High-Pressure Crystallization of Glass-Forming Liquids …
29
slow down or accelerate. Regrettably, in experimental studies, it is extremely difficult, if not impossible, to control and separate the individual contributions coming
from the kinetic and thermodynamic factors. Therefore, we are still missing some
key information related to the crystal formation allowing us to make use of it in a
fully aware manner.
In contrast to α-relaxation time, μ cannot be measured directly. At most, it can
be calculated from volumetric and calorimetric data with the use of the following
expression:
μ(T, p) = −
T
T m (0)
S(T, p)dT +
p
p 0
V (T m (0), p)d p
(4)
where V = V liq − V cry (in cm
3 /g) and S = S liq − S cry [in J/(K g)] are the
differences in the specific volumes and entropies of the liquid and crystalline states,
respectively. Thus, calorimetric and volumetric data are required to determine the
T-p evolution of the thermodynamic driving force towards a crystallization. Values
of S(T, P) can be estimated with the use of the following formula:
S(T, P) =
T
T K (P)
C p (T, P)
T
dT
(5)
where C p is the difference in the specific heat between liquid and crystalline phases
(or the glassy state, if needed). The temperature T K in the integration limit is Kauzmann temperature. It can be estimated on the basis of Vogel temperature T 0 , obtained
from the fitting of the isobaric τ α (T ) dependences with the use of Vogel–Fulcher–
Tammann equation (VFT). Since high-pressure calorimetric data are not available in
many cases, C p (T, p) values can be estimated from the heat capacity measurements
at ambient pressure C p (T, p 0 ) and pressure–volume–temperature (PVT) data by
using Maxwell’s thermodynamic relations:
c p (T, p) = c p (T, p 0 ) − T
p
p 0
∂
2 V
∂ T 2
d p
(6)
where c p (T, p 0 ) = a ∗ T + b.
The rates of nucleation and crystal growth form characteristic bell-shaped curves
with the maxima located within the melting temperature T m and the glass transition
temperature T g , see Fig. 2. Depending on the intensity and the extent of overlap of
both curves we can determine good or bad glass-forming ability on cooling from
the melt so as when reheating the material from the glassy state. For example, when
nucleation and growth maxima are well-separated from each other crystallization
can be omitted on cooling. However, on subsequent heating from the glassy state
29
slow down or accelerate. Regrettably, in experimental studies, it is extremely difficult, if not impossible, to control and separate the individual contributions coming
from the kinetic and thermodynamic factors. Therefore, we are still missing some
key information related to the crystal formation allowing us to make use of it in a
fully aware manner.
In contrast to α-relaxation time, μ cannot be measured directly. At most, it can
be calculated from volumetric and calorimetric data with the use of the following
expression:
μ(T, p) = −
T
T m (0)
S(T, p)dT +
p
p 0
V (T m (0), p)d p
(4)
where V = V liq − V cry (in cm
3 /g) and S = S liq − S cry [in J/(K g)] are the
differences in the specific volumes and entropies of the liquid and crystalline states,
respectively. Thus, calorimetric and volumetric data are required to determine the
T-p evolution of the thermodynamic driving force towards a crystallization. Values
of S(T, P) can be estimated with the use of the following formula:
S(T, P) =
T
T K (P)
C p (T, P)
T
dT
(5)
where C p is the difference in the specific heat between liquid and crystalline phases
(or the glassy state, if needed). The temperature T K in the integration limit is Kauzmann temperature. It can be estimated on the basis of Vogel temperature T 0 , obtained
from the fitting of the isobaric τ α (T ) dependences with the use of Vogel–Fulcher–
Tammann equation (VFT). Since high-pressure calorimetric data are not available in
many cases, C p (T, p) values can be estimated from the heat capacity measurements
at ambient pressure C p (T, p 0 ) and pressure–volume–temperature (PVT) data by
using Maxwell’s thermodynamic relations:
c p (T, p) = c p (T, p 0 ) − T
p
p 0
∂
2 V
∂ T 2
d p
(6)
where c p (T, p 0 ) = a ∗ T + b.
The rates of nucleation and crystal growth form characteristic bell-shaped curves
with the maxima located within the melting temperature T m and the glass transition
temperature T g , see Fig. 2. Depending on the intensity and the extent of overlap of
both curves we can determine good or bad glass-forming ability on cooling from
the melt so as when reheating the material from the glassy state. For example, when
nucleation and growth maxima are well-separated from each other crystallization
can be omitted on cooling. However, on subsequent heating from the glassy state