Download classical notions of heterogeneous freezing

Different heterogeneous routes of the
formation of atmospheric ice
Anatoli Bogdan
Institute of Physical Chemistry, University of Innsbruck
Austria
and
Department of Physics, University of Helsinki
Finland
Contents
Introduction
- ice in the atmosphere
Heterogeneous formation of ice
- contact freezing
- immersion freezing
- classical notions of heterogeneous freezing
- a new hypotheses of heterogeneous freezing
What are high-altitude cirrus clouds?
- sub-visible cirrus (SVC) clouds with optical depth<0.03
- thin cirrus clouds with optical depth < 0.3
SVC cloud in tropical tropopause region at about 17 km
(From Winker and Trepte, GRL, 25, 3351-3354 (1998).)
Short information about SVC clouds
Location of the SVC clouds:
- globally widespread near the tropopause region
Temperature range:
- between 210 and 185 K
Formation:
- freezing diluted H2SO4/H2O aerosol droplets
Origin of aerosol droplets in the upper troposphere:
- in situ gas-to-particles conversion
- deposition processes from the lower stratosphere
(Minnis, P. et al., Science, 259, 1411-1415 (1993); McCormick, M. P., et al.,
Nature, 373, 399-404 (1995))
Observed size of aerosol droplets:
- depending on background or volcanic conditions
diameter can be in the range of about 0.2 – 3 mm
(Hofmann & Rosen, Science, 222, 325-327 (1983).
Composition of aerosol droplets:
- < 30 – 35 wt % H2SO4
Why are SVC clouds important for the climate?
- they reflect and scatter incoming solar radiation
(cooling effect)
- they trap outgoing terrestrial infrared radiation
(warming effect)
- they supply surface for heterogeneous loss of ozone
that is an important greenhouse gas at high altitudes
(cooling effect)
Both radiative properties and the rate of ozone
loss depend on the microphysics of SVC clouds
Microphysical characteristics of SVC clouds
- Small ice water content (IWC) :
10-4 – 10-6 g/m3
- Small effective ice diameter :
< 20 – 30 mm
- Small ice particle density :
< 1 cm-3
- Different shape (habits) of ice crystals
Polar stratospheric clouds (PSCs)
Formation of ice in the atmosphere
Nucleation occurs in the interior of a uniform pure substance
(vapor of liquid), by a process called
homogeneous nucleation.
The creation of a nucleus implies the formation of an interface at
the boundaries of a new phase.
Heterogeneous freezing in the atmosphere
Nucleation induced by a foreign insoluble particle (nuclei)
is called by hetergogeous nucleation.
Fumed silica (SiO2) can be considered as
a representative of atmospheric ice nuclei
Fumed silica (SiO2) is fractal object
- SiO2 is widely encountered in meteoritic smoke
particles (MSPs) and combustion origin
Using of fumed silica in heterogeneous
freezing measurements of aqueous droplets
Weight (g) of H2SO4 in diluted H2SO4/H2O
drops as a function of size (0.2 - 2 mm) and
composition (10 – 25 wt % H2SO4)
Weight of H2SO4 is in the range
5*10-16 – 1.2*10-12 g
Initial thickness of an over-layer of initial
composition of 38 wt % H2SO4
Different routes of the formation of ice in the atmosphere
Contact freezing
occurs when a
super-cooled drop
comes in "contact"
with an ice nuclei
Immersion freezing
occurs when an ice
nuclei is inside of
a super-cooled drop
When an ice crystal
falls through
super-cooled
droplets then
accretion or riming
occurs, i.e.,
super-cooled drops
freeze onto ice crystal.
The resultant particle
is often referred to
as graupel
When an ice crystal falls
through other ice crystals
then aggregation occurs
which produce a snowflake
1.4 Some useful thermodynamic properties
dqrev
, is a thermodynamic property showing a direction of any spontaneous changes
T
occurring in nature. When a system is in thermal equilibrium with its surrounding and there is an exchange of heat dq between the
system and the surrounding then for any spontaneous changes
Entropy, thermodynamically defined as d S 
dq
dS  .
T
(1.28)
This expression is called the Clausius inequality. If the system is isolated then dq = 0 and the Clausius inequality implies that
dS  0 ,
i.e., in the isolated system entropy cannot decrease when a spontaneous processes takes place.
(1.29)
A system can exchange a heat with its surrounding either when volume is constant or when pressure is constant. When heat is
transferred to a system at constant volume then, in the absent of any non-expansion work (for example, chemical reactions), we have
dqv = dE, and consequently the Clausius inequality reads as
dS 
dE
0
T
(1.30)
or
dE – T dS ≤ 0.
(1.31)
The last expression allows us to define a new thermodynamic property called the Helmholtz energy
A = E – TS.
(1.32)
Eqs. (1.31) and (1.32) show that, at constant T and V, the criteria of the spontaneous change is
dAT,V ≤ 0,
i.e., the system tends towards a lower Helmholtz energy.
When a heat is transferred to a system at constant pressure, and there is no non-expansion work, then dqp = dH, and the Clausius
inequality can be written as
dH
dS 
0
T
(1.33)
dH – T dS ≤ 0 .
