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Drying of virus-containing aerosol particles: Modelling effects of droplet origin and composition

Virus-containing aerosol droplets emitted by breathing, speech or coughing dry rapidly to equilibrium with ambient relative humidity (RH), increasing in solute concentration with effects on virus survival and decreasing in diameter with effects on sedimentation and respiratory uptake. In simulations starting from drying data on mixed NaCl/KCl aerosols, the supersaturated salt concentrations were shown to reach 20-25M at the efflorescence RH of 40-55%, depending on the K:Na ratio. These salt concentrations may inactivate some viruses. The dependence on K:Na ratio implies that the evaporation curves differ between aerosols derived from saliva and from airway surfaces. Differences in drying behaviour are consequently predicted between breathing, speech and coughing emissions and between droplet size fractions within these. The direct effect of liquid droplet size through the Kelvin term was shown to be smaller and restricted to breath emissions. Comparative simulations starting from osmotic pressure measurements on airway surface liquid showed that salts are the primary determinants of drying equilibria down to the efflorescence RH, and macromolecules at lower RH.


Introduction
Some respiratory viruses can be transmitted in aerosol form, as well as in larger droplets and surface deposits 1 . Aerosols are conventionally defined as droplets or particles less than 5-10 m in diameter that, according to Stokes' Law, remain suspended in still air for minutes or longer 2 . It has been argued 3,4 that the size range should be extended to 50-100 m because turbulence, either in a cough jet 5 or due to draughts 6 or convection 7 , keeps larger particles airborne for longer than is predicted by Stokes' Law.
Despite initial doubts, it is now quite widely accepted that certain viruses including SARS-CoV-2 are transmitted in aerosols, particularly from asymptomatic subjects 8,9 . Aerosol transmission is most likely in enclosed spaces such as schools, public buildings and transportation 10,11 . Aerosols can transmit viruses from person to person with transiently inadequate social distancing 4 , but they can also build up, over minutes to hours, throughout the air in an enclosed space so that the risk of infection depends on the duration of emission and exposure, not on distancing 12 . In these circumstances the risk depends on Wells-Riley dynamics 13 and is reduced by ventilation with fresh or filtered air and by anything that decreases the viable lifetime of the virus 11,12 .
The viable lifetime of airborne viruses varies. SARS-CoV-2 remains viable indoors for minutes to hours [14][15][16][17] . The rate of inactivation is enhanced by sunlight 18,19 and increases with temperature 19,20 . There is also a significant effect of humidity 14,19 , but the nature and mechanism of the humidity effect are still poorly understood.
The effect of humidity on virus inactivation also differs between viruses. For some viruses inactivation is faster at intermediate levels of relative humidity, around 40%, than at high or very low humidity 21 . It has been suggested that the common cold virus HRV-16, which follows this pattern, loses activity in the supersaturated salt solution when dried from the high RH of emission to just above the precipitation (efflorescence) zone of the salts present, roughly 40% RH 22 . In contrast, for SARS-CoV-2 in aerosol form, the limited data available suggest that the rate of inactivation is low at all RH values tested up to about 50% and increases at higher RH 19 . The rate of inactivation of surface deposits of SARS 23 and SARS-CoV-2 24,25 is lower than in aerosols but likewise rises with increasing humidity 26 . This implies that the SARS viruses are more resistant than, for example, HRV-16 to very high salt concentrations, and that whatever its mechanism the inactivation of SARS-CoV-2 is dependent on available water.
Free water is present in aerosol droplets and surface deposits above the efflorescence RH of the solutes present. Water availability, expressed as the water activity aw, increases with the ambient RH but is also modulated by the Kelvin effect in small aerosol droplets.
These observations suggest that that the drying process for aerosol-sized droplets deserves closer examination. The same is true for surface deposits, particularly as it is not known why viral viability is enhanced in that form 16,25 . The drying of aerosols when emitted into ambient air has been quite extensively studied [27][28][29] and modelled 27,28,30,31 , often with the aim of predicting when droplets, initially large enough to sediment in still air, will shrink enough to remain suspended. In the biomedical literature, a droplet that has dried to equilibrium with the ambient air is called a droplet nucleus 1 . Depending on the solids present and the moisture that their hygroscopicity retains, a droplet nucleus may consist of a very concentrated solution, a polycrystalline salt precipitate, a protein gel or amorphous solid, or a combination of these phases; presenting very different environments in which viruses may be inactivated 21 . For example, thermal denaturation depends on mobile water 32 . Also polycrystalline or other solids may refract or absorb daylight, which is known to inactivate SARS-CoV-2 19 .
Much of the published experimentation on droplet drying (cited by 27 ) has made use of simplified analogues of the respiratory fluids in which viruses are emitted by breathing, speaking, singing, coughing or sneezing. Sometimes just NaCl solutions have been used. The detailed salt composition of natural aerosols has a profound effect on their drying behaviour 33 . Metzger et al. 34 give an accessible description of the underlying physics (Kohler theory) as an appendix: their nomenclature is adopted here. In respiratory droplets, proteins and glycoproteins have been recognised to contribute volume to the dried droplet nuclei, but little attention has been paid to other ways in which these polymers might influence the drying process 27,35 .
Because of the very high salt and polymer concentrations that can be reached when biological aerosols dry at low RH, classical colligative relationships like Raoult's Law become increasingly unsatisfactory approximations and the relevant physical chemistry becomes necessarily more empirical. In these circumstances direct experimental measurements using real biological fluids may be more informative than theoretical prediction 27 . However, experimentation on aerosols is technically challenging 27,36,37 and during the pandemic time is short. This paper describes simulations of the effects of some of the main variables in the composition of virus-containing aerosols on drying equilibria. In view of the above comments, these simulations do not aim at quantitative descriptions of complex, real-life bioaerosols, but may serve to suggest some simplifying assumptions and to prioritise variables that deserve experimental investigation.

