Size Distributions

Notation

By convention blackboard bold characters denote size distributions

Info

Blackboard bold font: 𝕒, 𝕓, 𝕔, 𝕕, 𝕖, 𝕗, 𝕘, ...

Histogram Representation

The size distribution is represented as a histogram. It is a composite data type SizeDistribution that has the fields Dp, De, ΔlnD, S, and N.

𝕟.Dp: Geometric midpoint diameters
𝕟.De: Bin edge diameters
𝕟.ΔlnD: Log bin spacing, ΔlnD = ln(Dup/Dlow)
𝕟.S: Spectral density
𝕟.N: Number concentration in the bin

The size distribution can be constructed various ways. The easiest is to use one of the constructor functions. For example, the lognormal function creates a lognormal size distribution. The following example creates a lognormal size distribution with number concentration equals 200 cm-3, geometric mode diameter of 80 nm, geometric standard deviation of 1.2, with 10 size bins between 30 and 300 nm. The result is placed in a DataFrame for display purposes. The r function is to round the results for clarity. The output illustrates that the SizeDistribution type is simply a histagram table.

r(x) = round.(Int,x)    # Function to round and convert to Int
𝕟 = lognormal([[200, 80, 1.2]]; d1 = 30.0, d2 = 300.0, bins = 10);
DataFrame(
    Dlow = r(𝕟.De[1:end-1]),
    Dup = r(𝕟.De[2:end]),
    ΔlnD = round.(𝕟.ΔlnD, digits = 2),
    Dp = r(𝕟.Dp),
    S = r(𝕟.S),
    N = r(𝕟.N),
)

10 rows × 6 columns

	Dlow	Dup	ΔlnD	Dp	S	N
	Int64	Int64	Float64	Int64	Int64	Int64
1	30	38	0.23	34	0	0
2	38	48	0.23	42	1	0
3	48	60	0.23	53	37	9
4	60	75	0.23	67	276	64
5	75	95	0.23	85	418	96
6	95	119	0.23	106	128	30
7	119	150	0.23	134	8	2
8	150	189	0.23	169	0	0
9	189	238	0.23	212	0	0
10	238	300	0.23	267	0	0

Manipulating Size Distributions

Size distributions can be intuitively manipulated through operators. For example, the sum of two size distributions (𝕩 = 𝕟₁ + 𝕟₂) is the superposition.

# Example addition of size distributions
𝕟₁ = lognormal([[120, 90, 1.20]]; d1 = 10.0, d2 = 1000.0, bins = 256)   # size distribution
𝕟₂ = lognormal([[90, 140, 1.15]]; d1 = 20.0, d2 = 800.0, bins = 256)    # size distribution
𝕩 = 𝕟₁ + 𝕟₂

The package implements a list of Operators for size distribution manipulation. Check out the Tutorial Session 1 and/or Notebook S3 in the Notebooks section for visualizations.