EY is known as an effective temperature probe and has been used as such in several studies,

though with measurements based on its fluorescence intensity (Koegl et al. 2020b). Our results and those extracted from Koegl et al. (2020b) in Fig. 7 suggest that EY's temperature dependence in fluorescence signal arises from the absorption cross section at the laser wavelength rather than the fluorescence quantum yield, which is proportional to the fluorescence lifetime τ.

## 2.5 Effect of the dye concentration

As the solvent evaporates from a droplet, the dye molecules remain in the liquid phase since they are not volatile. This results

<figure><img src="image_8.png" /><figcaption>Fig. 7 Influence of temperature on the emission and absorption spectra of EY in water. Data extracted from Koegl et al. (2020b)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6 Influence of temperature on the fluorescence lifetime of RhB and EY in various solvents. A Case of RhB at a concentration of 10⁻⁶ mol/L, B Case of EY at a concentration of 10⁻⁵ mol/L</figcaption></figure>
in a gradual increase in dye concentration, which can eventually affect the fluorescence signal through processes such as fluorescence reabsorption, self-quenching and dimerization of the fluorescent molecules. These effects are briefly discussed below for several dyes, in particular, EY.

i. **Self-quenching:** Energy transfer between neighboring fluorescent molecules is an important process that becomes predominant as the distance between them decreases. A donor fluorophore in the excited state transfers energy to an acceptor molecule through non-radiative dipole-dipole coupling, resulting in fluorescence quenching, as the acceptor has a limited probability of photon emission afterward. To investigate this effect, two-photon excitation was used to position the excitation spot near the interface between the fluorescent solution and the glass cell, minimizing potential reabsorption effects. The laser wavelength was tuned to 980 nm for EY and 800 nm for RhB and Rh6G. Additionally, fluorescence detection was conducted within the 570-590 nm spectral band, which is unaffected by reabsorption. As the dye concentration approaches $10^{-3}$ mol/L, a marked decrease in fluorescence lifetime was observed for RhB, Rh6G and EY and attributed to energy transfer between fluorescent molecules (Fig. 17 in Appendix 3). The efficiency $E$ of energy transfer can be evaluated using the following expression:

$$E = 1 - \tau/\tau_0, \qquad (4)$$

where $\tau_0$ is the fluorescence lifetime at low concentration, when self-quenching is negligible. Considering self-quenching a form of Förster Resonance Energy Transfer between identical fluorophore molecules (homo-FRET), the efficiency $E$ can be expressed as (Lakowicz 2006; Berezin and Achilefu 2010):

$$E = \frac{R_0^6}{R_0^6 + r^6}, \qquad (5)$$

where $R_0$ is the Förster distance. The distance $r$ separating the fluorescent molecules can be evaluated by $0.1 \cdot (C N_{Av})^{-1/3}$, where $N_{Av}$ is the Avogadro number and $C$ is the dye concentration in mol/L. In Fig. 8A, a reasonably good agreement can be reached with this theoretical description. The solvent nature is expected to have minimal impact on the energy transfer efficiency $E$, as it is primarily determined by the distance between donor and acceptor fluorescent molecules. Since the temperature sensitivity of RhB originates from its quantum yield, a change in the temperature dependence of RhB is expected as self-quenching becomes significant. As illustrated in Fig. 8B, self-quenching in RhB/water solutions leads to a marked reduction in the temperature sensitivity of the fluorescence lifetime. Collisional de-excitation with solvent molecules, which is the dynamic quenching responsible for the temperature dependence of RhB, is replaced by self-quenching when the distance between

<figure><img src="image_10.png" /><figcaption>**Fig. 8** A Self-quenching efficiency $E$ as a function of the distance between the dye molecules in water. Dotted lines correspond to the theoretical model assuming $R_0=6.5$ nm, 7.7 nm and 5.6 nm for RhB,</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Rh6G and EY, respectively. B Evolution of the fluorescence lifetime of RhB dissolved into water at two concentrations. Dotted lines are used to highlight the trends</figcaption></figure>
fluorescent molecules approaches the critical distance $R_0$.

Assuming a critical concentration $C_{sq} = 10^{-3}$ mol/L for the apparition of self-quenching, a critical droplet size, $d_{min}$, can be evaluated as a function of the initial droplet size $d_0$:

$$d_{min} = (C_0/C_{sq})^{1/3} d_0. \qquad (6)$$

ii. **Fluorescence reabsorption:** When absorption and emission events occur successively in a fluorescent medium, the time distribution of fluorescent photons reaching the detection system outside the medium can become extended. While the intrinsic fluorescence lifetime remains unchanged at the molecular level, each cycle of fluorescence reabsorption and re-emission introduces an additional delay, thereby lengthening the apparent fluorescence decay (Berezin and Achilefu 2010).

A simplified model serving as foundation for understanding the contribution of secondary fluorescence (photons of fluorescence emitted after the absorption of a photon of fluorescence) is detailed in Appendix 4. This model aims to estimate the relative magnitude of secondary fluorescence in comparison to primary fluorescence. For the sake of simplicity, we consider a 1D medium rather than a droplet, where photons travel in three dimensions. However, the model is expected to account for the effects of liquid thickness L and the dye concentration C. Figure 9 illustrates some of the results obtained for EY. The evolution of the ratio of secondary fluorescence to primary fluorescence, denoted as $R_{21}$ hereafter, is presented as a function of the initial dye concentration and liquid thickness. In Fig. 9, $R_{21}$ increases with rising concentration for a given liquid thickness, while in Fig. 10, $R_{21}$ increases with increasing liquid thickness at fixed concentrations. These evolutions are strongly nonlinear, as they are derived from Beer's law (Appendix 4).

