Chemistry Reference
In-Depth Information
directions. The non-covered regions correspond to the bare KCl(100) surface. Note
the amazing fortress-like shape of the growing microcrystals and that the material
retained inside exhibits round-shaped forms, indicative of incomplete growth.
And what about our long forgotten guiding N
2
molecule? It turns out that, when
growing on HOPG at coverages above two layers in the temperature range 10
<
20 K, N
2
incompletely wets the surface (Seguin
et al
., 1983). The system thus
seems to prefer the Volmer-Weber mechanism of growth over the other mechanisms
for the particular growth conditions.
T
sub
<
5.2 Dynamic scaling theory
The 2D to 3D transition implies changes in morphology and thus in roughness, as
can be inferred from Fig. 5.1. Because of the complexity of such time-fluctuating
systems a statistical approach becomes mandatory. Many systems of diverse nature
exhibiting rough interfaces in their evolution process, e.g., fluids invading porous
media, fire fronts, crystal growth, growing tumours, etc., have been successfully
described by means of scaling analysis, a powerful mathematical tool used in the
study of fractal geometry (Family, 1990; Barabasi & Stanley, 1995; Krug, 1997).
The dynamics of such systems can be characterized by a set of critical exponents
obtained from scale-invariant properties of certain physical quantities. Systems
exhibiting the same critical exponents belong to the same universality class, sharing
a common mathematical description.
Let us describe the essential trends of the dynamic scaling theory (DST) adapted
to the field of film growth. We first assume that the surface morphology can be
described by a continuous function
h
(
x
y
), where
h
stands for the height and
x
and
y
represent the in-plane 2D Cartesian coordinates (a perfect flat surface would
be represented by a constant
h
value). In many cases the surface morphology of
thin films is, within certain spatial ranges, invariant under scale transformations.
This invariance implies that there exist rescaling factors
s
1
and
s
2
such that
h
(
x
,
,
y
)
is statistically indistinguishable from
s
1
h
(
s
2
(
x
,
y
)). This property is known as self-
affinity.
The magnitude most commonly employed to statistically characterize the sur-
face morphology for a given growing system of size
L
observed at time
t
is the
roughness,
t
). This magnitude represents the height distribution of the surface
and is a measure of the width or dispersion of the real surface.
ς
(
L
,
is defined as
the mean square deviation of the local height with respect to the mean height,
ς
ς
)
2
]
1
/
2
, where
represents the average
value of the quantity within brackets for the length scale
L
. The surface rough-
ness can be described by the dynamic scaling expression
(
L
,
t
)
=
[
(
h
(
x
,
y
,
t
)
−
h
(
x
,
y
,
t
)
···
L
α
f
(
t
L
z
),
ς
(
L
,
t
)
=
/
v
β
for
where the function
f
(
v
) scales as
v
1 and is constant for
v
1. The
exponents
α
,
β
and
z
are known as the roughness, growth and dynamic exponents,
