Size Distributions• Many processes and properties depend on particle size

– Fall velocity– Brownian diffusion rate– CCN activity– Light scattering and absorption– Others

• There are a number of quantitative ways to represent the size distribution– Histogram– Number Distribution– Number Distribution function– Volume, area, mass distributions– Cumulative distributions– Statistics of size distributions: Median, mode, averages, moments…– Others

• We will review a number of these next

• Our primary goal is to explain the Number Distribution Function, which is the most widely used

Size DistributionsThe Histogram

NB = Number of size binsDi = Lower-bound particle diameter for bin i Di+1 = Upper-bound particle diameter for bin i Ni = concentration of particles in bin i (cm-3)

NiDi –Di+1

BNi . . . 2, 1,

Simplest form of distribution – Very instrument-based

Lots of structure at small sizesFew particles at largest sizes

Size DistributionsCumulative Properties from Histogram

NiDi –Di+1

BN

iit NN

1

Total Concentration

Total Surface Area

Total Volume

Total Mass

BN

iiiit NDDS

11

BN

iiiit NDDV

1

2/3

16

BN

iiiiit NDDM

1

2/3

16

i = density of aerosol substance in bin i

Note: We don’t have an “average” diameter for the bin – only the bin boundaries. Above I use the geometric mean.

Sometimes it makes sense to estimate where the particles are within the bin based on the concentrations of neighboring bins, and then calculate the effective mean diameter.

N

ii

NN

iig

AN

AA

1

/1

1

ln1

exp

Size DistributionsCumulative Distributions

NiDi –Di+1

i

jji NDN

11)(

1

11)(

i

jjjji NDDDS

1

1

2/3

16)(

i

jjjji NDDDV

1

1

2/3

16)(

i

jjjjji NDDDM

A Cumulative distribution gives the concentration (or some other property) of particles smaller than diameter Di

• Cumulative values are properties at bin boundaries, not bin centers!• They are monotonically increasing in size

• N(DNB+1) = Nt

•

• Different instruments should report the same function, just sampled differently

1

1

)(i

jji NDNCum. Concentration

Cum. Surface Area

Cum. Volume

Cum. Mass

)()( 1 iii DNDNN

Size DistributionsThe Number Distribution

ni = aerosol number distribution for bin iDi = Di+1–Di is the bin widthNi = niDi

ni = has units of (cm-3 m-1)

NiDi –Di+1 in

i

jji NDN

01)(

i

jjji DnDN

01)(

Area under the curve = total aerosol concentration, N

More uniform way to present instrument data

Instruments with different Di would produce very different histograms, but similar number distributions

Histogram Number distribution

Size DistributionsThe Number Distribution

Small bin width at small sizes leads to amplification of concentrations here relative to histogram

Size DistributionsThe Log Number Distribution

Aerosol distributions span orders of magnitude in size, and are often best shown as a function of log-diameter. Now, the area under curve is NOT equal to total concentration.

To remedy this, we can create a log number distribution (not shown above)

in

ii

i

i

ii DD

N

D

Nn

/loglog 1

0

Size DistributionsThe Number Distribution Function

Distributions are often represented in models or analytically, as continuous functions of diameter. This is as if we had an number distribution with perfectly precise resolution

in

dD

DdND

Dd

DdNDn p

)(303.2

log

)()(0

ii

i

DDp DD

NDn

ii

11

lim)(

This looks a lot like the definition of the derivative.If we use the cumulative distribution, we get…

ii

ii

DD DD

DNDN

dD

DdNDn

ii

1

1

1

lim)(

)(

We think of the number distribution function as the derivative with diameter of the cumulative distribution

When n(D) is plotted vs. D (NOT logD), then the area under the curve = total concentration

The log-diameter distribution is the derivative of the cumulative distribution with log of diameter

Size DistributionsOther Distribution Functions

Aerosol distributions span orders of magnitude in size, and are often best shown as a function of log-diameter. We must use the identity

This lowers the power of Dn in the functions above.

Note the “shifting of the peaks” from number area volume

dD

dNDn )(Number Distribution

Surface Distribution

Volume Distribution

Mass Distribution

)()( 2 DnDdD

dSDnS

)(6

)( 3 DnDdD

dVDnV

)()(6

)( 3 DnDDdD

dMDnM

)(303.2)(log0 DDnDn NN

Statistics of Size distributions

The “moments” will come in when you do area, volume distributions

2/1

11

1

BN

iiii

t

DDNN

DMean Diameter

Standard Deviation

Geometric Mean

nth moment

Histogram Discrete distribution Continuous dist.

2/1

11

1

BN

iiiii

t

DDDnN

D dDDDnN

Dt

)(1

BN

iiii

tD DDDN

N 1

22/11

1 dDDnDDN t

D )()(1 2

BN

iiii

tg DDN

ND

11ln

2

11exp

dDDnD

ND

tg )()ln(

1exp

2/

11

1nN

iiii

tn

B

DDNN

D

dDDnDN

D n

tn )(

1

More Statistics

In-class…

• Power-law distributions

• Log-normal distributions

• Properties of each