A thermodynamic model for the melting of supported metal nanoparticles
A thermodynamic model for the melting of supported metal
nanoparticles
S. C. Hendy∗
Industrial Research Ltd, Lower Hutt, New Zealand and
MacDiarmid Institute for Advanced Materials and Nanotechnology,
School of Chemical and Physical Sciences,
Victoria University of Wellington, New Zealand
(Dated: November 4, 2018)
Abstract
We construct a simple thermodynamic model to describe the melting of a supported metal
nanoparticle with a spherically curved free surface both with and without surface melting. We
use the model to investigate the results of recent molecular dynamics simulations, which suggest
the melting temperature of a supported metal particle is the same as that of a free spherical
particle with the same surface curvature. Our model shows that this is only the case when the
contact angles of the supported solid and liquid particles are similar. This is also the case for the
temperature at which surface melting begins.
http://arxiv.org/abs/0704.0393v1
INTRODUCTION
Despite decades of study, the melting of nanoparticles continues to generate interest
[1, 2, 3, 4]. In general, the melting temperature of spherical nanoparticles has been found
to decrease in proportion to the surface area to volume ratio of the particle [5], as the
surface free energy of a molten droplet is less than that of the corresponding solid particle.
Although free nanoparticle calorimetry has advanced considerably in recent years [6, 7],
most experimental determinations of the melting points of nanoparticles are conducted with
supported particles (gold [8], lead [9] and tin [10], for example). The melting of supported
nanoparticles is also important in carbon nanotube growth and other catalytic processes
[11, 12], and for the stability of devices assembled from nanoparticles [13, 14]. Thus it is of
interest to study the effect of the substrate on the melting point of a supported nanoparticle.
Recent molecular dynamics simulations [15] of supported iron nanoparticles with a
strongly interacting substrate found that the melting point of the particles was reduced
in inverse proportion to the equilibrium surface curvature that results as they relax to wet
the substrate. This statement also holds in the free particle limit since the curvature of a
free spherical particle of radius a is proportional to its surface to volume ratio, 3/a. Inter-
estingly, the simulations in Ref. [15] found that the constant of proportionality between the
decrease in melting point and the surface curvature did not depend on whether the particle
was supported or free. In other words, the melting temperature of a supported particle that
has a free surface with radius of curvature a, was found to be the same as that of a free
spherical particle with the same surface curvature. The simulated nanoparticles in Ref. [15]
also exhibited surface melting prior to complete melting. Surface melting is phenomena
thought to occur both on certain bulk metal surfaces [16] and in certain metal nanoparticles
[17, 18].
In this paper we use a simple thermodynamic model to investigate the role of the sub-
strate in both melting and surface melting of metal nanoparticles. Our model suggests that
the result in Ref. [15], that the relative decrease in melting point is proportional to the solid
particle surface curvature, only holds when the contact angles of the supported solid and
liquid particles with the substrate are similar. We also show that supported clusters will
exhibit surface melting under certain circumstances, and that the surface melting tempera-
ture in free and supported particles in clusters with same surface curvature is the same only
when the contact angles of the supported solid and liquid phases coincide.
GEOMETRY OF SUPPORTED PARTICLES
We start by considering a solid nanoparticle, initially spherical with radius a, that is
placed on a flat substrate. We neglect the effect of faceting, curvature dependent surface
energies and internal strains due to epitaxial mismatch with the substrate. Furthermore, we
will assume that the particle has relaxed to its equilibrium geometry i.e. that the nanopar-
ticle has relaxed to ”wet” the substrate. Provided the nanoparticle is heated sufficiently
slowly, the particle should relax to this geometry prior to melting. With these simplifying
assumptions, the geometry of the relaxed particle can be approximated by a spherical cap,
as shown in Fig. 1, with dimensions parameterised by either the cap height H , or radius of
curvature of the free surface R, which minimizes the surface energy of the nanoparticle and
substrate.
The surface energy Γ of the system can be written as:
Γ = 2πRHγs + πH(2R−H)(γsb − γb) + Γb (1)
where γs is the surface energy density of the free particle surface, γb is the surface energy
density of the substrate, γsb is the particle-substrate interfacial energy density and Γb is the
total energy of the bare substrate. We will assume that the density of the particle ρs does
not depend on the contact angle so that the volume of the supported particle remains the
same as that of the free particle.
