Functions for Internally Mixed Black Carbon Aggregates

Theory

This package computes the core phase shift parameter (\({\rho}\)BC) 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, \({\rho}\)BC is constrained by determining whether the measured mass absorption cross-section (MACBC,meas) is significantly less than that given by:

\({MAC_{BC,pred}=MAC_0\left (\frac{\lambda}{\lambda_0} \right)^{-AAE}\left[1+\frac{AC^{-B}\Gamma(B+1,C)}{C}-\frac{A\left(\frac{M_{tot}}{M_{BC}}\right)^{B}\left(\frac{M_{tot}}{M_{BC}}\right)^{-B}\Gamma\left(B+1,C\frac{M_{tot}}{M_{BC}}\right)}{C}\right]}\).

  • \({A=-1.189\pm0.029}\)

  • \({B=-0.674\pm0.006}\)

  • \({C=0.043\pm0.0007}\)

  • \({MAC_0=6.819\pm0.131}\)

  • \({AAE=1.231\pm0.005}\)

If MACBC,meas is within 10% of MACBC,pred, then \({\rho}\)BC can be constrained to 0 < \({\rho}\)BC < 1, but cannot be exactly calculated. If MACBC,meas is less than 90% of MACBC,pred, then \({\rho}\)BC 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 \({e_{[1,2,3,4]}}\).

  • \({d_1=5.679\pm0.027}\)

  • \({d_2=1.066\pm0.058}\)

  • \({d_3=0.264\pm0.010}\)

  • \({d_4=11.421\pm0.137}\)

  • \({e_1=2.440\pm0.017}\)

  • \({e_2=0.593\pm0.024}\)

  • \({e_3=0.418\pm0.020}\)

  • \({e_4=10.106\pm0.131}\)

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 \({\rho}\)BC 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 \({\left(\phi\right)}\) 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, \({\rho}\)BC is given by:

\({\rho_{BC}=\frac{4\pi R_g}{\lambda}\left|m_{eff}-1\right|}\),

where \({m_{eff}}\) 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

abs2shape_SP(diameter, coating, absorption, wavelength[, k_coat=0.0, abs_error=0.0, mode='MtotMbc', r_monomer=20, asDict=True, ReturnPlot=True, PlotPoint=True])

Black carbon mass-equivalent diameter, coating amount, and MACBC are input and morpholgy is inferred using the procedure outlined above. The particle mass is used to infer the number of monomers, assuming the density of black carbon is 1.8 g/cm3.

Parameters

diameterfloat

Black carbon mass-equivalent diameter with units of nm.

coating:

Coating amount with units matching that of the optional ‘mode’ input. Default is ratio of total particle mass to black carbon mass.

absorption:

MACBC with units of m2/g.

wavelengthfloat

The wavelength of incident light, in nanometers.

k_coatfloat

Imaginary part of coating refractive index.

abs_errorfloat, optional

Error associated with measurement of MACBC, in m2/g.

modestring, optional

  • ‘Mtot_Mbc’ : ratio of total particle mass to black carbon mass.

  • ‘Rbc’ : ratio of coating mass to black carbon mass.

  • ‘OC:EC’ : ratio of organic carbon mass to black carbon mass.

  • ‘percent_BC’ : percentage of total particle mass which is attributed to black carbon.

r_monomerfloat, optional

Radius of monomers, in nanometers.

asDictbool, optional

If true, returns dict of output variables.

ReturnPlotbool, optional

If true, returns figure and axes with morphology retrival plot.

PlotPointbool, optional

If true, shows measured particle on morphology retrival plot.

Returns

fig, axfigure, axes

Figure and axes with morphology retrival plot. If PlotPoint==True, then particle is shown on morphology retrieval plot.

massfloat

Mass of particle, in fg.

rho_lowerfloat

Lower limit of core phase shift parameter, based on average MACBC and MACBC errors.

rho_avgfloat

Average core phase shift parameter, based on average MACBC.

rho_upperfloat

Upper limit of core phase shift parameter, based on average MACBC and MACBC errors.

Forward function for single particle

shape2abs_SP(dp, coating, collapse, wavelength[, k_coat=0.0, mode='MtotMbc', r_monomer=20, asDict=True])

Black carbon mass-equivalent diameter, coating amount, and morphology are input and MACBC is calculated using the procedure outlined above.

Parameters

dpfloat

Black carbon mass-equivalent diameter with units of nm.

coatingfloat

Coating amount with units matching that of the optional ‘mode’ input. Default is ratio of total particle mass to black carbon mass.

collapsestring

  • ‘fresh’ : black carbon morphology matches fresh soot with fractal dimension of 1.8.

  • ‘partial’ : black carbon core has partially collapsed, fractal dimension of 2.5.

  • ‘full’ : black carbon core has fully collapsed, fractal dimension of 3.0.

wavelengthfloat

The wavelength of incident light, in nanometers.

k_coatfloat

  • Imaginary part of coating refractive index.

modestring, optional

  • ‘Mtot_Mbc’ : ratio of total particle mass to black carbon mass

  • ‘Rbc’ : ratio of coating mass to black carbon mass

  • ‘OC:EC’ : ratio of organic carbon mass to black carbon mass

  • ‘percent_BC’ : percentage of total particle mass which is attributed to black carbon.

r_monomerfloat, optional

Radius of monomers, in nanometers.

asDictbool, optional

If true, returns dict of output variables.

