INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 16 (2005) 1290–1293
doi:10.1088/0957-4484/16/8/051
Melting point oscillation of a solid over
the whole range of sizes
Chang Q Sun1,3, C M Li1 , H L Bai2 and E Y Jiang2
1
School of Electrical and Electronic Engineering, Nanyang Technological University,
639798, Singapore
2
Institute of Advanced Materials Physics and Faculty of Science, Tianjin University, 300072,
People’s Republic of China
E-mail: [email protected]
Received 2 February 2005, in final form 28 April 2005
Published 7 June 2005
Online at stacks.iop.org/Nano/16/1290
Abstract
It is intriguing that a solid containing ∼101 atoms of Ga or IV-A elements
(C, Si, Ge, Sn, and Pb) melts at temperatures that are higher than the melting
point of the corresponding bulk solid (Tm,b ) though the Tm of a solid in the
100–2 nm size range drops universally with the sample size. Consistent
insight into the phenomenon of Tm oscillation (suppression followed by
elevation as the solid size is reduced from bulk to subnanometre size) over
the whole range of sizes remains a scientific challenge. Here we show that
the Tm oscillation arises from the joint effect of bond order loss and its
consequence on bond strength gain of small clusters in particular for Ga and
atoms of the IV-A elements.
1. Introduction
The melting of a nanosolid (or nanocluster, nanoparticle,
etc) has been a long historical mystery, which has attracted
tremendously renascent efforts recently.
An enormous
contribution has been made from various perspectives to
understanding the underlying mechanism [1–4]. It has been
well established both experimentally and theoretically that
the melting point (Tm (K j )) of a solid in the size range of
fashion, where K j =
nanometres drops with size in a K −1
j
R j /d is the dimensionless form of size that equals the number
of atoms lined along the radius of a spherical solid with radius
R j or thin film of R j thick and d is the atomic diameter in
the bulk. It has been clear that the melting of an impurityfree solid proceeds predominately through a mechanism of
liquid shell nucleation and growth [5–8]. The Tm suppression
in nanometre size material is often explained with either the
Lindermann’s criterion [9] of abrupt magnitude increase of
atomic vibration and its derivatives of surface lattice/phonon
instability [10–12] or Born’s criterion [13] of shear modulus
disappearance and its derivatives of surface energy. It has
been shown [14] that the criteria of Lindermann and Born are
very much the same in numerically matching to the measured
melting point suppression.
3 http://www.ntu.edu.sg/home/ecqsun/
0957-4484/05/081290+04$30.00 © 2005 IOP Publishing Ltd
However, it is surprising that sophisticated experimental [15, 16] and theoretical [17–19] efforts have uncovered
recently that a free-standing nanosolid at the lower end of
the size limit, or clusters containing 10–50 atoms of Ga+ or
IV-A elements, melt at temperatures that are 10–100% or even
higher than the bulk melting point, Tm,b . For example, Ga +39−40
clusters were measured to melt at about 550 K, while the Ga+17
cluster does not melt even up to 700 K compared with the Tm,b
of 303 K [15]. Small Sn clusters with 10–30 atoms melt at least
50 K above the Tm,b of 505 K [16]. Advanced ab initio density functional molecular dynamics simulations suggest that
Ga+13 and Ga +17 clusters melt at 1400 and 650 K [17], and Snn
(n = 6, 7, 10, and 13) clusters melt at 1300, 2100, 2000, and
1900 K, respectively [19]. For a Sn10 cluster, the structural
transition is calculated to happen at 500 and 1500 K [20], and
the structural transition of a Sn20 cluster is calculated to happen
at 500 and 1200 K [21]. Calculations [18] also suggested that
the IV-A elements, Cn , Sin , Gen , Snn (n ∼ 13) clusters melt
at temperature higher than their bulk Tm,b . The C13 cluster
prefers a monocyclic ring or a tadpole structure with the most
probability to appear in the simulated annealing when the temperature is between 3000 and 3500 K. Although the Tm may
be overestimated to some extent for the smallest clusters [19],
the calculated Tm elevation follows the trend of measurement.
The Tm elevation of the smallest Ga and Sn nanosolid is
attributed either to the bond-nature alteration from covalent-
Printed in the UK
1290
Melting point oscillation of a solid over the whole range of sizes
(a)
ci(z) = 2/{1+exp[(12-z)/(8z)]}
Goldschmidt
Feibelman
EC(m = 1)
EC(m = 3)
EC(m = 5)
20
∆d/d; ∆EC/EC (%)
metallic to pure covalent with slight bond contraction [17, 22]
or to the heavily geometrical reconstruction as Ge, Si, and
Sn clusters are found to be stacks of stable tricapped trigonal
prism units [23]. However, consistent insight into the Tm
oscillation in the whole range of sizes (from single atom
to the bulk) is still lacking, though numerous outstanding
models have been developed especially for the Tm elevation or
suppression. Here we show that the intriguing phenomenon
is within the expectation of the recent bond-order-lengthstrength (BOLS) correlation mechanism which indicates the
bond-strength gain as a consequence of atomic coordination
(CN) imperfection [24, 25].
