Quiet nucleation of atmospheric aerosol and intermediate ions Quiet nucleation of atmospheric aerosol and intermediate ions

Hannes.Tammet@ut.ee

Laboratory of Environmental Physics Institute of Physics, University of Tartu

Quiet nucleation of atmospheric aerosol and intermediate ions

Quiet nucleation of atmospheric aerosol and intermediate ions

15th Finnish-Estonian air ion and atmospheric aerosol workshop Hyytiälä 20110524

Sources of knowledge about growth and charging of nanoparticles

Kerminen, V.-M., and Kulmala, M.: Analytical formulae connecting the

“real” and the “apparent”

nucleation rate and the nuclei number

concentration for

atmospheric nucleation events, J. Aerosol Sci., 33, 609–622, 2002.

Tammet H. and

Kulmala M.: Simulation tool for atmospheric aerosol nucleation bursts, J. Aerosol Sci., 36: 173–196, 2005.

Verheggen, B. and Mozurkewich, M.:

An inverse modeling procedure to determine particle growth and

nucleation rates from measured aerosol size distributions, Atmos. Chem. Phys.,

Long quiet periods may happen between burst events. The particles of secondary

aerosol are mortal and would disappear when no supply. How they are regenerated?

Many research papers are written about burst events of atmospheric aerosol nucleation. Not so much

about nucleation during quiet periods between the burst events. Why?

A reason: concentration of intermediate ions sufficiently exceeds the noise level of

common instruments

only during burst events.

Extra noise as in BSMA, lowest contour of 100 cm^–3

Measurement with SIGMA, noise from BSMA

Extra noise as in BSMA, lowest contour of 20 cm^–3

Measurement with SIGMA, noise from BSMA

Measurement with SIGMA, lowest contour of 20 cm^–3

Measurement with SIGMA without extra noise

Low noise instrument

SIGMA:

Tammet, H. (2011) Symmetric inclined grid mobility analyzer for the measurement of charged clusters and fine nanoparticles in atmospheric air. Aerosol Sci. Technol., 45, 468–479.

http://dx.doi.org/10.1080/02786826.2010.546818

Air inlet

Air outlet through Repelli

ng electro

de Attracti

ng electro

des Sheath

air filter

Repelli ng electro

de

Sheath air filter

Attracti ng electrod

es

Repellin g electrod

e

Shield electrod

e Inlet

gate

Air ion trajecto

ry

Electrometricfilter for positive ions Filter batteries Electrometricfilter for negative ions Filter batteries

Shield electro

de

Repellin g electrod

e WORSE HALF OF

MEASUREMENTS

BETTER HALF OF MEASUREMENTS

NOISE

(10 min cycles)

Charged nanoparticles are air ions

Particles and cluster ions

Ion or particle

Molecule or growth unit

Quantum retardation of sticking:

internal enegy levels of a cluster will not be excited and the impact

is elastic-specular

Particle or molecular cluster ? to grow, or not to grow ?

does not grow, molecules will

bounce back

grows,

molecules will stick

1.5 or 1.6 nm

CLUSTER PARTICLE

Introduction to modeling

An aim is to make the mathematical model easy to understand.

GDE is not used and equations will be derived from scratch.

Empiric information is coming from measurements of

intermediate ions. Quiet periods are characterized by very low concentration of nanoparticles and nearly steady state of

aerosol parameters. This allows to accept assumptions:

 the size range is restricted with d = 1.5 – 7.5 nm,

 the nanoparticles can be neutral or singly charged,

 attachment flux of ions does not depend on polarity,

 nanoparticle-nanoparticle coagulation is insignificant,

 all processes are in the steady state.

