Chemistry Reference
In-Depth Information
functional forms
D
c
(θ)
[
4
]. This in turn implies different deviations of the profile
shape from the ideal error function-based profiles of (
3.2
). Profiles which are steeper
(than the error function) in the vacant region (
r
r
0
) lead to larger derivative in
the denominator of (
3.4
) and therefore reduced
D
c
(θ)
<
(such dependence indicates
predominantly attractive interactions); while profiles which are flatter in the region
r
<
r
0
have smaller derivative in the denominator of (
3.4
) and larger
D
c
(θ)
(such
dependence indicates predominantly repulsive interactions).
The features listed previously as characteristic of classical diffusion (i.e., the
spreading of the initial steep profile, the dependence on the scaled variable
r
/
√
t
,
and the decreasing edge velocity) are also true for the case of coverage-dependent
D
c
(θ)
. They follow simply from the nature of the diffusion process that a more pro-
nounced spatial variation (i.e., the second derivative
2
r
2
) causes weaker tem-
∂
θ/∂
poral variation (i.e., first derivative
∂θ/∂
t
). Concerning possible functional forms
of
D
c
(θ)
for classical diffusion, in general the coverage-dependent diffusion is a
smooth function of
θ
. Only close to a phase transition can
D
c
(θ)
have more structure
(usually in the form of either maxima or minima with
T
or
) as a result of changes
in spatial and temporal correlations of the diffusing atoms in the system. Close to
a second-order phase transition, critical fluctuations generated thermodynamically
are so large (
θ
∂μ/∂θ
→∞
as
T
→
T
c
) that no diffusion currents can eliminate
them. This effectively means that
D
c
(
T
c
, known as critical
slowing down [
5
]. Close to an order-disorder transition, the coverage dependence
D
c
(θ)
T
)
goes to 0 as
T
→
(for fixed temperature
T
<
T
c
) can show maxima as the ideal coverage
of the ordered phase
θ
c
the perfect defect-free
phase is least compressible (it costs large energy to generate adatoms out of this
ideal structures,
θ
→
θ
c
is approached, since at
θ
→
θ
c
). The amplitude of the diffusion maxima
depends on the nature of the order parameter describing the phase transition; the
maxima have larger amplitude when the order parameter is the coverage
∂θ/∂μ
→
0as
(i.e., for
first-order phase transition in systems with attractive interactions). In all cases, these
thermodynamic effects can produce narrow maxima or minima in
D
c
(θ)
θ
vs.
θ
,but
not a sharp step-like jump at a critical coverage
θ
c
, within a very narrow transition
region
θ <<
0
.
1ML such that
D
c
jumps from one value for
θ<θ
c
toadifferent
value that differs by orders of magnitude for
θ>θ
c
.
An unusual diffusion profile resembling somehow such
dependence has been
observed in multilayer diffusion (that involves at least two layers adsorbed on the
surface with very different diffusion coefficients in each layer) [
6
]. If diffusion on
top of a layer is fast while diffusion within the layer is slow, then the slow layer
advances predominantly through the fast transfer of material from the top to the
bottom layer. Over the timescale of the slow diffusion it seems that the lower layer
θ
moves at a constant speed, not the expected
√
t
scaling behavior. This is the so-
called carpet unrolling mechanism and will be discussed in more detail below in
connection with Li diffusion on metal surfaces. In essence, it is the simplest case of
“reaction”-limited kinetics where diffusion is the fastest process but the timescale
of the observations is set by a slow “reaction,” i.e., mass transfer from a higher to a
lower level. Even under these conditions the profile disperses and there is no critical
dependence of the refilling time
τ
on
θ
.
