General Concepts of Crystallization: Some Recent …
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p α − p β ∼ = g(T, p)
g(T, p) =
k
i=1
ρ iα
μ iβ
T, p,
x iβ
− μ iα (T, p, {x iα })
(5)
Accounting in addition to the assumed independence of the properties of the
critical clusters on pressure and temperature, for both stoichiometric and nonstoichiometric crystallization, the thermodynamic driving force can be written then
as:
g(T, p) ∼ = h m
T m − T
T m
1 − γ T (T m , p m )
(T m − T )
2T m
+ p m v m
p − p m
p m
1 − γ p (T m , p m )
( p m − p)
2 p m
(6)
Here (T m , p m ) are temperature and pressure at a particular equilibrium state along
the melting curve. Specific properties of the system under consideration are reflected
here by the melting entropy, s m , or the melting enthalpy, h m , and the differences
of the volumes between liquid and crystal phases per unit volume of the crystal phase:
s(T, p) =
S liquid
T, p,
x iβ
− S crystal (T, p, {x iα })
V crystal (T, p, {x iα })
v(T, p) =
V liquid
T, p,
x iβ
− V crystal (T, p, {x iα })
V crystal (T, p, {x iα })
(7)
The parameters γ T , and γ p are defined via:
γ T (T m , p m ) =
c p (T m , p m )
s m
, γ T p (T m , p m ) =
p m κ T (T m , p m )
v m
(8)
Here c p is the specific heat per unit volume and κ T is the isothermal compressibility, given by:
c p = T
ds
dT
p
, ,c p (T m , p m ) = c
(liquid)
p
(T m , p m ) − c
(crystal)
p
(T m , p m )
κ T = −
1
V
dV
d p
T
, ,κ T (T m , p m ) = κ
(liquid)
T
(T m , p m ) − κ
(crystal)
T
(T m , p m )
(9)
Employing these results and the basic relations utilized in their derivation, it has
been shown by us that at the Kauzmann temperature [11], corresponding to states
where the specific entropies of glass-forming melt and crystal coincide, the thermodynamic driving force has a maximum in dependence on temperature. In addition,
similarly to the mentioned well-known notation of a Kauzmann temperature, the