Ž is the roughness exponent that describes the fractal properties of the surface Chow,
. 1998 . The correlated noise is due to the microstructure of rough surfaces in agreement
with experimental data. In the case where there is no correlated noise the right-hand side Ž .
Ž .
of Eq. 5 can be written as 1 q su r2 j .
3. Classical theory of heterogeneous nucleation
In the present paper, we are using the classical theory of heterogeneous nucleation to evaluate the effect of surface roughness on the activation of atmospheric insoluble
aerosol particles. As we have already argued, surface roughness reduces the value of the contact angle and consequently, will increase the nucleation capability of the aerosol
particles. The classical theory of heterogeneous nucleation is based on a macroscopic description of an embryo in contact with a substrate using the concept of the contact
angle u. Under the capillarity approximation, the free energy of formation of a liquid embryo from a single vapor on a substrate can be expressed as:
DG s ynkT ln S q S g q g y g S
6
Ž .
Ž .
lv lv
ls vs
ls
where n is the number of molecules in the droplet, S is the saturation ratio, g and S
i j i j
are the surface tension and surface area of the interface between phases i and j. The interaction between the liquid embryo and the solid substrate has been described
in the classical model with the help of the Young’s equation. The critical droplet composition is given by the composition at the saddle point. It is defined by the extremal
Ž . Ž
. conditions Gibbs–Thomson equations
Lazaridis and Drossinos, 1997 . Ž
. The critical value for the Gibbs free energy is obtained as Lazaridis et al., 1992 :
2
2
DG s p r
g f m, x , 7
Ž .
Ž .
lv
3 Ž
. Ž
. Ž where f m, x is a function of the cosine of the contact angle and x s R rr
R
p p
denotes the radius of the pre-existing aerosol particle, m is the cosine of the contact .
angle and r is the radius of the critical nucleating cluster .
In the classical nucleation theory, we can derive a steady-state expression for the Ž
heterogeneous nucleation rate that can be written as Lazaridis and Drossinos, 1997; .
Lazaridis, 1998 : r
v
J s S
n , n R
k exp yb F ,
8
Ž .
Ž .
Ž .
12 1
2 aa
1r2
2p l l l s
Ž .
1 2
3 e
where l refer to the eigenvalues that result from a rotation of a matrix, that includes the
i
impingement rates of the two components, from the coordinate system that has as coordinates the number of molecules in the cluster and energy to a new one in such a
Ž way that the mathematical manipulations of the equations are easier see also Barrett,
. 1994; Langer, 1969; Lazaridis and Drossinos, 1997 . Furthermore, the term k is the
Ž .
absolute value of a growth rate, S n , n
is the surface area of the nucleating droplet
12 1
2
Ž .
including n molecules of the specie i , s is the mean-square energy fluctuations, b
i ´
is the Boltzmann’s constant and F is the Helmholtz free energy of formation of a
critical cluster. The energy fluctuations contribution arises from the fact that a condens- ing or evaporating monomer not only alters the composition of the forming droplet, but
it also adds or removes the latent heat of condensation. The dependence of the nucleation rate on the contact angle arises from the terms R ,
a a
Ž .
S n , n
and F . For a detailed description of the kinetics and energetics of the
12 1
2
Ž .
Ž .
heterogeneous nucleation, we refer to Fletcher 1969 and Lazaridis et al. 1991 . However, there are many aspects missing from the simple version of the theory such
Ž as the concepts of surface diffusion and line tension Joanny and De Gennes, 1986;
. Talanquer and Oxtoby, 1996; Lazaridis, 1993, 1994; Lazaridis and Ford, 1993 . An
important aspect of the interaction between the liquid nucleating embryo and the substrate is the consideration of the long-range forces that exist between the liquid and
solid. An approach which has been used is the introduction of the line energy that arises from the line discontinuity between two or more surface phases. In the present paper, we
are mainly concerned with the effect of surface roughness on the heterogeneous nucleation rate and we will not be concerned with the effects of surface diffusion and
line tension. Surface diffusion and negative values for the line tension may result in even higher values for the heterogeneous nucleation process as shown in a number of
Ž .
previous studies De Gennes, 1985; Lazaridis, 1993 . In the following, the correction factor we derived for the contact angle due to
existence of a rough surface will be evaluated. The sensitivity of the nucleation flux on the value of the contact angle will also be evaluated for the H SO –H O binary system.
2 4
2
In Fig. 2, we show the effect of the ratio x s s C rj on the contact angle. Increased
a
values of the ratio x result in a further decrease of the unperturbed value of the contact angle in an almost linear way. Therefore, surface roughness enhances wetting and leads
to higher nucleation rates for the solid fraction of the atmospheric aerosols. The initial contact angle used in the calculations was 308.
Fig. 3 shows the nucleation dependence on the contact angle for the H SO –H O
2 4
2
binary system. The nucleation rate is very sensitive to small changes in the value of the contact angle and its value decreases as the contact angle increases. Of course, this
strong dependence is a consequence of the parametrization of the interactions between the solid surface and the liquid droplet through the contact angle. Since surface
roughness can decrease the value of the contact angle by several degrees, it is expected that it will also modify the nucleating properties of the aerosol surfaces and enhance the
nucleation flux. In the calculations, the radius of the particle is chosen to be 1 mm, the temperature equal to 262.15 K, the water activity equal to 0.178 and the sulfuric acid
activity equal to 2.86 = 10
y2
. The above values correspond to atmospheric conditions at altitude of 4 km.
The effect of surface roughness on the binary heterogeneous nucleation of the sulfuric acid–water system at atmospheric conditions was further examined with respect to the
probability of nucleation occurrence on the surface of the aerosol particles. In the simulations, we have chosen a contact angle of 708 to correspond to the case of
smooth surface and contact angles of 688 and 668 to describe the situation of rough surface where the contact angle is reduced by 28 and 48. This corresponds to values for
the function x of 2 and 4.5, respectively, as can be seen from Fig. 2. The value of 708 for the contact angle has been chosen since it corresponds to contact angle between
Ž .
Fig. 2. Effect of the factor x s s C rj on the value of the contact angle. The y-axis shows the decrease of
a
Ž .
the contact angle in degrees due to the effect of surface roughness.
Ž .
sulfuric acid and solid sulfur Hamill et al., 1982 . However, there is a considerable variability of the contact angle between various surfaces due to chemical composition of
the liquid embryo and the particle surface. For example, the contact angle of water on Ž
. over 100 low energy organic surfaces ranges between 608 and 1208 Hamill et al., 1982 .
In the calculations, the radius of the particle is also chosen to be 1 mm. In Fig. 4, we show the effect of the ambient temperature on the heterogeneous
nucleation probability of atmospheric aerosols. The background aerosol concentration was chosen equal to 300 cm
y3
and the background concentrations of water, sulfuric acid and temperature similar to the conditions at altitude of zero kilometers. Change of 158 in
the ambient temperature can lead to considerable changes in the activation process of atmospheric aerosols. Again, the shape of aerosol particles has a crucial impact on their
activation potential. The results of Fig. 4 highlight the importance of processes such as isobaric and adiabatic cooling as effective mechanisms of aerosol activation and cloud
formation.
However, we have to point out that the above simulations correspond only to hypothetical scenarios in the atmosphere since in reality neither the size of the particles,
relative humidity, acid concentration neither the contact angle values are fixed. The simulations performed aim mainly to highlight the implications of enhanced nucleation
rates that may result due to the surface roughness effect.
Fig. 3. Dependence of the nucleation rate on the contact angle.
Fig. 4. Heterogeneous nucleation probability of insoluble aerosols as a function of ambient temperature.
4. Conclusions
