Generalized Surface-Area-Difference model for cohesive energy
of nanoparticles with different compositions
W.H. Qi
Received: 29 January 2005 / Accepted: 5 October 2005 / Published online: 20 June 2006
Ó
Springer Science+Business Media, LLC 2006
Abstract The Surface-Area-Difference (SAD) model has
been generalized to account for the cohesive energy of
nanoparticles with different compositions (NDC) in dif-
ferent shape, where the particle shape is described by the
shape factor. It is found that the cohesive energy of NDC
depends on the particle size, the particle shape and the
atomic percent of each composition, which can be simply
regarded as mathematical mean values of the cohesive
energies of all the compositions.
The cohesive energy of nanoparticles depends on the par-
ticle size, which has been confirmed by recent experiments
[1] and explained by different models [2–7]. The first
experimental values of the cohesive energy of pure Mo and
W nanoparticles have been reported in 2002 [1], where the
cohesive energy is determined by measuring the oxidation
enthalpy of the corresponding oxidations of nanocrystals.
Some researchers have developed different models to
explain the size dependent cohesive energy [2–7]. One of
these models is called as Surface-Area-Difference (SAD)
model [3, 4], which is based on the basic concept of cohe-
sive energy. The SAD model is developed by our group,
which has been introduced in a review article by professor
Sun [8]. However, the published SAD model can only ac-
count for the cohesive energy of pure metallic nanoparti-
cles, but not for that of nanoparticles with different
compositions (NDC). In this letter, we will generalize SAD
model to account for the cohesive energy of NDC.
Since the cohesive energy of a material is the energy to
divide the material into isolated atoms, in other words, the
direct result of cohesive energy is to create new surface.
The increased surface energy should equal the cohesive
energy of the material, which results from the surface area
difference between the total isolated atoms and the mate-
rial. This is the basic concept of SAD model [3, 4]. In SAD
model, the surface of the material approximately denotes
the first layer of the material. In the following, we will give
the generalized SAD model.
We assume that a nanoparticle consists of m composi-
tions, i.e., A
i
ði ¼ 1; 2; ÁÁÁmÞ, where A
i
denotes the com-
position i. The atomic percent of each element A
i
is
x
i
ði ¼ 1; 2; ÁÁÁmÞ, where
P
m
i¼1
x
i
¼ 1. The total number of
atoms is n, and the number of atoms of A
i
is x
i
n. The atomic
diameter of A
i
is denoted as d
i
, and the surface energies per
unit area of A
i
is c
i
. The surface energy of A
i
with nx
i
atoms
is nx
i
p d
i
2
c
i
, and then the total surface energy of n atoms is
P
m
i¼1
nx
i
pd
2
i
c
i
.
If the surface of the NDC is spherical, its surface energy
is 4p D
2
c, where D is the diameter of the nanoparticle and c
is the surface energy per unit area. Without considering the
surface segregation, the surface energy per unit area can be
estimated by the following equation, i.e.
c ¼
X
m
i¼1
xc
i
ð1Þ
For a non-spherical nanoparticles, the surface energy
can be written as paD
2
c
i
, where a is shape factor. Shape
factor is defined as the surface area ratio of non-spherical
W.H. Qi (&)
School of Materials Science and Engineering, Jiangsu
University, Zhenjiang, Jiangsu 212013, China
e-mail: weihong.qi@gmail.com
W.H. Qi
School of Materials Science and Engineering, Central South
University, Changsha Hunan 410083, China
J Mater Sci (2006) 41:5679–5681
DOI 10.1007/s10853-006-0251-0
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