Considering only primary and secondary fluorescence photons (higher orders resulting from further reabsorption/re-emission are neglected), an estimate of the fluorescence decay can be obtained as follows

$$F(t)/F(t = 0) \approx (1 - R_{21})f(t) + R_{21}(f * f)(t), \quad (7)$$

where $f(t)$ is the decay for primary fluorescence (such that $f = 1$ at $t = 0$). Based on Eq. (1), this equation gives us:

$$F(t)/F(t = 0) = (1 - R_{21})e^{-t/\tau} + R_{21} \frac{t}{\tau} e^{-t/\tau}, \quad (8)$$

<figure><img src="image_12.png" /><figcaption>Fig. 9 Variation of the ratio of secondary fluorescence to primary fluorescence as a function of EY concentration for several liquid depth. The star symbols consider a droplet with an initial diameter of 2 mm and an initial dye concentration of $5 \times 10^{-7}$ mol L$^{-1}$. They indicate the concentration C and the diameter d as water evaporates and the droplet size reduces. The red dashed line indicates the critical concentration of $10^{-3}$ mol L$^{-1}$, beyond which self-quenching is expected to become predominant. B</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10 Evolution of the ratio $R_{21}$ of secondary fluorescence to primary fluorescence and the resulting variation of fluorescence lifetime as a function of the liquid depth for different concentrations in EY</figcaption></figure>

Finally, parameter estimation based on the least squares allows for the determination of $a$, $\tau_1$ and $\tau_2$ for a time window of 12.5 ns (matching the laser pulse
repetition period), and the equivalent lifetime is calculated using Eq. (3). According to Fig. 9, the ratio $R_{21}$ never exceeds 16%, and the lifetime extension due to reabsorption/re-emission is limited to approximately 0.2 ns at most. Indeed, the fluorescence quantum yield, fixed at $\phi = 0.23$ in the case of EY, imposes an asymptotic limit for $R_{21}$ in the regime of strong reabsorption, as discussed in Eq. (S13) in Appendix 4.

According to Fig. 10, when very small liquid thickness is considered, very high dye concentrations are necessary for obtaining a significant amount of secondary fluorescence. Extrapolating this observation to droplets on the order of 10 µm (typical in spray applications), it can be observed that variations in fluorescence lifetime become significant only at concentrations exceeding $10^{-3}$ mol L$^{-1}$. This value is close to the critical concentration at which self-quenching becomes dominant. Given the significant reductions in lifetime induced by self-quenching (spanning several orders of magnitude on $\tau$), the effects of secondary fluorescence are likely to manifest as a second-order phenomenon in comparison. In general, self-quenching imposes a more stringent limitation on measurements compared to secondary fluorescence. In the experiments on acoustically levitated droplets described in Sect. 3, the initial concentration of dye was set to $5 \times 10^{-7}$ mol/L, while $d_0$ is typically around 2 mm, leading to a critical diameter for self-quenching to become important to be $d_{min} = 158$ µm based on Eq. (6). For a droplet of this size and a concentration in EY of $10^{-3}$ mol L$^{-1}$, the variation in fluorescence lifetime due to secondary fluorescence is limited to about 0.04 ns. It must be also said that it is difficult to steadily maintain a droplet of such a small size and capture its evolution, its evaporation time becoming extremely short in comparison to the acquisition time of the measurements.

iii. **Dimerization:** As the dye concentration increases, self-association of dye molecules can be encountered owing to the strong intermolecular attractive forces between the molecules:

$$M + M \leftrightarrow D, \qquad (9)$$

The dimer D exhibits distinct spectroscopic properties compared to the monomer M. The effect of dimerization has been examined in more detail for EY, drawing on the literature. Enoki and Katoh (2018) determined the absorption coefficient, fluorescence quantum yield and fluorescence lifetimes of both monomeric and dimeric forms of EY. For the monomer, the quantum yield in water is 0.23, and its fluorescence lifetime is around 1.1 ns, a value close to our measurements. In contrast, the dimer's fluorescence yield is

about 0.005, with a lifetime of approximately 20 ps. Molar absorption coefficients are similar in value for the two forms. We believe that the fluorescence decay of the dimer is simply too fast to be resolved by our TCSPC system, meaning that lifetime measurements primarily reflect the monomer's fluorescence decay. However, it should be noted that the dimer becomes the predominant form of EY at dye concentrations greater than $10^{-5}$ mol/L, according to the equilibrium constant reported by Chakraborty and Panda (2011) for the chemical equilibrium in Eq. (7).

## 2.6 Others effects affecting the fluorescence signal

Photodegradation of dye molecules has been reported by various authors (Watanabe et al. 1992; Chakraborty and Panda 2011), a critical concern for LIF measurements. We studied this phenomenon in the case of EY. While the fluorescence signal intensity significantly decreases due to photodegradation, the measured fluorescence lifetime value remained unchanged. Further details on the results obtained are provided in Appendix 5.

# 3 Evaporation of bicomponent droplet under acoustic levitation

## 3.1 Description of the experiment

To validate the measurement technique, composition of bicomponent droplets was investigated while they were acoustically levitated. The schematic of the experimental setup is provided in Fig. 11. The acoustic levitator (Tec 5 GmbH) consists of a transducer vibrating at a frequency of 58 kHz and a concave-shaped reflector. As the droplet is introduced into the acoustic field, its position stabilizes below one of the pressure nodes of the standing wave formed between the transducer and reflector. The acoustic pressure is sufficient to levitate droplets with a size slightly below 3 mm, approximately half the wavelength of the acoustic field. Shadow images of the levitated droplet are obtained using a flashlamp directed through a flexible waveguide and diffuser, which illuminates the droplet's background. A camera captures these images with a spatial resolution of approximately 8 µm per pixel, at a frequency of about one image every 8 s.

As shown in Fig. 11C, the acoustically levitating droplet appears squeezed under the effect of the acoustic radiation pressure, which exerts greater force on the droplet's top and bottom surfaces compared to its edges. Analyzing these images allows for determining the time evolution of the droplet's surface area and volume.

