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# Respiratory Drop Dynamics: Evaporation, Sedimentation, and Implications for Airborne Disease Transmission

Benoit Haut, Omer Atasi, Benjamin Sobac

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Benoit Haut, Omer Atasi, Benjamin Sobac. Respiratory Drop Dynamics: Evaporation, Sedimentation, and Implications for Airborne Disease Transmission. Journal of Colloid and Interface Science, 2025, pp.137715. 10.1016/j.jcis.2025.137715. hal-05074024

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Submitted on 28 Jul 2025

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Regular Article

# Respiratory drop dynamics: Evaporation, sedimentation, and implications for airborne disease transmission

Benoit Haut<sup>a, ID</sup>, Omer Atasi<sup>a,b</sup>, Benjamin Sobac<sup>b, ID, *</sup>

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GRAPHICAL ABSTRACT

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ARTICLE INFO

Keywords:

Sedimentation

Evaporation

Respiratory drop

Aerosol

Airborne transmission

Infectious disease

ABSTRACT

* Hypothesis: Understanding and predicting the dynamics of respiratory drop evaporation and sedimentation are critical for addressing airborne disease transmission and devising effective prevention strategies. Existing models often face limitations due to the complexity of the evaporation process, oversimplified assumptions, or incomplete knowledge of respiratory fluid properties.

* Theory: We present a novel theoretical model combining gravity-driven sedimentation under the Stokes flow approximation with a quasi-steady, diffusion-limited evaporation framework in a non-isothermal gas phase. The model incorporates essential respiratory fluid properties, including the sorption isotherm and solute diffusion coefficient, and is validated against experimental data. It applies to drops with radii < 50 µm, which are relevant for airborne transmission, and provides compact analytical expressions suitable for direct prediction and integration into larger-scale numerical models.

* Findings: The model elucidates the physicochemical mechanisms governing drop dynamics, considering the effects of drop size, ambient conditions, and solute presence. Sedimentation times are predicted across a wide range of ambient conditions (relative humidity 0–100%, temperature 0–40°C, and pressure 0.5–1 atm), emphasizing the critical role of environmental factors in transmission risks. We show that, over this full range, respiratory drop dynamics can be effectively described using a simple, analytical model based on pure water equations, complemented by the final drop size derived from the sorption isotherm. The model can also evaluate evaporation-induced pH changes resulting from ion concentration variations.

<footnote><sup>a</sup> Transfers, Interfaces and Processes (TIPs Lab), Universite libre de Bruxelles, 1050 Brussels, Belgium</footnote>

<footnote><sup>b</sup> CNRS, Université de Pau et des Pays de l'Adour, E2S UPPA, LFCR, 64600 Anglet, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: benjamin.sobac@cnrs.fr (B. Sobac).</footnote>

<footnote>https://doi.org/10.1016/j.jcis.2025.137715</footnote>

<footnote>Received 13 January 2025; Received in revised form 17 April 2025; Accepted 25 April 2025</footnote>

<footnote>Available online 7 May 2025</footnote>

<footnote>0021-9797/© 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 1. (a) Cumulative fraction of the total volume of expelled respiratory drops and (b) its derivative with respect to the drop radius, according to the BLO model of Johnson et al. [19] for speaking and coughing. (c) Illustration of the problem and main parameters.</figcaption></figure>

# 1. Introduction

The COVID-19 pandemic, caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has raised awareness of the importance of respiratory particles in the spread of infectious diseases [1–12]. Transmission of this disease is primarily airborne, through the inhalation of virus-laden respiratory particles released by an infected individual [1,2]. The infectivity of these particles depends on both the viability of the virus they carry [13] and the time they remain suspended in the air, facilitating transmission between individuals.

Respiratory particles are commonly classified as either aerosols or droplets, depending on their presumed transmission routes. In this work, we adopt a physical perspective and use the term respiratory drops to refer to all respiratory particles emitted during expiratory activities, regardless of their size or fate [14]. In addition, we characterize their size by their radius, following standard practice in fluid mechanics. While the historical threshold between aerosols and droplets was typically set at 2.5 µm, recent studies have proposed a critical radius of 50 µm to distinguish between airborne and contact transmission modes [15–17]. Drops smaller than this threshold remain suspended in air for minutes to hours, while larger ones typically settle to the ground in less than 5 seconds. This shift in perspective echoes debates from the early stages of the pandemic, when mounting evidence for airborne transmission prompted a re-evaluation of particle size thresholds [18].

Respiratory drops are produced when a person breathes, vocalizes (speaks, hums, or sings), coughs, or sneezes. To date, three modes of emission, each with a specific site of origin and mechanism of production, have been identified and their resulting size distributions characterized [14,19–23]. The bronchiolar mode consists of the smallest respiratory drops (radius < 2 µm), while drops in the laryngeal mode are slightly larger (radius between 0.5 and 5 µm). Finally, the so-called oral mode produces the largest drops (radius > 10 µm). Fig. 1(a-b) shows the trimodal drop size distribution given by the BLO model of Johnson et al. [19] for speaking and coughing (cumulative fraction of the total ex-

pelled volume and its derivative with respect to the drop radius). This widely used model is based on a meta-analysis of experimental measurements collected using complementary techniques over a broad size range. Fig. 1(a) shows that the bronchiolar and laryngeal modes show size distributions centered on radii of about 0.9 and 2 µm, respectively, while the oral mode produces a distribution of larger drops centered on a radius of about 150 µm. As sketched in Fig. 1(c), these drops undergo simultaneous evaporation and sedimentation once emitted.<fnref n="1" />

In order to focus on the sole airborne route of disease transmission, only respiratory drops with a radius of less than 50 µm are considered in this study, in line with the above delimitation. It is important to emphasize that although these drops represent only about 1% of the total volume of liquid expelled by a person (as shown in Fig. 1(a)), they alone contribute to airborne transmission. This is because, as mentioned above and assessed below, larger drops fall to the ground too quickly. Drops with a radius less than 50 µm are strictly spherical because their Bond number,<fnref n="2" /> which measures the importance of gravitational forces relative to surface tension forces, is much less than 1.

The sedimentation of a spherical object in air is a classical problem [24,25]. The balance between the drag force and gravity acting on a spherical drop of radius R (m) falling through the air is given by

$$6\pi\mu_{\text{air}} R U = \frac{4}{3}\pi R^3 \rho_d g \quad (1)$$

<footnote n="1">¹ Note that these drops, actively produced within the human respiratory tract, are fundamentally different from those that may originate through spontaneous vapor condensation in the ambient environment, which are not biologically generated and are not relevant for airborne transmission of pathogens.</footnote>

<footnote n="2">² The Bond number is defined as $Bo = \frac{\rho_d g R^2}{\gamma}$, where g is the gravitational acceleration, R the drop radius, $\rho_d$ the density of the drop, and $\gamma$ the surface tension between the drop and air. For a water drop with R = 50 µm, $Bo \approx 3 \times 10^{-4} \ll 1$.</footnote>
where $U > 0$ (m/s) is the relative velocity of the drop with respect to the air, $\mu_{air}$ (kg/(m s)) the viscosity of the air, $\rho_d$ (kg/m³) the density of the drop, and $g = 9.806 \text{ m/s}^2$ the acceleration of gravity at the Earth's surface.

In this force balance, the drag force $F_d$ is described by Stokes' law $F_d = 6\pi\mu_{air}R U$ [26], which assumes that inertial forces are negligible compared to viscous forces. As shown in the Supplementary Material or Refs [27,14], for the ambient conditions analyzed in this study, the Reynolds number of respiratory drops with $R < 50 \text{ µm}$, defined as $Re = 2RU\rho_{air}/\mu_{air}$, where $\rho_{air}$ (kg/m³) is the density of the air, is low enough to ensure the validity of this assumption. Furthermore, buoyancy is negligible in this force balance since $\rho_{air} \ll \rho_d$. Finally, note that the velocity $U$ given by Eq. (1) is the terminal relative velocity between the drop and the air, obtained once the drag force balances gravity. However, in practice, a respiratory drop is emitted with an initial vertical velocity component relative to the air which may differ significantly from $U$. The drop must therefore accelerate (or decelerate) toward this terminal regime. This transient has a characteristic timescale equal to $(2\rho_d R^2)/(9\mu_{air})$, which is about a few dozen of milliseconds for the largest drops considered here ($R = 50 \text{ µm}$), much shorter than their sedimentation time (about 5 s, see below). This justifies the assumption of an instantaneous terminal velocity within the present framework.

Considering an environment with no global convective air movement (stagnant air), $U$ can be replaced by $-h$ (m/s), the time derivative of the vertical distance between the drop and the ground $h$ (m). Thus, the force balance Eq. (1) can be rewritten as

$$ \dot{h} = -\frac{2}{9} \frac{\rho_d g}{\mu_{air}} R^2 \qquad (2) $$

Omitting evaporation, i.e., for a drop with constant $R$ (and thus constant $\dot{h}$), this allows us to deduce the sedimentation time of the drop $\tau_{sed}$ (s), given an initial vertical distance from the ground $h_0$:

$$ \tau_{sed} = -\frac{h_0}{\dot{h}} = \frac{9}{2} \frac{\mu_{air}}{\rho_d g} \frac{h_0}{R^2} = S \frac{h_0}{R^2} \qquad (3) $$

For a water drop in air at 20°C, the numerical prefactor is $S = 8.334 \times 10^{-9} \text{ m s}$ (with $\mu_{air} = 1.813 \times 10^{-5} \text{ kg/(m s)}$ and $\rho_d = 998.2 \text{ kg/m}^3$). Thus, in the absence of evaporation and with a radius of 0.5 µm, 5 µm or 50 µm, a drop initially at a height of $h_0 = 1.5 \text{ m}$ (average height above the ground for the mouth of a standing human adult) takes $50 \times 10^3 \text{ s}$ (about 14 h), 500 s (about 8 min) and 5 s, respectively, to fall to the ground.

Evaporation affects the sedimentation dynamics of a drop with a volatile solvent. As the solvent evaporates, the drop radius decreases over time, thus changing both the gravitational ($\propto R^3$) and drag ($\propto R$) forces unevenly, leading to a decrease in the settling velocity over time. Therefore, modeling the sedimentation dynamics of a respiratory drop requires accounting for evaporation, which is itself strongly influenced by ambient conditions including temperature, humidity, and atmospheric pressure.

