Low temperature growth of sub 10 nm particles by ammonium nitrate condensation

Low temperature growth of sub 10 nm particles by

ammonium nitrate condensation

Neil M. Donahue,

Mao Xiao,

Ruby Marten,

Mingyi Wang,

Weimeng Kong,

Meredith Schervish,

Qing Ye,

Victoria Hofbauer,

Lubna Dada,

Jonathan Duplissy,

Henning Finkenzeller,

Hamish Gordon,

Jasper Kirkby,

Houssni Lamkaddam,

Vladimir Makhmutov,

Maxim Philippov,

Birte Rörup,

Rainer Volkamer,

Dongyu Wang,

Stefan K. Weber,

Richard C. Flagan,

Dominik Stolzenburg

and Imad El Hadad

Co-condensation of nitric acid and ammonia vapors to form ammonium nitrate transforms from a fully

semi-volatile behavior when it is relatively warm (273 K and above, typical of the seasonal planetary

boundary layer) into e

ectively non-volatile and irreversible uptake for the limiting vapor when it is cold

(well below 273 K, typical of the upper troposphere and occasionally the wintertime boundary layer). This

causes the system to switch in character from the one governed by semi-volatile equilibrium (how it is

usually portrayed) to the one governed by irreversible reactive uptake to even the smallest particles.

Uptake involves an activation diameter, which can be as small as 1 nm for typical vapor concentrations,

and subsequent growth rates can be very high, exceeding 1000 nm h

−

. In addition to this somewhat

surprising behavior, the system provides an exemplary case for semi-volatile reactive uptake within the

context of volatility and saturation ratios.

Environmental signi



cance

Ammonium nitrate is a major constituent of submicron aerosol particles, and with rapidly decreasing sulfur emissions, it is likely to become even mor

important to submicron aerosol composition in the future. Though it is o



en treated with equilibrium thermodynamics in chemical transport models, when it

is very cold, the condensation of the less abundant (rate limiting) vapor, either ammonia or nitric acid, becomes irreversible. This can drive growth

of extremely

small particles and may be a prominent growth mechanism in the parts of the upper troposphere such as the Asian tropopause aerosol layer, where nitrate

known to be a major aerosol constituent.

1 Introduction

In atmospheric aerosols, ammonium nitrate (NH

$

)isin

many ways a quintessential example of semi-volatile reactive

uptake, with strong temperature and humidity dependent par-

titioning between the gas-phase acid

–

base pair and the

condensed-phase (aerosol) crystalline or dissolved salt.

Þþ

HONO

$

The equilibrium and volatility are a

ected by both water

activity (for an aqueous solution) and of course temperature (for

any aerosol phase). Both ammonia and nitric acid are by

themselves highly volatile; nitric acid has a saturation vapor

pressure of 117 Pa at a triple point of 235 K,

while ammonia has

a saturation vapor pressure of 6063 Pa at a triple point of 195 K.

Consequently, by themselves, neither would be found in the

condensed phase under any realistic conditions of Earth's

atmosphere. It is only the proton transfer reaction to make the

ionized salt solution that renders them at least semi volatile.

The system thus serves as a canonical example of a condensed-

phase reaction (proton transfer) that drives volatility.

Carnegie Mellon University, Department of Chemistry, Pittsburgh, PA, USA. E-mail:

nmd@andrew.cmu.edu; Tel: +1 412 268-4415

Laboratory for Atmospheric Chemistry, Paul Scherrer Institute, Villigen, Switzerland

University of California, Irvine Department of Chemistry, Irvine, CA, USA

University of Chicago Department of the Geophysical Sciences, Chicago, IL, USA

Technical University of Vienna, Vienna, Austria

Institute, for Atmospheric and Environmental Sciences, Goethe University Frankfurt,

Altenhöferallee 1, 60438 Frankfurt am Main, Germany

Institute for Atmospheric and Earth System Research/Physics, Faculty of Science,

University of Helsinki, 00014 Helsinki, Finland

Division of Chemistry and Chemical Engineering, California Institute of Technology,

Pasadena, CA 91125, USA

Department of Chemistry, University of Colorado, Boulder, CO 80305, USA

CERN, Geneva, CH-1211, Switzerland

P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991 Moscow,

Russia

Cite this:

Environ. Sci.: Atmos.