The TCSPC technique, previously applied in Sect. 2 to examine solvent effects on the fluorescence lifetime of
<figure><img src="image_14.png" /><figcaption>**Fig. 11** Experimental setup used for studying the evaporation of bicomponent droplets in acoustic levitation. The figure shows the arrangement of optics for observing the droplet and measuring</figcaption></figure>

fluorescence lifetime from both a top view (A) and a side view (B). Example images in (C) display a pure water droplet at $T_{amb}=293$ K and relative humidity RH=40%

various dyes, is applied here to determine the composition of the droplets. The femtosecond laser beam is directed to the levitated droplet using different mirrors. The acoustic levitator is mounted on a lab jack with translation stages, allowing precise positioning of the droplet relative to the laser beam. Additionally, a beam expander, made up of an afocal doublet (focal lengths $f_1=30$ mm and $f_2=150$ mm), creates a laser beam with a 6 mm diameter to achieve volumetric illumination of the droplet. A receiving optic, positioned at a right angle to the laser beam, transmits the fluorescence signal to an optical fiber linked to the PMT, which counts the fluorescence photons. Following the discussions in Sect. 2, Eosin Y (EY) was chosen to characterize the composition of bicomponent droplets, and a band-pass filter [570–590 nm] was selected for fluorescence detection. This choice is motivated by the fact that the lifetime of EY is highly sensitive to solvent polarity and does not depend on temperature. This sensitivity allows for effective monitoring of changes in the droplet’s composition during the evaporation process without bias from temperature variations within the droplet. Furthermore, in order to strongly limit the effects of the dye on the droplet evaporation while reducing the undesirable effects discussed in Sect. 2, a concentration of $10^{-7}$ mol L$^{-1}$ was chosen.

As noted in Appendix 6, ray tracing of the laser beam and fluorescence rays within the droplet reveals that the central

region contributes more strongly to the fluorescence signal than regions near its surface. However, this uneven distribution of signal origin does not affect the measurement results, as the composition within the measurement volume remains representative of the entire droplet. The evaporation of acoustically levitated droplets has been extensively studied in the literature. Theoretical and experimental studies, such as those by Brenn et al. (2007) and Yarin et al. (1998), have demonstrated that an internal flow with a nonzero average velocity develops within the acoustically levitated droplet. According to Yarin et al. (1998), this flow typically exhibits velocity scales on the order of several cm/s, which is sufficient to promote rapid mixing of the liquid components. Without this rapid mixing, measurement bias could favor the composition at the droplet’s center.

## 3.2 Results and discussion

Experimental investigations were conducted to study the evaporation process of several binary mixtures formed from the solvents considered in Sect. 1. Since the experiments were performed in a room where the air is not dried, water present in the air may condense or be absorbed on the surface of the droplets, potentially becoming a component of the droplet even if it was not initially present. For simplicity, the focus was placed on mixtures of water with species
of significantly different volatilities: 2-propanol (boiling point $T_b$=355.4 K), 1-propanol ($T_b$=370 K) and ethylene glycol ($T_b$=470.4 K). The ambient temperature and relative humidity are recorded using temperature and humidity sensors placed approximately 5 cm from the levitator. Their measurements are expected to provide ambient conditions with an accuracy of ± 1 K and ± 2%, respectively.

### (a) Characterization of the fluorescence lifetime in the studied mixtures

To determine the composition of a mixture, a preliminary study is conducted in a cell at room temperature. Component 1 is added step by step to a cell containing component 2, with the mixture stirred after each addition. The mass of the solution was measured using a precision balance, and the fluorescence decay was recorded. The same approach was performed by initially placing component 1 in the cell and gradually adding component 2. Finally, the fluorescence lifetime, evaluated using Eq. (3), was plotted against the mass and volume fractions of components 1 and 2. As the mixtures are not ideal, care was taken for the calculation of the volume fraction to take into account excess volumes of the component. Figure 12 shows the calibration curves obtained for the three mixtures mentioned above. These calibration curves are, in principle, independent of the measurement system, as the fluorescence lifetime is an intrinsic property of the fluorescent species in its solvent. However, the values of $\tau$ might vary slightly depending on the chosen detection band and laser excitation wavelength.

In Fig. 12, the fluorescence lifetime does not vary proportionally with the volume fraction of the components. We also verified that using mass or molar fractions instead of volume fraction results in even less linear curves. Polynomial interpolations of the calibration curves in Fig. 12 were derived to convert fluorescence lifetime measurements into volume fractions of water:

Water/ethylene glycol:

$$ \varphi_w = 3.19\tau^6 - 32.62\tau^5 + 137.9\tau^4 - 309.2\tau^3 \\ + 386.9\tau^2 - 256.8\tau + 71.73 $$

Water/2-propanol:

$$ \varphi_w = 0.736\tau^6 - 8.36\tau^5 + 38.88\tau^4 - 94.86\tau^3 \\ + 128.4\tau^2 - 91.86\tau + 28.24 $$

Water/1-propanol:

$$ \varphi_w = 0.04\tau^6 + 0.35\tau^5 - 5.54\tau^4 + 22.65\tau^3 \\ - 41.83\tau^2 + 36.20\tau - 22.65 $$

In the above formula, the volume fraction of water $\varphi_w$ is defined between 0 and 1 and the fluorescence lifetime $\tau$ is

<figure><img src="image_15.png" /><figcaption>Fig. 12 Evolution of the fluorescence lifetime of EY in A water/ethylene glycol, B water/2-propanol and C water/1-propanol mixtures as a function of the volume and mass fraction of water</figcaption></figure>

expressed in ns. Appendix 1 explains the propagation of uncertainty from $\tau$ to $\varphi_w$. The precision in evaluating the volume fraction improves with the slope $|d\varphi_w/d\tau|$. Consequently, certain composition ranges are more favorable for accurately determining the mixture composition. In general, measurements are more precise when water predominates in volume
<table><caption>**Table 3** Experimental conditions for the study of water/ethylene glycol droplets</caption><thead><tr><th>Initial water volume fraction φ<sub>w,0</sub></th><th>d<sub>0</sub> [mm]</th><th>T<sub>amb</sub> [°C]</th><th>RH [%]</th></tr></thead><tbody><tr><td>1</td><td>1.53</td><td>22.4</td><td>26</td></tr><tr><td>0.55</td><td>1.79</td><td>22.2</td><td>26</td></tr><tr><td>0</td><td>1.05</td><td>21.6</td><td>27</td></tr></tbody></table>

within the mixture. The uncertainty in φ<sub>w</sub> generally does not exceed 0.05 and can sometimes drop below 0.01.