The Schmidt number of water vapor in air, written $Sc$ and defined as the ratio of the kinematic viscosity $\mu_{air}/\rho_{air}$ to the diffusion coefficient of water vapor in air $D_{va}$, is close to 1. Consequently, the Péclet number of a respiratory drop, defined as $Pe = 2RU/D_{va} = Sc Re$, is approximately equal to its Reynolds number. This Péclet number compares a time scale of water vapor diffusion away from the drop, $(2R)^2/D_{va}$, with a time scale of convective renewal of the air around the drop, $2R/U$. As previously discussed, for all respiratory drops relevant for airborne transmission (radii below 50 µm), the Reynolds number remains sufficiently low to neglect inertial forces, and accordingly, the Péclet number is also small enough to consider that the convective flow induced by the relative motion of the drop with respect to the air has a negligible influence on its evaporation, compared to diffusion. The evaporation process of a respiratory drop can therefore be adequately modeled by considering a “suspended” drop, thereby neglecting the relative motion of the drop with respect to the air [27].

Today, numerous models have been developed to accurately predict the evaporation of a pure liquid drop [28–34], see also books and reviews [35–39]. These models have progressively incorporated interfacial equilibrium, thermal effects, and transport phenomena in the surrounding gas phase. In particular, Sobac et al. [32] have recently proposed a general analytical model for the evaporation of a suspended liquid drop. This model includes both diffusive and Stefan-convective transport mechanisms in a non-isothermal gas phase. It considers thermodynamic equilibrium at the liquid–gas interface and accounts for the local dependence of the physicochemical properties of the gas on temperature and water vapor concentration. The authors have provided a detailed analysis of the problem, focusing on discussing the validity of the assumptions and the influence of ambient conditions. They have also proposed simplified versions of the model. Under the present conditions of interest (in terms of initial drop radius and ambient conditions), they have shown that a quasi-stationary diffusion-controlled evaporation model, taking into account a thermodynamic equilibrium at the liquid–gas interface and a non-isothermal gas phase with constant physicochemical properties evaluated at the conditions far away in the surrounding air, is sufficient to accurately describe the evaporation of a pure water drop.

However, respiratory drops contain nonvolatile solutes such as salts, phospholipids, mucins, and enzymes [40–42]. These nonvolatile solutes have been estimated to make up for approximately 1% of the mass of a respiratory drop [40,43]. Therefore, additional effects must be considered when predicting the evaporation and sedimentation times of a respiratory drop. Through complex physicochemical interactions that may include liquid-solute coupling and internal structuring of the drop, the presence of the nonvolatile solutes reduces the water vapor partial pressure at the liquid–gas interface [44], and thus the evaporation rate, compared to a pure water drop. This decrease in water vapor partial pressure is accentuated during evaporation, as nonvolatile solutes become increasingly concentrated in the drop. This has an important consequence: a respiratory drop does not evaporate completely into the air, but has an equilibrium radius depending on its composition, the ambient conditions, and its initial radius [45]. This reduction of the water vapor partial pressure at the interface is classically characterized by the so-called sorption isotherm [46], which expresses a relationship between the water activity of the drop (i.e., the ratio of the water vapor partial pressure at the liquid–gas interface to the saturation pressure of a pure water drop at the same temperature) and its water content, at overall thermodynamic equilibrium between the drop and its environment. Note that this reduction in the water vapor partial pressure at the interface is also influenced by the competition between the receding interface and the diffusion of the solutes in the drop, resulting in the accumulation of these solutes near the interface [47].

Recently, several studies have attempted to model the dynamics of evaporation and sedimentation of respiratory drops by incorporating these effects, in order to better capture the complexity of real respiratory fluids in the context of viral transmission [27,48–50]. Netz [27] developed an analytical model to predict the sedimentation time of respiratory drops. He treated sedimentation using Stokes' law and considered the evolution of the drop radius due to evaporation, modeled as a quasi-steady diffusion-limited process in stagnant air. He included evaporative cooling (i.e., the decrease in temperature at the drop interface due to the energy consumed in evaporating the liquid) and the decrease in water vapor partial pressure at the interface due to the presence of nonvolatile solutes, whose concentration was assumed to be homogeneous in the drop. In a later study, Rezai and Netz [49] added to their model an equation describing the diffusion of the solutes within the drop and developed a heuristic analytical expression for the evaporation time based on a numerical solution of the equations of the problem. Although these studies have resulted in a comprehensive analysis, they lack detailed physicochemical information about respiratory drops, especially regarding the sorption isotherm, which was assumed to be a linear relationship between water activity and water mass fraction, or the diffusion
coefficient of solutes inside the drop. In addition, a linear equation was used in Refs [27,48,49] to express the saturation concentration of water as a function of temperature, which prevents the models from accurately predicting the evaporation rate and hence the sedimentation time over a wide range of ambient conditions. These works were followed by the study of Lieber et al. [45], who experimentally measured the equilibrium radius of an evaporating respiratory drop and showed that it is about 20% of the initial radius over a wide range of relative humidities. They proposed a numerical solution of the equations governing the evaporation and sedimentation of a respiratory drop taking into account this equilibrium radius, using mass and heat transfer correlations to calculate the evaporation flux, based on the model of Abramzon and Sirignano [30]. To our knowledge, this was the first study to use the correct equilibrium radius of a respiratory drop to predict its sedimentation time. Recently, Merhi et al. [43] continued the experimental characterization by measuring the full sorption isotherm of human saliva. They showed that it deviates significantly from a linear relationship between water activity and water content, and that this deviation has a strong influence on the sedimentation time of a respiratory drop. In addition, in the same study, the authors measured an overall diffusion coefficient for nonvolatile solutes present in saliva and used this measurement to assess the effect of concentration gradients in a saliva drop on its evaporation and sedimentation times. However, they did not take evaporative cooling into account, even though it has a significant impact on the dynamics of the process.

Despite recent advances, a detailed understanding and accurate prediction of the coupled evaporation and sedimentation processes in realistic respiratory drops have yet to be achieved. This lack of insight hinders the reliable integration of these effects, whose impact on drop dynamics can be considerable, into larger-scale models, such as Computational Fluid Dynamics (CFD) studies aimed at supporting public health strategies (see for example [51]). In this context, our goal is to develop a theoretical model that captures the essential physicochemical mechanisms governing evaporation and sedimentation of respiratory drops, within the size limit of interest for airborne transmission (initial radius less than 50 µm), and for ambient conditions corresponding to the wide range of natural atmospheric conditions found on Earth. We aim to provide precise, compact analytical expressions that can be used in a variety of settings, including direct predictions of evaporation or sedimentation times under diverse environmental conditions, parametric studies of drop behavior, and integration with numerical tools such as CFD codes. Our methodology consists in modeling only the relevant physicochemical mechanisms governing the evaporation of respiratory drops, starting from the evaporation model developed by Sobac et al. [32] for pure liquid drops and then taking into account the presence of non-volatile solutes using the aforementioned experimental results of Merhi et al. [43] regarding the equilibrium at the liquid-gas interface and the diffusion coefficient of these solutes. Thus, our model overcomes the limitations of the previous models while incorporating recent experimental results. We discuss in the detail the assumptions and the minimum level of complexity that must be considered to correctly calculate the evaporation and sedimentation times of a respiratory drop. The validity of the model is verified by comparison with experimental results from the literature. In addition to a comprehensive analysis of the problem, the influence of drop size, ambient conditions, and the presence of solutes is discussed. We further extend the scope of the model by quantifying the pH changes induced by evaporation-driven ion concentration, providing a relevant physicochemical indicator that may inform future studies on virus viability. The Supplementary Material includes a Wolfram Mathematica file for reproducing most of the results presented in the article, and values of physicochemical parameters used to produce these results (or information on how to calculate these parameters).

## 2. Evaporation and sedimentation of a pure water drop

### 2.1. Evaporation

The present analysis builds on the drop evaporation model developed in our previous works [32,33]. This model considers an isolated spherical drop of pure water suspended in a surrounding gas phase of infinite extent, with no relative motion between the drop and the gas. The gas phase is treated as an ideal mixture of air and water vapor, with homogeneous and constant total pressure $P_g$ (Pa), temperature $T_∞$ (K), and vapor mole fraction $X_∞$. Its physicochemical properties are assumed to be homogeneous and constant. Thermal transients being negligible, the process is quasi-steady, which implies in particular that the drop temperature remains uniform. Since heat transfer in the gas phase limits the evaporation process, evaporative cooling is significant and the non-isothermal nature of the gas must be taken into account. Under the conditions considered, the contributions of Stefan flow, radiative heat transfer, and Knudsen and Kelvin effects are negligible [32]. Therefore, the present model is valid for drop radii ranging from 100 nm (lower limit set to neglect Kelvin effects) up to 50 µm (upper limit set by the onset of convective effects), under ambient conditions representative of natural environments on Earth. As shown in Fig. 1(a), drops with a radius smaller than 100 nm constitute only a tiny fraction of the population involved in airborne transmission, so that their exclusion does not affect the validity of the present analysis.

Based on these assumptions and using the conservation of mass and heat, the following equations can be written for the molar flux of evaporation $j$ (expressing the number of moles of water evaporated per second and per unit area of the drop interface) and the temperature at the liquid-gas interface $T_i$ (K) at any time during the drop evaporation:

$$jR = c_g D_{va} (X_i - X_\infty) \quad (4)$$

$$T_i = T_\infty - \frac{\mathcal{L}(T_i)jR}{\lambda_g} \quad (5)$$

where $R$ (m) is the instantaneous drop radius, $c_g = P_g/(\hat{R} T_∞)$ (mol/m³) the molar concentration of the gas phase, with $\hat{R}$ (J/(mol K)) the universal gas constant, $D_{va}$ (m²/s) the diffusion coefficient of water vapor in air, $X_i$ (-) the mole fraction of vapor at the liquid-gas interface, $\mathcal{L}(T_i)$ (J/mol) the latent heat of vaporization of water evaluated at temperature $T_i$, and $\lambda_g$ (W/(m K)) the thermal conductivity of the gas phase.

Assuming local thermodynamic equilibrium at the liquid-gas interface, the vapor mole fraction at the interface $X_i$ is related to the interfacial temperature $T_i$ as

$$X_i = \frac{P_{\text{sat}}(T_i)}{P_g} = \frac{P_{\text{sat}}(T_{\text{ref}})}{P_g} \exp \left( - \frac{\mathcal{L}(T_{\text{ref}})}{\hat{R}} \left( \frac{1}{T_i} - \frac{1}{T_{\text{ref}}} \right) \right) \quad (6)$$

where $P_{\text{sat}}(T)$ (Pa) is the saturation pressure of water at temperature $T$ (K), expressed using the Clausius-Clapeyron equation with a reference temperature $T_{\text{ref}} = 293.15$ K (and a corresponding saturation pressure $P_{\text{sat}}(T_{\text{ref}}) = 2340$ Pa). As shown in the Supplementary Material, this gives an excellent comparison with experimental data in the temperature range 0-40°C. $\mathcal{L}(T_{\text{ref}})$ (J/mol) is the latent heat of vaporization of water evaluated at the temperature $T_{\text{ref}}$.

