2. Surface roughness
We examine first the effect of small surface inhomogeneities on the nucleation Ž
process. The focus is on the formation of liquid embryos from atmospheric vapors such .
as the sulfuric acid–water binary system onto a curved rough surface which corre- sponds to an atmospheric insoluble aerosol particle. Binary nucleation is important in the
atmosphere, where usually various condensible vapors exist at low supersaturations. The classical theory of heterogeneous nucleation will be used for the description of vapor
association on pre-existing surfaces and for calculating the effect of contact angle changes on the nucleation probability of atmospheric insoluble particles. The interaction
between the liquid embryo and the solid substrate has been described in the classical
Ž Ž
. .
model with the help of Young’s equation cosu s g y g rg
where u is the
vs ls
lv
Ž contact angle between the substrate and the liquid embryo Fletcher, 1969; Lazaridis et
. al., 1991 and g refers to the surface tension. The sub-terms l, s, v refer to liquid, solid
and vapor, respectively. As we can see, the classical theory describes the interactions between the embryo and the substrate by the use of the macroscopic quantity of contact
angle.
Furthermore, we proceed with studying the effect of surface roughness on the mechanism of heterogeneous nucleation. Actually, we examine the effect of the surface
roughness on the contact angle. Since the nucleation flux is very sensitive to changes in the value of the contact angle, small changes in the contact angle can result in large
differences in the nucleation rate.
Ž Rough surfaces of aerosols can be viewed as a Aself-affine fractalB Barabasi and
. Stanley, 1994; Chow, 1997 . Self-affine fractals are defined as the fractal objects that are
Ž .
invariant under anisotropic transformations Barabasi and Stanley, 1994 , a description which is generally valid for many surfaces. The same approach will be adopted in the
current paper.
2.1. Weak fluctuations–chemical contamination Here we consider the effect of small inhomogeneities on the surface of aerosol
particles for the value of the contact angle. The situation corresponds to the case of weak fluctuations that may result from small chemical contamination on the surface of a
smooth aerosol surface. The case corresponds to an ideal situation but it is an interesting Ž
. example to consider first. We follow here the approach of De Gennes 1985 where the
chemical contamination can be described in terms of the local interfacial energies g ,
sv
g . The new local contact angle can be written as:
sl
² :
g cosu s H q g y g 1
Ž .
lv sv
sl
² :
where the unperturbed angle u can be expressed as g cosu s g y g
and H
lv sv
sl
corresponds to the fluctuating value of the term g cosu. After performing a Taylor
lv
expansion at u s u for small H, we get:
g cosu s g cosu q g u y u
ysinu 2
Ž . Ž
. Ž .
lv lv
lv
Ž . Ž
. which can be expressed as u y u s y Hrg sinu
De Gennes, 1985 .
vl o
Since the above equation holds for small values of H, we can conclude that the Ž
. changes in the contact angle are minimal much lower than a degree and would not
change significantly the process of heterogeneous nucleation. However, in the case that the surface contamination has a size similar to that of the nucleating embryo, the
situation changes and we may expect considerable differences in the nucleation rate
Ž .
compared to the case of an energetically homogeneous surface Lazaridis et al., 1992 . In the next section, we examine the effect of rough surfaces on the heterogeneous
nucleation mechanism.
2.2. Rough surfaces The main objective in this section is to evaluate changes of the value of contact angle
between a smooth and a rough solid surface of aerosol particles. The current approach does not assume that the nucleation is controlled by the equilibrium contact angle. We
begin with the ideal situation of smooth surface, where the equilibrium contact angle is Ž
. adopted but we introduce spatial
x, y dependent interfacial energy densities to
Ž determine a generalized contact angle for describing the local wetting phenomena see
. also Chow, 1998; Barabasi and Stanley, 1994 . As we have already mentioned above, in
the case of rough substrates, the Young equation has to be modified in order to include the changes in the interfacial energies g , g and contact angle in the description of the
sv sl
Ž .
wetting behavior of the aerosol particle Barabasi and Stanley, 1994 : g y g s g cos u x , y
s g cos u x , y y f x , y
3
Ž .
Ž .
Ž .
Ž .
sv sl
lv lv
Ž .
where x, y are the spatial coordinates on the surface of the aerosol particle and f x, y is the change of the contact angle because of roughness. The geometry of the system
under investigation is shown in Fig. 1. As shown in Fig. 1, the contact line is expressed Ž .
in the form x s l y and the contact free energy is the sum of the vapor-substrate and liquid-substrate interactions.
The rough surface of the particle can be described as a self-affine fractal and the change of the height correlation can be obtained with the solution of the following
Ž .
stochastic Langevin-like differential equation Chow, 1997, 1998 : d D h
D h
Ž .
s y q h r
4
Ž . Ž .
d r 2 j
Ž where D h s h y h
h describes the unperturbed height of the nucleating liquid cluster .
on the surface of the aerosol particle , r is the position vector of the reference surface Ž
. vertical distance equal to zero , j is the function which describes the fluctuations
Ž . parallel to the surface and h r is the noise term that is the source of fluctuations of D h
Ž .
Ž . Barabasi and Stanley, 1994 . The first term on the right side of Eq. 4 is the average
local slope. During the growth process, all sites on the particle surface are uncorrelated. However, there is a typical distance over which the heights of neighbor sites are
Ž . correlated, and this correlated length is denoted by j . The noise term h r
is an Ž² :
. uncorrelated random number that has zero configuration average
h s 0 and the
Ž .
Fig. 1. Definitions of contact angle and contact line adopted from Chow, 1998 .
² Ž . Ž
.: Ž
. second moment is given by: h r h r
s constant = d r s r , which implies that
1 2
1 2
the noise has no correlations in space. Ž .
The solution of Eq. 4 provides the change of the contact angle as function of the Ž
. unperturbed contact angle u x, y , the function s that is the standard deviation of the
Ž .
fluctuations normal to the surface and the correlation length j Chow, 1998 : cos u x , y
y f x , y C su
Ž .
Ž .
a
, 1 q .
5
Ž .
cosu x , y 2 j
Ž .
Ž . The above derivation is based on the assumption that the noise term h r
has a long-range slope correlation. The non-dimensional term C , which can be viewed as a
a
correction factor, includes the effect of the correlated noise due fluctuations that interact between them. The function C can be written with the help of Gamma functions as
a
wŽ Ž Ž
...ŽŽ Ž
.. Ž Ž
...x
1r2
Ž C s 1r 2 2 a y 1
G 2 a q 1 r G 2 a y 1 for 1r2 - a - 1. The term a
a
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
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