1
Material Properties and characterization
D. Poulikakos (2nd Week)
On the Melting of Nanoparticles
(N. Bieri and D. Poulikakos)
1.1 Introduction
Driven by the continuous need for advancement in the state of the art of micro- and
nanomanufacturing in a host of fields, exemplified prominently by electronics, new
methodologies and processes need to be developed, after overcoming formidable scientific
challenges, to engineer nanostructures with desired properties and functionality. At the sane
time, the emergence of consistent manufacturing methods of ultra fine particles (UFP) (or,
after the more popular terminology-nanoparticles) of many materials, is creating a vivid
activity related to their utilization. The potential they possess through remarkable
thermophysical properties (significantly different than those of their bulk counterparts), can
play a pivotal role in the development of emerging technologies of the kind mentioned above.
The reason for using nanoparticles in this work lies in the significant melting temperature
depression compared to their bulk counterpart (Buffat and Borel, 1976).
The handling of nanoparticles by themselves is difficult because a dust free and inert
atmosphere is needed. Schaefer et al. (1995) and Ben Ali et al. (2002) manipulated single
nanoparticles with an atomic force microscope whereas Krinke et al. (2001) directly
deposited nanoparticels on a Si substrate from the gas phase attracted by charge patterns
(created by contact charging) and Xu et al. (2004) used laser-based direct deposition of
nanoparticles from the gas phase on a substrate to build one- and two-dimensional structures.
Another method to handle nanopraticle is their dispersion in a carrier-liquid, a so called
nanoparticle suspension or nanoink. Many applications utilize nanoparticle suspensions with a
volatile solvent as carrier-liquid which evaporates naturally or by heating leaving the
nanoparticles in the desired two- or three-dimensional structures, either by self assembling
(Maillard et al., 2000; Stowell and Korgel, 2001; Shah et al., 2003) or in combination with
2
pre-patterning the substrate (Liu et al., 2002; Santhanam et al., 2003), or by using
prefabricated traces or moldings to fill with the nanoparticle suspension (Rogers et al., 1998;
Iwashige et al., 2001) or by printing the nanoink into the desired pattern (He et al., 2000;
Fuller et al., 2002; Szczech et al., 2002; Huang et al., 2003; Magdassi and Ben Moshe, 2003;
Voit et al., 2003). The self assembling technique is restricted by the limited number of
possible patterns without pre-patterning and the pre-patterning itself counteracts the simplicity
of the process. Printing nanoink with drop-on-demand (DOD) ink jet technology was
accomplished by several authors who attempted different methods to produce conductive
patterns and to manipulate the feature size. Szczech et al. (2002) baked the printed nanoink
pattern in the oven. The achievable feature size of O (120 µm) is limited by the smallest
droplet-size of the DOD ink jet printer. In order to reduce the features size of the printed line,
Fuller et al. (2002) heated the substrate to evaporate the carrier-liquid upon impact and
subsequently baked the nanoparticles in the oven. He et al. (2000) fabricated designed
architectures of Au nanoparticles with printed self-assembled monolayer as templates.
Organic-encapsulated gold nanoparticles are printed and subsequently annealed at
temperatures < 200 °C to form low resistance conductor pattern by Huang et al. (2003).
The reason for using nanoparticles lies in the scientifically reduced melting temperature
of approximately 400 °C for single nanoparticles with a mean diameter of 4 nm compared to
1063 °C for bulk gold (Buffat and Borel, 1976). Microfabrication of electronics and
mechanical structures are commonly accomplished with lithography which is a time
consuming expensive process. The use of DOD ink jet printer is an alternate approach of
depositing conducting material on a substrate with the advantage of the precise drop wise
deposition of small amounts of liquid at desired locations saving expensive materials. In
lithography a metal coating is applied over the entire substrate, followed by a photoresist. A
mask of the line pattern is placed on the photoresist. After exposure, the developed photoresist
prevents the covered metal from being etched away in a chemical bath. This chemical etching
produces highly toxic waste which is not an issue of the introduced method. Drop-on-demand
ink jet printing is successfully used for MEMS packaging with molten solder picoliter
droplets (Hayes et al., 1992; Attinger et al., 2000; Haferl et al., 2000; Haferl and Poulikakos,
2003), metallo-organic decomposition ink (Teng and Vest, 1988) and organic light emitting
materials (Bharathan and Yanga, 1998; Hebner et al., 1998; Percin G, 1998; Sirringhaus et al.,
2000; Calvert et al., 2002; Blanchet et al., 2003; Burns et al., 2003) but the usage of molten
3
gold
requires
temperatures
higher
than
the
melting
point
of
bulk
gold
(1063 °C) which exceeds the temperature range of ink jet technology.
