This package computes the core phase shift parameter (ρBC) to infer the morphology of fractal black carbon aggregates using the particle mass-equivalent diameter (dp), mass absorption cross-section (MACBC), and mixing state (Mtot/MBC). It should be noted that in this package, MACBC is defined as the absorption cross-section per unit mass of black carbon. First, ρBC is constrained by determining whether the measured mass absorption cross-section (MACBC,meas) is significantly less than that given by:
- ${MAC_{BC,pred}=MAC_0\left (\frac{\lambda}{\lambda_0} \right)^{-AAE}\left[1+\frac{AC^{-B}\Gamma(B+1,C)}{C}-\frac{A\left(\frac{M_{tot}}{M_{BC}}\right)^{B}\left(\frac{M_{tot}}{M_{BC}}\right)^{-B}\Gamma\left(B+1,C\frac{M_{tot}}{M_{BC}}\right)}{C}\right]}$.
- A = − 1.189 ± 0.029
- B = − 0.674 ± 0.006
- C = 0.043 ± 0.0007
- MAC0 = 6.819 ± 0.131
- AAE = 1.231 ± 0.005
If MACBC,meas is within 10% of MACBC,pred, then ρBC can be constrained to 0 < ρBC < 1, but cannot be exactly calculated. If MACBC,meas is less than 90% of MACBC,pred, then ρBC is calculated by solving:
${MAC_{BC,meas}=MAC_0\left (\frac{\lambda}{\lambda_0} \right)^{-AAE}\left[\frac{D}{E+1}\left(\rho_{BC}^{1-E}-1\right)+\frac{D}{1-2E}\left(\rho_{BC}^{1-2E}-1\right)\right]+MAC_{BC,pred}}$ .Where D and E are sigmoid functions, given by:
${X=x_1+\frac{x_2-x_1}{1+\text{exp}\left[x_3\left(\rho_{BC}-x_4\right)\right]}}$ .Here X represents D or E, and x[1, 2, 3, 4] represents d[1, 2, 3, 4] or e[1, 2, 3, 4].
- d1 = 5.679 ± 0.027
- d2 = 1.066 ± 0.058
- d3 = 0.264 ± 0.010
- d4 = 11.421 ± 0.137
- e1 = 2.440 ± 0.017
- e2 = 0.593 ± 0.024
- e3 = 0.418 ± 0.020
- e4 = 10.106 ± 0.131
Details on the derivation of the above equations are available here.
The morphology of of the measured black carbon aggregates can be determined by comparing the calculated ρBC to three cases. The first case is that of freshly emitted black carbon, which has fractal dimension (Df) of 1.8. The second case is black carbon which has partially collapsed, and has Df of 2.5. The final case is black carbon which has fully collapsed (but not sintered), and has Df of 3.0.
The core phase shift parameter of black carbon aggregates with morphologies outlined above is found by first determining their radius of gyration Rg, given by:
${R_g=a\left(\frac{m_p}{m_1 k_f}\right)^{\frac{1}{D_f}}}$ ,where a is the monomer radius, kf is the fractal prefactor (fixed at 1.2), mp is the black carbon mass, and m1 is the mass of a BC monomer. Both the BC mass and the monomer mass are determined assuming BC density of 1.8 g/cm3. Next, the monomer packing fraction (ϕ) is found using:
${\phi=k_f\left(\frac{D_f+2}{D_f}\right)^{-\frac{3}{2}}\left(\frac{a}{R_g}\right)^{3-D_f}}$ .Finally, ρBC is given by:
${\rho_{BC}=\frac{4\pi R_g}{\lambda}\left|m_{eff}-1\right|}$ ,where meff is given by:
${\phi\left(\frac{m^2-1}{m^2+2}\right)=\left(\frac{m_{eff}^2-1}{m_{eff}^2+2}\right)}$ .Here, m is the refractive index of black carbon, 1.95 + 0.79i. The core phase shift parameter and mass of the measured black carbon aggregates is then compared to the three cases described above, allowing for inference of particle morphology.
In a similar manner, users can supply the morphology, mixing state, and particle diameter, and the mass absorption cross-section can be calculated based on the core phase shift parameter.
Inverse function for single particle ---------------------------------
Forward function for single particle ---------------------------------
Inverse function for black carbon size distribution ---------------------------------
Forward function for black carbon size distribution ---------------------------------