Formation of Fractal aggregates and Why Df should not be constant with aeorosol size
Studies of Titan hydrocarbon aerosols have assumed that all aggregates, regardless of size, have a fractal dimension of 2 (S7 - S11). This value is in general agreement with theoretical calculations of diffusion limited cluster-cluster aggregation in both the ballistic and continuum regimes. (S12, S13). Though valid as a first approximation, the assumption of a constant fractal dimension across all size bins misses an important nuance of fractal aggregate microphysics. Terrestrial carbonaceous fractal aggregate aerosols have been observed to restructure, becoming more compact (higher Df) as the number of monomers contained in the aggregate increases (S14 - S17). The fractal dimension of early Earth (and Titan) haze aggregates therefore should not be considered constant, but rather will vary across the aggregate size distribution. Hydrocarbon aerosols initially condense into nanometer sized spherical particles. The initial growth phase is characterized by a high fractal dimension (D ~f 3), as monomers are built molecule by molecule forming spherical particles with radii ranging from 10 to 100 nm. In the secondary growth phase, spherical monomers coagulate into short linear chains of low fractal dimension ( nmon <100, Df ~ .5). As aggregates grow larger structuring becomes important. Chain-like aggregates tend to be electrically charged. During the restructuring process oppositely charged aggregate limbs attach resulting in more compact arrangements. Laboratory experiments have shown that electrical restructuring increases the fractal dimension of carbonaceous aggregates to Df ~ 2.4 S17). Aggregate restructuring is also triggered by Brownian motion of monomers within an aggregate and by surface energy minimization of condensed water trapped within an aggregate (S14, S17). Restructuring is parameterized in the microphysical code by incorporating a size bin dependent fractal dimension (Fig. S2). The end result of the complex restructuring process is that the fractal dimension increases with aggregate mass. 2) \citep{Wolf_2010}
Mean Field Theory Limitation
The optical properties of fractal aggregates were determined using the mean-field
approximation for Mie scattering by aggregates composed of identical spheres described by Botet t al. (S20). This method has been validated against exact solutions to Maxwell’s equations for scattering by aggregates composed of multiple spheres as described by Xu (S22). However, a validation was only conducted for a silica aggregate having nmon = 64, a monomer optical size parameter of 3.9, and a complex refractive index of 1.4 + 0.0001i. A rigorous validation for large absorbing aggregates typical of this study has not been conducted. It should be noted that Botet’s method follows directly from that of Xu, constructed with the same guiding principles but with an interest to increase computational speed. Xu’s solutions have been validated by laboratory experiments, but only for monomer optical size parameters of ~5, and monomer numbers of 2, 8, and 27 (S23). \citep{Wolf_2010}
From \citep{Sorensen_2001}
Figure 3. Diagram of light incident from the left with wave The difference Eki ¡ Eks detector with scattering wave vector Eks at a scattering angle µ. vector Eki scattering from a scattering element at Er toward theis Eq, the scattering wave vector
Why random orientation scattering solution does not depend azimuthal angle \(\phi\)
".... practical applications often involve a collection of randomly oriented scatterers; consequently, scattering properties introduced above need to be averaged over orientation (therefore eliminating the dependence on orientation and ϕ"\citep{Zhao_2009}
Pair correlation function and structure factor ( how to calculate them)
"To calculate pair correlation a monomer in the cluster was chosen to be the center for shells of radius r and thickness dr progressing outward. For each shell the number of monomers was counted. These counts were divided by 4pr2dr. The next monomer was then made the center for shells and monomers in those shells were counted again. This was done for all monomers in the cluster to obtain an ensemble average pair-correlation for the cluster. Structure factor for an aggregate was calculated from squaring the Fourier transform of the spatial coordinates of the monomers and normalizing by N2 \citep{Sorensen_2001} ".\citep{Heinson_2012}
Calculation of the correlation function \(g\left(r\right)\) From \citep{Lattuada_2003}
On the other hand, the radius of the smallest sphere en-compassing the cluster, Rs , cannot be calculated from the particle–particle correlation function, but requires the spe- cific knowledge of the maximum value of the interparticle distance. Also from an experimental point of view, only an electron-microscopy analysis of a real cluster allows the de- termination of this quantity. Therefore, it would be interest- ing to see if Rs can be related to other more easily measur- able lengths, like the radius of gyration Rg. The procedure followed to compute the particle–particle correlation function deserves some special care. First, all the particle–particle distances rmj in all the generated clusters of the same mass have been computed. The total number of these distances is given by the product between the number of clusters and the number of distances in a single cluster, cles in the cluster. Then the value of g(r) at the distance r is the latter being equal to i(i −1), with i the number of parti- obtained by counting the number of values of rmj falling in 0.1Rp. The chosen value ofdr is also important: it cannot be the interval between r −dr/2and r +dr/2, with dr equal to too small, or the g(r) function becomes very irregular due to the discrete nature ofthe data, but it cannot also be too large, or a poor approximation of the continuous g(r) function is obtained. The number ofdistances in each interval is divided by the volume of the spherical shell of thickness dr,that is, 4πr2 dr, and then by the number of clusters and the number of particles per cluster. The procedure described above is re- peated for several values of r, until all the particle–particle distances have been assigned to an interval. According to the above definition, the function g(r) is normalized in such a way that, when integrated over the whole space, the result is the number of particles in the cluster minus one [24]:
Ideas
- would be interesting to show and ( if possible ) compare the spectral plots of our model and some others , such as the plot below ( from \citep{McKay_2001} )