## (b) Results and discussions

In the following, the diameter *d* corresponds to the volume equivalent diameter of the droplet:

$$d = (6/\pi V)^{1/3}, \qquad (10)$$

where the droplet volume *V* is obtained from the shadow images of the droplet assuming axial symmetry of the droplet. It is worth noting that the square diameter of a pure liquid droplet is expected to follow a *d*<sup>2</sup> law:

$$d^2/d_0^2 = 1 - Kt/d_0^2 \qquad (11)$$

where *K* is the evaporation rate, *t* represents time and *d*<sub>0</sub> the droplet equivalent diameter at *t* = 0. Equation (11) implies a linear decrease of the square diameter with time for a droplet evaporating in a steady-state environment.

### 3.2.1 Water/ethylene glycol droplets

Experiments conducted with the water/ethylene glycol mixture include: a droplet of pure ethylene glycol, a droplet initially composed of 55% water and 45% ethylene glycol in volume, and a droplet of pure water. The experimental conditions for these measurements are provided in Table 3, and the time evolutions of the square diameter and the composition are presented in Fig. 13.

* In the case of φ<sub>w,0</sub> = 0, measurements reveal an increase in the droplet diameter of about 8%, corresponding to a 25.9% increase in droplet volume. This increase can be explained by the hygroscopic nature of ethylene glycol, which absorbs water vapor from the ambient air. Fluorescence lifetime measurements indicate an increase in the water volume fraction of approximately 22%, which aligns well with the observed change in droplet volume (25.9/125.9 = 0.21).

* For φ<sub>w,0</sub> = 0.55, the droplet diameter decreases over time due to the progressive evaporation of the water initially contained in the mixture. The slope of the square diameter is quite like that of the pure water droplet, indicating that the evaporation of water is indeed the main process occurring for this droplet. Over the studied period, the droplet diameter decreases by about 17.5%, corresponding to a volume loss of about 43.8%. Measurements based on the fluorescence lifetime indicate a decrease in the water volume fraction from approximately 55% to 25% over the same period. Letting V<sub>0</sub> be the initial volume of the droplet, the droplet consists initially of water (55/100

<figure><img src="image_16.png" /><figcaption>**Fig. 13** Evolution of the square diameter and the water volume fraction measured from the fluorescence lifetime in the case of droplets composed of water and ethylene glycol. Same conditions as in Table 3</figcaption></figure>
<table><caption>**Table 4** Experimental conditions for the study of water/2-propanol droplets</caption><thead><tr><th>Initial water volume fraction φ<sub>w,0</sub></th><th>d<sub>0</sub> [mm]</th><th>T<sub>amb</sub> [°C]</th><th>RH [%]</th></tr></thead><tbody><tr><td>1</td><td>1.53</td><td>22.4</td><td>26</td></tr><tr><td>0.75</td><td>1.79</td><td>22.2</td><td>26</td></tr><tr><td>0.5</td><td>1.05</td><td>21.6</td><td>27</td></tr><tr><td>0.25</td><td>1.16</td><td>22.1</td><td>26</td></tr><tr><td>0</td><td>1.45</td><td>22.2</td><td>26</td></tr></tbody></table>

$V_0$) and ethylene glycol (45/100 $V_0$). The entire 43.8% loss in volume can be attributed to the evaporation of water. Thus, the final state of the droplet consists of water ((55–43.8)/100 $V_0$) and ethylene glycol (45/100 $V_0$). This leads to a water volume fraction (55–43.8)/45 = 0.249, which aligns closely with the value of $φ_w$ obtained from the fluorescence lifetime measurements. This consistency reinforces the reliability of the measurement technique.

* For the pure water droplet, the fluorescence lifetime and, consequently, the water volume fraction do not change significantly during the measurements, as expected.

### 3.2.2 Water/2-propanol droplets

The study of water/2-propanol droplets was conducted for mixtures prepared with initial water volume fractions $φ_{w,0}$=0, 0.25, 0.5, 0.75 and 1. The experimental conditions for these measurements are summarized in Table 4, while

the time evolutions of the square diameter and the composition are presented in Fig. 14.

In Fig. 14A, the evaporation rate $K$ slows down considerably as the initial water volume fraction increases. This behavior is expected, given that 2-propanol is significantly more volatile than water. In Fig. 14B, a droplet initially composed of pure 2-propanol is observed to gradually transformation into a droplet of pure water. This evolution is driven by the condensation of water vapor from the ambient air, triggered by the decrease in the droplet's surface temperature resulting from the rapid vaporization of 2-propanol. The water volume fraction at the first measurement point shows a higher value than the initial mixture composition prepared before the experiment (see Table 4). This discrepancy is attributed to the fact that it takes approximately one minute to place the droplet in the levitator, adjust the sound pressure level and position the droplet relative to the optics. During this time, evaporation of 2-propanol and water condensation occurs, increasing the initial water content. For $φ_{w,0}$ = 0 and $φ_{w,0}$ = 0.25, water volume fraction increases moderately during the first 120 s, after which the droplet rapidly transitions to pure water in less than 30 s. This rapid change in composition is closely correlated with a change in the slope of the squared diameter. The non-ideality of the water/2-propanol mixture likely plays a role in this effect, accelerating the evaporation of 2-propanol. As anticipated, once the droplet has entirely transformed into water, the evaporation rate $K$ appears approximately the same across all cases, given that the ambient air temperature and relative humidity are nearly identical in each experiment.

<figure><img src="image_17.png" /><figcaption>**Fig. 14** Evolution of the square diameter and the water volume fraction measured from the fluorescence lifetime in the case of droplets composed of water and 2-propanol. Same conditions as in Table 4</figcaption></figure>
<table><caption>**Table 5** Experimental conditions for the study of droplets made up of water/1-propanol of different initial compositions</caption><thead><tr><th>Water initial volume fraction, φ<sub>w,0</sub></th><th>d<sub>0</sub> [mm]</th><th>T<sub>amb</sub> [°C]</th><th>Rh [%]</th></tr></thead><tbody><tr><td>1</td><td>1.72</td><td>22.1</td><td>26</td></tr><tr><td>0.75</td><td>1.98</td><td>22.1</td><td>26</td></tr><tr><td>0.5</td><td>1.52</td><td>21.6</td><td>26</td></tr><tr><td>0.25</td><td>1.74</td><td>22.1</td><td>26</td></tr><tr><td>0</td><td>1.70</td><td>22.1</td><td>26</td></tr></tbody></table>

### 3.2.3 Water/1-propanol droplets

Experimental conditions for the study of droplets made up of water/1-propanol mixtures are summarized in Table 5. Time evolutions of the square diameter and the composition are presented in Fig. 15.

