1 Introduction to Laser Micro-to-Nano Manufacturing
7
particle! According to thermodynamics estimation, a large surface/volume leads to
a size depression effect of the particle melting temperature [39, 40]. Accordingly
to classical Gibbs-Thomson equation, the melting point of particles (T m (d)) can be
expressed as follows [41]
T m (d) = T (∞)
1 −
4σ sl
H f ρ s d
(1.1.5)
where d stands for the particle diameter, T (∞) is the bulk melting point, σ sl is the
solid-liquid interface energy, H f is the bulk heat of fusion, ρ s density of solid and d is
the particle size. In a practical situation, particle surfaces have a high concentration
of defects [40]. As a result, the surface atomic mobility is higher than the inner atoms.
Nanoparticles can even demonstrate more activated behaviors at a temperature lower
than its melting temperatures. Shi et al. indiced a surface melting model to describe
the thermal stability of nanoparticles and/or a 2D system [42],
T m (d) = T m (∞)exp
−
α − 1
d
3h
− 1
(1.1.6)
where h is the monolayer height of the surface atoms, r is the particle diameter, α is
a material constant, relevant to the surface vibration energy. Considering a surface
diffusion is a thermally activated, thus D(d, T ) can be written in the Arrhenius law
D(d, T ) = D o (d)exp
−
E(d)
RT
(1.1.7)
where D 0 is the intrinsic diffusion coefficient, R is the ideal gas constant. E(d) is the
diffusion barrier. Assuming D 0 (d) ~ D 0 (∞) and
E(d)
E(∞)
≈
T m (d)
T m (∞)
Combining with the Arrhenius law, one can obtain the thermally activated
diffusion coefficient as a function of particle diameter [43]
D(d, T ) = D 0 (d)exp
−
E(∞)
RT
exp
−
α − 1
d
3h
− 1
(1.1.8)
This indicates the diffusion is remarkably enhanced at a nanoscale. Equations (1.1.5) and (1.1.8) will significantly influence the nanomanufacturing. In brief,
(1.1.5) indicates a the nanomanufacturing does not require a high energy. For
example, compared to a welding at the macroscale is a Kilo Joule to Mega Joule
procedure, a nanojoining only needs a nanojoule energy [44]. Soldering or even
brazing at a nanoscale can be even realize by a innovative self-heating procedure
7
particle! According to thermodynamics estimation, a large surface/volume leads to
a size depression effect of the particle melting temperature [39, 40]. Accordingly
to classical Gibbs-Thomson equation, the melting point of particles (T m (d)) can be
expressed as follows [41]
T m (d) = T (∞)
1 −
4σ sl
H f ρ s d
(1.1.5)
where d stands for the particle diameter, T (∞) is the bulk melting point, σ sl is the
solid-liquid interface energy, H f is the bulk heat of fusion, ρ s density of solid and d is
the particle size. In a practical situation, particle surfaces have a high concentration
of defects [40]. As a result, the surface atomic mobility is higher than the inner atoms.
Nanoparticles can even demonstrate more activated behaviors at a temperature lower
than its melting temperatures. Shi et al. indiced a surface melting model to describe
the thermal stability of nanoparticles and/or a 2D system [42],
T m (d) = T m (∞)exp
−
α − 1
d
3h
− 1
(1.1.6)
where h is the monolayer height of the surface atoms, r is the particle diameter, α is
a material constant, relevant to the surface vibration energy. Considering a surface
diffusion is a thermally activated, thus D(d, T ) can be written in the Arrhenius law
D(d, T ) = D o (d)exp
−
E(d)
RT
(1.1.7)
where D 0 is the intrinsic diffusion coefficient, R is the ideal gas constant. E(d) is the
diffusion barrier. Assuming D 0 (d) ~ D 0 (∞) and
E(d)
E(∞)
≈
T m (d)
T m (∞)
Combining with the Arrhenius law, one can obtain the thermally activated
diffusion coefficient as a function of particle diameter [43]
D(d, T ) = D 0 (d)exp
−
E(∞)
RT
exp
−
α − 1
d
3h
− 1
(1.1.8)
This indicates the diffusion is remarkably enhanced at a nanoscale. Equations (1.1.5) and (1.1.8) will significantly influence the nanomanufacturing. In brief,
(1.1.5) indicates a the nanomanufacturing does not require a high energy. For
example, compared to a welding at the macroscale is a Kilo Joule to Mega Joule
procedure, a nanojoining only needs a nanojoule energy [44]. Soldering or even
brazing at a nanoscale can be even realize by a innovative self-heating procedure