,2025,

Received 17th August 2024

Accepted 19th October 2024

DOI: 10.1039/d4ea00117f

rsc.li/esatmospheres

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Because the individual constituents are so volatile and the

equilibrium partitioning between the vapor and aerosol phases

is so sensitive to temperature, most treatments of nitrate par-

titioning assume that (submicron) particles equilibrate rapidly,

though supermicron particles (o



en with di

erent cations)

require a dynamical treatment.

Because of this, ammonium

nitrate has rarely been regarded as important to nucleation and

growth of nanoparticles. However, when it is cold enough and

when there is a su

ffi

cient quantity (production or emission rate)

of ammonia and nitric acid, ammonium nitrate will participate

in the growth of extremely small particles and eventually

nucleation itself.

–

Our objective here is to explore the transi-

tion of ammonium nitrate partitioning from an essentially

semi-volatile nature to an e

ectively non-volatile (irreversible in

the limiting vapor) behavior.

2 Setup

We are interested in the transition of ammonium nitrate par-

titioning from being qualitatively

“

semi volatile

”

at relatively

high temperature to qualitatively

“

irreversible

”

at lower

temperature. In this case the individual constituents remain

volatile under all relevant atmospheric conditions, but the

equilibrium reaction to an e

ectively non-volatile salt becomes

ectively irreversible and ultimately rate-limited by uptake of

the limiting vapor. Because we are interested in the low

temperature behavior, here we shall consider only the solid,

crystalline phase of ammonium nitrate. We need to consider

three fully reversible processes:

(v)

(s)

(R1)

HONO

(v)

HONO

(s)

(R2)

(s) + HONO

(s)

$

(s)

(R3)

A full treatment, especially with other ions and for aqueous

solutions, would break reaction R3 into its elementary steps as

well.

HONO

(s)

−

(s) + H

+NO

−

$

(s)

However, as we are considering pure, solid ammonium

nitrate we omit these reactions here.

The experimental evidence motivating this analysis is

observations from the CERN CLOUD experiment demonstrating

that systems containing mixtures of sulfuric acid, nitric acid,

and ammonia (and in some cases oxidized organics) show clear

evidence for nucleation followed by activation and very rapid

growth (20

–

2000 nm h

−

) of small particles with 1.7 nm <

15 nm. The experiments were designed to mimic conditions of

a wintertime megacity and were thus carried out at 263 K and

278 K, with 1 ppbv <

< 3 ppbv and 10 pptv <

HONO

< 1000

pptv. The observed activation diameters and growth rates are

shown in Fig. 1.

Both the activation diameters and the growth

rates showed an empirical relationship to the product

HONO

$

. This is expected for activation as it is fundamentally

related to the saturation ratio of the vapors, and for this second-

order process the saturation ratio involves the product of the

vapor concentrations (activities). However, the quadratic

growth-rate dependence is more of a surprise, as one might

expect the growth rate to depend only on the concentration of

the limiting vapor to an expected stoichiometric uptake.

3 Thermodynamics

For our calculations we need to know the saturation vapor

pressure of ammonia,

;

the saturation vapor pressure,

;

of nitric acid,

HONO

;

and the equilibrium constant for

ammonium nitrate formation in a condensed phase containing

pure saturated ammonium nitrate,

. From this we can

determine activities,

, and the reactive activity product in the

vapor (v) and suspended (s) phases. The gas-phase activity at

pressure,

, and temperature,

, is the saturation ratio with

Fig. 1

Activation diameter (top) and activated growth rate (bottom) of

particles

via

ammonium nitrate condensation observed in the CERN

CLOUD experiment. Both are proportional to the product of nitric acid

and ammonia vapors, and both shift by roughly a factor of 10 at 263 K

(green circles)

vs.

