General Concepts of Crystallization: Some Recent …
7
studies as discussed in detail in [7, 12, 17–19]. Again, employing the relation for the
dependence of pressure on temperature along the melting curve obtained as described
above based on Eq. 6, Eq. 10 results in an expression for the determination of the
surface tension along the melting curve.
Quite frequently, the dependence of the surface tension of critical clusters is treated
not in terms of its dependence on pressure and temperature as expressed by above
given relation but in dependence on the size of the critical clusters or its curvature.
A first relation in this respect has been derived already by Gibbs [8] in application to
condensation. It was advanced later by Tolman [20] resulting in an equation of the
form:
σ (R) =
σ ∞
1 +
2δ
R
, σ ∞ = σ ∞ (T m , p m ), δ = δ ∞ (T m , p m )
(12)
Here δ is the Tolman parameter. In accordance with its original definition by
Tolman, it has to be considered as a property of the interface liquid-solid for an
equilibrium coexistence of both phases at a planar interface, i.e., δ = δ ∞ (T m , p m ) is
a function of melting pressure and temperature, σ = σ ∞ (T m , p m ) is the value of the
surface tension for the respective state. However, both Gibbs [8] and Tolman [20] did
not consider phase formation caused by variation of temperature but by variation of
pressure. As mentioned by Tolman: “We shall be concerned with the effect of changes
in radius on surface tension in the case of droplets and vapor composed of a single
substance maintained at some given constant temperature.” Consequently, strictly
speaking, it was not clear so far whether the Tolman equation can be really utilized
at all for the description of melt crystallization if the process is caused by variations
of temperature. This open problem was resolved by us in two recent publications
[7, 19].
It was shown that the Tolman equation can be employed for the description of the
curvature dependence of the surface tension of critical crystallites in one-component
systems if either pressure or temperature is varied. This relation holds also for crystallization in multi-component systems provided the composition and shape of the
critical crystal clusters do not change in dependence on pressure and temperature.
As discussed here earlier this independence of the properties of critical clusters on
pressure and temperature is a basic assumption of CNT. Consequently, employing
basic ideas of CNT, the Tolman equation is a quite appropriate tool for the description of the curvature dependence of the surface tension of critical crystallites if either
pressure or temperature is changed. However, the values of the Tolman parameter
differ for both cases and are given by:
δ
(T )
∞
∼ = σ ∞
1 + γ T (T m , p m )
h m
at p = p m
(13)
respectively:
7
studies as discussed in detail in [7, 12, 17–19]. Again, employing the relation for the
dependence of pressure on temperature along the melting curve obtained as described
above based on Eq. 6, Eq. 10 results in an expression for the determination of the
surface tension along the melting curve.
Quite frequently, the dependence of the surface tension of critical clusters is treated
not in terms of its dependence on pressure and temperature as expressed by above
given relation but in dependence on the size of the critical clusters or its curvature.
A first relation in this respect has been derived already by Gibbs [8] in application to
condensation. It was advanced later by Tolman [20] resulting in an equation of the
form:
σ (R) =
σ ∞
1 +
2δ
R
, σ ∞ = σ ∞ (T m , p m ), δ = δ ∞ (T m , p m )
(12)
Here δ is the Tolman parameter. In accordance with its original definition by
Tolman, it has to be considered as a property of the interface liquid-solid for an
equilibrium coexistence of both phases at a planar interface, i.e., δ = δ ∞ (T m , p m ) is
a function of melting pressure and temperature, σ = σ ∞ (T m , p m ) is the value of the
surface tension for the respective state. However, both Gibbs [8] and Tolman [20] did
not consider phase formation caused by variation of temperature but by variation of
pressure. As mentioned by Tolman: “We shall be concerned with the effect of changes
in radius on surface tension in the case of droplets and vapor composed of a single
substance maintained at some given constant temperature.” Consequently, strictly
speaking, it was not clear so far whether the Tolman equation can be really utilized
at all for the description of melt crystallization if the process is caused by variations
of temperature. This open problem was resolved by us in two recent publications
[7, 19].
It was shown that the Tolman equation can be employed for the description of the
curvature dependence of the surface tension of critical crystallites in one-component
systems if either pressure or temperature is varied. This relation holds also for crystallization in multi-component systems provided the composition and shape of the
critical crystal clusters do not change in dependence on pressure and temperature.
As discussed here earlier this independence of the properties of critical clusters on
pressure and temperature is a basic assumption of CNT. Consequently, employing
basic ideas of CNT, the Tolman equation is a quite appropriate tool for the description of the curvature dependence of the surface tension of critical crystallites if either
pressure or temperature is changed. However, the values of the Tolman parameter
differ for both cases and are given by:
δ
(T )
∞
∼ = σ ∞
1 + γ T (T m , p m )
h m
at p = p m
(13)
respectively: