Supplementary material

6.1.1. Solvent evaporation of a single µm sized droplet

For an isothermal, quasi-steady state evaporation of a spherical droplet, Maxwell obtained the time rate of change of the droplet mass dm/dt, the equation ¹⁷⁸

𝑑𝑚 𝑑𝑡 = (¹

2𝜋𝑑_𝑑²) 𝜌^𝑑𝑑𝑑

𝑑𝑡 = 2𝜋𝑑_𝑑𝐷_𝑣(𝑐_𝑤,∞− 𝑐_𝑤,𝑠), (6S.1)

where dd is the diameter of the droplet, ρ is the droplet density, cw,∞ and cw,s is the concentration of water in the gas and surface of the droplet, respectively, and Dv is the diffusivity of water vapor in air as a function of temperature and pressure given as Dv=(0.211/P)(T/273)^1.94 in cm² s^-1. Evaporation of sub-micron droplets is in the Knudsen transition regime, hence, non-continuum effects should be included by introducing a modified diffusivity Dv’:

𝐷_𝑣^′ = ^𝐷𝑣

[1+^2𝐷𝑣 𝛼𝑐𝑑𝑑(^{2𝜋𝑀𝑤}

𝑅𝑇𝑎 )^1/2]

, (6S.2)

where 𝛼_𝑐 is the water accommodation coefficient, Mw is the molar mass of droplet and R is the gas constant. Equation 6S.1 is derived on the assumption that the interfacial concentration cw,s is constant (equivalent to assuming the interfacial temperature Ta is constant). When the initial temperature of the droplet is different from the surrounding continuum and a large heat of vaporization is involved, the interfacial temperature cannot be expected to be constant, especially during the short time after vaporization begins ¹⁷⁹. Therefore, the heat and mass transfer processes are coupled. The steady-state temperature distribution around a particle is governed by

𝑢_𝑟^𝑑𝑇

𝑑𝑟 = 𝛼 ¹

𝑟² 𝑑

𝑑𝑟(𝑟^{2 𝑑𝑇}

𝑑𝑟), (6S.3)

where α =k/ρcp is the thermal diffusivity of air and ur is the mass average velocity. The first term is the convective term called Stefan flow, which can be neglected on the assumption that the bulk

flow of vapor is small compared with diffusional flux of vapor. Hence, an energy balance on the droplet can be written as

2𝜋𝐷_𝑝𝑘_𝑎^′(𝑇_∞− 𝑇_𝑎) = −∆𝐻_𝑣(^𝑑𝑚

𝑑𝑡). (6S.4)

Ta is the surface temperature, T∞ is the environment temperature, ΔHv is the latent heat of vaporization and ka’ is the modified form for the thermal conductivity of air accounting for non- continuum effects and is given by

𝑘_𝑎^′ = ^𝑘𝑎

[1+ ^2𝑘𝑎

𝛼𝑇𝐷𝑝𝜌𝑐̃𝑝(^{2𝜋𝑀𝑎} 𝑅𝑇 )^1/2]

, (6S.5)

where 𝑘_𝑎 = 10⁻³(4.39 + 0.071𝑇) in Jm^-1s^-1K^-1, and Ma, ρ and cp are the air molar mass, density and heat capacity, respectively. 𝛼_𝑇 is the thermal accommodation coefficient, which is also uncertain and often set equal to the value of the mass accommodation coefficient 𝛼_𝑐. Equation 6S.4 simply states that at steady state the heat released during water condensation is equal to the heat released to the droplet surroundings ¹⁸⁰.

There are two types of factors that affect the kinetics of droplet growth and evaporation given in Equation 6S.1; namely, those that provide resistance to heat and vapor flow and those that change the equilibrium vapor pressure of the droplet. The thermal and water accommodation coefficient belong to the first group representing resistance to heat and vapor flow. Two opposing factors belong to the second type - the Kelvin effect and the solute effect. The Kelvin effect of the droplet surface curvature tends to cause an increase in vapor pressure as the droplet decreases and the solute effect causes a decrease in vapor pressure as radius decreases. One or the other of these effects is dominant depending on the droplet radius and solute content of the drop. Substituting ideal gas law, Equation 6S.1 can be written as

𝐷_𝑝^𝑑𝐷𝑝

𝑑𝑡 =^{4𝜋𝐷𝑣′𝑀𝑤𝑝0(𝑇∞)}

𝜌𝑅𝑇_𝑎 ( ^𝑝𝑤,∞

𝑝⁰(𝑇_∞)− ^{𝑝𝑤,𝑠}

𝑝⁰(𝑇_∞)), (6S.6)

where Mw is the molar mass of the solution, 𝑝⁰(𝑇_∞) is the vapor pressure at T, 𝑝_𝑤,∞/𝑝⁰(𝑇_∞) is the environmental saturation ratio and is equal to unity (relative humidity is equal to 100%) when the partial pressure in the atmosphere 𝑝_𝑤,∞ is equal to saturation vapor pressure 𝑝⁰(𝑇_∞).