2. Principle
0
-20
-40
-60
12
ci = 2/{1 + exp[(12 − z i )/(8z i )]}
Tm,i ∝ z i E c,i = z i ci−m E c,b .
(1)
Equation (1) formulates well, as shown in figure 1(a),
the bond-order-length premises of Goldschmidt [30] and
Feibelman [31], who indicated that if taking z b = 12 as
standard, the atomic radius will shrink by 3%, 4%, 12%, and
30% when the z i is reduced to 8, 6, 4, and 2, respectively.
Figure 1(a) also shows the CN and m value dependence of
the atomic cohesive energy that sums the binding energy over
all the shortened z i bonds of the lower-coordinated atom.
The atomic cohesive energy is important as it determines the
thermal dynamic behaviour of a solid such as phase transition,
self-assembly growth, structural deviation, activation energy
for atomic diffusion and dislocation in a solid.
As consequences of the BOLS correlation, the CN
imperfection induced bond contraction and potential-well
deepening localize the electrons, which enhances the charge
10
8
6
4
2
Atomic CN (z)
(b)
150
+
Ga 13-17
Sn39-40
Sn-dot
100
∆Tm/Tm(%)
The BOLS correlation mechanism, as detailed in [24–26]
suggests that the CN (or z i ) imperfection of an atom denoted
i at a site surrounding a defect (voids, stacking of fault,
etc) or near the surface edge causes the remaining bonds
of the lower-coordinated atom to contract (with coefficient
ci = di /d, where di is the bond length of the specific i th
atom) spontaneously disregarding the nature of bond or the
structural phase even if it is a liquid. It has been found [27]
recently that a high-density and ordered atomic layer forms
at the free surfaces of liquid Sn, Hg, Ga, and In with a
10% contraction of the spacing between the first and second
atomic surface layers, relative to that of subsequent layers.
The spontaneous process is associated with the intensity of
the rise in bond energy. Consequently, the potential well is
deepened (E i /E b = ci−m , where E is the cohesive energy
and the subscripts denote the specific i th atom and the bulk
atom) and the cohesive energy of the lower-coordinated atom
will change (E c,i /E c,b = z i /z b ci−m , where E c,i = z i E i is the
atomic cohesive energy) correspondingly. The index m is a key
parameter that represents the nature of the bond. For Au, Ag,
and Ni metals, m ≡ 1; for alloys and compounds m is around 4;
for C and Si the value of m has been optimized to be 2.56 [28]
and 4.88 [29], respectively. The m value may vary if the bond
nature evolves with the atomic CN. The CN dependence of the
bond contraction coefficient ci and the critical temperature for
melting are given as [24]:
+
Ga 19-31
Sn500
m=6
m=5
m=3
m=1
m = 1 + 12/{1+exp[(z-2)/1.5]}
50
0
-50
-100
-0.5
0.0
0.5
1.0 1.5
Log (K)
2.0
2.5
Figure 1. (a) BOLS correlation showing the atomic CN dependence
of the bond length and its effect on the atomic cohesive energy,
E c = z i E i , and hence the Tm ∝ E c , that varies with bond nature
indicated with m. Scattered data are from Goldschmidt [30] and
Feibelman [31]. (b) Comparison of the predicted Tm oscillation with
those measured from Ga+13−17 [15, 17], Sn19−31 [16], Ga +39−40 [15],
Sn500 [43] and Sn nanosolid on Si3 N4 substrate [38]. The numerical
fit with the function m(z) = 1 + 12/{1 + exp[(z i − 2)/1.5]} to let m
transit from 7(z i = 2) to 1(z i > 4) indicates the bond-nature
alteration (the m value changes with atomic CN) of the smaller
clusters considered.
(This figure is in colour only in the electronic version)
density in the relaxed region. Bond strengthening enhances
the energy density per unit volume in the relaxed region,
which perturbs the Hamiltonian of an extended solid and the
associated properties such as the band and band-gap widths,
core-level shift, Stokes shift (electron–phonon interaction),
and dielectric suppression. On the other hand, the competition
of bond-order loss and the associated bond-strength gain
dictates the mechanical strength and thermodynamic process
of the solid such as atomic vibration, chemical reactivity, and
phase and thermal stability. Most strikingly, incorporating the
BOLS correlation to the freedom of physical size has enabled
us to elucidate quantitative information of single energy levels
of an isolated (Si, Pd, Au, Ag and Cu) atom and its shift
1291
C Q Sun et al
upon bulk and nanosolid formation [24], and the vibration
frequency of a (Si–Si) dimer bond [32] and bonding identities
such as the length, strength, extensibility, and thermal and
chemical stability in gold monatomic chains [33] and in carbon
nanotubes [28].
