Aerosol synthesis: Structure and evaporation rates
Supplementary material
SUPPLEMENTAL INFORMATION
Modelling size and structure of nanoparticles during droplet-phase aerosol synthesis
Bandyopadhyay, Arpan; Pawar, Amol; Venkataraman, Chandra; Mehra, Anurag*
SI 1: Droplet Shrinkage Regime
During the process of evaporationoccur simultaneously: diffusion of solvent from the surface of the droplet to ambient, the shrinkage of the droplet leading to solute build up at surface, change in the droplet temperature resulting from evaporative cooling and diffusion of solute toward the center of the droplet. This regime is primarily concerned with the evaporation of a solution droplet as a coupled heat and mass transfer problem leading to shrinkage of the droplet to the point when the crust starts forming. The solvent evaporation rate, derived from the Fick’s law, is written in terms of the rate of change of the droplet diameter, as
(1)
wheredp denotes the droplet diameter, Dv is diffusivity of solvent vapors in air, Mis molecular weight of solvent, Pd is vapor pressure over the droplet, Tdis temperature of the droplet, P∞ is partial pressure of solvent in the suspending medium, T∞is temperature of the suspending medium is the non-continuum correction factor and R is the universal gas constant.
As the diameter of the droplet approaches the mean free path of the carrier gas molecules, the continuum assumption may no longer be valid as the concept of gradients breaks down in the region within one mean free path of the droplet surface and the transport of vapor molecules is controlled by kinetic processes (Fuchs and Sutugin 1970). It was shown that for submicron size drops the droplet evaporation rate and temperature history changed due to the non-continuum effects (Eslamian et al. 2006a).To account for the non-continuum effects, Fuchs’ correction factor is applied to Equation (1), which is given by,
(2)
where Kn is Knudsen number (), is the mean free path of air, dp is the droplet diameter and is the mass accommodation coefficient or sticking probability for vapor molecules impinging on the drop surface to enter the liquid phase (Fuchs and Sutugin 1970).
The vapor pressure over a solution droplet (Pd) is affected by the activity of solvent and thedroplet curvature (Kelvin effect). The concentration of the non-volatile species (solute) in the solutiondroplet increases as the solvent evaporates from the droplet surface. A decrease in the solvent molefractioninside the evaporating droplet decreases the solvent activity. To account for the decreasein solvent activity, the activity coefficient of the solvent is calculated by the UNIFAC groupcontribution method. Equation (3), is used to compute the vapor pressure over a droplet (Pd)relative to solvent vapor pressure (Psat), considering the effect of activity of the solvent anddroplet curvature (Pruppacher and Klett 1997)
(3)
where asis the activity of the solvent, M is the molecular weight, σs/a is the surface tension of the solvent and ρsis the density of solution.The activity of the solvent (as) was calculated by the UNIFAC (Universal Functional Activity Coefficient) group contributionmethod.
As the dropletcontinues to evaporate, the solvent vapor molecules accumulates in the suspending gas, whichresults in an increasing solvent partial pressure (P∞) and reducesthe driving force fordroplet evaporation (Pd - P∞).The solvent accumulation in the suspending gas is accounted by,
(4)
where N is number of drops and V is volume of suspending gas.
Apart from transfer of solvent molecule from the droplet to the suspending gas, heat transfer also accompanies evaporation. With the evaporating solvent, there is continuous loss energy from the droplet in form of latent heat of vaporization, which leads to evaporative cooling. The lowering of the droplet temperature is compensated by sensible heat transfer from the surrounding gas. The overall heat balanceequation is given by,
(5)
whereHis the enthalpy of vaporization, Cplis theheat capacity of the solvent,k is thermal conductivity of air, and are non-continuum correction factors for thermal and mass transfer respectively which are given by Equation (2), with respective accommodation coefficients. The thermal accommodation coefficient,, accounts for the incomplete equilibration of impinging air molecules with droplet the surface temperature (Knudsen 1911). Experimental data on thermal and mass accommodation coefficients suggest that the values of and tend to unity (Winkler et al. 2006; Tsuruta and Nagayama 2004; Vieceli et al. 2004; Shaw and Lamb 1999).
The temperature of the surrounding gas was obtained by considering heat transfer between the suspending gas and drops, as follows
(6)
WhereCpgis the heat capacity of the surrounding gas.
SI 2: Fitting values for critical supersaturation
- The lower bound and upper bound for SC was determined. Lower and upper bound for SA-CYC were 1 and 36 respectively. The upper bound was determined by
- For a particular droplet size, the model solutions from the droplet shrinkage regime were obtained at gas temperature, Tg = 25 °C (final particle size, solute distribution, droplet temperature).
- Insights on the particle structure were obtained from the information on internal solute distribution using the percolation theory (Jayanthi et al. 1993).
- The density of the particle was calculated using Equation 4.
- Steps 2 to 4 were repeated for different initial droplet sizes and SC values.
- A 3-D plot was constructed with the initial droplet size, final particle size, and SC values.
- The feasible region of SC values was obtained from the information on experimentally observed particle sizes, typical initial droplet diameters from atomizer ratings, the calculated density of the particle and literature.
- The feasible region for the SA-CYC marked in Figure 3 and the value of SC was chosen to be 10.
- This value of SC was then corrected for other droplet temperatures using Equations 9-11.
FIGURE S1: Computed solute concentration distribution inside a droplet generated from stearic acid solution (10mg/cm3) in cyclohexane at various evaporation times.
FIGURE S2: Comparison of droplet temperature history for droplets generated from stearic acid solution (10mg/cm3) in cyclohexane, at different carrier gas temperatures 25 °C and110 °C.
FIGURE S3: Internal solute distribution for a droplet generated from stearic acid solution in cyclohexane, with different stearic acid concentrations (1-35 mg/cm3) at different gas temperatures (a) 25 °C and (b)110 °C.
Table S1: Physical properties of precursor solvent (cyclohexane, CYC) and solute (stearic acid, SA) at 25 °C used in model.
Solvent Properties (cyclohexane, CYC)Solvent vapour pressure (Pa) / 12992
Surface Tension (N/m) / 0.024
Solvent diffusivity in air (m2/s) / 8.91×10-6
Solvent density (kg/m3) / 778.5
Solvent Specific heat capacity (J/kg K) / 1843
Conductivity (W/m K) / 0.13
Solute Properties (stearic acid, SA)
Solute crystal density (kg/m3) / 847
Solute diffusivity in solvent (m2/s) / 9.01×10-10
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
Table S2: Characteristic time constants for droplets generated from stearic acid solution in cyclohexane relative to solute diffusion time constant.
Droplet Shrinkage Regime / Shell Growth RegimeTime Scale
(s) / Relative time scales / Time Scale
(s) / Relative time scales
Solvent Vapour Diffusion / / 1.4 x 10-08 / ~10-4 / / 1.1 x10-08 / ~10-3
Droplet Shrinkage / / 3.4 x 10-04 / ~100 / / 2.7 x 10-03 / ~102
Solute Diffusion / / 1.4 x 10-04 / ~100 / / 2.5 x 10-05 / ~100
Heat conduction in Droplet / / 1.4 x 10-06 / ~10-2 / / 2.5 x 10-07 / ~10-2
DL, diffusivity of the solute in droplet (= 9.01x10 -10 m2/s), Dv,diffusivity of the solvent vapours (= 8.91x10 -6 m2/s), Dcr, diffusivity of the solvent vapours through the solid crust (= 1.38x10-12 m2/s), dp, initial droplet diameter (= 350 nm), x∞, mass fraction of accumulated vapour (= 0.01), di,liquid core diameter (= 150 nm), d*,final particle diameter (= 160 nm), ρL, solvent density (= 780 kg/m3) and ρg, solvent vapour density (= 3.43 kg/m3), κL ,thermal diffusivity of the liquid core (= 9x10-8 m2/s)
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
Table S3: Variation of computed critical supersaturation with droplet temperature as a consequence of increasing gas temperature.
Tg (°C) / Td (°C) / CSS25 / 2.1 / 10
50 / 11.3 / 8.2
75 / 18.9 / 7.0
110 / 28.7 / 4.2
Table S4: Estimated values of drop evaporation times of Zirconium hydroxylchloride (ZHC) in water from the current model as against values recently reported in literature (Eslamian et al. 2006).
S.No / Temperature of suspending gas (oC) / Particle diameter(μm) / Evaporation time (ms)
Reported by Eslamian et al. 2006 / Present model
1 / 100 / 1.5 / 4.2 / 4.1
2 / 200 / 1.6 / 1.8 / 1.7
3 / 300 / 1.7 / 1 / 1.0
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