Extra comment:





























 

 









 c c

N c

c c I

N c

c c I

0 0

0

0

 









Assumption: all surfaces are away Law of balance:

genesis = destruction

Flux of ions

to particles

Particle growth through a diameter margin

dd = GR(d) dt d

_o

d

J = GR n

Symbols:

diameter crossing rate,

–

apparent nucleation rate, transit rate, cm^–3s^–1

dt d dN

J ( ) 

→ dN / dt = GR n dN = n dd = n GR dt d –

particle diameter (d =

d

_p^{), nm,}

dd d d dN

n ( )

)

( 

density of concentration distribution, cm^–3nm^–1

–

dt d dd

GR ( )  –

growth rate, nm s^–1,

N(d) –

number concentration of particles in diameter range of

0...d

^{, cm–3,}

(a well known equation)

NB: particle growth rate may essentially differ from the population growth rate.

c –

concentration of small ions, cm^–3

Particle growth through a diameter interval

d

_a

= d – h/2 d

_b

= d + h/2

Inflow Leakage Outflow

d

Extrasource

(analog: classic problem about water tank and pipes)

Steady state balance:

Inflow + Extrasource – Outflow – Leakage = 0

or

Outflow = Inflow + Extrasource – Leakage

(GDE : Inflow + Extrasource – Outflow – Leakage = Increment)

Equation of steady state balance

Inflow J(d

_a

) = GR(d

_a

) n(d

_a

), Outflow J(d

_b

) = GR(d

_b

) n(d

_b

), Leakage = , 

d^d_a^b

S ( d ) n ( d ) dd Extrasource = 

d^d_a^b

E ( d ) dd



 ^





^b

a b

a

d d d

a d a

b

b

n d GR d n d E d dd S d n d dd

d

GR ( ) ( ) ( ) ( ) ( ) ( ) ( )

General steady state balance equation (integral form):

d

_a

= d – h/2 d

_b

= d + h/2

Inflow Leakage Outflow

d

Extrasource

Outflow = Inflow + Extrasource – Leakage dt

d dn d

d n

S ( )

) ( ) 1

(  

relative depletion rate or

sink

of particles s^–1,

(incl. CoagS as a component)

–

state = CST^Charging

Comparison with Lehtinen et al. (2007)



 ^





^b

a b

a

d d d

a d a

b

b

n d GR d n d E d dd S d n d dd

d

GR ( ) ( ) ( ) ( ) ( ) ( ) ( )

Balance equation:

Substitute GR n with J, assume E = 0, consider d_a = const & d_b = argument:







^d

const

S d n d dd const

d

J ( ) ( ) ( )

Equation (4) in Lehtinen et al. (2007):

J GR

d CoagS dd

dJ

_p

p

)

 (



Differences: different notations of sink and two simplifications

E = 0 &

additional components of sink are neglected, substitute

n

^with

J/GR

^:

) ) (

( ) ( )

( J d

d GR

d S dd

d

dJ   )

( ) ) (

( S d n d

dd d

dJ  

calculate derivative:

Sink of nanoparticles on background aerosol

The background aerosol can be replaced with an amount of monodisperse particles in simple numerical models. The diameter of particles is

assumed d_bkg = 200 nm that is close to the maximum in the distribution of coagulation sink. The concentration N_bkg can be roughly estimated

according to the sink of small ions. The coagulation sink is calculated as

S

_bkg

= K(d, d

_bkg

) N

_bkg

The coagulation coefficient

K (d, d

_bkg

)

depends on the nanoparticle charge and the sink could be specified according to the charge.

Notations: neutral nanoparticles – index 0, charged nanoparticles – index 1.

Sink of neutral nanoparticles

S

_bkg0

= K

₀

(d, d

_bkg

) N

_bkg

Sink of charged nanoparticles

S

_bkg1

= K

₁

(d, d

_bkg

) N

_bkg

Difference is small and neglecting of the charge would not induce large errors.