Origins and composition of emitted droplets
Aerosols and larger droplets emitted during breathing 38,39 , speech 37,40 , coughing 39 and other activities 41 originate by aerodynamic disruption of the mucosal lining 42 in different zones in the respiratory tract 36,43,44 , leading to different droplet size distributions 45,46 . The principal zones in which droplets are generated are the bronchioli (modal droplet diameter 1-2 m) the laryngeal region (modal droplet diameter 1-2 m), and the oral cavity and nasal region (modal droplet diameter >100 m) 46 . Multiple sites of origin lead to bimodal, trimodal or broad continuous ranges of droplet diameter for each mode of emission 46 . With droplet diameters covering several orders of magnitude, caution is needed in the interpretation of modal figures because they may be derived by several experimental methods with differing size limitations 42,46 and because number-weighted and volume-weighted distributions are very different: volume-weighted distributions are more relevant to viral load 35 .
Droplets emitted in normal breathing originate primarily from the bronchial zone and have diameters in the submicron to m range 46,47 , while droplets emitted in talking or coughing are derived partly from the laryngeal and oral zones, with a preponderance of larger particles 36,39,46 . The viral load of the mucosa in each region 48,49 varies with disease progression 50 and between individuals 39,40 .
The drying of emitted droplets depends on their ionic and polymer composition. It has not been well recognised that the composition of the droplets differs according to their site of origin 27 , and therefore also differs with droplet diameter.
A key function of the airway lining throughout the respiratory tract is to sustain hydration 51 and freedom of motion for the underlying cilia 52 , defects in hydration leading to disorders such as cystic fibrosis 53 . Equilibrium hydration depends on osmolytes in a very similar way to water retention by emitted droplets 51 , although the RH within the respiratory system is much higher. The cation composition of the airway surface liquid is dominated by sodium, with Na + :K + molar ratios typically around 4:1 53-55 : the precision is lower for K + due to the difficulty of sampling without cellular damage 56 . The principal anion is Cl -, with a much smaller amount of HCO3 -53, 54 . In health the total osmolarity is approximately 300 mM 54 , increasing from the lower respiratory tract to the nasal region 53 and increasing substantially in conditions such as chronic bronchitis 51 . Fluid harvested from human bronchial epithelial (HBE) cell cultures is rather similar in ionic composition to native airway fluids but with lower protein content 55 .
In contrast, saliva has much lower ion concentrations and osmolarity 57 . In the normal (resting) state, the total osmolarity averages 50 mM and the main cation is K + with a K + :Na + ratio of about 3:1 58 . The main anion is Cl -. On stimulation, water secretion is driven by an increase in Na + and Clconcentrations. The Na + :K + ratio is therefore variable and can exceed unity 58 . The large difference in solute concentrations between saliva and airway surface fluids means that sputum varies in composition between these extremes 55 . Similarly, emitted droplets are predicted to have an overall K + :Na + ratio that depends on the saliva contribution and is highest for speech 46 , whereas the aerosol droplets emitted by breathing originate mainly from the lining of the lower respiratory tract and are dominated by Na + . In emissions of mixed origin, large droplets 46 are likely to be dominated by K + and small droplets 46 by Na + , leading to differences in their evaporation equilibria.

