3.7 Kinetic Analysis of SANS Data
A more in-depth analysis of TR-SANS data collected during the first freezing step
(Fig. 16a) indicates that the structural changes that occur at correlation distances
between 60–300 nm and 7–60 nm are different and involve different characteristic
times [77]. Therefore, the scattering data collected in the relevant q ranges were
analyzed separately by evaluating the Lorentz-corrected integral Ψ of the scattering
cross-section in the q range between q 1 ¼ 0.022 and q 2 ¼ 0.108 nm
À1 for Ψ low (t)
and between q 1 ¼ 0.108 and q 2 ¼ 0.881 nm
À1 for Ψ high (t) as a function of time.
This kind of integral corresponds to the scattering invariant when the integration
limits in q are extended from zero to infinite. For this reason, the function Ψ
corresponds to a sort of “reduced scattering invariant.” A working hypothesis is
that, in the early stages of gelation, the SANS patterns contain additive contributions from the aggregated and non-aggregated phases and that the interaction
between these two phases may be neglected. Thus, changes in the reduced scattering invariant Ψ low (t) probe structural changes occurring in the initially homogeneous solution during permanence at subzero temperatures at length scales of the
order of hundreds of nanometers, essentially due to water crystallization [77]. By
contrast, changes in the reduced scattering invariant Ψ high (t) probe structural
changes occurring at length scales of the order of nanometers, essentially due to
PVA crystallization inside the unfrozen liquid microphase [77].
The values of the reduced scattering invariant calculated from the SANS data
collected during the first freezing step of PVA/D 2 O solutions are reported in Fig. 17
as a function of the permanence time of the solutions at À13
C. In Fig. 17a
0 (inset),
the values of Ψ low (t) of pure D 2 O are also shown for comparison.
The sigma shape of the curves describing the structural changes occurring at a
length scale of hundreds of nanometers (Fig. 17a) reflects the formation of ice
crystals of pure water (Fig. 17a
0 ), which takes about 10–15 min regardless of PVA
concentration [77].
By contrast, the values of the reduced scattering invariant calculated in the high
q region Ψ high (t) increase with the permanence time of the PVA solutions at subzero
temperature according to a smoothed sigma shape (Fig. 17b), and reflect structural
changes at length scales of tens of nanometers due to the crystallization of PVA in
the unfrozen liquid microphase. It is apparent that the values of Ψ high (t), regardless
of PVA concentration, increase smoothly in the first 30 min, present an upturn at
around 30–40 min, and increase smoothly again in the successive 120–330 min of
freezing, without reaching any plateau value, even for prolonged times of permanence at À13
C. This suggests that after completion of the crystallization of the
solvent (which takes about 10–15 min), the PVA chains, which are mostly segregated in the unfrozen liquid microphase, tend to form precrystalline or crystalline
aggregates, probably because the concentration of the liquid microphase reaches
the eutectic composition [77]. However, although the temperature of the system
(À13
C) is probably below the eutectic temperature, the full crystallization of PVA
may not be achieved and is slowed down due to the fact that the unfrozen liquid
Kinetic Analysis of Cryotropic Gelation of Poly(Vinyl Alcohol)/Water. . .
189
A more in-depth analysis of TR-SANS data collected during the first freezing step
(Fig. 16a) indicates that the structural changes that occur at correlation distances
between 60–300 nm and 7–60 nm are different and involve different characteristic
times [77]. Therefore, the scattering data collected in the relevant q ranges were
analyzed separately by evaluating the Lorentz-corrected integral Ψ of the scattering
cross-section in the q range between q 1 ¼ 0.022 and q 2 ¼ 0.108 nm
À1 for Ψ low (t)
and between q 1 ¼ 0.108 and q 2 ¼ 0.881 nm
À1 for Ψ high (t) as a function of time.
This kind of integral corresponds to the scattering invariant when the integration
limits in q are extended from zero to infinite. For this reason, the function Ψ
corresponds to a sort of “reduced scattering invariant.” A working hypothesis is
that, in the early stages of gelation, the SANS patterns contain additive contributions from the aggregated and non-aggregated phases and that the interaction
between these two phases may be neglected. Thus, changes in the reduced scattering invariant Ψ low (t) probe structural changes occurring in the initially homogeneous solution during permanence at subzero temperatures at length scales of the
order of hundreds of nanometers, essentially due to water crystallization [77]. By
contrast, changes in the reduced scattering invariant Ψ high (t) probe structural
changes occurring at length scales of the order of nanometers, essentially due to
PVA crystallization inside the unfrozen liquid microphase [77].
The values of the reduced scattering invariant calculated from the SANS data
collected during the first freezing step of PVA/D 2 O solutions are reported in Fig. 17
as a function of the permanence time of the solutions at À13
C. In Fig. 17a
0 (inset),
the values of Ψ low (t) of pure D 2 O are also shown for comparison.
The sigma shape of the curves describing the structural changes occurring at a
length scale of hundreds of nanometers (Fig. 17a) reflects the formation of ice
crystals of pure water (Fig. 17a
0 ), which takes about 10–15 min regardless of PVA
concentration [77].
By contrast, the values of the reduced scattering invariant calculated in the high
q region Ψ high (t) increase with the permanence time of the PVA solutions at subzero
temperature according to a smoothed sigma shape (Fig. 17b), and reflect structural
changes at length scales of tens of nanometers due to the crystallization of PVA in
the unfrozen liquid microphase. It is apparent that the values of Ψ high (t), regardless
of PVA concentration, increase smoothly in the first 30 min, present an upturn at
around 30–40 min, and increase smoothly again in the successive 120–330 min of
freezing, without reaching any plateau value, even for prolonged times of permanence at À13
C. This suggests that after completion of the crystallization of the
solvent (which takes about 10–15 min), the PVA chains, which are mostly segregated in the unfrozen liquid microphase, tend to form precrystalline or crystalline
aggregates, probably because the concentration of the liquid microphase reaches
the eutectic composition [77]. However, although the temperature of the system
(À13
C) is probably below the eutectic temperature, the full crystallization of PVA
may not be achieved and is slowed down due to the fact that the unfrozen liquid
Kinetic Analysis of Cryotropic Gelation of Poly(Vinyl Alcohol)/Water. . .
189