Eqs (4), (5) and (6) form a system of three equations with three unknowns ($jR$, $T_i$ and $X_i$) that can be easily solved numerically for given values of the ambient conditions $P_g$, $T_∞$, and $X_∞$. These equations show that there are unique values of $jR$, $T_i$ and $X_i$ for given values of the ambient conditions, i.e., $T_i$ and $X_i$ are independent of time $t$ and instantaneous drop radius $R$. To solve these equations, the physicochemical properties of the drop are evaluated at its temperature $T_i$, while those related to the gas phase are evaluated using the ambient conditions far
from the drop.<fnref n="3" /> It is also important to remember that the mole fraction of water vapor at infinity $X_∞$ is related to the relative humidity of the surrounding air RH (-) by

$$X_{\infty} = RH \times \frac{P_{\text{sat}}(T_{\infty})}{P_{g}}. \qquad (7)$$

According to a global mass balance on the drop, the time derivative of the number of liquid moles, $c_ℓ 4πR^3/3$, with $c_ℓ$ (mol/m<fnref n="3" />) the molar concentration of liquid water, is equal to the opposite of the evaporation flux j times the drop surface area $4πR^2$. Simplifying this gives

$$\dot{R} = -\frac{j}{c_ℓ} \qquad (8)$$

where $\dot{R}$ (m/s) is the time derivative of R. In the following, the density $ρ_d$ of a water drop is considered independent of temperature, since it has a relative variation of less than 1% between 0°C and 40°C [52], and equal to 998.2 kg/m<fnref n="3" /> (exact value at 20°C). Thus, $c_ℓ = ρ_d/M_{H_2O}$, with $M_{H_2O}$ the molar mass of water, is also independent of temperature.

By inserting Eq. (4) for j into Eq. (8) and integrating the resulting equation, we get

$$R^2 = R_0^2 - 2D_{\text{va}} \frac{c_g}{c_ℓ} (X_i - X_{\infty})t = R_0^2 - \mathcal{E}(X_i - X_{\infty})t \qquad (9)$$

where t (s) is the time, $R_0$ the initial value of R (at time t = 0), and the prefactor $\mathcal{E} = 2D_{\text{va}}c_g/c_ℓ$ (function of $T_∞$ via its numerator). The radius of the drop follows an $R^2$-law: the square of R decreases linearly with time.

From Eq. (9), we can directly derive the expression for the evaporation time of the drop (value of t that gives R = 0), written $τ_{ev}$ (s):

$$τ_{ev} = \frac{R_0^2}{\mathcal{E}(X_i - X_{\infty})} \qquad (10)$$

To provide useful information, we have numerically solved Eqs (4), (5) and (6) for a water drop evaporating in ambient air, at $P_g = 1$ atm and for a broad range of temperature ($T_∞ \in [0, 40]°C$) and relative humidity ($RH \in [0, 1]$). A polynomial fit of the results gives

$$T_i = T_{\infty} - (1 - RH)(3.312 + 84.58 RH - 0.4063 T_{\infty} + 2.621 RH^2 - 0.3121 RH T_{\infty} + 0.001531 T_{\infty}^2) \qquad (11)$$

with $T_i$ and $T_∞$ in K. This equation allows predicting the interfacial temperature $T_i$ with a good precision (average error less than 0.1°C when compared to the solution of Eqs (4)-(7), and maximal error of 0.8°C when $T_∞ = 40° C$ and RH = 0). Together with Eq. (6), this equation allows predicting the vapor mole fraction at the interface $X_i$ and thus, with Eqs (7) and (10), the evaporation time of a drop for given values of $T_∞$, RH and $R_0$. Note that Eq. (11) is only given here to be useful to a reader who would like to easily predict evaporation times. In what follows, this equation is not used and all results are obtained from the numerical solution of Eqs (4)-(7).

For a water drop evaporating into ambient air at $T_∞ = 20° C$, RH = 50% (equivalent to $X_∞ = 0.01155$), and $P_g = 1$ atm, we obtain $T_i = 13.35° C$, $X_i = 0.01516$, and $jR = 3.69 \times 10^{-6}$ mol/(m s). Additionally, the value of the numerical prefactor $\mathcal{E}$ in Eqs (9) and (10) is $\mathcal{E} = 3.689 \times 10^{-8}$ m²/s. Thus, we can estimate that a water drop with an initial radius $R_0$ of 0.5 µm, 5 µm, or 50 µm takes $1.877 \times 10^{-3}$ s,

0.1877 s and 18.77 s, respectively, to evaporate completely. To complete the illustration, Fig. 2(a,b) shows the profiles of the vapor mole fraction X and temperature T in the gas phase for a water drop evaporating in such ambient conditions. Since heat and mass transfers are controlled by diffusion in a spherical geometry, T and X are linearly related to R/r, with r the radial position in the gas phase.

Even under these classical environmental conditions, evaporative cooling causes a significant decrease in the temperature of the drop, with $T_∞ - T_i = 6.6° C$. This effect increases as $X_∞$ (or RH) decreases, reaching $T_∞ - T_i = 15.5° C$ in the dry air limit ($X_∞ = 0$). Thus, assuming an isothermal gas phase generally results in a significant overprediction of $X_i$ and, consequently, of the evaporation rate. Specifically, for the conditions in this example, it can be evaluated that the evaporation rate would have been overpredicted by more than a factor of three with the assumption of an isothermal gas phase. This substantial increase is intimately linked to the nonlinear relationship between $T_i$ and $X_i$ (see Eq. (6)).

Fig. 2(c,d) shows the time evolution of the radius of a water drop evaporating in air at various relative humidities RH and for different initial sizes. The results obtained with the model are compared with the experimental results from Refs [53–55]. These experiments were performed under conditions coherent with airborne transmission. Indeed, they considered a water drop evaporating in ambient air at $T_∞ \approx 20° C$, $P_g = 1$ atm, and for different values of RH. Moreover, they characterized the evaporation of an isolated micrometric drop ($R \approx 20–30$ µm), either trapped [54,55] or falling under the effect of gravity [53]. We see that our model accurately predicts these experimental data, whatever the relative humidity of the air and for the two experimental protocols.

## 2.2. Coupling with sedimentation

The calculated values of $τ_{ev}$ must be analyzed in the light of the sedimentation times $τ_{sed}$. The comparison between $τ_{ev}$ and $τ_{sed}$ determines whether the drop evaporates completely during its fall or whether it eventually reaches the ground. However, evaporation affects the sedimentation time in two ways. First and foremost, evaporation reduces the radius of the drop as it falls, thus increasing the sedimentation time. Second, evaporation cooling affects the temperature of the drop (causing it to change from $T_∞$ to $T_i$), reducing the viscosity of the air at its contact and thus the value of the numerical prefactor $S$ (see Eq. (3)). However, this remains marginal as it can be estimated that even under extreme conditions (hot and dry air), evaporative cooling induces a decrease in viscosity of less than 7%.

To describe the sedimentation of an evaporating water drop, it is necessary to replace $R^2$ in the force balance Eq. (2) by its expression in Eq. (9), which gives

$$\begin{aligned} \dot{h} &= -\frac{2}{9} \frac{\rho_d g}{\mu_{\text{air}}(T_i)} \left( R_0^2 - 2D_{\text{va}} \frac{c_g}{c_\ell} (X_i - X_\infty)t \right) \\ &= -\frac{1}{S(T_i)} (R_0^2 - \mathcal{E}(X_i - X_\infty)t) \end{aligned} \qquad (12)$$

where $\mu_{\text{air}}(T_i)$ indicates that the air viscosity<fnref n="4" /> is evaluated at $T_i$ instead of $T_∞$. Therefore, the sedimentation prefactor $S(T_i) = 9/2 \mu_{\text{air}}(T_i)/(\rho_d g)$ now depends on $T_i$.

Solving the last equation with $h(t = 0) = h_0$ gives the sedimentation time of an evaporating water drop $τ_{sed}$ (s), which is defined only if the drop reaches the ground before it is completely evaporated:

<footnote n="3">³ More precisely, $D_{\text{va}}$ is evaluated at pressure $P_g$ and temperature $T_∞$, using a fit of experimental data. $λ_g$ is considered to be the thermal conductivity of dry air at temperature $T_∞$, and is evaluated using an equation derived from the kinetic theory of gases. Note that even in the worst case of air at 40°C and saturated with water, the thermal conductivity of this humid air differs by only 2% from that of dry air at the same temperature (this can be evaluated using equations presented in Ref. [32]). A linear fit of experimental data allows the expression of $L(T)$. More details about the evaluation of the liquid and gas phase properties are provided in the Supplementary Material.</footnote>

<footnote n="4">⁴ Note that the viscosity of air can be estimated to a very good approximation as that of dry air at the same temperature. Indeed, even in the worst case of air at 40°C and saturated with water, it can be shown using equations presented in Ref. [32] that the kinematic viscosity of this humid air differs by only 3% from that of dry air at the same temperature.</footnote>
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

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

<figure><img src="image_9.png" /><figcaption>Fig. 2. (a,b) Profiles of temperature T and vapor mole fraction X in the gas phase surrounding an evaporating water drop in air at $T_\infty = 20^\circ C$, RH = 50%, and $P_g = 1$ atm. r is the radial position. (c,d) Evolution over time t of the radius R of an evaporating pure water drop under some airborne transmission conditions ($R_0 \approx 20-30$ µm, $T_\infty \approx 20^\circ C$, and $P_g = 1$ atm). Experimental data from the literature at (c) RH = 0% [53], RH = 90% [54], RH = 94% [55], and (d) 0% $\le$ RH $\le$ 90% [53] are compared with the model predictions. Note that, while the model predictions are shown as continuous lines in (c), a range of predictions considering the experimental uncertainties (+/- 1%) on the ambient relative humidity is also shown in light color. For clarity, the latter is not shown in (d).</figcaption></figure>

$$ \tau_{\text{sed}} = \tau_{\text{ev}} \left[ 1 - \left( 1 - \frac{2S(T_i)h_0}{\tau_{\text{ev}} R_0^2} \right)^{1/2} \right] \quad (13) $$

Note that, in the limit of a non-evaporating drop ($RH \to 1$, implying $\tau_{\text{ev}} \to \infty$ and $T_i \to T_\infty$), Eq. (3) is recovered ($1 - \sqrt{1 - 2x} \approx x$ if $x \to 0$).