Simulated effects of Na + :K + ratio
KCl is less hygroscopic than NaCl 59 . Therefore, neglecting the direct (Kelvin) effect of droplet size, large droplets with KCl as the predominant salt would be predicted to reach their efflorescence point at higher RH than small droplets with NaCl as the predominant salt. However, the behaviour of salt mixtures is not necessarily additive. Li et al. 59 measured droplet sizes of mixed KCl : NaCl aerosols as they became larger with increasing RH and smaller with decreasing RH, using a microscopy technique after impaction. Figure 1, calculated from their experimental data 59 , shows that in aerosol mixtures of KCl and NaCl, with no other solutes present, the relationship between efflorescence RH and K:Na ratio is non-linear, with a minimum efflorescence RH at about K mol fraction 0.4. Thus each salt tends to keep the other in supersaturated solution until both precipitate together at the efflorescence RH. Figure 2 shows that In the absence of other solutes, mixtures of NaCl and KCl can reach supersaturated concentrations up to 20 M or more. Such high concentrations have been suggested to reduce survival of susceptible viruses 22 . Somewhat higher maximal salt concentrations are reached when the major cation is potassium compared to sodium, with the maximum at higher RH. When the RH fell below the efflorescence point each droplet contracted abruptly to an irregular solid that remained constant in size down to RH = 5% 59 , from which it was assumed that the liquid phase disappeared at the efflorescence RH. The mixture with K mol fraction 0.2 is representative of the principal ion content of airway lining fluid 53,54 emitted mainly as small (<10 m) droplets 46 . The mixture with K mol fraction 0.8 is representative of the principal ion content of saliva 58 emitted mainly as larger (>10 m) droplets 46 . Ions other than Na + and K + are also present, of course, and their contribution to drying equilibria could be calculated 33 if comprehensive consensus values for their concentrations were available.

Droplet diameter: the Kelvin effect
The extent to which droplets dry at any RH is also influenced directly by their size due to the Kelvin effect. In Kohler theory, the increased surface curvature of small droplets leads to enhanced evaporation at a given RH, according to the diameter-dependent Kelvin term Ke in the expression for their drying equilibrium 34 Figure 2. Figure 3 shows that the effect of including the Kelvin term in the simulation is to displace the whole curve to higher RH. The Kelvin effect starts to become significant only at droplet diameters below about 0.1 m. In calculating the Kelvin term it is generally assumed that the droplet is wholly liquid and is spherical 34 . When a mixture of irregular solid and liquid phases is present, the local radius at protuberances may be less than calculated and the effect augmented.
The magnitude of the Kelvin effect depends also on sol, the surface tension, which in the simulation above was assumed constant and equivalent to that of water. However, surfactants reduce sol, 60 and are present in airway surface liquids 61 , influencing their fragmentation into aerosols 42 . Surfactant proteins 61 are best known from the lungs but are detectable elsewhere 62 and are accompanied by deacylated phosphatidyl choline 63 . The effect of surfactants on biogenic droplet drying has not been quantified but it can be assumed that they reduce the magnitude of the Kelvin term, to an uncertain and possibly large extent 60 . Figure 2 simulates the drying of NaCl/KCl mixtures with no other solutes present. However, viruscontaining aerosols contain larger amounts of proteins and glycoproteins 55,58 , which increase in concentration as the droplets dry until they constitute most of the mass of the droplet nucleus. It is often assumed that these macromolecules have no influence on water activity 28 , an assumption that might not be valid when there is more protein than remaining water.