278 K (purple squares). Growth rates can exceed

1000 nm h

−

with activation diameters as small as 1.5 nm.

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respect to partial pressure (

), vapor concentration (

), or vapor

mole fraction (mixing ratio,

HONO

(1)

The condensed-phase (suspended) activity is the product of

a fraction (mole,

; mass,

, and volume) and the appro-

priate activity coe

ffi

cient (mole,

; mass,

=

=

(2)

In practice, values are known for the overall equilibrium

between the condensed phase and the vapor, giving the vapor

activity product,

but it is informative to separate out the gas-

particle partitioning of the unreacted vapors and the particle-

phase proton-transfer reaction as we have done in order to

illustrate the overall process. All of the processes are

“

func-

tionally

”

reversible, making this a true semi-volatile system. We

need to know the physical and thermodynamic properties of all

constituents to describe the dynamics. Ultimately the accuracy

for this system depends on the overall (vapor to salt) equilib-

rium and so we ensure that the individual vapor pressures and

condensed-phase equilibrium constant are consistent with the

more accurate overall equilibrium constant.

3.1 Ammonia

Ammonia is extremely volatile. The molar mass is

=

17.03026

×

−

kg mol

−

Its triple point properties are

;

195

48 K

;

6063 Pa

;

733

74 kg m

3.1.1 NH

functions.

The density and saturation vapor

pressure of ammonia were reported by Haar and Gallagher at

NIST (NBS) in the 1970s.

¼½

636

kg m

atm

3684

7798 K

428787

02893289

4798128

2219845

or since 1 atm

=

1.01325

×

Pa,

/Pa

=

31.9549. This all

gives the vapor pressure plot under atmospheric conditions

shown in Fig. 2. That is such an enormous value it is worth

considering the gas-phase activity of a reference level of

ammonia. For this we shall consider a mole fraction of 1 ppbv,

meaning a partial pressure of

=

−

Pa. Fig. 3 shows the

gas-phase activity

of 1 ppbv of ammonia.

Even at 200 K it is 10

−

and at 280 K it is just above 10

−

. The

gas-phase activity of 1 ppbv of ammonia at +5 °C is 2.0

×

−

while at

−

10 °C it is 3.5

×

−

3.2 Nitric acid

Nitric acid is also quite volatile, with a molar mass

HONO

=

63.012

×

−

kg mol

−

Combining values from Mozurkewich

and the saturation

vapor pressure from Duisman.

;

HONO

235 K

;

HONO

117 Pa

;

HONO

1513 kg m

Fig. 2

Saturation vapor pressure of ammonia over a pure liquid.

Fig. 3

Saturation ratio (gas-phase activity) of 1 ppbv of ammonia over

a pure liquid.

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3.2.1 HONO

functions.

The density and saturation vapor

pressure of nitric acid were reported by Duisman at Union

Carbide.

HONO

1510 kg m

HONO

torr

230

Þþ

1486

238

61628

HONO

torr

230

273

Þþ

15 K

Þþ

HONO

15 K

Þþ

8928

5091

Nitric acid has a lower vapor pressure than ammonia but not

by all that much, as we see in Fig. 4. Thus, again for a reference

mixing ratio of 1 ppbv, the gas-phase activities in Fig. 5 are

about 2

×

−

at 280 K and 4

×

−

at 200 K (over a super-

cooled liquid solution, not nitric acid trihydrate or anything like

that). The gas-phase activity of 1 ppbv of ammonia at +5 °C is 2.1

×

−

while at

−

10 °C it is 3.2

×

−

3.3 Ammonium nitrate

Finally, ammonium nitrate has the following properties:

$

044

kg mol

$

1725 kg m

Given the density of ammonium nitrate we can easily

calculate the number of NH

$

per particle

vs.

diameter, as

shown in Fig. 6.

