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 evaporation occur 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
dd p
4Dv M Pd P¥
(1)
( - )f
dt
Rr d p Td T¥ M
where dp denotes the droplet diameter, Dv is diffusivity of solvent vapors in air, M is molecular
weight of solvent, Pd is vapor pressure over the droplet, Td is temperature of the droplet, P∞ is
partial pressure of solvent in the suspending medium, T∞ is temperature of the suspending
medium f M 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,
0.75a M (1+ Kn)
(2)
fM = 2
Kn + Kn + 0.283Kn + 0.75a M
where Kn is Knudsen number ( Kn = 2l d p ), l is the mean free path of air, dp is the droplet
diameter and a M 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
the droplet curvature (Kelvin effect). The concentration of the non-volatile species (solute) in the
solution droplet increases as the solvent evaporates from the droplet surface. A decrease in the
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
solvent mole fraction inside the evaporating droplet decreases the solvent activity. To account for
the decrease in solvent activity, the activity coefficient of the solvent is calculated by the
UNIFAC group contribution 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 and droplet curvature (Pruppacher and Klett 1997)
4 M s s/a
Pd
(3)
= as exp(
)
Psat
rs RTd p
where as is the activity of the solvent, M is the molecular weight, σs/a is the surface tension of the
solvent and ρs is the density of solution. The activity of the solvent (as) was calculated by the
UNIFAC (Universal Functional Activity Coefficient) group contribution method.
As the droplet continues to evaporate, the solvent vapor molecules accumulates in the
suspending gas, which results in an increasing solvent partial pressure (P∞) and reduces the
driving force for droplet evaporation (Pd - P∞).The solvent accumulation in the suspending gas is
accounted by,
 Pd P 
d  P   N 
(4)
     2 d p Dv    M
dt  T   V 
 Td T 
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 balance equation is given by,
dTd
D MH Pd P¥
12k
(5)
=
((T¥ - Td )fT - v
( - )f )
2
dt rl C pl d p
Rk Td T¥ M
where H is the enthalpy of vaporization, Cpl is the heat capacity of the solvent, k is thermal
conductivity of air, fT and f M 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, a T , 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 a M and a T 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
dT¥
æ N ö æ 2p kd p ö
(6)
= -ç ÷ ç
÷ T -T f
dt
è V ø è r g C pg ø ¥ d T
(
)
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
Where Cpg is the heat capacity of the surrounding gas.
SI 2: Fitting values for critical supersaturation
1. 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
SC ´ ES = rsolute
2. 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).
3. Insights on the particle structure were obtained from the information on internal solute
distribution using the percolation theory (Jayanthi et al. 1993).
4. The density of the particle was calculated using Equation 4.
5. Steps 2 to 4 were repeated for different initial droplet sizes and SC values.
6. A 3-D plot was constructed with the initial droplet size, final particle size, and SC values.
7. 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.
8. The feasible region for the SA-CYC marked in Figure 3 and the value of SC was chosen
to be 10.
9. This value of SC was then corrected for other droplet temperatures using Equations 9-11.
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
FIGURE S1: Computed solute concentration distribution inside a droplet generated from stearic
acid solution (10 mg/cm3) in cyclohexane at various evaporation times.
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
FIGURE S2: Comparison of droplet temperature history for droplets generated from stearic acid
solution (10 mg/cm3) in cyclohexane, at different carrier gas temperatures 25 °C and 110 °C.
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
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.
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
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
Time Scale
Shell Growth Regime
Time Scale
(s)
Relative time
scales
(s)
Relative time
scales
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
1.4 x 10-06
~10-2
2.5 x 10-07
~10-2
Solvent Vapour
Diffusion
Heat conduction in
Droplet
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)
CSS
25
2.1
10
50
11.3
8.2
75
18.9
7.0
110
28.7
4.2
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Aerosol synthesis: Structure and evaporation rates
Supplementary material
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
Evaporation time (ms)
(μm)
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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