Returns

dpfloat

Mass-equivalent diameter of particle in nm.

coatingfloat

Amount of coating with same units as input.

MACfloat

MACBC with units of m2/g.

Inverse function for black carbon size distribution

abs2shape_SD(dpg, sigma_g, coating, absorption, wavelength[, k_coat=0.0, abs_error=0.0, mode='MtotMbc', r_monomer=20, asDict=True, ReturnPlot=True])

Black carbon mass-equivalent lognormal size distribution, coating amount, and MACBC are input and morpholgy is inferred using the procedure outlined above. The particle mass is used to infer the number of monomers, assuming the density of black carbon is 1.8 g/cm3.

Parameters

dpgfloat

Black carbon geometric mean mass-equivalent diameter of lognormal distribution with units of nm.

sigma_gfloat

Geometric standard deviation of black carbon lognormal size distribution.

coating:

Coating amount with units matching that of the optional ‘mode’ input. Default is ratio of total particle mass to black carbon mass.

absorption:

MACBC with units of m2/g.

wavelengthfloat

The wavelength of incident light, in nanometers.

k_coatfloat

  • Imaginary part of coating refractive index.

abs_errorfloat, optional

  • Error associated with measurement of MACBC, in m2/g.

modestring, optional

  • ‘Mtot_Mbc’ : ratio of total particle mass to black carbon mass.

  • ‘Rbc’ : ratio of coating mass to black carbon mass.

  • ‘OC:EC’ : ratio of organic carbon mass to black carbon mass.

  • ‘percent_BC’ : percentage of total particle mass which is attributed to black carbon.

r_monomerfloat, optional

Radius of monomers, in nanometers.

asDictbool, optional

If true, returns dict of output variables.

ReturnPlotbool, optional

If true, returns figure and axes with morphology retrival plot.

Returns

fig, axfigure, axes

If ReturnPlot==True, figure and axes with morphology retrival plot.

lower_massfloat

Average-standard deviation of mass of particles, in fg.

avg_massfloat

Average mass of particles, in fg.

upper_massfloat

Average+standard deviation of mass of particles, in fg.

rho_lowerfloat

Lower limit of core phase shift parameter, based on average MACBC and MACBC errors.

rho_avgfloat

Average core phase shift parameter, based on average MACBC.

rho_upperfloat

Upper limit of core phase shift parameter, based on average MACBC and MACBC errors.

Forward function for black carbon size distribution

shape2abs_SD(dpg, sigma_g, coating_avg, coating_stdev, collapse, wavelength[, k_coat=0.00, mode='MtotMbc', r_monomer=20, DataPoints=False, ShowPlots=True])

Black carbon mass-equivalent lognormal size distribution, coating distribution, and morphology are input and distribution of MACBC is calculated. Black carbon mass-equivalent diameter and coating amount are randomly sampled per the input distributions, and MACBC is calculated using the procedure outlined above.

Parameters

dpgfloat

Black carbon geometric mean mass-equivalent diameter of lognormal distribution with units of nm.

sigma_gfloat

Geometric standard deviation of black carbon lognormal size distribution

coating_avgfloat

Average value of coating amount, assuming a Gaussian normal distribution. Units should match that of the optional ‘mode’ input, default is ratio of total particle mass to black carbon mass.

coating_stdevfloat

Stabdard deviation of coating amount, assuming a Gaussian normal distribution.

collapsestring

  • ‘fresh’ : black carbon morphology matches fresh soot with fractal dimension of 1.8.

  • ‘partial’ : black carbon core has partially collapsed, fractal dimension of 2.5.

  • ‘full’ : black carbon core has fully collapsed, fractal dimension of 3.0.

wavelengthfloat

The wavelength of incident light, in nanometers.

k_coatfloat

  • Imaginary part of coating refractive index.

modestring, optional

  • ‘Mtot_Mbc’ : ratio of total particle mass to black carbon mass

  • ‘Rbc’ : ratio of coating mass to black carbon mass

  • ‘OC:EC’ : ratio of organic carbon mass to black carbon mass

  • ‘percent_BC’ : percentage of total particle mass which is attributed to black carbon.

r_monomerfloat, optional

Radius of monomers, in nanometers.

DataPointsbool, optional

If true, returns dict of output variables.

ShowPlotsbool, optional

If true, shows histograms of input parameters and calculated MACBC.

Returns

dpfloat

  • If DataPoints==True, mass-equivalent black carbon diameters used in calculations, in nm.

  • If DataPoints==False, average and standard deviation of mass-equivalent black carbon diameters used in calculations, in nm.

coatingfloat

  • If DataPoints==True, coating amounts used in calculations, with units matching those of ‘mode’ option.

  • If DataPoints==False, average and standard deviation of coating amounts used in calculations, with units matching those of ‘mode’ option.

MACfloat

  • If DataPoints==True, calculated MACBC values, in m2/g.

  • If DataPoints==False, average and standard deviation of calculated MACBC values, in m2/g.