By equating Eqs (10) and (13), we can obtain the critical initial radius below which a drop evaporates completely before reaching the ground, written $R_0^c$:

$$ R_0^c = [2S(T_i)\varepsilon h_0(X_i - X_\infty)]^{1/4} \quad (14) $$

For $T_\infty = 20^\circ C$, RH = 50%, $P_g = 1$ atm, and $h_0 = 1.5$ m, we get $S(T_i) = 8.184 \times 10^{-9}$ m s and $R_0^c = 42.52$ µm. Therefore, under these conditions, all pure water drops initially smaller than 42.52 µm evaporate completely before reaching the ground. Returning to our illustrative drops with initial radii of 0.5, 5, and 50 µm, we see that only the largest drop reaches the ground with an evaluated sedimentation time of $\tau_{\text{sed}} = 5.809$ s, which is about 15% longer than the sedimentation time predicted if evaporation is neglected.

To extend the analysis, Fig. 3 shows the sedimentation time of an evaporating drop $\tau_{\text{sed}}$ (Eq. (13)) as a function of its initial radius $R_0$, for an initial height of $h_0 = 1.5$ m, and different relative humidities of the ambient air at $T_\infty = 20^\circ C$ and $P_g = 1$ atm. In the limit RH = 1, no evaporation takes place and the result of Eq. (3) is obtained (black line). The evaporation time $\tau_{\text{ev}}$ (Eq. (10)) is also plotted (thin continuous lines). The sedimentation and evaporation times intersect at the critical initial radius $R_0^c$, given by Eq. (14) and indicated by a dashed line. $R_0^c$ decreases monotonically with increasing relative humidity. If $R_0 < R_0^c$, the drop

evaporates completely during its fall in time $\tau_{\text{ev}}$, while if $R_0 > R_0^c$, the drop reaches the ground in time $\tau_{\text{sed}}$, with a size that can be determined by Eq. (9) evaluated at time $t = \tau_{\text{sed}}$. Furthermore, for a given relative humidity, the closer $R_0$ is to $R_0^c$, the further the sedimentation time deviates from its prediction neglecting the effect of evaporation (black line). Using Eqs (3), (13) and (14), we can easily deduce that when $R_0 = R_0^c$, there is a ratio $2S(T_i)/S(T_\infty) \approx 2$ between the sedimentation times of a water drop predicted with and without taking into account its evaporation. All this demonstrates the significant influence of evaporation and ambient air conditions on the sedimentation of a water drop. Depending on $R_0$ and RH, the sedimentation time of a water drop varies considerably, from a minimum of 5 s for a drop with an initial radius of 50 µm, to an infinite duration when $R_0 < R_0^c$ or as $R_0 \to 0$.

Note that Fig. 3 and several of the figures presented in the following sections show evaporation and sedimentation times for drops with initial radii up to 60 µm, slightly beyond our maximum radius of interest of 50 µm. For $R_0 \in [50-60]$ µm, inertial effects on the sedimentation time and convective effects on the evaporation time become non-negligible, so the parts of the curves shown between 50 µm and 60 µm should be considered only qualitative and not fully quantitative.

### 3. Evaporation and sedimentation of a respiratory drop

#### 3.1. Full problem

We now turn to the full problem of the coupled evaporation and sedimentation of a respiratory drop, containing nonvolatile solutes. In the following, the initial mass fraction of these solutes in the drop is
<figure><img src="image_10.png" /><figcaption>**Fig. 3.** Evaporation time $\tau_{ev}$ (thin lines) and sedimentation time $\tau_{sed}$ (thick lines) of an evaporating pure water drop, as functions of its initial radius $R_0$, according to Eqs (10) and (13), for an initial height of $h_0 = 1.5$ m. Results are shown for air at $T_\infty = 20^\circ\text{C}$, $P_g = 1$ atm, and different relative humidities RH: 0, 30, 50, 70, 90, 99, and 100%. In the limit RH = 1 (saturated air), no evaporation occurs and Eq. (3) is recovered (black line). The critical initial radius $R_0^c$ defined at the intersection of the two curves is plotted according to Eq. (14) (dashed line).</figcaption></figure>

assumed to be $\phi_0 = 1\%$, consistent with data reported for saliva [40]. Furthermore, the molar concentration of water within the drop $c_f$ and its density $\rho_d$ are taken to be those of pure water as used previously, regardless of the ambient conditions and the solute concentration.

First, as far as the sedimentation of the drop is concerned, Eq. (2) remains valid to describe its settling, with $\mu_{air}$ evaluated at the interfacial temperature $T_i$.

Regarding the evaporation of the drop, Eqs (4) and (5) are still valid to express the evaporation flux $j$ and the interfacial temperature $T_i$, since they describe phenomena taking place at quasi-steady state in the gas phase. In addition, the mass balance for the water in the drop is still described by Eq. (8), since water is the only evaporating component. However, as mentioned in the introduction, the nonvolatile solutes present in respiratory fluids, such as salts, phospholipids, mucins, and enzymes, reduce the water vapor partial pressure at the liquid-gas interface (i.e., they modify the value of $X_i$), through complex physicochemical interactions that may include liquid-solute coupling and internal structuring of the drop. As a result, Eq. (6) is no longer valid in their presence, and the evaporation model must be adapted accordingly.

These effects are classically globally captured using a sorption isotherm, which expresses a macroscopic relationship between the water activity of the drop (i.e., the ratio of the water vapor partial pressure at the liquid-gas interface to the saturation pressure of a pure water drop at the same temperature) and its water content, at overall thermodynamic equilibrium between the drop and its environment. Merhi et al. [43] recently provided experimental data on this sorption isotherm for human saliva. As shown in Fig. 4(a), these data can be fitted to a good approximation by the following equation (similar to the so-called GAB isotherm [56]):

$$ \psi = I(a_w) = \frac{(1 + c_1 + c_2)a_w}{1 + c_1 a_w + c_2 a_w^2} \qquad (15) $$

where $\psi$ (-) is the mass fraction of water in the drop, $a_w$ (-) is its water activity, and $c_1$ (-) and $c_2$ (-) are constants, fitted to 0.3951 and -1.079, respectively. Although Merhi et al. [43] performed experiments at room temperature, we assume that this equation is valid for the temperature range of airborne transmission. In the following, this equation is considered to be representative of any respiratory drop.

As the drop evaporates, water loss leads to an increase in solute concentration within the drop, resulting in a decreased water vapor partial pressure at the interface. This concentration increase is most pronounced at the receding interface, where solutes tend to accumulate [43,49]. Consequently, a concentration gradient forms within the drop, which is attenuated by the diffusion of solutes toward the center of

the drop. Despite the complex interplay between solute accumulation at the receding interface and diffusion within the drop, it has been demonstrated that for accurately predicting sedimentation time, it is sufficient to assume a homogeneous solute concentration in the drop [43,49]. This implies that the competition between the receding interface dynamics and solute diffusion can be neglected. This point is further detailed in the Supplementary Material, where we compare the predictions of the full problem presented in this section, assuming homogeneous solute concentration, with those of a more detailed model that relaxes this assumption. The detailed model incorporates a diffusion equation to account for solute concentration gradients within the drop, using the diffusion coefficient provided by Merhi et al. [43]. The comparison reveals that both approaches yield very similar sedimentation time predictions, with a relative difference of less than 2.5% under all tested ambient conditions, including extreme scenarios.

If the concentration of solutes in the drop is homogeneous, a simple mass balance can be used to relate the mass fraction of solutes in the drop $\phi$ at any time during evaporation to the instantaneous value of the drop radius $R$:

$$ \phi = \frac{R_0^3}{R^3} \phi_0 \qquad (16) $$

where $\phi_0$ is the initial mass fraction of solutes in the drop.

Now, still assuming that there is an instantaneous and local thermodynamic equilibrium at the liquid-gas interface, the partial pressure of water at this interface (which is $P_g \times X_i$) is at any time the product of the saturation pressure of water at the interface temperature $P_{sat}(T_i)$ and the water activity $a_w$ corresponding to a water mass fraction $\psi = 1 - \phi$. Thus, the equilibrium equation Eq. (6) is rewritten as

$$ X_i = a_w(\phi) \frac{P_{sat}(T_i)}{P_g} = a_w(\phi) \frac{P_{sat}(T_{ref})}{P_g} \exp \left( - \frac{\mathcal{L}(T_{ref})}{\hat{R}} \left( \frac{1}{T_i} - \frac{1}{T_{ref}} \right) \right) \qquad (17) $$

with the expression of $a_w$ as a function of $\phi$ obtained by inverting Eq. (15):

$$ a_w(\phi) = I^{-1}(1 - \phi) = \frac{1 + c_2 + c_1 \phi - \sqrt{(1 + c_2 + c_1 \phi)^2 - 4c_2(1 - \phi)^2}}{2c_2(1 - \phi)} \qquad (18) $$

For $\phi = 0$, $a_w = 1$. Thus, Eq. (6) is recovered for a drop of pure water.

It is important to emphasize that as a respiratory drop evaporates, it tends toward a state of equilibrium with its environment, characterized by a final value of the mass fraction of solutes in the drop $\phi_f$ and a
<figure><img src="image_11.png" /><figcaption>Fig. 4. (a) Sorption isotherm of a drop of human saliva: mass fraction of water in the drop $\psi$ as a function of its water activity $a_w$. The circles are data from Merhi et al. [43]. The continuous line is Eq. (15), with $c_1 = 0.3951$ and $c_2 = -1.079$. A linear isotherm $\psi = a_w$ is also shown. (b) Ratio of the final drop radius $R_f$ to its initial radius $R_0$, as a function of the relative humidity RH of the ambient air, for a drop with an initial solute mass fraction $\phi_0 = 1\%$. The circles are obtained from the experimental isotherm data shown in sub-figure (a), using Eq. (20) to convert them to $R_f/R_0$ as a function of RH. The continuous line is obtained by combining Eqs (19) and (20) using the fitted values of $c_1$ and $c_2$. The dashed line is obtained using a linear isotherm $\psi = a_w$, which gives $R_f/R_0 = (\phi_0/(1 - RH))^{1/3}$. Finally, the dotted horizontal line gives the approximation $R_f/R_0 = \phi_0^{1/3}$.</figcaption></figure>

final value of the drop radius $R_f$. When this equilibrium is reached, $T_i = T_\infty$ and $X_i = X_\infty$, which implies that the water activity of the drop $a_w$ is equal to the relative humidity of the ambient air RH (see Eqs (7) and (17)). Thus, the mass fraction of solutes in the drop when it is in equilibrium with its environment is

$$ \phi_f = 1 - I(RH) = 1 - \frac{(1 + c_1 + c_2)RH}{1 + c_1RH + c_2RH^2} \quad (19) $$

This equation shows that unless the air is completely dry ($RH = 0$), $\phi_f$ is less than 1. This means that the respiratory drop, in equilibrium with its environment, still contains a certain amount of water. For this reason, it is somewhat preferable to speak of the *drying* of a respiratory drop rather than its *evaporation*. Note that this final drop is frequently referred to in the literature as the “nucleus” [14,57]. $R_f$ is directly related to $\phi_f$ using Eq. (16):

$$ R_f = \left( \frac{\phi_0}{\phi_f} \right)^{\frac{1}{3}} R_0 \quad (20) $$

The final radius of the drop depends on its initial composition $\phi_0$, its initial radius $R_0$, and the relative humidity of the ambient air RH (which defines $\phi_f$, see Eq. (19)). If the drop has not fallen to the ground at the end of its drying, this final radius dictates the end of its sedimentation. Note that, strictly speaking, this final radius is only reached asymptotically, since the driving force for evaporation $X_i - X_\infty$ goes to zero when $\phi \to \phi_f$.

