chaPter 7 nanomaterials: Properties
212
surface phases in addition to the typical volume phases. In addition, for 0-D and 1-D nanomaterials, the curvature of the surface
is usually very pronounced. Consequently, for nanomaterials the
melting temperature is size dependent. In general, surface effects
can be expressed mathematically by introducing an additional
term (ΔG Surface ) into the total energy change (ΔG Total ) resulting from
the solid-liquid transformation. This is given by
∆
∆
∆
G
G
G
Total
Bulk
Surface
=
+
(7.9)
where ΔG Bulk is the free energy of the bulk material given by
∆G
L T T
T
V
Bulk
L
=
−
(
)
0 0
0
(7.10)
and L o is the latent heat of melting, T o is the melting temperature
of the bulk material, T is the melting point of the extended system,
where surface effects are included, and V L is the volume of liquid.
When the surface of a body is increased, the change in surface
energy is given by
∆
∆
G
A
Surface = γ
(7.11)
where γ is the surface tension and ΔA is the increment in surface
area. Evidently, at the melting temperature, a layer of liquid with
thickness t is formed on the surface and moves at a certain rate into
the solid. During the change, a new liquid surface and liquid/solid
interface are created, whereas the solid surface is destroyed (see
Figure 7.13). In other words, ΔG surface can be written as
∆G
A
A
A
Surface
L L
SL SL
S S
=
+
−
γ
γ
γ
(7.12)
where A L is the new liquid surface area, σ L is the surface energy
of the liquid per unit area, A SL is the new liquid/solid interfacial
area, γ SL is the solid/liquid interfacial energy per unit area, A S is the
surface area of the solid destroyed, and γ S is the solid surface energy
per unit area.
At equilibrium, the solid core of radius r has the same chemical
potential as the surrounding liquid layer of thickness t, which occurs
when the differential ∂ΔG Total /∂t = 0. For a sphere, this happens
when
L T T
T
r t
O
SL
0
0
2
−
(
) = −
γ
(7.13)
Assuming t → 0, which represents the appearance of the first
melting, the upper melting temperature for a sphere can be found
from the expression
Figure 7.13
Upon formation of a liquid layer on the
nanoparticle’s surface, an interface between the
liquid layer and the solid core develops.
Solid
Solid
Nanoparticle
t
Liquid surface
212
surface phases in addition to the typical volume phases. In addition, for 0-D and 1-D nanomaterials, the curvature of the surface
is usually very pronounced. Consequently, for nanomaterials the
melting temperature is size dependent. In general, surface effects
can be expressed mathematically by introducing an additional
term (ΔG Surface ) into the total energy change (ΔG Total ) resulting from
the solid-liquid transformation. This is given by
∆
∆
∆
G
G
G
Total
Bulk
Surface
=
+
(7.9)
where ΔG Bulk is the free energy of the bulk material given by
∆G
L T T
T
V
Bulk
L
=
−
(
)
0 0
0
(7.10)
and L o is the latent heat of melting, T o is the melting temperature
of the bulk material, T is the melting point of the extended system,
where surface effects are included, and V L is the volume of liquid.
When the surface of a body is increased, the change in surface
energy is given by
∆
∆
G
A
Surface = γ
(7.11)
where γ is the surface tension and ΔA is the increment in surface
area. Evidently, at the melting temperature, a layer of liquid with
thickness t is formed on the surface and moves at a certain rate into
the solid. During the change, a new liquid surface and liquid/solid
interface are created, whereas the solid surface is destroyed (see
Figure 7.13). In other words, ΔG surface can be written as
∆G
A
A
A
Surface
L L
SL SL
S S
=
+
−
γ
γ
γ
(7.12)
where A L is the new liquid surface area, σ L is the surface energy
of the liquid per unit area, A SL is the new liquid/solid interfacial
area, γ SL is the solid/liquid interfacial energy per unit area, A S is the
surface area of the solid destroyed, and γ S is the solid surface energy
per unit area.
At equilibrium, the solid core of radius r has the same chemical
potential as the surrounding liquid layer of thickness t, which occurs
when the differential ∂ΔG Total /∂t = 0. For a sphere, this happens
when
L T T
T
r t
O
SL
0
0
2
−
(
) = −
γ
(7.13)
Assuming t → 0, which represents the appearance of the first
melting, the upper melting temperature for a sphere can be found
from the expression
Figure 7.13
Upon formation of a liquid layer on the
nanoparticle’s surface, an interface between the
liquid layer and the solid core develops.
Solid
Solid
Nanoparticle
t
Liquid surface