3.3.1 Equilibrium.

Thermodynamic data are known for

ammonium nitrate equilibrium directly between the vapor and

condensed phase

(v) + HONO

(v)

$

(s)

However, we can consider the equilibrium of reaction R3,

where in terms of activity the equilibrium can be written as

$

$

HONO

x

$

HONO

Here, we make use of the negligibly small activities of the

undissociated molecules in the suspended particle phase at

equilibrium with the vapor partial pressures we have already

calculated. For our purposes, over a



at surface the gas-particle

system is at equilibrium for 100 pptv of ammonia and about 2

ppbv of nitric acid at +5 °C and about 20 pptv of nitric acid at

−

10 °C.

The overall equilibrium between vapor and a saturated

condensed phase of ammonium nitrate can be expressed

via

a dissociation constant for

$

(v) + HONO

(v)

Fig. 4

Saturation vapor pressure of nitric acid over a pure liquid.

Fig. 5

Saturation ratio (gas-phase activity) of 1 ppbv of nitric acid over

a pure liquid.

Fig. 6

Molecules per particle of pure ammonium nitrate.

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NHO

$

HONO

$

NHO

Þ¼

NHO

Mozurkewich

gives an overall dissociation constant for

solid-vapor (in nb

=

−

)of

118

24 084

025 ln

100

24 084

025 ln

An e

ective Arrhenius activation energy for the equilibrium

coe

ffi

cient is 18 150 K; it is extremely temperature dependent.

It is important to emphasize again that this overall equilib-

rium is well constrained, and the individual condensed-phase

equilibrium coe

ffi

cient and the vapor pressures only provide

useful insights. Fig. 7 shows that most of the temperature

dependence comes from the ammonium nitrate dissociation,

which spans 18 orders of magnitude over the atmospheric

temperature range (compared w

ith 2 or 3 decades for the vapor

pressures). This is why the potential for relatively rapid growth

(rather than equilibrium ammonium nitrate formation in

larger particles) is con



ned to low temperature regions of the

atmosphere.

4 Microphysics

With the thermodynamics (vapor pressures and equilibrium

coe

ffi

cients) known, we can now consider the microphysics of

collisions. We will use a microphysical formulation grounded in

the kinetic regime of molecular (and particle) collisions, rather

than the bulk



uid continuum regime of di

usion to and from

particles. This is because we are most interested in condensa-

tion to particles with 2

(

100 nm, where the Knudsen

number Kn

1 and microphysics are more kinetic than not.

The net condensation



ux of a species with vapor concen-

tration

and saturation concentration

a suspension of identical particles

with number concentration

and (physical) diameter

depends fundamentally on the

collision cross section,

, the collision speed,

;

and the

thermodynamic excess concentration,

This can also be

expressed in terms of the (e

ective spherical) particle surface

area,

and the (e

ective perpendicular) collision speed of

molecules with that surface,

, but with several modifying

factors that we de



ne below.

Note that the diameter is the physical diameter, not the

mobility diameter (which is typically

mob

x

+ 0.3 nm).

The



ux of the condensing species per unit particle surface

area is

;

(3)

This depends on the collision speed, the vapor concentra-

tion, and an uptake coe

ffi

cient,

, that in turn depends on the

activities in both phases as well as a Kelvin term,

for

curvature e

ects over very small particles. The



ux is either

number per unit area or mass per unit area depending on the

concentration units.

The collision speed,

, is the e

ective line-of-centers

velocity of each vapor toward the particle surface, accounting

for enhancements (

i.e.

van der Waals, vapor size, reduced mass)

and limitations (

i.e.

accommodation, gas-phase di

usion near

the particle). It is derived from the average molecular speed,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

van der Waals interactions between the

vapor and particle (charge-dipole, dipole-induced dipole, and

induced dipole-induced dipole) can be represented as an

Fig. 7

Equilibrium constant for ammonium nitrate formation in the

condensed phase.