Fig. 4(b) shows the ratio $R_f/R_0$ as a function of the relative humidity RH of the ambient air for a respiratory drop. The circles are derived from the experimental data shown in Fig. 4(a), using Eq. (20) to convert them to $R_f/R_0$ as a function of RH. The continuous line is obtained by combining Eqs (19) and (20), using the fitted values of $c_1$ and $c_2$. It can be observed that the equilibrium state is reached with a quasi-constant ratio $R_f/R_0$ over a wide range of relative humidity, namely as long as RH is less than about 70%. This is related to the downward concavity of the isotherm (see Fig. 4(a)): $I(RH)$ remains well below 1 when RH < 70%, leading to Eqs (19) and (20) showing that $R_f/R_0 \approx (\phi_0)^{1/3} = 0.22$ (shown as a horizontal dotted line in Fig. 4(b)). Thus, for a wide range of ambient relative humidities, the radius of an evaporating respiratory drop has decreased by a factor of nearly 5 when it reaches a state of equilibrium with its environment. These results are consistent with the experimental data of Lieber et al. [45], who reported a final radius of 20% of the initial radius, regardless of relative humidity in the range 6–65%. Consequently, the experimental data from Merhi et al. [43] accurately reflect the behavior observed in the literature, thus validating their use in our modeling with respect to the final equilibrium state. Above a relative humidity of about 70%, the ratio $R_f/R_0$ increases rapidly.

The complete set of equations describing the full problem of drying and sedimentation of a respiratory drop has now been established. It includes Eq. (2) for the sedimentation of the drop, Eqs (4), (5), and (8) for the drying of the drop, supplemented by Eqs (17) and (18) for the thermodynamic equilibrium at the liquid-gas interface and by Eq. (7) to evaluate $X_\infty$ as function of the relative humidity RH and temperature $T_\infty$ of the ambient air, and Eq. (16) for the evolution of the solute concentration with time. Eqs (19) and (20) allow the calculation of the final state of the drop when it reaches equilibrium with its environment. Since $\phi$ varies with time, so do $X_i$, $jR$, and $T_i$ (see Eqs (17), (4), and (5)). Therefore, the full problem presented here does not allow the derivation of simple algebraic expressions as for the coupled evaporation and sedimentation of a pure water drop.

Fig. 5 is the strict equivalent of Fig. 3, but for a respiratory drop. The drying and sedimentation times are obtained by numerically solving the equations of the full problem described above in this section. Since the final radius of the drop is reached asymptotically, the drying time $\tau_{cv}^*$ (s) is defined as the time required to obtain $(R - R_f)/(R_0 - R_f) < 10^{-3}$. A sedimentation time is calculated for each initial value of the drop radius (i.e., even if drying is complete before sedimentation), since the presence of solutes means that the drop does not disappear completely due to evaporation, but rather reaches a final size in equilibrium with the environment. The results presented in Fig. 3 for pure water are also shown as dashed lines for comparison, and the result for a drop in the non-evaporation limit is added as a solid black line.

We can again introduce the critical initial radius of a respiratory drop under given ambient conditions, written $R_0^{c,*}$, which is such that $\tau_{sed} = \tau_{ev}^*$. In Fig. 5 and for a given value of the ambient relative humidity RH, this critical radius is therefore read as the abscissa of the intersection of the corresponding drying and sedimentation time curves. The results presented in Fig. 5 show that, except for high values of RH (> 90%), the critical radius of a respiratory drop is very close to the critical radius of a pure water drop under the same ambient conditions ($R_0^c$, given by Eq. (14)). Consistently, the sedimentation time of a respiratory drop with $R_0 > R_0^{c,*}$ is very close to that of a pure water drop under the same conditions. This is explained by the shape of the sorption isotherm (downward concavity, see Fig. 4(a)), which still yields high values of water activity even when the drying of the drop has already significantly
<figure><img src="image_12.png" /><figcaption>Fig. 5. Drying time $\tau_{ev}^*$ (thin lines) and sedimentation time $\tau_{sed}$ (thick lines) of a respiratory drop as functions of its initial radius $R_0$, for an initial height of $h_0 = 1.5$ m and $\phi_0 = 1\%$. Results are shown for air at $T_\infty = 20^\circ C$, $P_g = 1$ atm, and different relative humidities RH: 0, 30, 50, 70, 90, 99%. The intersections of the corresponding drying and sedimentation time curves are indicated by full circles. The “non evaporating” curve gives the sedimentation time of a respiratory drop with a constant radius equal to its initial value $R_0$ and with $T_i = T_\infty$ ($\tau_{sed}$ given by Eq. (3)). The dashed curves are the results obtained under the same conditions for a drop of pure water and are already shown in Fig. 3.</figcaption></figure>

concentrated the solutes (i.e., reduced the water content $\psi$), and by the rapid decrease of the drop radius at the end of its drying (see Fig. 2(c,d)). Therefore, for $R_0 > R_0^{c,*}$, the same comment applies as for pure water: the closer the initial radius is to the critical radius (by higher values), the longer the sedimentation time will be compared with the sedimentation time of an equivalent object that does not evaporate; and it is almost twice as long if $R_0 = R_0^{c,*}$. On the other hand, when $R_0 < R_0^{c,*}$, the sedimentation time of a respiratory drop is significantly different from that of a pure water drop since it remains finite. When the relative humidity is less than 70%, the final radius $R_f$ is about 1/5 of the initial radius $R_0$ (see Fig. 4(b)). Consistently, Fig. 5 shows that for $R_0 < R_0^{c,*}$ and $RH < 70\%$, the sedimentation time rapidly becomes about 25 times longer than that of an equivalent object that does not evaporate. This is because the settling velocity is proportional to the square of the radius (see Eq. (2)) and because most of the fall occurs at $R_f$ when $R_0$ is significantly less than the critical radius. As the relative humidity gets closer to 100%, the sedimentation time approaches that of a non-evaporating object. However, even at $RH = 99\%$, it can be significantly longer, by more than a factor of 2 when $R_0 < R_0^{c,*}$. Globally, the results presented in Fig. 5 show the crucial importance of considering evaporation and the influence of solutes on it when estimating the sedimentation time of a respiratory drop. While the sedimentation time of a respiratory drop increases as $R_0$ and $RH$ decrease, it varies significantly, from a minimum of 5 s for a drop with an initial radius of 50 µm to an infinite duration as $R_0 \to 0$.

As a complement, an updated version of the diagram originally proposed by Wells in his 1934 article [57] is provided in the Supplementary Material and graphical abstract. This updated diagram, based on our model, illustrates the height above the ground at which a respiratory drop completes its drying for different air relative humidities and initial drop sizes.

## 3.2. Simplified model

Here we propose a simplified model in which the respiratory drop evaporates in the same way as a pure water drop would under identical conditions (i.e., with $X_i = P_{sat}(T_i)/P_g$), but with the evaporation process stopping instantaneously when the drop reaches its equilibrium radius $R_f$, provided that this occurs before the drop hits the ground. The equations of such a simplified model can be summarized as follows.

First, Eqs (4), (5), and (6), which describe the evaporation of a pure liquid drop, can be used to compute the local evaporation flux $j$, the interfacial vapor mole fraction $X_i$ and the interfacial temperature $T_i$,

for a given set of ambient air conditions (i.e., $T_\infty$, $P_g$, and $X_\infty$ linked to RH via Eq. (7)).

Then Eqs (19) and (20) allow predicting the final radius $R_f$ of the drop, and the drying time needed to reach that radius $\tau_{ev}^*$ can be easily derived from Eq. (9):

$$R_f = \left( \frac{\phi_0}{1 - I(RH)} \right)^{\frac{1}{3}} R_0 \qquad (21)$$

$$\tau_{ev}^* = \frac{R_0^2 - R_f^2}{\mathcal{E}(X_i - X_\infty)} = \tau_{ev} \left[ 1 - \left( \frac{\phi_0}{1 - I(RH)} \right)^{\frac{2}{3}} \right] \qquad (22)$$

where, as a reminder, $\mathcal{E} = 2D_{va}c_g/c_\ell$ and $\tau_{ev} = R_0^2/[\mathcal{E}(X_i - X_\infty)]$ is the evaporation time of a pure water drop of radius $R_0$ under the same ambient conditions (see Eq. (10)).

Regarding sedimentation, there are two possible scenarios. First, if the sedimentation time $\tau_{sed}$ is less than the drying time $\tau_{ev}^*$, all sedimentation occurs during evaporation, at a temperature $T_i$, and the previous result given in Eq. (13) holds:

$$\tau_{sed} = \tau_{ev} \left[ 1 - \left( 1 - \frac{2S(T_i)h_0}{\tau_{ev} R_0^2} \right)^{1/2} \right]$$

where, as a reminder, $S(T_i) = 9/2 \mu_{air}(T_i)/(\rho_d g)$.