In Fig. 15, two distinct evolutions based on the initial water volume fraction can be observed. For $φ_{w,0}=0.5, 0.75$ and 1, the droplet evolves toward a pure water droplet due to a faster evaporation of 1-propanol compared to water. In contrast, for $φ_{w,0} = 0$ and 0.25, the droplet evolves toward a mixture that contains an excess of 1-propanol. Differences are also evident in the evolution of the squared diameter: for $φ_{w,0}=0$ and 0.25, the diameter shows no slope change and follows the $d^2$ law throughout the observed period, unlike the other cases. In Fig. 15B, the initially pure 1-propanol droplet ultimately stabilizes with 7% water by volume. Although this value is low, it is significant given our measurement uncertainty. Measurements suggest that water condensation

occurs in the first seconds (not visible, as measurements start after a few tens of seconds), after which the droplet composition stabilizes. This behavior is somewhat reminiscent of the azeotropic nature of the 1-propanol/water mixture, albeit under non-boiling conditions. Further investigation, particularly through numerical simulation, would help elucidating this process by accounting for various factors: the acoustic streaming that enhances heat and mass transfer, the non-ideal behavior of the mixture affecting liquid-vapor equilibrium, the minimal volatility difference between the two components, and air humidity, which can slow water evaporation and induce water condensation.

## 4 Conclusion

This study demonstrates the feasibility of measuring the composition of binary-component droplets based on the fluorescence lifetime. The fluorescence decay is measured using time-correlated single-photon counting (TCSPC) with a femtosecond laser source. For several dyes, the primary factors influencing the fluorescence lifetime are examined in detail: solvent nature, temperature and dye concentration. The study shows that for Eosin Y, fluorescence lifetime varies with solvent polarity while being minimally sensitive to temperature variations. This makes Eosin Y a promising choice for characterizing droplet composition during evaporation, as it allows avoiding measurement biases caused by simultaneous changes in composition and temperature. Additionally, the self-quenching threshold of this dye is around $10^{-3}$ mol/L, while measurements are possible

<figure><img src="image_18.png" /><figcaption>**Fig. 15** Evolution of the square diameter and the water volume fraction measured from the fluorescence lifetime in the case of droplets composed of water and 1-propanol. Same conditions as in Table 5</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Fig. 16** Evolution of the water mass fraction and the estimated error on this parameter as a function of the fluorescence lifetime for the mixture. A Water/ethylene glycol, B water/2-propanol and C water/1-propanol</figcaption></figure>

at much lower concentrations on the order of $10^{-7}$ mol/L. This allows the droplet composition to be characterized throughout nearly the entire evaporation process. The practical implementation of the technique is detailed for studying droplet evaporation in an acoustic levitator for three binary mixtures: water/1-propanol, water/2-propanol and water/ethylene glycol. A preliminary study is necessary to accurately establish the relationship between the fluorescence lifetime and the volume fraction of the components in the mixture. The measurement technique proves highly effective in detecting composition changes resulting from the evaporation of mixture components, as well as the condensation and absorption of water from the humidity in ambient air

## Appendix 1: Estimation of the measurement uncertainties

To evaluate the measurement uncertainties, the derivative of $\varphi_w$ with respect to $\tau$ is used:

$$ \Delta\varphi_w = \left| \frac{d\varphi_w}{d\tau} \right| \Delta\tau $$

where $\Delta\tau$ is the uncertainty on the measurement of the fluorescence lifetime. This error depends on several parameters such as the number of photons collected and the signal-to-noise ratio. To estimate $\Delta\tau$, repeated measurements were conducted on levitated droplets. These measurements suggest that $\Delta\tau$ is on the order of 0.03 ns under the conditions used in the acoustic levitator. This value is also consistent with the example shown in Fig. 13, where an ethylene glycol droplet remains levitated for an extended period. Figure 16 presents the uncertainty $\Delta\varphi_w$ across the entire range of volume fractions for the three mixtures considered in the droplet measurements. The measurements are particularly precise in mixtures containing an excess of water by volume.

<table><caption>**Table 6** Estimated values for the parameters $E_A$ and $A$ in the Arrhenius law (Eq. S1) for RhB in different solvents</caption><thead><tr><th></th><th><i>E</i><sub>a</sub>[J/mol]</th><th><i>A</i>[-]</th></tr></thead><tbody><tr><td>Water</td><td>19,830</td><td>-7.612</td></tr><tr><td>Acetone</td><td>16,190</td><td>-4.885</td></tr><tr><td>Ethanol</td><td>10,150</td><td>-3.202</td></tr><tr><td>Propan-1-ol</td><td>7483</td><td>-1.951</td></tr><tr><td>2-Propanol</td><td>8890</td><td>-2.581</td></tr><tr><td>3-Pentanone</td><td>14,200</td><td>-4.885</td></tr><tr><td>Ethylene glycol</td><td>12,130</td><td>-3.964</td></tr></tbody></table>

The reported values for $E_a$ are consistent with previous studies conducted on RhB (Mehdi et al. 2021; Wang et al. 2022). The solvent type primarily influences the pre-exponential factor $A$, while the activation energy $E_a$ exhibits moderate changes.

## Appendix 2: Effect of the temperature on the fluorescence lifetime of rhodamine B

According to Mehdi et al. (2021), the effect of temperature on fluorescence decay of RhB can be modeled using an Arrhenius law:

$$ \tau = A e^{E_a/RT}. \qquad (S1) $$

Table 6 provides the values of the pre-exponential coefficient $A$ and the activation energy $E_a$ for RhB in different solvents, based on the fits of the curves shown in Fig. 6A.