;

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

cross section

;

|fflfflffl{zfflfflffl}

speed

;

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

excess conc

;

|fflfflfflffl{zfflfflfflffl}

area

;

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

collision speed

;

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

excess conc

;

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

flux per unit surface area

;

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enhancement over hard-sphere collisions,

=

(

), where

is the Hamaker constant in joule.

12,13

The non-zero size of

the vapor is accounted for by

=

(

, and the non-



nite mass of small clusters gives a reduced-mass correction

;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The mass accommodation coe

ffi

cient,

is likely 1.

Gas-phase di

usion limitations emerging

for small Knudsen numbers are given by

10,15

The excess concentration, sometimes also known as the

thermodynamic driving force, can be written as

;

(4)

The Kelvin term

scales with a

“

Kelvin diameter

”

for

decadal change,

K10

. We use base-10 because it is a little easier

to think in decades; for particles with

=

K10

, condensed-

phase activity is increased by a factor of 10.

log

;

K10

.

K10

434

;

(5)

This relates the bulk properties (

) to the molecular

properties of clusters

via

the correspondence principle

(potentially with higher-order terms). It is perhaps not widely

recognized how small

K10

actually is; it is the diameter where

the e

ective saturation concentration increases by 1 decade,

and for typical surface tensions and densities,

K10

x

–

5nm.

Still, for

=

1nm,

=

–

, which is why, in classical

nucleation theory, very high supersaturations can be required

to surmount the critical cluster. This is crucial to the activa-

tion behavior observed in ammonium-nitrate condensation in

Fig. 1.

We can express the excess concentration as the product of

the vapor concentration and an uptake coe

ffi

cient,

(the

fraction of collisions resulting in uptake), de



ned in eqn (3). We

can re-write this in terms of the saturation ratio (including

curvature),

, and the excess saturation ratio,

=

−

;

(6)

The uptake coe

ffi

cient is simply the fractional excess satu-

ration ratio over a small, curved particle: the ratio of the

“

Kelvin

adjusted

”

suspended-phase activity,

, to the equilibrium

suspended-phase activity, which would be

. For example, for

water vapor during cloud droplet activation, the fractional

excess saturation ratio is typically 0.002

–

0.01, so between 1%

and 0.2% of water vapor collisions are taken up (assuming

a mass accommodation coe

ffi

cient,

=

1). If the system is at

equilibrium,

=

0, whereas for kinetic uptake on every

collision with the surface,

=

1. The uptake coe

ffi

cient is thus

just the fractional excess saturation ratio (at the surface

–

neglect any condensed phase di

usion limitations, which in

any event are irrelevant for this pure model system).

The growth rate,

, is proportional to the



ux per unit

surface area,

v,s

. Assuming a spherical particle (or a spherical

equivalent

), the growth rate due to condensation of a given

species is

;

volume

;

mass volume

(7)

The factor of 2 is because growth extends the diameter by

extending the radius symmetrically about the sphere. The

molecular volume in the particle is

=

; we retain the

{

}

“

species within particle

”

designation to account for the

form of the species within the particle. Speci



cally in the case

we are considering here, one mole of either ammonia or nitric

acid condensation will result in one mole of ammonium nitrate,

and so the appropriate mass and density are those of ammo-

nium nitrate. More generally, we could write

as the change

in particle volume arising from condensation of this species;

here we can also treat condensation of water vapor for deli-

quesced and/or aqueous particles.

