4.1. Droplet Temperature and Concentration Fields
Rosebrock et al. [
8] investigated single droplet combustion of aluminium nitrate (
), iron nitrate (
) and zinc nitrate (
) solution droplets. They found for
the droplet-to-particle route as the preferred mechanism, whereas
and
solution droplets undergo microexplosions and the resulting particles indicate gas-to-particle conversion. To evaluate the mass transfer model for particle synthesis from low cost metal nitrate precursors, the simulation results in this chapter are compared qualitatively to the experiments from Rosebrock et al. [
8]. To our knowledge, there is no measurement method to resolve concentration fields of burning droplets during flame spray synthesis. Therefore, we evaluate if calculated temperature and concentration fields provide an explanation for the different combustion behaviour of the metal nitrate precursor solutions as observed by Rosebrock et al. In the following, temperature and concentration fields are simulated for an initially 100
sized droplet containing
/
,
or
ethanol solutions. Droplets are burning at 2300 K.
In
Figure 6, the size evolution for all three materials is shown. Immediately, liquid-phase precipitation begins in
and
solutions, since they are oversaturated at the beginning. Then, all materials follow initially the nearly identical surface regression. This is a result of preferential evaporation of ethanol. Around
, all three materials experience superheating in the region below the surface as shown in
Figure 7a). When the droplet surface is depleted of ethanol, droplets obtain
of their respective precursor. This is accompanied with a change of regression rate in the region below
. The results also show that liquid-phase precipitation precedes precursor decomposition. In
Figure 6b), the influence of droplet relative velocity
is shown. Due to the increased regression rate,
is reached only at the very end of droplet lifetime. If, in the case of
, droplet-to-particle formation is preferred as reported by [
8], the resulting oxide particles will be smaller.
Figure 7a,b show the temperature fields and concentration fields at the onset of superheating. Superheating occurs in proximity to the droplet surface, enabling homogeneous vapor nucleation inside the droplet for all three precursors. On the other side, only
and
solution droplets reach supersaturation at onset of superheating. The supersaturation is high at the droplet surface und low in the interior. This indicates that precipitates will form preferentially on the surface, meeting the requirement for shell formation, whereas precipitates inside the droplet enable heterogeneous vapor nucleation. Nucleated vapor bubbles then expand while being heated up, leading to a rupture of the surface layer and causing droplet fragmentation. The combination of a diffusional barrier of surface precipitates and heterogeneous and homogeneous vapor bubble nucleation could explain why micro explosions are observed in
and
, while these are absent in the case of
[
8]. If vapor bubble nucleation occurs in the case of
, the bubbles can leave the droplet without causing disruption, since no pressure build up takes place due to a lack of a shell.
The interpretations above differ from those of Rosebrock et al. [
8,
26] and Li et al. [
9] in the form that precursor decomposition is not required to form a viscous shell. Especially in the case of nitrates with comparatively low solubility in ethanol, such as
and
, precipitates may act as nucleation sites for vapor bubbles and form a viscous shell.
To conclude the discussion above, droplet fragmentation event can be decided in the framework of DPBMC simulation when two consecutive events take place. First, liquid-phase precipitation or precursor decomposition needs to occur. Then, internal boiling needs to happen in the form of heterogeneous or homogeneous vapor nucleation. Widiyastuti et al. [
27] modeled spray pyrolysis under low-pressure conditions and concluded, from the obtained very high evaporation rates, that droplets are ruptured nearly instantaneously, releasing all monomers at once. This might prove a feasible approximation in the case of violent droplet disruptions.
4.2. Droplet Population Evolution
To investigate the influence of the DSD on particles, nanoparticle synthesis in a flame spray burner is simulated. The ambient pressure and temperature are 1 bar and 2300 K with initial droplet parameters given in
Appendix A. As a model system,
in ethanol is chosen. The relative velocity is approximated by Equation (
23) for droplets between 5
and 100
and set to 0
/
for particles below 5
and to 80
/
for particles bigger than 100
. This is close to Phase Doppler Anemometry (PDA) measurements [
3]
To illustrate the influence of droplet polydispersity on particle formation, only the heat conduction model following from Equations (
5) with (
7) is applied, while a uniform precursor concentration inside the droplet is assumed. All DSDs are assumed to be initially lognormal distributed. The initial mean droplet diameter
equals 10
in all simulations, while the initial geometric standard deviation
of the DSD is set to 1.1, 1.3 and 1.7. Flame spray synthesis of nanoparticles follows three distinct stages. First, only droplets are present. Second droplets and particles coexist in the lower flame region. In the third stage, only particles are left. However, earlier publications in PB modeling of FSP applied restrictions on droplet behaviour. Mueller [
6] assumed instant droplet evaporation, while Widiyastuti et al. [
4] used constant droplet concentrations. With these approaches, these three distinct stages can not be discriminated. By applying the DPBMC technique, no further restrictions or simplifications regarding the droplet phase have to be imposed. The resulting overall particle and droplet dynamics are exemplary shown in
Figure 8 for an initial
.
