1 Introduction to Laser Micro-to-Nano Manufacturing
27
Fig. 1.21 sintering of two
particles with the identical
radiums
x
r
surf
=
4.17δ D s γ sv t
RT r 4
1/6
(1.3.4)
where δ is the surface layer thickness and is estimated as 0.3 nm for Ag [105])
D s is the surface diffusion coefficient calculated by (2), δ sv is the bulk solid-vapor
interfacial surface energy and is estimated as 1.14 J/m
2 for Ag [106]) and is
the molar volume (10.3 cm
3 /mol for Ag). If sintering is driven by grain boundary
diffusion, (1.3.3) becomes
x
r
gb
=
32ωD gb γ sv t
RT r 4
1/6
(1.3.5)
where w is the grain boundary width and is estimated as 0.5 nm for Ag [105]) and
D gb is the grain boundary diffusion coefficient, which can be calculated by (1.1.8).
For lattice diffusion sintering, (1.3.3) becomes
x
r
l
=
42.05D l γ sv t
RT r 3
1/4.78
(1.3.6)
where D l is the lattice diffusion coefficient calculated by (1.1.8). D s = 5 × 10
3 m
2 /s)
for surface diffusion, D gb = 1.2 × 10
5 m
2 /s) for grain boundary diffusion, and D l
= 4.4 × 10
5 m
2 /s) for volume diffusion [107]. E(∞) is bulk activation energy and
is equal to (E(∞) = 2.661 × 10
5 J/mol), (E(∞) = 9 × 10
4 J/mol), and (E(∞) =
18.5 × 10
4 J/mol) for surface diffusion, grain boundary diffusion, and lattice diffusion, respectively [107, 108]). Dependent on different materials, particle size, particle
shape and local temperatures, one of these or two, even three of them may involve the