Second, if the drying of the drop is complete before it hits the ground ($\tau_{sed} > \tau_{ev}^*$), sedimentation is a two-step process: First, the drop falls while evaporating down to the equilibrium radius $R_f$, then it continues its fall with this constant radius, at a temperature $T_\infty$, giving a total sedimentation time (see Eqs (22) and (3)):

$$\tau_{sed} = \tau_{ev}^* + S(T_\infty) \frac{h_{int}}{R_f^2} \qquad (23)$$

where $h_{int}$ is the height $h$ reached at the end of the drying step (i.e., at $t = \tau_{ev}^*$). Using Eqs (10), (12), (21), and (22), it can be evaluated that

$$h_{int} = h_0 - \frac{\tau_{ev}}{2} \frac{R_0^2}{S(T_i)} \left[ 1 - \left( \frac{\phi_0}{1 - I(RH)} \right)^{\frac{4}{3}} \right] \qquad (24)$$

The critical initial radius of the drop $R_0^{c,*}$, for which $\tau_{sed} = \tau_{ev}^*$, can be obtained directly by setting $h_{int} = 0$ in Eq. (24) and rearranging:
<figure><img src="image_13.png" /><figcaption>Fig. 6. Drying time $\tau_{cv}^*$ (thin lines) and sedimentation time $\tau_{sed}$ (thick lines) of a respiratory drop as functions of its initial radius $R_0$, for an initial height of $h_0 = 1.5$ m and $\phi_0 = 1\%$. Results are shown for air at $T_\infty = 20^\circ C$, $P_g = 1$ atm, and different relative humidities RH: 0, 30, 50, 70, 90, 99%. The dashed curves are obtained using the simplified model described in this section and the full lines, already presented in Fig. 5, are the results obtained solving the full problem described in the previous section.</figcaption></figure>

$$R_0^{c,*} = \left[ \frac{2S(T_i)h_0\mathcal{E}(X_i - X_\infty)}{1 - \left(\frac{\phi_0}{1 - I(RH)}\right)^{\frac{4}{3}}} \right]^{\frac{1}{4}} = \frac{R_0^c}{\left[ 1 - \left( \frac{\phi_0}{1 - I(RH)} \right)^{\frac{4}{3}} \right]^{\frac{1}{4}}} \quad (25)$$

where $R_0^c$ is the critical initial radius of a pure water drop giving, under the same conditions, equal evaporation and sedimentation times of that drop (see Eq. (14)). This equation explains why the critical radius of a respiratory drop $R_0^{c,*}$ is very close to the critical radius of a pure water drop $R_0^c$ under the same ambient conditions. Indeed, the combination of the small value of $\phi_0$ (about 1%), the downward concavity of the isotherm (which means that $I(RH)$ remains significantly less than 1 even for large values of RH), and its exponent $1/4$, results in a denominator on the right-hand side of the equation that remains relatively close to one over a wide range of RH values. In fact, it can be evaluated that even for $RH = 0.99$, $R_0^{c,*} = 1.023 R_0^c$ (for $\phi_0 = 1\%$). Note that the exponent $1/4$ on the denominator of the right-hand side of Eq. (25) arises because the evaporation time of a pure liquid drop is proportional to the square of its initial radius, while the sedimentation time of a non-evaporating object is inversely proportional to the square of its radius (and to express the critical radius, the evaporation and sedimentation times are equalized).

Recall that Eq. (15) provides a fit of experimental data to express the isotherm $I$ and that Eq. (11) gives a polynomial expression of $T_i$ as a function of $T_\infty$ and $RH$, at $P_g = 1$ atm, which facilitates the application of the equations of this simplified model.

In Fig. 6, the results from the simplified model (dashed curves) show excellent agreement with those from the full problem (continuous curves), validating the proposed approach. The maximum relative difference between the two models occurs near the critical initial radius and remains below 10% for relative humidities in the range $RH \in [25\%, 75\%]$. Even at extreme humidities ($RH = 0\%$ or 99%), the average relative difference stays under 4% in the range $R_0 \in [10-50]$ µm, further demonstrating the robustness of the simplified model.

### 3.3. Comparison with other models

This section compares the present model (full problem) with several recent models from the literature. Rather than juxtaposing separate models with differing assumptions and parameter choices, risking inconsistencies, a systematic approach is adopted: common simplifications from the literature are introduced directly into the full problem, each considered independently to modify only one hypothesis at a time. This

ensures a rigorous comparison within a consistent modeling framework using identical parameter values.

Three simplifications frequently encountered in previous studies have been identified and we demonstrate how they can lead to significant inaccuracies in predicting the sedimentation time of respiratory drops. For each case, the underlying causes of these discrepancies are analyzed. In Fig. 7, each panel compares the results obtained using one of these simplifications (dashed curves) with those from the full problem (solid curves).

First, Fig. 7(a) shows the results obtained under the assumption that the gas phase is isothermal (i.e., imposing $T_i = T_\infty$). This assumption, which is often used (see for example [43]), simplifies the evaporation problem. However, as mentioned above, it leads to an overestimation of the evaporation rate because it neglects evaporative cooling. Consequently, the sedimentation time is also overestimated because the drop radius decreases too rapidly. As shown in Fig. 7(a), this leads to significant inaccuracies in the estimation of the sedimentation time. Over the range $R_0 \in [10-50]$ µm and for any relative humidity, the sedimentation time is calculated on average to be more than twice the value obtained by solving the full problem. For certain values of the initial drop radius, this discrepancy can reach almost an order of magnitude.

Second, the evaporation modeling is sometimes simplified by approximating the saturation concentration of water in air $C_{sat}(T)$, using a linear relationship with temperature, instead of the nonlinear Clausius-Clapeyron equation. For example, Ref. [27] used the linear relationship: $C_{sat}(T) = (0.2062 + 0.0330(T - 273.15))$, with $T$ in Kelvin. While this provides a good approximation of $C_{sat}(T)$ for $T < 15^\circ C$, it significantly underestimates experimental data at higher temperatures, with deviations up to 50% at $T = 40^\circ C$, as shown in the Supplementary Material. This underestimation directly affects the predicted values of $X_\infty$ (Eq. (7)) and $X_i$ (Eq. (17)), leading to inaccuracies in the evaporation rate and sedimentation dynamics. As a result, the sedimentation dynamics is miscalculated, especially at high temperatures. This is illustrated in Fig. 7(b), where even at a moderate ambient temperature of $T_\infty = 20^\circ C$, the sedimentation times predicted using the linear relationship presented in [27] are significantly longer than those from the full problem. For $RH = 30\%$, the relative difference reaches 60% near the critical initial radius, and for $RH = 90\%$, it exceeds 100%. These discrepancies arise from an underestimated $X_\infty$, leading to an overestimation of the evaporation rate, causing the drop to shrink faster and settle more slowly. As shown in the Supplementary Material, these effects are even more pronounced at higher temperatures. For $T_\infty = 40^\circ C$, the relative differences between the two approaches grow significantly
<figure><img src="image_14.png" /><figcaption>**Fig. 7.** Drying time $\tau_{ev}^*$ (thin lines) and sedimentation time $\tau_{sed}$ (thick lines) of a respiratory drop as functions of its initial radius $R_0$, for an initial height of $h_0 = 1.5$ m and $\phi_0 = 1\%$. Results are shown for air at $T_\infty = 20^\circ C$, $P_g = 1$ atm, and different relative humidities RH: 30, 50, 70, 90%. In each of these plots, the full problem (continuous curves) described in Sec. 3.1 is compared with recent models incorporating assumptions such as: (a) an isothermal gas phase, (b) linearity of the saturation concentration of water in air with temperature, and (c) linearity of the sorption isotherm with solute volume fraction.</figcaption></figure>

over a wider range of the initial drop radius, in some cases exceeding 200%.

Finally, in the absence of specific information about the isotherm, it may be tempting to approximate it using a linear relationship: $\psi = a_\omega$, as in [27]. Fig. 4(a) shows that using such a linear isotherm underestimates the water activity for a given solute mass fraction. Therefore, the partial pressure of water vapor at the interface is also underestimated. As a result, the evaporation rate is reduced, the drop remains too large, and the final equilibrium radius $R_f$ is overestimated, as shown in Fig. 4(b). This leads to an underestimation of the sedimentation time, a

trend that becomes particularly pronounced when $R_0 < R_0^{c,*}$, as shown in Fig. 7(c). For $R_0 \ll R_0^{c,*}$, the calculations show that the assumption of a linear isotherm leads to sedimentation times that are 16, 29, 43, and 61% shorter than those obtained by solving the full problem for RH = 30, 50, 70, and 90%, respectively. It is worth noting that a simple calculation can also give these values. As discussed earlier, most of the fall occurs at $R_f$ when $R_0$ is significantly less than $R_0^{c,*}$. Since the settling velocity is proportional to the square of the radius (see Eq. (2)), $\tau_{sed}$ becomes $\propto 1/R_f^2$ when $R_0$ is sufficiently small. In the full problem, $R_f$ is calculated with Eqs (19) and (20), using the values of $c_1$ and $c_2$
<caption>**Table 1**</caption>

Comparison of predicted sedimentation times $\tau_{sed}$ for respiratory drops with various initial radii $R_0$, considering the full model presented in this article, its simplified version, and recent models incorporating assumptions such as an isothermal gas phase [43], linearity of the saturation concentration of water in air with temperature, and linearity of the sorption isotherm with solute fraction [27,48,49]. The relative difference compared to the full model is indicated, with differences greater than 10% highlighted in red. Results are presented for a respiratory drop falling from an initial height of $h_0 = 1.5$ m, with an initial concentration $\phi_0 = 1\%$, into air at $T_\infty = 20^\circ C$, $RH = 50\%$, and $P_g = 1$ atm. The table also includes results from models omitting evaporation and considering evaporation with saliva modeled as pure water under the same conditions. (For interpretation of the colors in the table, the reader is referred to the web version of this article.)

<table><thead><tr><th></th><th>0.5 µm</th><th>1 µm</th><th>5 µm</th><th>10 µm</th><th>20 µm</th><th>30 µm</th><th>40 µm</th><th>50 µm</th></tr></thead><tbody><tr><td>Full model</td><td><strong>263.92 h</strong></td><td><strong>65.97 h</strong></td><td><strong>2.64 h</strong></td><td><strong>39.48 min</strong></td><td><strong>9.46 min</strong></td><td><strong>3.42 min</strong></td><td><strong>44.21 s</strong></td><td><strong>5.84 s</strong></td></tr><tr><td>Simplified model</td><td>264.19 h<br>(+0.11%)</td><td>66.06 h<br>(+0.13%)</td><td>2.64 h<br>(+0.11%)</td><td>39.52 min<br>(+0.08%)</td><td>9.47 min<br>(+0.07%)</td><td>3.42 min<br>(0.00%)</td><td>43.96 s<br>(-0.57%)</td><td>5.81 s<br>(-0.56%)</td></tr><tr><td>Model with $T_g = \text{cst}$</td><td>263.92 h<br>(0.00%)</td><td>65.98 h<br>(+0.01%)</td><td>2.64 h<br>(+0.02%)</td><td>39.55 min<br>(+0.18%)</td><td>9.76 min<br>(+3.16%)</td><td>4.10 min<br>(+19.69%)</td><td>1.93 min<br>(+162.60%)</td><td>44.45 s<br>(+4660.89%)</td></tr><tr><td>Model with linear $C_{sat}(T)$</td><td>263.92 h<br>(+0.00%)</td><td>65.98 h<br>(+0.01%)</td><td>2.64 h<br>(+0.01%)</td><td>39.49 min<br>(+0.02%)</td><td>9.51 min<br>(+0.51%)</td><td>3.53 min<br>(+3.20%)</td><td>55.92 s<br>(+26.50%)</td><td>6.01 s<br>(+2.84%)</td></tr><tr><td>Model with $a_w(\phi)$</td><td>188.33 h<br>(-28.64%)</td><td>47.08 h<br>(-28.63%)</td><td>1.88 h<br>(-28.64%)</td><td>28.17 min<br>(-28.65%)</td><td>6.75 min<br>(-28.62%)</td><td>2.45 min<br>(-28.47%)</td><td>32.20 s<br>(-27.16%)</td><td>5.82 s<br>(-0.32%)</td></tr><tr><td>Non-evap. Model</td><td>13.89 h<br>(-94.74%)</td><td>3.47 h<br>(-94.74%)</td><td>8.33 min<br>(-94.74%)</td><td>2.08 min<br>(-94.72%)</td><td>31.25 s<br>(-94.50%)</td><td>13.89 s<br>(-93.23%)</td><td>7.81 s<br>(-82.33%)</td><td>5 s<br>(-14.41%)</td></tr><tr><td>Pure water Model</td><td>$\infty$<br>($\infty$)</td><td>$\infty$<br>($\infty$)</td><td>$\infty$<br>($\infty$)</td><td>$\infty$<br>($\infty$)</td><td>$\infty$<br>($\infty$)</td><td>$\infty$<br>($\infty$)</td><td>$\infty$<br>($\infty$)</td><td>5.81 s<br>(-0.56%)</td></tr></tbody></table>