At HAB = 0 mm, only droplets exist. The particle nucleation mode is formed when droplets first start to evaporate and produce monomers at HAB = 0.1 mm. Evaporation of small droplets leads to a decrease of droplet concentration and a broadening of the DSD between HAB of 0.1 mm to 5 mm (
Figure 8a). At the same HAB, self-coagulation of the nucleation mode leads to the formation of first larger particles (
Figure 8b). Continuous nucleation and coagulation continues until the majority of droplets has burned off at approximately HAB = 40 mm and thus halts nucleation. Shortly before this point,
reaches its maximum value. Continuous build up of the nucleation mode with simultaneous coagulation results in a bimodal PSD and consequently a high
. From here, no droplets exist anymore and particle coagulation dominates the temporal evolution of the PSD. In the end, the PSD reaches the self-preserving form.
The influence of transient heating on a droplet population is shown in
Figure 9. Small droplets heat up first and start evaporating. Following that, bigger droplets heat up and start to evaporate at around 50 mm above the burner. This results in a broadening of the DSD, while the droplet concentration continuously decreases. In the end, only very few droplets between 30
and 100
exist. Therefore, if the DSD consists mainly of large droplets, as is the case for
, the longer heating time will delay droplet evaporation and particle formation.
The resulting mean droplet diameter
and the normalized spray load
over HAB are shown in
Figure 10a,c for all
. Broader DSDs contain more large droplets, which have a longer lifetime than small droplets. This is also shown by
Figure 10b as SMD stays unchanged and then increases as the weighted average shifts to larger droplets. In contrast to that, SMD decreases for
= 1.1 initially. This is the expected behaviour if only one size class for the droplet phase is simulated, as droplets can only decrease in size due to combustion. Heine and Pratsinis [
3] show PDA measurements where SMD increases initially during
synthesis by FSP. According to them, the initial increase of SMD is due to evaporation of smaller droplets, while droplet coagulation or droplet dispersion also might affect SMD. The simulation results in
Figure 10c show that this behaviour can be explained by combustion of small droplets, although it is highly dependent on
.
Finally,
Figure 11a–d show the PSD properties over time for the each initial DSD. Particle concentration is initially the lowest for
= 1.7 due to the lowest nucleation rate in the case of
= 1.7, whereas, for
= 1.1 and 1.3, particles nucleate in a burst-like fashion as shown in
Figure 11b. The spikes in
are caused by the formation of a nucleation mode, which are more pronounced for
= 1.3 and 1.7. A prolonged nucleation of monomers thus result in particles with higher polydispersity. After nucleation slows down, coagulation dominates the growth process until
is reached. This is expected when the self-preserving form is reached in the free molecular regime. At around HAB = 800–1000 mm particle concentration,
and
reach the same value for all three simulations as a direct consequence of attaining the self-preserving form.
Depending on the particle residence time in the reactor, the PSD may be quenched by exhaust gas before coagulation depletes the nucleation mode, resulting in a bimodal PSD. Otherwise, the PSD becomes independent of the initial DSD. Similar results were found by Heine and Pratsinis [
3] for monodisperse droplet calculations. Large droplets of 40
release monomers longer over time because of their higher lifetime compared to smaller droplets and consequently lead to broader PSDs. They calculated for 40
droplets a
over 2.2 in the simulated time. They also found that self-preserving form is reached relatively quickly once all droplets evaporate. It can be concluded that prolonged monomer release promotes the formation of bimodal PSDs. To describe the monomer release and nucleation rate correctly, the droplet polydispersity has to be taken into account.
For all simulations, isothermal flame temperature is assumed. To test this assumption, simulations with flame temperatures of 1700 K, 2000 K and 2300 K are calculated and evaluated with respect to
and single droplet results regarding surface temperature
and surface regression rate
for an initially 100
droplet. In all three cases, the droplet reaches its maximum temperature at around HAB = 300 mm (
Figure 12b)), which is slightly below ethanol boiling temperature. Here, the droplet also attains its maximal
in each case. For the flame temperature 1700 K and 2300 K, the maximum
equals approximately
and
respectively, which is fairly similar considering 600 K difference in flame temperature. It can be concluded that
is not sensitive regarding the flame temperature.
However, the flame temperature has a major influence on primary particle size as shown by Heine and Pratsinis [
3]. When particles experience a temperature drop after leaving the flame region, primary particle growth stops while agglomerate growth continues. In the case of ZnO synthesis, the flame may exceed the decomposition temperature of ZnO (2250 K). In this case, ZnO particles dissociate into Zn and oxygen and may form ZnO again. Rosebrock et al. [
8] state this to be the reason for different morphology of particles produced in single droplet experiments compared to particle synthesis by FSP. As far as only
is concerned, the isothermal assumption can be applied leading to a negligible error in
.