Charging and discharging of particles

0 +

–

+

–

+

–  ₁

 ₀

 ₁  ₀

ion-to-neutral-particle attachment coefficient

(a special case of

ion-to-opposite-charged-particle attachment coefficient

or the recombination coefficient

TWO

TWO ONE ONE

Sink of nanoparticles due to the small air ions

When a neutral particle encounters a small air ion then it converts to a charged particle and number of neutral

particles is decreased. We expect concentrations of positive and negative ions c equal and the sink is

S

_ion0

= 2 β

_o

(d) c

A charged particle can be neutralized with an ion of

opposite polarity. The sink of charged nanoparticles on small ions is

S

_ion1

= β

₁

(d) c

Extrasource of nanoparticles

Some amount of neutral particles appear as a result of

recombination the charged nanoparticles of the same size with small ions of opposite polarity:

E

₀

(d) = 2 β

₁

(d) c n

₁

(d)

The ion attachment source of charged particles of one polarity is

E

₁

(d) = β

₀

(d) c n

₀

(d)

E

₀ is usually a minor component in the balance of neutral particles while

E

₁ is an important component in the balance of charged particles.

If the rate ion-induced nucleation is zero, then all charged

nanoparticles are coming from the extrasource.

Numerical solving of balance equations



 ^





^b

a b

a

d d d

a d a

b

b

n d GR d n d E d dd S d n d dd

d

GR ( ) ( ) ( ) ( ) ( ) ( ) ( )

b ab

a a

b ab

d

d^b

Y d dd Y d d d d d d

a







 ^{( )}  ^{( )( ) where}

2

) ( )

) (

(

_ab

Y d

^a

Y d

^b

d

Y  

A small step can be made using the midpoint rule and few iterations:

) ( )

( )

1 Y d

_b

 Y d

_a

b

a

d

d )

...

3, 2,

The first mean value theorem states for any continuous

Y = Y(d):

d_a d_b d_a d_b d_a d_b d_a

d_a d_b d_a d_b d_a d_b

Step by step: d

GR

^or

n

can be computed step by step moving upwards or downwards

a

b

d

d

h   G

_0a

 GR

₀

( d

_a

) G

_1b

 GR

₁

( d

_b

)

Abbreviations:

, , , etc.

Itemized numerical model of steady state growth of nanometer particles

h n

c S

h cn

n G

n

G

_{0b 0b}



_{0a 0a}

 2 

_{1ab 1ab}

 (

_bkg0ab

 2 

_0ab

)

_0ab

h n

c S

h cn

n G n

G

_{1b 1b}



_{1a 1a}

 

_{0ab 0ab}

 (

_bkg1ab

 

_1ab

)

_1ab

Equations:

Example of a specific problem:

Given –

nucleation rates

J

₀ ^and

J

₁ or values of distribution functions

n

₀ ^and

n

₁ at first diameter, and growth rates

GR

₀ at all sizes.

) , (

) ,

(

_{1 0}

1

0

K d d G K d d

G

_u



_u

Two degrees of freedom

Growth rates or values of a distribution function can be computed step by step starting form four initial values of G

₀

, G

₁

, n

₀

, and n

₁

. If the distribution of intermediate ions is measured then one initial value (n

₁

) is known. The ratio G

₀

/G

₁

is always known and the

number of unknown initial values is reduced to two. These two may be presented by G

₀

and n

₀

at some point or by any pair of parameters that are unambigyosly related with G

₀

and n

₀

.

Some examples of necessary initial information:

 growth rate at a certain size and a nucleation rate,

 growth rates at two different sizes,

 ratio of growth rates for two sizes and a nucleation rate.

 ratio of growth rates for two sizes

and value of n

₀

at a certain size.

Test data

characteristic of quiet nucleation

Measurements with the SIGMA in the city of Tartu (April 2010 – February 2011) were sorted by the

instrumental noise and the worse half of data was

deleted. Next the data were sorted by concentration of intermediate ions and the half of measurements with high concentration was deleted. Remained 16240 five-minute records are expected to belong to the quiet phase of nucleation.

d : nm dN₁/dd : cm^–3nm^–1

(average of 16240 records of both + and– intermediate ions)

N noise

OK

Fitting the measurements by means of the numerical model

J₀ = 5.0 cm^–3s^–1,

J₁ = 0.00133 cm^–3s^–1, d_bkg = 200 nm,

N_bkg = 2224 cm^–3.

nm S_bkg:1/h GR CST 1.5 7.38 1.26 0.402 2.0 4.48 2.85 0.686 2.5 3.11 3.74 0.693 3.0 2.33 3.73 0.679 3.5 1.82 3.55 0.680 4.0 1.47 3.54 0.693 4.5 1.22 3.55 0.711 5.0 1.03 3.56 0.751 5.5 0.88 3.59 0.770 6.0 0.76 3.64 0.790 6.5 0.67 3.70 0.809

d : nm dN/dd : cm^–3nm^–1

(average of + and– ions)

N_bkg is estimated according to the small ion depletion. J₀ and J₁ are chosen by method of trial and error.