Writing the volume of the particle as a function of H and R, it is then straightforward
to show that Γ is minimized if H = − (∆γsb/γs)R where ∆γsb = γb − γs − γsb. We note
that ∆γsb is often called the spreading parameter in the context of wetting phenomena: if
∆γsb > 0 then the particle will relax to fully wet the substrate. Here we are interested in the
case where the particle does not fully wet the substrate (contact angles greater than zero)
i.e. when ∆γsb < 0 and H/R = −∆γsb/γs > 0 at equilibrium. In fact this minimum value
of Γ can be written as
Γ∗ = 2πγs
a2 + Γb, (2)
where R∗s is the corresponding radius of curvature of the supported solid nanoparticle, given
R∗s =
(3 + ∆γsb/γs)
. (3)
Thus Γ∗ and R∗s are the equilibrium surface energy and radius of curvature of the particle
respectively. Note that the contact angle of the particle can range from 0 to 180 degrees
depending on the value of the spreading parameter ∆γsb.
MELTING AND SURFACE MELTING
In what follows we will assume that the density of the solid and liquid phases are identical
i.e. ρs = ρl = ρ. We first consider the situation in which there is no surface melting. In this
case, melting will occur at a temperature when the free energy of the solid particle wetting
the substrate is equal to that of the corresponding liquid droplet wetting the substrate.
If γl is the surface energy density of the free liquid droplet and R
l is the corresponding
equilibrium radius of curvature, then the difference in free energy between the solid and
liquid is
fs − fl + 3
where fs (fl) is the bulk free energy density of the solid (liquid). Now, using fl − fs =
ρL (1− T/Tc), where L is the latent heat of melting and Tc is the bulk melting temperature,
we find the melting temperature Tm of the supported particle is given by:
Tm = Tc
= T freem (R
ρR∗sL
Tc (5)
Thus if R∗s = R
l = R
∗ then we recover the result of Ref. [15], namely that Tm = T
In other words, if the contact angles of the solid and liquid droplets are equal, the melting
temperature of the supported particle is the same as that of a free particle with an identical
surface curvature, a = R∗. However, if the curvature of the supported liquid particle is dif-
ferent from that of the supported solid particle, it can be seen that the melting temperature
will deviate from that found in Ref. [15].
Now we consider surface melting as illustrated in figure 1 which is thought to occur in
many metals prior to melting [16]. We are interested in determining the onset of melting,
when the solid particle is wet by a thin layer of melt (thickness δ) at the solid-vapor interface.
We will assume that this melt forms a layer of uniform thickness with a geometry like that
represented in figure 1 with δ = R− r = H − h. The total free energy of the surface melted
particle is then a function of δ: F (δ) = Vs(δ)(fs − fl) + V fl +Γ(δ) where Vs(l) is the volume
of the solid (liquid) and Γ is the thickness dependent surface energy. In particular
Γ = π (2RHγl + r(2r − h)γsb + δ(2R− δ)γlb + 2rhγsl(δ))
where γsl(δ) = γsl+∆γsl exp (−δ/ξ) and ξ is a correlation length that describes the thickness
dependence of the interfacial energy in thin metallic liquid films [16] (in Pb, for example, ξ
has been measured to be ∼ 0.6 nm [19]). As the surface melting proceeds, the curvature of
the particle will relax to minimize the free energy i.e R∗ = R∗(δ) where R∗ minimizes the
free energy F for a given δ.
In an isolated spherical nanoparticle of radius a, by minimizing the free energy F (δ) with
respect to δ and setting δ = 0, one can show that surface melting begins at a temperature,
Ts(a) given by
T frees (a) = Tc
(γs − γl)
. (6)
provided ∆γsl > 0 and a > ξ(γs − γl)/∆γsl (if a is less than this, full melting will precede
surface melting i.e. T frees > T
m [17], and equation (5) will hold).