We can solve eqn (7) for any given parameter, but for the

most part we will either want to know the growth rate for a given

vapor concentration

or mixing ratio

or conversely



either of those in terms of a given growth rate:

=

=

(8)

We also may want to identify the maximum possible growth

rate, when

=

Fig. 8

Collision speed

for NH

and HONO

vs.

the particle diam-

eter with

HONO

=

×

−

joule and

=

0 joule. Dashed hori-

zontal lines are asymptotic values equal to one-quarter of the average

gas speed. This is the so-called

“

kinetic

”

limit of nominally constant

growth; for

x

10 nm the NH

speed exhibits a modest plateau near

the

“

kinetic

”

value. For very small particles, the size of the molecule,

(small) mass of the particle, and the interaction potential between

them all enhance the speed in the free-molecular or

“

collision

”

limit.

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max

=

=

(9)

The concentration of all (air) molecules is

Fig. 8 shows the collision speeds for both NH

and HONO

including the

“

kinetic regime

”

asymptote

;

kin

This

assumes Hamaker constants of

HONO

=

×

−

joule and

=

0 joule, because HONO

has a larger dipole moment

than NH

. Values are based on growth rate observations for

(NH

)

$

(ref. 15) because the dipole moment of

HONO

vapor is similar to the dipole moment of H

vapor

and the relative values appropriate for (NH

)

$

vs.

(NH

)

$

are not known. The exact values are not overly

important to this example, but the di

erence between a highly

polar but relatively heavy acid and a non-polar but light base is

interesting and illustrative. As Fig. 8 shows, NH

has a speed

advantage above 5 nm, but for very small particles the van der

Waals e

ect (as well as the molecular size and reduced mass

ects) give HONO

a speed advantage. Growth by ammonium

nitrate formation ultimately means a 1 : 1 uptake frequency for

and HONO

, and this collision speed transition means that

in some circumstances a system can go from being NH

limited

for the smallest particles to HONO

limited for larger particles.

5 Semivolatile condensation

Let us consider a



ux balance for ammonium nitrate uptake.

For the sake of simplicity and generality of (mono)acid

–

base

condensation we shall refer to HONO

as A and NH

as B.

5.1 Setup

Because of the 1 : 1 stoichiometry enforced by the reactive

uptake, there will be a stoichiometric, molar,



ux balance:

;

(10)

Here we de



ne a collision ratio of A and B with the surface,

. This is the concentration ratio of the two species scaled by

the collision speed for each molecule seen in Fig. 8. Eqn (10) is

symmetrical, but it is easier to think of the case where

$

1so

that A is in excess and B is the rate-limiting contributor to

condensational growth. In the limit of

[

1, there is a large

excess of A and we expect A to be equilibrated (

), so

When the collision ratio

=

1, the system is at the

“

equi-

collision

”

point where with stoichiometric uptake we must have

=

. For large excess saturation,

=

1. In the real case we

are considering, NH

(B) is about twice as fast as HONO

(A), so

the equi-collision point occurs when

HONO

x

(A : B

x

2).

The maximal growth rates, for

=

1, form a chevron meeting

along this

“

equi-collision

”

line, as shown in Fig. 9 for

=

10 nm. This shows that a few 100 pptv of either NH

or HONO

have the potential to drive particle growth at

> 100 nm h

−

provided that there is enough driving force (supersaturation) for

this to actually occur. As a point of reference, growth rates

between 1 and 10 nm h

−

are typical of the continental

boundary layer when conditions permit accurate determination

of aerosol growth.

16,17

However, we also have the ammonium nitrate equilibrium to

consider. Here we assume rapid equilibration of the salt with

the undissociated molecules in the suspended particle phase

(slow particle phase kinetics is a completely di

erent limit,

mostly controlled by volatility). This imposes a balance on the

activity product in the particles.

(11)

The overall saturation of vapors toward the particle phase

can be written as

;

(12)

This is the product of the vapor activities (saturation ratios)

and suspended particle phase activities (including curvature)

and it is the fundamental imposed thermodynamic driver for

growth. The two Kelvin terms multiply, which we represent as

the square based on the average of the two Kelvin diameters,

K10,AB

=

0.5

×

(

K10,A

K10,B

The overall equilibrium between vapor concentrations and

the salt is what is best constrained by thermodynamic data and

calculations (without the Kelvin e

ect).