Macromolecular composition: proteins and mucin glycoproteins
In the intact airway lining, mucin glycoproteins have been stated to play a role in maintaining hydration 51 and have been studied with that function in mind 55 . Reduced water content and increased viscosity are well-known factors in cystic fibrosis and other pathological conditions 51,55 . A substantial rheological change attributed to gelation has been observed when the solids content and mucin content of normal airway fluid are doubled 55 , which would occur at RH > 90% during droplet drying. It would then follow that during much of the supersaturation part of the aerosol drying curve the mucin fraction, and perhaps other proteins, are in the gel state. Osmotic relations of polymer gels are difficult to handle, although for simplification it is often assumed that only low-molecular species -free salts and the counterions associated with any charges on the polymer -contribute to lowering water activity 64 . This assumption may not hold at high polymer concentrations or for very flexible polymers that undergo vigorous segmental motion.
In the case of airway fluid, these conceptual problems have been circumvented by direct measurement of the polymer-associated osmotic pressure using a membrane permeable to salts that are not associated with the polymer 51,52 . An empirical relationship of osmotic pressure to polymeric solids content (proteins plus glycoproteins) of airway mucus, above and below the sol-gel transition, was derived from the measurements of osmotic pressure in ref 52 . Water activity aw, and hence equilibrium relative humidity, can be connected to osmotic pressure using an appropriate form of the Van t'Hoff relationship 65 : Osmotic pressure  = -RTln(aw)/Vw Thus aw = exp(-Vw RT) Where RH is expressed as a fraction, R is the gas constant, T is absolute temperature and Vw is the molar volume of water.  (1-aw), during drying of airway surface liquid by the influence of salt, calculated from RH data for a NaCl/KCl mixture with initial total concentration 300 mM and K + :Na + molar ratio 0.2 (see Figure 2); and by the influence of the macromolecular (protein and glycoprotein) fraction, calculated from the osmotic pressure measurements of ref 52 . Note that the (1-aw) scale is logarithmic. Figure 4 compares (a) the reduction in water activity due to the salts, calculated (without including the Kelvin term) from the RH data of Li et al. 59 , with (b) the reduction in water activity due to proteins and glycoproteins calculated from the osmotic pressure data of Button et al 52 when solutions with ionic and protein/glycoprotein content representative of airway fluid were dried as far as equilibrium with 71% RH. Drying to this extent requires extrapolation of the osmotic pressure data somewhat beyond the measured protein concentration range 52 . Within the concentration range shown, the contribution of the protein and glycoprotein fraction to the reduction in water activity is two orders of magnitude lower than the contribution of the salt mixture, and so would for many purposes be negligible. Extrapolation further into the concentration range where the mucin components gel would be unsafe, bearing in mind the complexity of the osmotic properties of biomacromolecule gels.
To explore what might happen at lower RH, Figure  5 shows the remaining moisture relative to the mass of protein and glycoprotein (approximated as total solids -(NaCl + KCl)) for compositions representative of airway fluid 55 (0.2 mol fraction K, 40 g/L total solids) and saliva 58 (0.8 mol fraction K, 5 g/L total solids). The extent of drying was calculated from the salt content only, neglecting the hygroscopic effect of the polymer fraction. The simulated moisture curves for solutions representing airway fluid and saliva were similar and remained above 30% until the efflorescence point was reached. In general proteins at less than 20% moisture form hydrated solids, with the water in adsorbed form retained quite strongly to low RH and no separate liquid phase 66 . This would be expected to happen below the efflorescence point.