1.2 Melting point reduction of nanoparticles
The influence of the surface and in particularly its shape (curvature) on the melting point of a
material is gaining importance with decreasing size of the sample (particle) under
investigation. The effect of surface energy on the melting of small particle was first
considered in 1909 by Pawlow (1909a; 1909b) who predicted that there would be a change in
melting point from the bulk value inversely proportional to the particle radius. He identified
the melting temperature as the temperature (triple point) where a solid particle and a liquid
particle of identical mass and their vapor are in equilibrium. Technical difficulties prevented
an experimental proof until Takagi (Takagi, 1954) observed this effect on thin metal films in
1954. Several authors conducted experiments on different materials and particle sizes and
defined different theoretical models to explain the so called “thermodynamic size effect”
(Reiss and Wilson, 1948; Hanszen, 1960; Semenchenko, 1962; Wronski C. R. M., 1967;
Sambles, 1970; Sambles et al., 1970; Berman and Curzon, 1974; Peppiatt, 1975; Peppiatt and
Sambles, 1975; Buffat and Borel, 1976; Curzon et al., 1976; Hasegawa et al., 1980). Sambles
(1970) compared in his thesis his experimental results with three different theories assuming
a) equilibrium of a solid particle, a liquid particle and their vapor (Pawlow, 1909a; Pawlow,
1909b), b) equilibrium of a solid particle within an infinite liquid (Rie, 1923) and
c) equilibrium of a solid body surrounded by a thin liquid layer which in turn is surrounded by
vapor (Reiss and Wilson, 1948; Hanszen, 1960; Wronski C. R. M., 1967). He derived the
equations for the melting temperature reduction again and fit the unknown parameters in the
equations with values evaluated in his experimental data. He rejected the theory b) of the solid
particle within a liquid and found for the other two theories correspond well with his
experimental results.
The reduced melting temperature, T, for a simultaneous existence of a solid particle and
liquid particles of identical mass and a vapor phase reads (Pawlow, 1909a; 1909b; Sambles,
1970):
4
(1.1)
where TM is the melting temperature of the bulk phase, Δhls is the latent heat of melting,
subscripts s, l and v mean solid, liquid and vapor, ρ and γι is the density and the surface
tension, respectively. The theory of Pawlow predicts too high temperatures for small particles
(Borel, 1981).
The temperature reduction of a solid particle embedded in a thin molten layer
surrounded by a vapor phase is given by (Hanszen, 1960; Sambles, 1970):
(1.2)
where
is the radius which would characterize the particle at its melting point assuming that
it remains entirely in the solid form, δ is the liquid layer thickness and γsl is the solid-liquid
interfacial tension.
Buffat and Borel (1976) measured the melting temperature of gold particles having
diameters down to 20 Å and compared the results with the two investigated and accepted
phenomenological models by Sambles (1970). They derived the reduced melting temperature
for a simultaneous existence of a solid particle and liquid particles of identical mass and a
vapor phase of Pawlow again and extended the derivation by considering a higher-order
approximation of the chemical potential leading to:
(1.3)
where cp is the specific heat capacity at constant pressure, ηs = -(1/γ)∂γ/∂T, α and χ is the
linear-expansion coefficient and isothermal compressibility coefficient, respectively.
Neglecting the second order terms of Eq. (1.3) leads to the same equation as derived by
Pawlow (Eq. (1.1)).
5
The “thermodynamic size effect” is still investigated with new more accurate
experimental methods and improved phenomenological models (Borel, 1981; Solliard, 1984;
Frenken et al., 1986; Garrigos et al., 1989; Nenow and Trayanov, 1990; Wautelet, 1991;
Kofman et al., 1994; Lai et al., 1996; Bachels and Güntherodt, 2000) and with molecular
dynamics simulations (Ercolessi et al., 1991; Chushak and Bartell, 2001; Arcidiacono et al.,
2004).
Melting Temperature (TM), K
1400
Hanzen, Sambles
Pawlow, Sambles
1200
1000
800
600
400
0
50
100
150
200
Particle radius (rp), Å
250
Fig. 1.1: Melting temperature of gold particles as a function of particle radius for two
models: a) a simultaneous existence of a solid particle and liquid particles of identical mass
and a vapor phase (Pawlow, 1909b; Sambles, 1970) and b) solid particle embedded in a thin
molten layer surrounded by a vapor phase (Hanszen, 1960; Sambles, 1970). All needed
physical values are taken from Buffat and Borel (1976).
6
The reduced melting temperature of small particles can be derived in the following way.
Consider a one component capillary system consisting of two phases, a continuous α phase
and a β phase, which is a spherical drop of radius r within the phase α, in equilibrium. Phase
β is a pure liquid drop and the α phase is its vapor. Figure 1.1 shows a capillary system
consisting of two phases with liquid / vapor interface. The freely moving piston maintains a
constant pressure of phase α, pα.
Fig. 1.2: Concept of a capillary system of two phases with liquid / vapor interface. The freely
moving piston maintains a constant pressure pα inside the system.