Warming up an atom to the melting point, or loosening all
the bonds of the specific atom, requires energy that is a portion
of the atomic cohesive energy [34, 35], Tm ∼ z i E c,i (z i ). For
a spherical dot with radius R j = K j d, the relative change of
the Tm (K j ) is given as [26]:
Tm (K j )
E c (K j ) =
=
γi j (z i /z b ci−m − 1)
Tm,b
E c,b
3
.
(2)
γi j = 3ci /K j
γi j is the portion of atoms in the i th atomic layer which is
counted from the outmost shell to the centre of the solid up
to three. At the lower end of the size limit, γi j = 1, which
means that all the atoms suffer from CN imperfection. The
expression is valid for any size, even a monatomic chain [33].
According to the BOLS correlation, the Tm of an isolated
atom is 0 K because z i = 0. On the base of measurements,
described in [36] and [37], the effective CN of an atom in
a curved surface changes with the particle size in the form
of z 1 = 4(1 − 0.75/K j ) [32]. The CN of the second layer
is z 2 = 6, and z 3 = 12 for an interior atom in closely
packed structures. Geometrical reconstruction may vary the
atomic CN slightly, but with minimization of cohesive energy
(strongest bonding).
3. Results and discussion
Using the dimensionless forms, we compare the model
prediction with the measured Tm change of Sn and Ga+ clusters
in figure 1(b). It is seen that the Tm curves drop universally
at K j > 3 (log(K j ) > 0.5, or z i > 3) disregarding the m
values, and then the Tm curves bend up for higher m values.
The transition point varies depending on the m values. This
can also be seen in figure 1(a) for the atomic cohesive energy.
For an isolated atom (K j = 0.5 or z i = 0), the Tm is zero.
As the smallest clusters are not of spherical shape and they
may subject to structural erratic variation, the equivalent size
specified here might be subject to some modification. It is
surprising that the prediction with an m(z i ) transition from
7(z i = 2) to 1(z i > 4) matches closely the measurement of
Ga+17 , Ga+39−40 , Sn19−31 , and Sn500 clusters and Sn nanosolids
deposited on Si3 N4 substrate as well [38]. Calculations [19]
show that the Tm transition for Sn6−13 happens at Sn7 though
the estimated Tm is subject to experimental confirmation.
Results indicate that the bond nature of Sn–Sn and Ga–Ga
indeed evolves from metallic-covalent to pure covalent as
the atomic CN reduces to much lower values, agreeing with
that uncovered by Chacko et al [17]. This also complies
with theoretical findings that the Al–Al bond for lowercoordinated or distorted Al atoms at grain boundaries [39]
and at free surfaces [40] becomes shorter (∼5%) and stronger
with some covalent characteristics [41]. A metal–nonmetal
transition also occurs in Pd solid containing 101−2 atoms [42].
As demonstrated in [24], strong localization of charges in
1292
the surface region should take the responsibility for bondnature evolution/alteration, band-gap production [22], and
conductivity reduction of a small specimen. Results show that
it is stronger for a IV-A atom to bond with two neighbours
than to bond with three or more atoms due to the bondnature evolution, which may explain why a C13 cluster prefers
a ring or a tadpole structure with each atom having two
bonds, as theoretically predicted by Ho and co-workers [18],
rather than the densely packed tetrahedron structure. It is
expected that covalent Si (m = 4.88) and C (m = 2.56)
clusters should also show Tm elevation/bending at K i <
3, agreeing with theoretical predictions [18]. For a pure
covalent system, such as IV-A elements, the bond strength
increases significantly as the bond contracts without bondnature evolution. For Au, however, the value of m remains at
unity throughout the course of CN reduction upon monatomic
chain formation [33]. Therefore, the bond-nature evolution
may be the unique characteristics of the III-A or IV-A elements
with larger numbers of electrons.
4. Conclusion
In summary, the BOLS correlation premise has enabled
the observed Tm -oscillation over the whole range of solid
sizes to be reconciled to the effect of bond-order loss
and its consequences on the bond-strength gain or bondnature evolution. The modified cohesive energy of the
lower-coordinated system also determines the geometrical
reconstruction, surface lattice/phonon instability, and surface
energy. Therefore, as the physical origin, the BOLS correlation
favours all the existing models. Consistent understanding
evidences the significance of atomic CN imperfection and the
essentiality of the BOLS correlation.
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