## Appendix 3: Self-quenching effect

See Fig. 17.
<figure><img src="image_20.png" /><figcaption>Fig. 17 Evolution of the fluorescence lifetime τ normalized by τ₀ (the lifetime at weak concentrations) as a function dye concentration for RhB, Rh6G and EY, respectively. Solvent is water</figcaption></figure>

## Appendix 4: Fluorescence reabsorption and re-emission model in 1D medium

A simplified model has been developed to provide a foundation for assessing the contribution of secondary fluorescence. The proposed model accounts for fluorescence emitted immediately following the absorption of a laser photon (primary fluorescence, first order) as well as fluorescence re-emitted after a single reabsorption event (secondary fluorescence, second order). Processes of higher order, involving more than two successive reabsorptions and subsequent re-emissions, are neglected. A region of thickness L is considered for the fluorescent medium, and the model is 1D, meaning photons are assumed to propagate unidirectionally along the x-axis, perpendicular to the liquid/gas interfaces. Reflection phenomena at the liquid/gas interface (at x = 0 and x = L) are neglected to simplify the analysis. Finally, primary fluorescence is assumed to be uniformly distributed (Fig. 18).

### Step 1: Numbers of photons from primary fluorescence

Let φ represent the fluorescence quantum yield. The number of fluorescence photons $f_1$ (photons/m³/s/Hz) emitted locally per unit volume at frequency v is given by:

$$f_1(v) = A_0 P(v) \phi \quad (S2)$$

$A_0$ (photons/m³/s) is the excitation field for primary fluorescence, representing the number of laser photons absorbed per unit volume and time. $P(v)$ (in Hz⁻¹) represents the probability density of fluorescence emission at frequency v:

$$P(v) = S_p(v) / \int_0^{+\infty} S_p(v) dv \quad (S3)$$

where $S_p(v)$ is the emission spectrum of the dye measured at low concentration to minimize the effects of reabsorption (Fig. 1). The flux of primary fluorescence photons $F_1(v)$ (photons/m²/s/Hz) emerging from the liquid domain at frequency v is given by Beer's law:

$$F_1(v) = \int_0^L \exp(-C\epsilon(v)x)f_1(v) dx \quad (S4)$$

$$F_1(v) = f_1(v) \cdot \frac{\exp(-C\epsilon(v)L) - 1}{C\epsilon(v)} \quad (S5)$$

In this expression, $\epsilon(v)$ represents the absorption coefficient at frequency v. Finally, fluorescence photons reach the detection system. Considering all photons in the band $[\lambda_1 - \lambda_2]$ are detected and the others are not, the flux of detected primary fluorescence photons $S_1$ (photons/m²/s) is given by:

$$S_1 = \int_{\lambda_1}^{\lambda_2} F_1(\lambda) d\lambda \quad (S6)$$

### Step 2: Numbers of photons from secondary fluorescence

The flux of photons of primary fluorescence (in photons/m³/s/Hz) emitted from $M(x')$ and absorbed at a point $N(x)$ can be calculated using Beer's law:

<figure><img src="image_21.png" /><figcaption>Fig. 18 Illustration of the parameters of the fluorescence model</figcaption></figure>
$$j(|x - x'|, v) = \frac{1}{2} \exp(-C\epsilon(v)|x - x'|) \cdot f_1(v) \quad (S7)$$

The coefficient 1/2 arises from the fact that half of the photons are emitted in M(x') in the direction where N(x) is located.

The flux of photons of primary fluorescence (in photons/m²/s/Hz) absorbed at N(x) can be determined by:

$$J_1(x, v) = \frac{1}{2} \int_0^L \exp(-C\epsilon(v)|x' - x|) f_1(v) dx' \quad (S8)$$

Remembering the expression of $f_1(v)$ in Eq. (S5):

$$J_1(x, v) = \frac{1}{2} A_0 \phi \frac{P(v)}{\epsilon(v)C} \left[ 2 - \exp(-C\epsilon(v)x) - \exp(-C\epsilon(v)(L-x)) \right] \quad (S9)$$

The number of photons of primary fluorescence reabsorbed (in photons/m³/s/Hz) can be expressed as:

$$a_1(x, v) = J_1(x, v)C\epsilon(v) \quad (S10)$$

The excitation field of secondary fluorescence (photons/m³/s) is determined by:

$$A_1(x) = \frac{1}{2} A_0 \phi G(x), \quad (S11)$$

where $G(x) = \int_0^{+\infty} P(v) [2 - \exp(-C\epsilon(v)x) - \exp(-C\epsilon(v)(L-x))] dv$.

* For very high C: $\lim_{C \to \infty} G(x) = 2$ and $A_1 = A_0 \phi$ \quad (S12)

* For very low C: $G(x) = L C \int_0^{+\infty} P(v) \epsilon(v) dv + o(c^2)$ \quad (S13)

The flux of photons of secondary fluorescence (photons/m²/s/Hz) emitted at point N(x) is thus:

$$f_2(v, x) = A_1(x)P(v)\phi \quad (S14)$$

The flux of photons of secondary fluorescence $F_2(v)$ (in photons/m²/s/Hz) emerging from the liquid domain is:

$$F_2(v) = \int_0^L \exp(-C\epsilon(v)x) f_2(v, x) dx \quad (S15)$$

By considering only the photons within the spectral band $[\lambda_1 - \lambda_2]$, the detected signal $S_2$ (in photons/m²/s) for secondary fluorescence is given by:

$$S_2 = \int_{\lambda_1}^{\lambda_2} F_2(\lambda) d\lambda \quad (S16)$$

Figure 19 illustrates the evolution of the ratio between primary and secondary detected photons, for a thickness of

<figure><img src="image_22.png" /><figcaption>Fig. 19 Variation of the ratio of secondary fluorescence to primary fluorescence as a function of the dye concentration, for an optical depth L=2 mm considering EY as fluorescence dye. To ensure the convergence of Monte Carlo simulations, a total of 100 million photons were considered</figcaption></figure>

L = 2 mm and in the case of EY ($\phi=0.23$). It is observed that secondary fluorescence starts to become significant around $10^{-3}$ mol/L. In the limit of an optically thick medium, the secondary fluorescence reaches an asymptotic value close to 15–16%.