Assuming that a differentially small amount of the droplet evaporates under constant vapor
pressure, the Gibbs-Duhem equation for the two phases is:
Phase α:
(1.4)
Phase β:
(1.5)
Where Ni is the number of atoms in the i phase. The chemical potential, µi, includes the
surface energy of the phase.
The free energy change over the entire system, GT, at equilibrium is zero, because GT is a
minimum at equilibrium
(1.6)
7
(1.7)
The subtraction of Eq. (1.5) from Eq. (1.4) together with Eq. (1.6) yields:
(1.8)
The Laplace equation (with dpv= 0) is defined as:
(1.9)
Substituting Eq. (1.9) into Eq. (1.8) and using the latent heat of evaporation, Δhvl which is
defined as
, results in:
(1.10)
The accurate solution of Eq. (1.10) by integration from ∞ to rl with the assumption Δhvl ≠ f(r)
yields:
(1.11)
(1.12)
Equation (1.12) is well known as the Thompson equation where T0 is the temperature of the
bulk material.
Solving Eq. (1.12) by a Taylor series of the logarithm accurate to 1st order the equation results
in:
(1.13)
8
The influence of curvature on the equilibrium temperature of a solid particle with its vapor
can be similarly derived. The phase β' is a pure solid particle and the phase α' is its vapor. The
pressure pα' is again constant. The Gibbs-Duhem equations for the two phases are:
Phase α':
(1.14)
Phase β':
(1.15)
Again the change in the Gibbs energy of the whole system is zero and therefore the change in
the chemical potential is zero, resulting again in
.
Neglecting the crystalline structure of the solid by assuming a perfect spherical particle the
Laplace equation is (with dpv = 0):
(1.16)
Similar to the previous case Eq. (1.14) is subtracted from Eq. (1.15) and the latent heat of
sublimation, Δhvs, which is defined as,
together with Eq. (1.16) results in:
(1.17)
The accurate solution (integration from ∞ to rs) of Eq. (1.17) (with the assumption Δhvs ≠ f(r))
is:
(1.18)
Solving Eq. (1.18) by a Taylor series of the logarithm accurate to 1st order results in:
(1.19)
9
The equilibrium condition of a solid and a liquid particle of identical mass in vacuum can be
derived simply by combining the liquid / gas system with the solid / gas system by subtracting
Eq. (1.10) from Eq. (1.17), resulting in:
(1.20)
Using the latent heat of melting, Δhsl which is defined as
and the same
integration limits as before, the Eq. (1.20) results in:
(1.21)
In the case of melting T0 is the melting temperature, TM, of the bulk solid.
The exact solution of Eq. (1.21) and assuming
and hsl ≠ f(r) is:
(1.22)
which is called the Semenchenko equation.
Solving Eq. (1.22) by a Taylor series of the logarithm accurate to 1st order Eq. (1.22) results
in the Pawlow equation:
(1.23)
The work of Kofman et al. (1994) investigated the surface melting of spherical and nonspherical nanometric lead inclusions and proved experimentally the existence of a liquid layer
below the melting temperature at the surface of the inclusions whose thickness is much larger
than that observed on the bulk (zero curvature). The proposed phenomenological model is
based on the minimization of the Gibbs free energy in equilibrium and shows good agreement
with the experimental results, therefore the derivation of Kofman et al. (1994) is discussed.
10
In a first approach Kofman et al. (1994) showed the existence of surface melting for surfaces
with no curvature. Figure 1.2 shows the two possible configurations of a solid with a flat
surface in contact with a fluid at the melting temperature. In configuration (1) the solid is in
contact with vacuum or its vapor whereas in configuration (2) the solid is in contact with its
liquid melt. The liquid wets the solid well. He defines the change in surface energy between
configurations (1) and (2) as:
(1.24)
where γi are the surface tensions (s = solid, l = liquid and v = vapour). Because the liquid wets
the solid, configuration 2 is energetically favoured and S is positive.
Fig. 1.3: The two configurations: solid / external medium with (2) or without (1) an intermediate liquid layer (Fig. 10 of Kofman et al., 1994).
The Gibbs free energy of configuration (1) at constant pressure and temperature is:
(1.25)
where A, µs and N is the free surface area, the chemical potential of the solid phase and the
number of atoms of the solid.
In configuration (2), N' atoms are in the liquid state. Neglecting the volume change, the Gibbs
free energy is:
(1.26)
11
where Γ depends on the thickness δ of the liquid layer. For δ = 0, Γ = γsv and Γ = γsl + γlv if
the layer is a bulk liquid. For thin liquid layers the value Γ is an intermediate value which
accounts for the interactions between the two layers.
(1.27)
where ξ is the characteristic length describing the short range interactions in liquid metals.
The system equilibrium is obtained by the minimization of
(1.28)
Using the approximation (close to the melting temperature TM):
(1.29)
where V= Aδ is the volume of the melted layer, ρl the density of the liquid and
.
(1.30)
The liquid layer thickness δ in equilibrium at temperature T is given by:
(1.31)
12
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