Discussion
Supersaturation of salts in aerosol particles means that very high salt concentrations can potentially be reached in equilibrium with a rather narrow mid-range band of RH values ( Figure 2). It should be noted than anything that can nucleate crystallisation, such as particulate solids or contact with a surface, would truncate the maximal salt concentration at higher RH. As the RH falls past the efflorescence point, NaCl / KCl mixtures precipitate abruptly (Figure 2), although a more gradual decrease in dissolved salt concentration at low RH is possible with more complex mixtures 33 . Such high salt concentrations inactivate some viruses 22 , although SARS-CoV-2 appears to be relatively resistant 19 . Its survival is enhanced when droplets dry onto metal surfaces 24 , for reasons that are not clear. Water activity in surface deposits depends on ambient RH in the same way as in aerosol droplets larger than the threshold of the Kelvin effect. Some fractionation of droplet components may occur during surface drying 67 , and there may then be radial variation in salt and protein content within the dried deposit. There is a potential parallel in the stability of viruses adsorbed on atmospheric particulates 35 .
The nature of the cations present in viruscontaining emitted droplets deserves closer attention, since the ionic composition of saliva 58 differs from that of airway fluids 55 . The low salt content and high proportion of KCl in saliva droplets means that their maximal salt content is reached at higher RH than Na-rich airway fluids. Although a saliva-rich cough 46 and a sneeze 41 emit droplets with rather similar initial size ranges, their drying behaviour and droplet nucleus diameter are predicted to differ.
The dependence of drying equilibrium on droplet size through the Kelvin effect becomes noticeable when the liquid droplet diameter is less than 0.1 m (Figure 3). Diameter distributions on emission by normal breathing include large numbers of droplets <0.1 m 46 . However, the volumeweighted diameter distributions, which are more relevant to viral load, include only a small fraction of <0.1 m droplets, even for breathing 35 . Also, the magnitude of the Kelvin effect is reduced to an unknown extent 60 by the presence of pulmonary and other surfactants. Thus for many purposes it is a reasonable assumption to neglect the Kelvin term in drying simulations. This prediction has practical consequences. Variations in drying equilibria with droplet size are more likely to arise from differences in site of origin and composition, rather than from the Kelvin effect. More usefully, experiments on the hygroscopic properties of bulk biological fluids, e.g. measurements of osmotic pressure 51,52 , are relevant to the behaviour of aerosols so that in suitable cases, it may be possible to obtain informative experimental data without the technical challenges inherent in aerosol generation and measurement.
However it may be inadvisable to neglect the Kelvin effect for partially dried particles or surface deposits where polycrystalline salts are present, because liquid films may have small local radii overlying crystal vertices and negative (inward) radii in interstices, leading to variation in local water activity.
In the drying range up to about 20% solids content, the hygroscopic effect of proteins and of glycoproteins such as mucins is predicted to be negligible in comparison with the effect of the salts present ( Figure 4). Below the efflorescence RH the macromolecular fraction is likely to be essentially an amorphous solid holding any remaining water quite strongly ( Figure 5). In the 40% -80% solids range the mucin fraction at least is likely to form a gel, within which water relations are difficult to predict 64 . Other, salted-out proteins are likely to be interspersed with the mucin glycoproteins and protein aggregates may act as nucleation points for salt precipitation above the efflorescence RH. In this region of complex physical chemistry, experiments on real biological fluids 27 may be a better guide to the behaviour of the system than available theory.

Conclusions
The drying equilibria of aerosol and larger droplets containing infectious viruses are determined principally by the salt composition of the droplets. The salts present in saliva are K + -dominated, whereas the more concentrated salts present in airway surface liquids are Na + -dominated and, in the absence of other solutes, precipitate (effloresce) at lower RH. These differences mean that droplets emitted by breathing, speech, coughing and sneezing differ in drying behaviour according to their site of origin and that in emissions with multiple sites of origin, drying behaviour differs between small and large droplets.

Modelling mixed NaCl/KCl solutions
The published efflorescence equilibria 59 in the form of measured droplet area ratios Ra with the constant droplet area at RH <30% set as unity, were converted to volume ratios Rv = Ra 3/2 . This approach was justified by a close match (+/-<1%) with the efflorescence and deliquescence RH measured at bulk scale 59 . For RH below the efflorescence point Rv = R0. For RH above the efflorescence point Rv is equivalent to the growth factor as conventionally defined 34 . R0 was converted to mass using a solid density solid interpolated between the densities of KCl (1980 Kgm -3 ) and NaCl (2176 Kgm -3 ) according to the molar ratio, with a correction factor of 0.6 for void volume in the polycrystalline salt deposits. Solution concentrations were then calculated as R0. solid(Rv -1) approximating the liquid density as 1000 Kg m -3 .
The Kelvin term in the predicted drying equilibrium was calculated as a function of droplet diameter using the relationship RH = a w . K e = a w .exp(4M w  sol /(RT w D wet ) Where Mw is the molar mass of water, w is its density and  sol is the surface tension of the solution 34 .

Modelling water activity in macromolecular solutions
The water activity of protein/glycoprotein solutions without salt was calculated from published osmotic pressure data 52 using the following form of the Van t'Hoff relationship 65 : Osmotic pressure  = -RTln(aw)/Vw Rearranging, aw = exp(-Vw RT) Where RH is expressed as a fraction, R is the gas constant, T is absolute temperature and Vw is the molar volume of water.
All simulations were carried out in Microsoft Excel.
The .xlsx files are available from the author on request.

Conflict of interest statement
The author declares no conflict of interest.
Funding statement The preparation of this review was not supported by any funding body.