that allow fitting the data of Merhi et al. [43]. If a linear isotherm is used, $\phi_f = 1 - RH$ and thus $R_f = R_0(\phi_0/(1 - RH))^{1/3}$. With this information in hand, it is easy to show that $1/R_f^2$ computed by considering a linear isotherm is, for $RH = 30, 50, 70$, and 90%, 16, 29, 43, and 61% smaller, respectively, than $1/R_f^2$ computed in the full problem, in agreement with the previously mentioned values. In fact, in the absence of detailed information about the isotherm, a more appropriate assumption would be to approximate the final radius as $R_f = R_0(\phi_0)^{1/3}$, which is a good approximation for $RH < 70\%$, and to assume that the drop evaporates like pure water until it reaches this final radius, which is then fixed until the end of sedimentation.

To summarize this comparison between models, Table 1 gives the predicted sedimentation times $\tau_{sed}$ for respiratory drops with various initial radii $R_0$, considering the full problem presented in this article, its simplified version developed in the previous section, and recent models incorporating the assumptions discussed above, under classical ambient conditions of $T_\infty = 20^\circ C$, $RH = 50\%$, and $P_{atm} = 1$ atm. The relative differences compared to the full problem are indicated, with discrepancies over 10% highlighted in red. The table also includes results from models omitting evaporation and considering evaporation with saliva modeled as pure water under the same conditions. Notably, only the simplified model developed in this article compares satisfactorily across all drop sizes with the full problem, which incorporates the most detailed and accurate physicochemical description available.

## 4. Influence of ambient conditions on airborne transmission

This section investigates the influence of ambient conditions on the sedimentation time of respiratory drops, as well as on pH changes induced by ion concentration during their drying. All results presented are obtained with the simplified model, which is an effective tool for capturing the dynamics of a respiratory drop. By providing accurate and practical insights into drying and sedimentation dynamics under various conditions, the proposed model aims to improve the understanding of transmission risks in airborne contexts. Throughout this section, we continue to consider respiratory drops expelled with an initial height of $h_0 = 1.5$ m and an initial solute concentration of $\phi_0 = 1\%$.

### 4.1. Sedimentation time

Fig. 8 depicts the drying time $\tau_{ev}^*$ (thin lines) and the sedimentation time $\tau_{sed}$ (thick lines) of a respiratory drop as functions of its initial

radius $R_0$. The results are shown for air at $T_\infty = 5^\circ C$ (a) or $T_\infty = 35^\circ C$ (b), $P_g = 1$ atm, and various relative humidities $RH$: 0, 30, 50, 70, 90, 99%. The “non-evaporation” curve gives the sedimentation time of a respiratory drop with a constant radius equal to its initial value $R_0$ and with $T_i = T_\infty$. These results complement those presented in Fig. 5, which shows these curves for $T_\infty = 20^\circ C$.

The results presented in Fig. 8 provide a simple summary of the influence of ambient conditions on sedimentation time. In general, an increase in temperature shifts the sedimentation time curves to the right, because the increase in evaporation rate it induces leads to smaller drops, which therefore fall more slowly. The effect of temperature on sedimentation time is more pronounced at low relative humidity, and going from a temperature of $5^\circ C$ to $35^\circ C$ can increase $\tau_{sed}$ by almost an order of magnitude. An increase in $RH$ reduces $\tau_{sed}$ because it slows the evaporation rate and so the drops remain larger during their fall. Overall, the cooler and wetter the air, the faster the respiratory drops fall to the ground, thereby reducing their contribution to airborne disease transmission.

The effect of the relative humidity of the ambient air on the sedimentation time can be analyzed in more detail using Figs. 5 and 8. First, for a respiratory drop of a given initial size, the sedimentation time is minimized in the non-evaporation limit, where the radius of the drop remains constant over time and the system is isothermal. Then, if $R_0$ is large enough for $\tau_{sed}$ to be of the order of just a few seconds, the different sedimentation time curves tend to merge and become close to the non-evaporation curve, meaning that $RH$ has a weak influence on $\tau_{sed}$. Sedimentation has a time scale $\propto R_0^{-2}$, while evaporation has a time scale $\propto R_0^2$. Therefore, if $R_0$ is large enough, evaporation does not significantly reduce the drop radius, which remains nearly equal to $R_0$ throughout the sedimentation. On the other hand, if $R_0$ is small enough to result in sedimentation times greater than a few minutes, the transition from completely dry air to air nearly saturated with water can reduce $\tau_{sed}$ by more than a factor of 20. Furthermore, in this case the different sedimentation curves tend to converge for $RH < 70\%$, which can be explained as follows. As mentioned earlier, the downward concavity of the sorption isotherm implies that for $RH < 70\%$, $R_f \approx R_0 \phi_0^{1/3}$ (see Fig. 4(b)). Therefore, if $R_0$ is small enough, drying is almost instantaneous compared to sedimentation, and the sedimentation process takes place essentially at a constant radius equal to the final radius $R_f$. As a result, the sedimentation time can be approximated as (see Eq. (3))
<figure><img src="image_15.png" /><figcaption>Fig. 8. Drying time $\tau_{ev}^*$ (thin lines) and sedimentation time $\tau_{sed}$ (thick lines) of a respiratory drop as functions of its initial radius $R_0$, for an initial height of $h_0 = 1.5$ m and $\phi_0 = 1\%$. Results are shown for air at $T_\infty = 5^\circ C$ (a) or $T_\infty = 35^\circ C$ (b), $P_g = 1$ atm, and different relative humidities RH: 0, 30, 50, 70, 90, 99%. The “non-evaporation” curve gives the sedimentation time of a respiratory drop with a constant radius equal to its initial value $R_0$ and with $T_i = T_\infty$. The dashed curve gives the scaling for the sedimentation time when drying is much faster than sedimentation (Eq. (26)).</figcaption></figure>

$$ \tau_{sed} \approx S(T_{\infty}) \frac{h_0}{R_0^2 \phi_0^{2/3}} \qquad (26) $$

Fig. 8 shows that this equation, which does not include RH, provides a good scaling for the sedimentation time when it exceeds a few minutes and for RH < 70%. Moreover, just as the non-evaporation curve given by Eq (3) provides a lower bound for sedimentation times, it is seen that Eq (26) provides an upper bound. Both equations give $\tau_{sed} \propto R_0^{-2}$ which emphasizes the strong dependence of sedimentation time on size.

Although the study of individual respiratory drops is essential, actual emissions from breathing, coughing, or speaking consist of clouds of drops. Fig. 8 shows that a relevant range for assessing the influence of ambient conditions on airborne transmission is $[R_0^{\min} = 5 \mu m, R_0^{\max} = 50 \mu m]$. The upper limit of 50 $\mu m$ is confirmed, as larger drops settle within seconds under all ambient conditions (except in extremely hot and dry environments). Conversely, although drops smaller than 5 $\mu m$ remain airborne for more than 8 minutes across all ambient conditions, Fig. 1(a) shows that they account for only about 1% of the total volume of drops potentially involved in airborne transmission (i.e., with $R_0 < 50 \mu m$) and can therefore be excluded from the analysis. Since viral load scales with the drop volume, we introduce a volume-weighted average sedimentation time, $\bar{\tau}$, to characterize their behavior:

$$ \bar{\tau} = \frac{\int_{R_0^{\min}}^{R_0^{\max}} \tau_{sed}(R_0) \nu(R_0) dR_0}{\int_{R_0^{\min}}^{R_0^{\max}} \nu(R_0) dR_0} \qquad (27) $$

where $\nu(R_0)dR_0$ is the volume of the drops emitted with an initial radius in the range $[R_0, R_0 + dR_0]$. Although this parameter omits complexities arising from collective effects, $\bar{\tau}$ provides a global measure of the sedimentation behavior of drops within the 5 to 50 $\mu m$ range, reflecting both their physical dynamics and their biological relevance in terms of viral transport.

Fig. 9 shows $\bar{\tau}$ as a function of ambient temperature $T_\infty$ and relative humidity RH, for a coughing person. The maps are shown for atmospheric pressures of $P_g = 1$ atm (a) and $P_g = 0.5$ atm (b) to highlight the effect of altitude, from sea level (about 0 m) to high mountain elevations (about 5500 m in this case). The BLO model of Johnson et al. [19] is used to evaluate $\nu(R_0)$ (see Fig. 1(b)). Note that the equivalent map for a speaking person is provided in the Supplementary Material. Similar results are obtained for the two emission modes as their size distributions, shown in Fig. 1, are very close.