To validate the model presented above, a Monte Carlo algorithm was implemented due to its simplicity to solve the same problem. Without going into details, it involves launching primary fluorescence photons and probabilistically simulating the phenomena of reabsorption and re-emission of secondary fluorescence. We start by randomly selecting the emission location, the propagation direction (toward positive or negative x) and the wavelength of the primary fluorescence photons. Next, we simulate the absorption phenomenon by drawing a parameter $\eta$ between 0 and 1 such that the path length l of the photon is:

$$l = - \frac{\ln(\eta)}{\epsilon(v)C} \quad (S17)$$

Finally, primary fluorescence photons with a path length l short enough to be absorbed in the liquid medium are considered for emitted secondary fluorescence photons with a probability equal to $\phi$, the quantum yield of fluorescence. We randomly select the propagation direction and frequency of the secondary fluorescence photons. To be detected, a photon of primary or secondary fluorescence must meet the following conditions to have a propagation direction toward x < 0, a wavelength within the range $[\lambda_1, \lambda_2]$ and a path length l obtained from Eq. (S17). that allows it to leave the liquid domain of thickness L. Figure 19
<figure><img src="image_23.png" /><figcaption>**Fig. 20** Degradation of the fluorescence signal of EY from an acoustically levitated droplet of ethylene glycol. The droplet is illuminated by a laser beam at 490 nm and the laser power is *P* = 230 mW</figcaption></figure>

shows good agreement between the model based on Eqs. (S2) to (S16) and the Monte Carlo approach.

**Appendix 5: Influence of the photodegradation on the fluorescence lifetime measurement**

Initial tests within a cell revealed a decline in fluorescent signal after extended illumination of the liquid solution, though this impact was notably minimal, especially when the solution was stirred. In the context of small-volume droplets, comprising just a few microliters, the issue of photodegradation becomes more critical. Thus, a series of experiments were conducted to evaluate the impact of photodegradation on lifetime measurements. Using ethylene glycol droplets, we eliminated volume-induced variations in fluorescence intensity, as ethylene glycol does not evaporate at room temperature. Figure 20 depicts the temporal evolution of photon count rates under continuous laser beam illumination (maximum power *P* = 230 mW). It's important to note that the laser beam was not expanded and maintained at a diameter of 1.2 mm. To prevent photodetector saturation, a neutral optical density of 3 was interposed in front of the photomultiplier, with dye concentration raised to 10⁻⁵ mol/L. The fluorescence signal intensity (expressed in counts per second, Cps) undergoes swift reduction within the droplet. Over approximately 15 min, the intensity decreases by a factor of 10, followed by a slower, but continuous degradation. The logarithmic scale underscores the exponential nature of this signal decay, approximate to exp(-t/t₁), where t₁ = 500 s. Importantly, this degradation phenomenon is irreversible, meaning that once the laser beam illumination stops, the signal intensity does not fully recover to its initial level.

<figure><img src="image_24.png" /><figcaption>**Fig. 21** Evolution of the photon counting rate and the fluorescence lifetime under the same laser power illumination as the evaporation measurements</figcaption></figure>

Figure 20 presents fluorescence lifetime measurement on the same droplet. As observed, photodegradation does not influence the fluorescence lifetime, which fluctuates without any particular trend. The fluctuations exhibit a standard deviation of 0.026 ns, a value that aligns with the uncertainty inherent to fluorescence lifetime measurements (Mehdi et al. 2021; Wang et al. 2022).

By removing the neutral density of 3, the laser excitation power and concentration of EY could be considerably reduced, rendering photodegradation effects more negligible. In Fig. 21, a count rate approaching the pile-up limit was achieved at approximately 10 mW laser power and a dye concentration of 5 × 10⁻⁷ mol/L. Confirming our earlier findings, the fluorescence lifetime remains constant with minimal standard deviation at *P* = 5 mW and *P* = 10 mW. Moreover, it is important to highlight that the presented results involve continuous droplet illumination. In contrast, during evaporation experiments, the droplet is illuminated intermittently, once every 10 s for a duration of about 2–3 s.

**Appendix 6: Measurement volume in acoustic levitated droplet experiments**

The measurement volume can be defined as the region inside the droplet that is within the field of view of the receiving optics (collection volume) and simultaneously illuminated by the laser beam (excitation volume). To define properly the collection and excitation volumes, it is necessary to solve an optical problem considering the refraction and reflection of light rays at the surface of the droplet. The choice of method, whether Maxwell's equations or geometrical optics, leads to intricate calculations due to the droplet's non-spherical nature
<figure><img src="image_25.png" /><figcaption>**Fig. 22** Example of ray tracing to determine the regions within the droplet that contributes the most to the fluorescence signal. These results are only light rays that have undergone a single refraction at the surface of the droplet. The droplet is a sphere of radius 1 mm, and the refractive index is equal to *n* = 1.33. It is also assumed to be perfectly centered optically</figcaption></figure>

and its gradual shift in position as it evaporates. As a rough estimate, we consider a perfectly centered spherical droplet aligned with the optics. Ray tracing considers light focusing within the droplet, which restricts collection volume and fluorescence excitation. Figure 22 provides an illustration for this simplified scenario, depicting a 2-mm-diameter water droplet. The measurement volume roughly corresponds to the central region of the droplet, approximately one-third of its diameter in size. The regions near the surface contribute much less significantly.

Before each experiment involving levitated droplets, the droplet is precisely positioned relative to the laser beam's center and the receiving optics' axis. This optical alignment process involves vertically adjusting the entire levitator until achieving peak signal intensity (count rate). Once alignment is achieved, the positions of both the levitator and optics remain unchanged throughout the experiment, despite the droplet's gradual vertical movement toward the pressure node resulting from evaporation.

**Acknowledgements** The authors would like to acknowledge the financial support from China Scholarship Council (CSC, M. Wang), European Regional Development Fund (ENERBATIN CPER) and European FEDER program.

