Parabolic vs Linear Interface Shift on the Nanoscale
Dezso ˝ L. Beke and Zoltán Erdélyi
(Submitted July 13, 2005; in revised form August 12, 2005)
It was shown very recently that diffusion nonlinearity, caused by the strong composition de-
pendence of diffusion coefficients, can lead to surprising effects on the nanoscale: a nonparabolic
shift of interfaces (both in ideal and phase separating systems) and sharpening of an initially
diffuse interface in ideal systems. Some of these can not be interpreted even qualitatively from
Fick’s classic equations. For instance, the nonparabolic shift of an interface at the very begin-
ning is a consequence of the violation of Fick’s first equation on the nanoscale, and the transition
from this to the classic parabolic behavior depends on the strength of the nonlinearity and the
value of the solid solution parameter V (proportional to the heat of mixing). Experimental and
theoretical efforts to explore the above phenomena are summarized in this paper.
1. Introduction
Diffusion in nanostructures presents challenging features
even if the role of structural defects (dislocations, phase or
grain boundaries) can be neglected. This can be the case for
diffusion in amorphous materials or in epitaxial, highly
ideal thin films or multilayers, where diffusion along short
circuits can be ignored, and “only” fundamental difficulties,
related to nanoscale effects, arise. For example, the con-
tinuum approach cannot be automatically applied,
[1,2]
and
there is also a gradient energy correction to the driving force
for diffusion. This correction becomes important if large
changes in the concentration take place along distances
comparable with the atomic jump distance, a, and results in
an additional term in the atomic flux proportional to the
third derivative of the concentration. It was shown recently
by our group
[1-15]
that these effects can lead to unusual
phenomena, especially if there is a strong nonlinearity in the
problem, i.e., if the diffusion coefficient has strong concen-
tration dependence.
2. Basic Equations
To have a general expression for the atomic fluxes, valid
also on the nanoscale, one has to choose a proper micro-
scopic model. Let us start from a set of deterministic kinetic
equations,
[1,2,4,5,16]
obtained from Martin’s model,
[17]
in
which the effect of the driving forces can be generally de-
scribed by the
i
/kT parameter present in the expression of
atomic fluxes between the ith and (i+1)th atomic layers,
perpendicular to the x-axis;
J
i,i+1
= z
v
i,i+1
c
i
1 - c
i+1
-
i+1,i
c
i+1
1 - c
i
= z
v
i
c
i
1 - c
i+1
exp
i
kT - c
i+1
1 - c
i
exp
i
kT
(Eq 1)
In this exchange model
i,i+1
is the probability per unit
time that an A atom in layer i exchanges its position with a
B atom in the layer i+1. z
v
is the vertical coordination num-
ber, and c
i
denotes the atomic fraction of A atoms on plane
i. It is usually assumed
[1,2,4,5,16]
that the jump frequencies
have Arrhenius-type temperature dependence:
i,i+1
=
o
exp-E
i,i+1
kT =
i
exp-
i
kT
i+1,i
=
o
exp-E
i+1,i
kT =
i
exp
i
kT (Eq 2)
with
i
=
o
exp-E
o
-
i
kT =
o
exp
i
kT (Eq 3)
where
o
denotes the attempt frequency, k is the Boltzmann
constant, T is the absolute temperature, and E
i,i+1
E
o
-
i
+
i
and E
i+1,i
E
o
-
i
-
i
are the activation barriers (E
o
is a composition-independent constant including saddle
point energy as well), which must be chosen to fulfill the
condition of detailed balance under steady state (J
i,i+1
J
i+1,i
c
i
t 0). There are many choices of E
i,i+1
, which
fulfill this condition.
[17]
For instance, the following choices:
i
= z
v
c
i-1
+ c
i+1
+ c
i
+ c
i+2
+ z
l
c
i
+ c
i+1
V
AA
- V
BB
2
(Eq 4)
i
= z
v
c
i-1
+ c
i+1
- c
i
- c
i+2
+ z
l
c
i
- c
i+1
V (Eq 5)
satisfy it,
[1,2]
where V
ij
(<0) are the nearest neighbor pair
interaction energies of ij atomic pairs, z
l
is the lateral coor-
dination number, z 2z
v
+ z
l
, and V V
AB
-(V
AA
+ V
BB
)/2
is the solid-solution parameter proportional to the heat of
mixing. For phase separating systems V > 0. The parameter
M=mkT/2Z determines the strength of the composition
dependence of the transition rates
[6]
in a homogeneous
alloy. It can be estimated, e.g., from the nearest neigh-
bor pair interaction energies of ij atomic pairs, V
ij
, as
This article is a revised version of the paper printed in the Proceedings
of the First International Conference on Diffusion in Solids and Liq-
uids—DSL-2005, Aveiro, Portugal, July 6-8, 2005, Andreas Öchsner,
José Grácio and Frédéric Barlat, eds., University of Aveiro, 2005.
Dezso ˝ L. Beke and Zoltán Erdélyi, Department of Solid State Phys-
ics, University of Debrecen, H4010 Debrecen, Hungary. Contact
e-mail: dbeke@delfin.klte.hu.
JPEDAV (2005) 26:423-429
DOI: 10.1361/154770305X66466
1547-7037/$19.00 ©ASM International
Basic and Applied Research: Section I
Journal of Phase Equilibria and Diffusion Vol. 26 No. 5 2005 423