(1.34)
or
This expression allows us to introduce the Gibbs energy (sometimes called also free energy)
G = H –TS.
(1.35)
Eqs. (1.34) and (1.35) show that, at constant T and p, the criteria of the spontaneous change is
dGT,p ≤ 0,
i.e., a system tends towards a lower Gibbs energy. Since in the atmosphere, the different phisyco-chemical processes occur rather at
constant pressure than at constant volume, below we will deal mainly with the Gibbs energy.
The Gibbs energy of solids and liquids is almost independent of pressure, Fig.3. However, in the case of gases (both real and perfect)
the Gibbs energy is a strong function of pressure. Indeed, an infinitesimal change in the properties of H = E + pV, T, and S result in
that the Eq. (1.35) can be written as
dG = [(T dS – p dV) + p dV + V dp] – T dS – S dT = V dp – S dT.
(1.36)
(Here we used the expression dE = TdS – p dV.)
The Eq. (1.36) shows that a change in G is proportional to a change in p and T, i.e., the Gibbs energy may the best regarded as a
function of p and T. At constant temperature, dT = 0, integration of (1.36) gives
pf
G(pfinal) = G(pinitial) +  Vdp .
(1.37)
pi
It is seen from Fig. 3 that for solids or liquids, the change in volume with pressure is small, so V can be treated as a constant and the
integral can be considered approximately as ≈ V∆p. Since at normal pressure the value of V∆p is small, we can suppose that the Gibbs
energies of solids and liquids are independent of pressure. But in the case of gases, when the volume varies markedly with pressure
and the molar volume of the gas is large, the integral cannot be neglected. For the perfect gas, substituting V = nRT/p into the integral,
we get
pf
dp
G(pf) = G(pi) + nRT  = G(pi) + nRT ln
.
p
p
i
pi
pf
(1.38)
In physical chemistry, the initial pressure is taken as pi = p° =1 bar, which is called standard pressure. The (1.38) allows relating the
Gibbs energy at any given pressure p with its standard Gibbs energy G° (Gibbs energy at the standard pressure 1 bar, Fig. 4) by the
expression
p
G(p) = G° + nRT ln p 
(1.39)
The molar Gibbs energy is called the chemical potential μ. For the perfect gas it is expressed as
Gm = μ = μ° + RT ln
p
 ,
p
(1.40)
where μ° is the standard chemical potential of the perfect gas (i.e., the chemical potential at the standard pressure 1 bar). The chemical
potential expresses a capacity of a substance to bring about phase and chemical changes.
Phase equilibrium: Two systems are in equilibrium if they fulfill the conditions of
chemical equilibrium
μ1 = μ2 ,
(1.44)
T1 = T2 ,
(1.45)
p1 = p2.
(1.46)
thermal equilibrium
and mechanical equilibrium
If one of the system possesses a curved surface then the mechanical equilibrium expressed by the Laplace equation
p1 – p2 = 2γ/r,
where p1 and p2 are pressures of the two phases, γ surface tension of the system with curved surface, and r radius of curvature. For
example, two phases will be in the unstable equilibrium at the moment of the formation of a critical embryo of a new phase within a
mother phase. The unstable means that the new phase will grow infinitely when the radius of the embryo slightly exceeds the critical
embryo.
As temperature decreases, ice is formed in a super-cooled water
droplet when the pressure of ice becomes less than the pressure of
water, i.e., when a super-saturation with respect to ice appears.
This brings about a change in Gibbs free energy per unit volume, Gv,
between the water and ice phase. This change in free energy is
balanced by the energy gain to create a new volume (negative
change), and the energy cost due to creation of a new interface
(positive change). When the overall free energy change,
ΔG is negative, nucleation is favored.
If a formed ice nucleus is too small (known as an unstable nucleus or
"embryo"), the energy that would be released by forming its volume
(negative change) is not enough to create its surface (positive change)
then nucleation does not proceed. The formed nucleus should reach
some critical size (or radius), in order to be stable and the growth of
the ice phase proceeds.
In the classic theory, for a spherical ice cluster we cam write for the
overall Gibbs energy change
where the first term is negative and the second one positive.
The Gibbs free energy needed to form the ice cluster of critical
radius can be found from the conditions
The Gibbs free energy change needed to form the ice cluster of
critical radius is
and the critical radius
where Gv is a change in free energy per unit volume.
Large supercooling favors the freezing of water. We can relate
ΔG to supercooling ΔT = T – To, where = 273.15 K, and find r* and
ΔG * as a function of ΔT
and
The larger the supercooling ΔT = T – To, the smaller the critical
radius and the less energy needed to form it.
Heterogeneous nucleation
Heterogeneous nucleation occurs much more often than
homogeneous nucleation. The free energy needed for heterogeneous
nucleation is equal to the product of homogeneous nucleation and a
function of contact angle :
where
The barrier energy needed for heterogeneous nucleation is
reduced, and less super-cooling is needed.
In nature freezing usually occurs below the maximum
heterogeneous freezing temperature (which is melting
temperature of ice, 273.15 K) but above the
homogeneous freezing temperature which is about 233 K.
Water in between these two temperatures is said to be in
super-cooled state.
A new hypotheses of heterogeneous freezing