**Funding** Open access funding provided by Institut National Polytechnique de Toulouse.

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## References

Berezin MY, Achilefu S (2010) Fluorescence lifetime measurements and biological imaging. Chem Rev 110:2641–2684. https://doi.org/10.1021/cr900343z

Boens N, Qin W, Basarić N et al (2007) Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy. Anal Chem 79:2137–2149. https://doi.org/10.1021/ac062160k

Brenn G, Deviprasath LJ, Durst F, Fink C (2007) Evaporation of acoustically levitated multi-component liquid droplets. Int J Heat Mass Transf 50:5073–5086. https://doi.org/10.1016/j.ijheatmasstransfer.2007.07.036

Casadevall I, Solvas X, Srisa-Art M, de Mello AJ, Edel JB (2010) Mapping of fluidic mixing in microdroplets with 1 µs time resolution using fluorescence lifetime imaging. Anal Chem 82:3950–3956. https://doi.org/10.1021/ac100055g

Casey KG, Quitevis EL (1988) Effect of solvent polarity on nonradiative processes in xanthene dyes: Rhodamine B in normal alcohols. J Phys Chem 92:6590–6594. https://doi.org/10.1021/j100334a023

Chakraborty M, Panda AK (2011) Spectral behaviour of eosin Y in different solvents and aqueous surfactant media. Spectrochim Acta A Mol Biomol Spectrosc 81:458–465. https://doi.org/10.1016/j.saa.2011.06.038

Castanet G, Maqua C, Orain M, Grisch F, Lemoine F (2007) Investigation of heat and mass transfer between the two phases of an evaporating droplet stream using laser-induced fluorescence techniques: comparison with modeling. Int J Heat Mass Transf 50(17–18):3670–3683. https://doi.org/10.1016/j.ijheatmasstransfer.2006.08.038

Enoki M, Katoh R (2018) Estimation of quantum yields of weak fluorescence from eosin Y dimers formed in aqueous solutions. Photochem Photobiol Sci 17:793–799. https://doi.org/10.1039/c7pp0465f

Jiao L (2019) Functionalized BODIPYs as fluorescent molecular rotors for viscosity detection. Front Chem. https://doi.org/10.3389/fchem.2019.00825

Jiménez-García JC, Rivera W (2023) Exergy analysis of an experimental ammonia/water absorption cooling system. Case Stud Therm Eng 49:103167. https://doi.org/10.1016/j.csite.2023.103167

Koegl M, Pahlevani M, Zigan L (2020a) A novel approach for measurement of composition and temperature of N-decane/butanol blends using two-color laser-induced fluorescence of Nile Red. Sensors 20:5721. https://doi.org/10.3390/s20195721

Koegl M, Weiß C, Zigan L (2020b) Fluorescence spectroscopy for studying evaporating droplets using the dye Eosin-Y. Sensors 20:5985. https://doi.org/10.3390/s20215985

Kristoffersen AS, Erga SR, Hamre B, Frette Ø (2014) Testing fluorescence lifetime standards using two-photon excitation and time-domain instrumentation: Rhodamine B, Coumarin 6 and Luger Yellow. J Fluoresc 24:1015–1024. https://doi.org/10.1007/s10895-014-1368-1

Lakowicz JR (2006) Principles of fluorescence spectroscopy, 3rd edn. Springer, New York

Lemoine F, Castanet G (2013) Temperature and chemical composition of droplets by optical measurement techniques: a state-of-the-art review. Exp Fluids 54:1572. https://doi.org/10.1007/s00348-013-1572-9
Liu L, Liu Y, Mi M et al (2016) Evaporation of a bicomponent droplet during depressurization. Int J Heat Mass Transf 100:615–626. https://doi.org/10.1016/j.ijheatmasstransfer.2016.05.007

Maqua C, Depredurand V, Castanet G et al (2007) Composition measurement of bicomponent droplets using laser-induced fluorescence of acetone. Exp Fluids 43:979–992. https://doi.org/10.1007/s00348-007-0368-1

Maqua C, Castanet G, Lemoine F (2008) Bicomponent droplets evaporation: temperature measurements and modelling. Fuel 87:2932–2942. https://doi.org/10.1016/j.fuel.2008.04.021

Mehdi S, Yangpeng L, Hadrien C et al (2021) Fluorescence lifetime measurements applied to the characterization of the droplet temperature in sprays. Exp Fluids 62:174. https://doi.org/10.1007/s00348-021-03264-x

Mendels D-A, Graham EM, Magennis SW et al (2008) Quantitative comparison of thermal and solutal transport in a T-mixer by FLIM and CFD. Microfluid Nanofluid 5:603–617. https://doi.org/10.1007/s10404-008-0269-5

Mishra YN, Abou Nada F, Polster S et al (2016) Thermometry in aqueous solutions and sprays using two-color LIF and structured illumination. Opt Express 24:4949. https://doi.org/10.1364/OE.24.004949

Moog RS, Ediger MD, Boxer SG, Fayer MD (1982) Viscosity dependence of the rotational reorientation of rhodamine B in mono- and polyalcohols. Picosecond transient grating experiments. J Phys Chem 86:4694–4700. https://doi.org/10.1021/j100221a011

Owen JF, Chang RK, Barber PW (1982) Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles. Aerosol Sci Technol 1:293–302. https://doi.org/10.1080/02786828208958595

Pan L, Zhang P, Daengngam C et al (2022) A review of artificial intelligence methods combined with Raman spectroscopy to identify the

composition of substances. J Raman Spectrosc 53:6–19. https://doi.org/10.1002/jrs.6225

Poling BE, Prausnitz JM, O'Connell JP (2001) The properties of gases and liquids, 5th edn. McGraw-Hill, New York

Promvongsa J, Fungtammasan B, Gerard G et al (2017) A study on the evaporation of water-ethanol mixture using rainbow refractometry. J Energy Resour Technol 139:062002. https://doi.org/10.1115/1.4037157

Reichardt C (1994) Solvatochromic dyes as solvent polarity indicators. Chem Rev 94:2319–2358. https://doi.org/10.1021/cr00032a005

Stiti M, Wang M, Chaynes H, Isaiev M, Castanet G (2024) A fluorescence approach for assessing the composition of evaporating droplets. Appl Phys Lett 125(24):243906. https://doi.org/10.1063/5.0238178

Wang M, Stiti M, Chaynes H et al (2022) Two-photon fluorescence lifetime imaging applied to the mixing of two non-isothermal sprays: temperature and mixing fraction measurements. Exp Fluids 63:172. https://doi.org/10.1007/s00348-022-03515-5

Watanabe A, Salmribara T, Hiroshi SI et al (1992) Photo-degradation and electron-beam-induced degradation of poly[(pentamethyldisilyl)styrene sulfonels. Macromolecules 25:692–697

Yarin AL, Pfaffenlehner M, Tropea C (1998) On the acoustic levitation of droplets. J Fluid Mech 356:65–91. https://doi.org/10.1017/S0022112097007829

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