𝑝𝑤,𝑠

𝑝⁰(𝑇∞)=^{𝑝𝑤,𝑠(𝐷𝑝,𝑇𝑎)}

𝑝⁰(𝑇𝑎) ∗^{𝑝0(𝑇𝑎)}

𝑝⁰(𝑇∞) (6S.7)

The first term in Equation 6S.7 for a non-ideal solution of soluble and insoluble materials can be written as

𝑝_𝑤,𝑠(𝐷_𝑝,𝑇_𝑎)

𝑝⁰(𝑇𝑎) = 𝑒𝑥𝑝 [^{4𝑀𝑤𝜎𝑤}

𝜌𝑅𝑇𝑎𝐷𝑝+ 𝑙𝑛𝛾_𝑤 − ^{6𝑛𝑠𝑣̃𝑤}

𝜋(𝐷_𝑝³−𝑑_𝑢³)], (6S.8)

where ns is the number of moles of solute, σw is the droplet surface tension, γw is the activity coefficient, 𝑣̃ is the partial molar volume of water and d_𝑤 u is the diameter of insoluble particles in the droplet solution. γw was estimated with the Davies equation for a 10g/L cesium di-hydrogen phosphate solution in water as 0.8. The first two terms in Equation 6S.8 are the Kelvin effect, giving the dependence of vapor pressure on surface tension and droplet diameter, the third term in Equation 6S.8 accounts for the solute effect from Raoults’s law.

The second term in Equation 6S.7 is a ratio of water saturation pressures at Ta and T∞ and is given by the Clausius-Clapeyron equation as

𝑝⁰(𝑇𝑎)

𝑝⁰(𝑇_∞)= 𝑒𝑥𝑝 ቂ^{∆𝐻𝑣𝑀𝑤}

𝑅 (^{𝑇𝑎−𝑇∞}

𝑇_𝑎𝑇_∞)ቃ. (6S.9)

Substituting into Equation 6S.6 we have the complete droplet evaporation given as

𝑑𝐷𝑝

𝑑𝑡 =^{4𝜋𝐷𝑣′𝑀𝑤𝑝0(𝑇∞)}

𝜌𝑅𝑇_𝑎 [ ^𝑝𝑤,∞

𝑝⁰(𝑇_∞)− 𝑒𝑥𝑝 [^{4𝑀𝑤𝜎𝑤}

𝜌𝑅𝑇_𝑎𝐷_𝑝+ 𝛾_𝑤− ^{6𝑛𝑠𝑣̃𝑤}

𝜋(𝐷_𝑝³−𝑑_𝑢³)+^{∆𝐻𝑣𝑀𝑤}

𝑅 (^{𝑇𝑎−𝑇∞}

𝑇_𝑎𝑇_∞)]]. (6S.10)

Equation 6S.10 and Equation 6S.4 are coupled and must be solved numerically to describe the evaporation rate of a single droplet. The set of parameters used to solve the evaporation are given as follows; T∞ =373K, and the concentration of the solute of choice (cesium di-hydrogen phosphate) is 10g/L. The solvent parameters molar mass, surface tension, density correspond to values for water. Figure 6.S.1 shows the evolution of a single droplet of initial droplet size of 1μm to its dry solute size 100nm for varying relative humidity and ambient temperature and a water accommodation coefficients (αc) and the thermal accommodation coefficient (αT) of unity.

Figure 6.S.1. Evolution of a single 1µm droplet to 100nm dry CsH2PO4 particle in 20-99%

relative humidity and ambient temperature of (a) 25⁰C and (b) 100⁰C.

The evaporation coefficient (αC) is defined as the fraction of molecules that hit the liquid surface and condense. This is also called the sticking probability and corresponds to the water accommodation coefficient in the derivations above. Polar liquids like water vaporize at significantly lower rates than the maximum rate predicted by kinetic theory using the equilibrium value of vapor pressure, and therefore have a smaller value of evaporation coefficient than non- polar liquids. An increase in water accommodation coefficient (sticking probability of water to the droplet) decreases the modified diffusivity of the water molecule in air. On the other hand, increase in the thermal accommodation coefficient (transfer of vapor molecules across the interface) increases the modified thermal conductivity of air.

The magnitude of the non-continuum correction depends strongly on the value of the water accommodation coefficient (αc) and the thermal accommodation coefficient (αT). In Figure 6.S. 1, αc=αT. The value of αc has been subject to be debate, a value of 0.045 was used by Pruppacher and Klett, while ambient measures suggest a value closer to unity ^{180, 181}. Fukuta et al explain the

𝑝𝑤,∞

𝑝^𝑜(𝑇)= 0.2 − 0.9, 0.99 α=1

𝑝𝑤,∞

𝑝^𝑜(𝑇)= 0.2 − 0.9, 0.99 α=1

difference between thermal accommodation coefficient and evaporation coefficient (water accommodation coefficient) ¹⁸². They define the thermal accommodation coefficient as the transfer of heat energy by molecules arriving and leaving the interface between a gas and a condensed phase, and consider it to be related to the transfer of vapor molecules across the interface.

Knudsen’s definition of accommodation coefficient is ¹⁸²

∝_𝑇= ^{𝑇2′−𝑇1}

𝑇2−𝑇1 , (6S.12)

where T1 is the temperature of the gas molecule (T1=373K), T2 is the temperature of the condensed phase and 𝑇₂^′ is the temperature of the gas molecule leaving the condensed phase. If we assume T2

is the initial droplet temperature, 298K, and 𝑇₂^′ is the geometric mean, 336K, this corresponds to a thermal accommodation coefficient (αT) of 0.5.

The combined effect is shown in Figure 6.S.2 where the change in droplet evolution time for constant relative humidity of 40% is shown for varying αC = αT. It is evident that, αc and αT do not have a significant effect on the overall prediction of evaporation rate in this setup.

Figure 6.S.2. For αT = αC ranging from 0.05-1, the evolution of a single droplet of initial droplet size 1μm at 40% relative humidity and T=100 °C.

The results above indicate that the drying time of a micron sized droplet is in the millisecond range. Since the particles have a substantially long flight time to the analyzer (22sec), the particles sizes reported in Section 6.1. are of dry CsH2PO4 nanoparticles.

αT = αC

0.05-1