For surface melting to occur in a supported solid nanoparticle with equilibrium curvature
Rs, a minimum in the free energy F (δ) must appear at δ = 0. It is straightforward to show
that a minimum in F at δ = 0 occurs at the temperature Ts:
Ts (Rs) = T
s (Rs) +
γs∆γlb − γl∆γsb
= T frees (Rs) +
cos θs − cos θl
1− cos θs
where θs and θl are the contact angles for solid particle and liquid particle respectively
(defined via Young’s relation γs(l) cos θs(l) = γb − γs(l)b). Once again, if the contact angles of
the solid and liquid droplets are equal, then the temperature at which surface melting occurs
is identical to that of a free particle with the same surface curvature, Rs i.e. Ts = T
s (Rs).
Further, if cos θs > cos θl, so that the substrate favors contact with the solid over that with
the liquid, the corresponding Ts increases and vice versa.
Complete melting will occur once the free energy of the surface melted particle, F (δ),
equals that of the corresponding liquid droplet, Fl i.e. at the temperature Tm and liquid film
thickness δm which satisfy F (δm) = Fl. It is not possible to obtain an analytic expression
for δm or Tm, but numerical solutions to the resulting equations are shown in figure 2 as
a function of Rs for Pb particles. The figures clearly show the strong dependence of the
melting temperature on the liquid droplet contact angle: a difference of ∼ 10o in the molten
particle contact angle can shift the melting point by ∼ 100 K. Note that the melting point
of a free particle with radius Rs no longer coincides with that of a supported particle with
radius of curvature Rs when cos θs = cos θl, as in general the radius of curvature of the
critical surface melted droplet will not be that of the solid particle (although the curves lie
close to each other).
CONCLUSION
We conclude that the melting temperature (and surface melting temperature, if the par-
ticle exhibits surface melting) of supported nanoparticles depends on the radius of curvature
(or the contact angle) of both the supported solid and liquid particles. In general, we do
not expect these curvatures to be the same: on a non-ideal solid substrate for example,
epitaxial effects may favor one phase over the other. It is likely that the ideal substrate used
in Ref [15] resulted in very similar solid and liquid particle contact angles. We have shown
that is only in this ”ideal” case that the melting temperature of free and supported particles
with the same curvature is coincident, whether they exhibit surface melting or otherwise.
Thus, results from free particle melting, where the curvature of the solid and liquid particles
remain substantially the same, have only limited applicability to supported particle melting.
∗ Electronic address: s.hendy@irl.cri.nz
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FIG. 1: The model for the geometry of a supported nanoparticle in equilibrium. We assume that
the particle is a spherical cap of height H and radius of curvature R (left - the dashed lines simply
illustrate the radius of curvature). At the onset of surface melting, we assume that the geometry is
close to that of the solid particle in its equilibrium geometry and that that the solid particle (radius
of curvature r and height h) is initially wet by a molten layer of uniform thickness δ = R−r = H−h
(right).
2 4 6 8 10 12 14
γlb=0.05
γlb=0.15
γlb=0.10
Rs (nm)
2 4 6 8 10 12 14
cos θs=cos θl
γlb=0.05
γlb=0.15
γlb=0.10
Rs (nm)
2 4 6 8 10 12 14
γlb=0.05
γlb=0.15
γlb=0.10
Rs (nm)
2 4 6 8 10 12 14
cosθs=cosθl
γlb=0.05
γlb=0.15
γlb=0.10
FIG. 2: The melting temperature Tm and critical liquid film thickness δm for supported Pb clusters
as a function of the radius of curvature Rs of the relaxed solid particle for γlb = 0.05, 0.10, 0.15 J
m−2 and in the case where cos θs = cos θl (γlb ≃ 0.13 J m
−2). Also shown is the melting temperature
of a free particle with radius Rs. Other surface energies used are γsv = 0.61, γlv = 0.48, γsl = 0.05,
γb = 0.25 and γsb = 0.1 J m
−2 giving a contact angle of 75.8o for the solid supported cluster, and
contact angles for the liquid droplets ranging from 78.0o to 65.4o respectively. Other parameters
used were ξ = 0.63 nm, ρ = 10950 kg m−3, L = 22930 J kg−1 and Tc = 600.65 K [9].
	Introduction
	Geometry of supported particles
	Melting and surface melting
	Conclusion
	References