NB: the method does not provide unambiguous results.

Alternative approach

Use any numeric model of nanometer aerosol dynamics, decide

steady state conditions, adjust growth parameters, and integrate over a long period at least of few hours Example (simulation tool)

J₀= 13 cm^-3s^-1, J₁ = 0.07 cm^-3s^-1 d = 1.5 2.5 4.5 6.5 nm GR = 0.8 3.6 3.5 3.8 nm/h dN₁/dd : cm^–3nm^–1

Automated fitting of intermediate ion measurements

Given: measurements of intermediate ions

n

₁

(d)

on a set of diameters

(d

₁

, d

₂

,…, d

_m

)

Assume and iterate 2…5 times:

ⁿ

_0b

^{ n}

_0a

 ⁿ

_1b

^{/ n}

_1a



 ^{G n c n h S c n h} 

G n

_{a a ab bkg ab}

b

b 1 1 0 0 1 1 1

1

1

 1    (   )

1 1

0 0

) , (

) ,

( G

d d K

d d G K

u



u

 ^{n G cn h S c n h} 

n G

_{a a ab bkg ab}

b

b 0 0 1 1 0 0 0

0

0

 1  2   (  2  )

2 2 ,

1 1 1

0

0 0 a b

b ab ab a

n n n

n

n  n   

Fitting the measurements adjusting the growth rate

J₀ = 5 cm^–3s^–1, J₁ = 0.013 cm^–3s^–1, d_bkg = 200 nm,

N_bkg = 2224 cm^–3. nm Sb:1/h CST 1.5 7.38 0.401 2.0 4.48 0.682 2.5 3.11 0.686 3.0 2.33 0.669 3.5 1.82 0.668 4.0 1.47 0.678 4.5 1.22 0.692 5.0 1.03 0.728 5.5 0.88 0.742 6.0 0.76 0.758 6.5 0.67 0.772

d : nm GR : nm h^–1

WARNING: the solution is ambiguous. Different

assumptions about

J

₀ ^and

J

₁ are possible

Restrictions on the free parameters

(when fitting the test distribution)

PRIOR INFORMATION?

ANALOG OF REGULARIZATION?

3 variants of GR₀(d₁) 3 variants of J₀(d₁) 3 variants of GR₀(d₁)

3 variants of J₀(d₁)

Effect of guess about J

₀

(1.5 nm)

while required relation is

GR

₀

(3 nm) = GR

₀

(7 nm)

(fitting the test distribution)

Sink, growth rate and transit rate compared with Lehtinen et al. (2007)

d : nm S : 1/h,

GR₀ : nm/h J₀(d) : cm^–3s^–1

Conclusions



SIGMA provides low-noise measurements of intermediate ions.



The integral equation of steady state balance derived in a straigth-

forward way enables to design correct numerical algorithms with ease.



Measurement of intermediate ions is not sufficient to get

unambiguous solution of balance equation. Additionally the values of two scalar parameters are required. Some combinations are:

 growth rate at a certain size and a value of

n

for neutral particles,  growth rates at two different sizes,

 ratio of growth rates at two different sizes and a nucleation rate.



The nucleation of 3 nm neutral particles at Tartu about J = 0.5 cm^–3s^–1

is considerable contribution into the atmospheric aerosol generation.



The nucleation rate of 3 nm charged particles at Tartu about 0.002…0.005 cm^–3s^–1 indicates the minor contribution

of ion-induced nucleation during periods of quiet nucleation.



The growth rate of fine nanometer particles during quiet phase of aerosol nucleation at Tartu is estimated about 3…9 nm/h.

for

Attention Thank

You

Thank

You