Fig. 9(a,b) shows that $\bar{\tau}$ varies significantly with ambient conditions, with important implications for disease transmission. In cool and humid environments, $\bar{\tau}$ is short: most of the volume emitted in the 5–50 $\mu m$ range settles within about 10 seconds, leaving only the sub-5 $\mu m$ fraction airborne, approximately 1% of the total volume below 50 $\mu m$. In contrast, in hot and dry conditions, $\bar{\tau}$ exceeds two minutes, meaning that the majority of the 5–50 $\mu m$ volume remains airborne for biologically relevant durations. Consequently, shifting from cool and humid to hot and dry conditions increases the airborne volume of viral particles by a factor of about 100. This striking contrast underscores the critical role of ambient conditions in determining exposure risk.
<figure><img src="image_16.png" /><figcaption>Fig. 9. (a-b) Maps of $\bar{\tau}$, as a function of ambient temperature $T_\infty$ and relative humidity RH, for atmospheric pressures of $P_g = 1$ atm (a) or $P_g = 0.5$ atm (b), for a coughing emission mode. (c) Map of ΔpH relative to (a) with $R_0 = 35$ µm. If drying is complete before the drop hits the ground, ΔpH is given by Eq (30). Otherwise it is given by Eq (29), with $t = \tau_{sed}$. An initial height of $h_0 = 1.5$ m and $\phi_0 = 1\%$ are considered for the calculations.</figcaption></figure>

Additionally, the pressure difference between sea level ($P_g = 1$ atm) and high altitudes ($P_g = 0.5$ atm) significantly affects sedimentation, with longer times observed at lower pressures. As shown in Eq. (22), drying time is proportional to $P_g$. Specifically, $X_i$ and $X_\infty$ are both $\propto P_g^{-1}$ at fixed relative humidity (see Eqs. (6) and (7)), while $\mathcal{E} = 2D_{va}c_g/c_f$ remains pressure-independent since $c_g \propto P_g$ and, according to the kinetic theory of gases, $D_{va} \propto P_g^{-1}$. Thus, a decrease in pressure shortens the evaporation time at constant $T_\infty$ and RH, causing drops to shrink faster and sedimentation times to increase. As a result, the risk of airborne viral transmission may be higher at elevated altitudes, all other factors being equal.

## 4.2. Evolution of the pH of respiratory drops induced by evaporation

Respiratory drops are naturally slightly acidic, with a typical pH of approximately 6.5 [40]. As a drop evaporates, the concentration of nonvolatile solutes increases, leading to a reduction in pH that can contribute to virus inactivation [58,59]. It is important to note that water loss is not the only factor influencing pH evolution: interactions with acidic or basic trace gases present in indoor air (such as nitric acid or ammonia) can also play a role [58]. Here, we focus exclusively on the physicochemical mechanism by which evaporation concentrates dissolved ions and alters pH. While we do not attempt to capture the full complexity of biochemical effects on virus viability, we show that our model provides a tractable way to quantify the pH shift induced by evaporation alone, thereby offering a useful indicator of microenvironmental changes within airborne drops.

If the pH of an emitted respiratory drop is changed only by the increase in ion concentration as it dries, the temporal evolution of the pH change (in absolute value) is simply described by

$$ \Delta \text{pH} = \text{pH}_0 - \text{pH} = \log_{10} \left( \frac{R_0^3}{R_f^3} \right) \qquad (28) $$

with $pH_0$ the initial pH of the drop. Using Eqs (9) and (10), Eq. (28) can be rewritten as

$$ \Delta \text{pH} = -\frac{3}{2} \log_{10} \left( 1 - \frac{t}{\tau_{\text{ev}}} \right) \qquad (29) $$

with $\tau_{ev}$ the evaporation time of a drop of pure water with initial radius $R_0$ under the same ambient conditions (see Eq. (10)). This equation is only valid for $t < \tau_{ev}^*$, since for $t > \tau_{ev}^*$, the radius of the drop has reached its final value $R_f$, in equilibrium with its environment, and so has the pH.

The final value of the pH change when the drop has reached its final radius can be expressed with Eqs (19) and (20):

$$ \Delta \text{pH}_f = \log_{10} \left( \frac{R_0^3}{R_f^3} \right) = \log_{10} \left( \frac{1 - I(RH)}{\phi_0} \right) \qquad (30) $$

As mentioned previously, if $RH < 70\%$, $R_f \approx R_0 \phi_0^{1/3}$ (see Fig. 4(b)). Hence, if $\phi_0 = 10^{-2}$, $\Delta \text{pH}_f \approx -\log_{10}(\phi_0) = 2$.

Fig. 9(c) shows the maximum pH change within a sedimenting respiratory drop, as a function of ambient temperature $T_\infty$ and relative humidity RH, under the conditions $R_0 = 35$ µm and $P_g = 1$ atm. This ΔpH corresponds either to the final pH shift, $\Delta \text{pH}_f$ (expected close to 2 if $RH < 70\%$), because drying is complete before the drop reaches the ground, or to an intermediate value if drying remains incomplete at the time of impact (given by Eq. (29), with $t = \tau_{sed}$).

The map reveals the existence of two distinct zones separated by a narrow transitional band. In the first zone, corresponding to low-to-moderate humidity levels, evaporation dominates over sedimentation, and the drop dries entirely during its airborne time. As a result, there is a substantial acidification with a pH shift approaching two units. In the second zone, found at higher humidities, the drop reaches the ground before having lost a significant fraction of its initial water content, and
the corresponding ΔpH remains close to zero. The boundary between these zones appears as an abrupt transition on the map. This sharp transition arises from the nonlinear dynamics of evaporation, governed by the relation $R_0^2 - R^2 \propto t$. This implies that the most significant reduction in drop volume, and hence the steepest changes in solute concentration and pH, occurs during the final moments of the drying process. If the drop touches the ground before reaching this stage, the acidification remains minimal. As temperature increases, evaporation becomes more efficient, and so does the critical RH delimiting the two regions: this threshold rises from about 50% at 0°C to nearly 80% at 40°C.

A pH decrease of approximately two units, as predicted in a large part of the parameter space, is potentially significant from a virological perspective. While our model focuses exclusively on the physicochemical dynamics of evaporation and sedimentation, such acidification may contribute to virus inactivation. For example, it has been shown that influenza viruses can lose viability within minutes when exposed to a pH decrease of this magnitude [59]. In contrast, SARS-CoV-2 appears to be much more resilient to acidic conditions, showing minimal loss of infectivity under similar pH variations [58]. These biological responses, although outside the scope of the present analysis, illustrate the potential relevance of the predicted pH shifts. Notably, the entire subregion defined by $T_\infty > 15°C$ and $RH < 70\%$, corresponding to typical indoor air conditions, lies well within the first zone identified in Fig. 9(c). In this zone, $\Delta pH \approx 2$ is thus systematically reached, indicating that acidification of this magnitude may be a common scenario in indoor environments.

A closer look at Figs. 9(a) and 9(c) reveals an important interplay between ambient conditions and transmission risks. The hot and dry environments that cause respiratory drops to remain airborne for several minutes, thus potentially increasing the probability of exposure, are also leading to significant acidification, with $\Delta pH \approx 2$. For viruses such as influenza, which are rapidly inactivated under such acidic conditions, this results in two competing effects: enhanced airborne persistence on one hand, and reduced viral viability on the other. Although our model does not account for viral inactivation kinetics, it offers a robust physicochemical framework for identifying ambient conditions under which these opposing mechanisms may significantly influence the balance of airborne transmission.

## 5. Conclusion

In this article, we present a novel theoretical model that predicts the evaporation and sedimentation dynamics of respiratory drops with initial radii smaller than 50 µm, a size range relevant for airborne transmission. The model integrates gravity-driven sedimentation [24,25,27] with diffusion-limited evaporation [32], incorporating key respiratory fluid properties [43]. Validated against experimental data from the literature [45,53–55], it addresses limitations of previous models that neglected evaporation-induced cooling [43] and the nonlinear dependencies of the sorption isotherm and the water saturation pressure on solute mass fraction and temperature, respectively [27,48,49].

The results demonstrate the significant impact of ambient conditions on sedimentation times and, consequently, the risk of airborne disease transmission. Drops with an initial radius larger than 50 µm settle to the ground in less than 5 seconds under most conditions, while drops smaller than < 5 µm can remain airborne for tens of minutes or even hours, depending on the ambient conditions. The study explores ambient conditions spanning relative humidity (0 – 100%), temperature (0 – 40°C), and pressure (0.5 – 1 atm), demonstrating that the number of viral particles remaining airborne for a few minutes can differ by a factor of 100 depending on these parameters. Notably, colder and more humid air at lower altitudes (higher atmospheric pressure) promotes faster sedimentation times, thereby reducing airborne transmission risk. This underscores the importance of ambient parameters in modulating transmission risks, as drops staying aloft for extended periods are more likely to contribute to airborne spread.

Another notable result is that a simplified analytical model, based on the dynamics of sedimentation and evaporation of a pure water drop and complemented by the sorption isotherm of saliva to determine the final drop radius, provides sufficiently accurate predictions for a respiratory drop across the entire explored parameter range.

Additionally, the model enables the quantification of evaporation-induced pH changes resulting specifically from solute concentration effects. Under certain ambient conditions, particularly at relative humidities less than 70%, this mechanism alone can lead to an acidification of up to two pH units. While this shift may contribute to the inactivation of certain viruses, such as influenza, as suggested by recent studies, its effect on more resilient viruses like SARS-CoV-2 remains uncertain. We hope that the present framework can support future investigations into virus viability by providing reliable physicochemical predictions of drop evolution under various ambient conditions.

In order to facilitate further use, a Wolfram Mathematica implementation of the model is provided, serving as a predictive tool for respiratory drop evaporation and sedimentation.

While the present work provides an accurate description of the key physicochemical processes governing respiratory drop dynamics, it intentionally omits certain complexities, such as collective effects, ambient airflow, and variations in initial emission velocity (see references [11,12,14]). These aspects, while beyond the scope of this study, represent perspectives for future research. In particular, we hope this study will inspire future efforts to integrate the present drop-scale description into larger-scale models, incorporating additional layers of complexity inherent to real-world scenarios. For instance, Eq. (11) allows direct calculation of the surface temperature of an evaporating drop; this quantity can then be used to estimate the evaporation rate (using Eqs. (4) and (17)) and, consequently, the drop radius as a function of time (with Eq. (8)). Similarly, Eq. (20) gives the final equilibrium radius of a drop as a function of the relative humidity of the ambient air. These expressions could serve as submodels in CFD simulations, offering a promising route to account for fine physicochemical effects within comprehensive frameworks aimed at simulating realistic transmission scenarios.

### CRediT authorship contribution statement

**Benoît Haut:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Omer Atasi:** Writing – review & editing, Writing – original draft, Visualization, Validation, Investigation, Data curation.
**Benjamin Sobac:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

### Declaration of competing interest

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

### Acknowledgements

OA gratefully acknowledges the financial support from the Fonds de la Recherche Scientifique - FNRS for his postdoctoral research position.

## Appendix A. Supplementary material

Supplementary material related to this article can be found online at
https://doi.org/10.1016/j.jcis.2025.137715.

### Data availability

We have included our data and code file as Supplementary Material.
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