As Fig. 10 shows, for

Fig. 9

Maximum growth rates of 10 nm particles at 263 K for

ammonium nitrate formation. Growth rates are shown as contour

labels. The maximum depends sharply on the limiting vapor, with

ammonia limitation causing horizontal contours at relatively high nitric

acid and nitric acid limitation causing vertical contours at relatively

high ammonia. The spine is o

set from 1 : 1 based on the relative vapor

speeds and depends on particle size.

Environ. Sci.: Atmos.

,2025,

,67

–

81 |

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sub-freezing conditions (263 K) the saturation isopleths over



at surface cut through typical ambient (urban) mixing ratios

of NH

and HONO

We have two constraints on the actual growth rate. One is the

maximum possible growth rate shown in Fig. 9, and the other is

the saturation ratio,

AB,

, shown in Fig. 10, both at 263 K. These

two combine to give the overall constraints to growth, as shown

in Fig. 11. We expect the actual growth rate to be at the

maximum value for

AB,

[

1, but obviously much lower for

AB,

1, where the natural quantity is the excess saturation,

AB,

−

1. Of course, for

AB,

< 1 there is no growth, and the

dynamics turn to evaporation.

The collision rates and thus the maximum growth rates are

only a weak function of temperature, whereas the saturation

ratios (at a



xed concentration or mixing ratio) are extremely

strong functions of temperature. The saturation limits will thus

sweep from the lower le



of Fig. 11 toward the upper right as

temperature increases.

5.2 Activation diameter

Before anything grows, the growth must start. This is activation,

and it is characterized by an activation diameter,

act

. Unac-

tivated particles are made of something, and here we assume

that ammonium nitrate (acid and base) does not interact with

the unactivated particles beyond the Kelvin e

ect. The activa-

tion diameter simply depends on supersaturation and so strictly

scales with the square of the gas-phase concentrations (and

thus activities) of the acid and base.

;

K10

;

act

K10

;

log

(13)

As stated above we treat the Kelvin diameter(s) as an

adjustable parameter given th

e very small number of mole-

cules in the activating clusters. However, we expect the values

to lie somewhere between those of water (

K10,H

x

1nm)

and organics (

K10,org

x

6nm).Ifthegrowingparticleis

a quasi aqueous solution with

x

75 mN m

−

,wemight

Fig. 10

Saturation ratio over a pure ammonium nitrate crystalline

surface at 263 K

vs.

nitric acid and ammonia mixing ratios. The solid

green diagonal shows

=

1. Regions to the lower left are undersatu-

rated and particles will evaporate; those to the upper right, with

saturation ratios indicated on the dashed contours, are super saturated

and particles will grow. The saturation threshold moves toward the

lower left as temperature falls and towards the upper right as it

increases.

Fig. 11

Phase space combining maximum growth rates and the

saturation ratio over an ammonium nitrate surface at 263 K. Only

systems in the saturated region to the upper right of the solid green

saturation contour (

=

1) will grow, and growth near

=

1 will be

(much) slower than the maximum possible values.

Fig. 12

Activation diameters at 263 K for solid ammonium nitrate with

K10

=

3 nm. Diameters are shown with contour labels, and

act

=

the saturation line

=

1. To the lower left of this saturation line

particles will not activate. Unlike the growth rates, the activation

diameters depend exponentially on the saturation ratios; for example,

the 5 nm activation diagonal from upper right to lower left corre-

sponds to

x

20, whereas the smallest diameter shown (2 nm)

corresponds to

x

1000.

Environ. Sci.: Atmos.

,2025,

,67

–

Environmental Science: Atmospheres

Paper

Open Access Article. Published on 21 October 2024. Downloaded on 2/21/2025 12:34:07 AM.

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