1.1 equilibrium and kinetic phase diagrams · chapter 1: general introduction 1 chapter 1 general...
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Chapter 1: General Introduction
1
Chapter 1
General Introduction
1.1 Equilibrium and kinetic phase diagrams A phase diagram is a common way of presenting the various phases of a substance or a
mixture, which coexist in equilibrium at given temperature and pressure. In case of multi-
component mixtures, beside the temperature and pressure, an additional important variable is
composition. As for the mixed crystalline state of two components, the solid-liquid phase
diagram is usually plotted in the TX plane, assuming a fixed value for the pressure. The
equilibrium states of the liquid and solid phase are determined by the minimum of the total
Gibbs free energy functions at the given temperature (Fig. 1). The compositions of the
coexisting phases plotted in the TX plane correspond to positions on the equilibrium curves,
the liquidus and solidus, which enclose the region where the liquid and solid phase stand in
equilibrium. Basically, the equilibrium state of a mixed system is completely fixed
thermodynamically once the pure component properties and Gibbs free energies are known.
The Gibbs free energy function consists of the thermodynamic properties of the pure
components, the ideal mixing term and an excess term that represents the deviation from ideal
mixing behaviour in a given phase. The excess properties determine to what extent the
components will mix in the phases and they practically determine the shape of the phase
diagram. As shown in Ref. [1], the influence of a positive excess term can be such that the
Gibbs energy curve is not convex over the whole composition range. Consequently, the result
is a phase diagram with a region of demixing, where the miscibility of the components in the
solid phase is limited. Clearly, the knowledge of the excess Gibbs energy is crucial for the
description of the phase behaviour of a system. However, the determination of correct excess
properties is not an easy task.
One step toward obtaining the information about the excess properties is the
measurement of the phase diagram. Many efforts have been put in the development of
experimental methods that could provide near equilibrium conditions during measurements.
Chapter 1: General Introduction
2
Nowadays, phase diagrams are commonly measured by means of DSC, X-ray diffraction and,
occasionally, by adiabatic calorimetry.
Figure 1. The equilibrium compositions of mixed liquid and solid phases, corresponding to
the points of contact of the common tangent line on the Gibbs free energy curves of the phases
at temperature θ, are projected as the points on liquidus and solidus lines in the phase
diagram.
Although these methods increase the experimental accessibility of phase diagram data,
care must be taken regarding the reliability and the accuracy of the data, due to the role of
kinetics. Typically, the determination of the equilibrium solidus is disturbed since the overall
equilibrium is usually not achieved during the measurements. The required equilibrium state
can be achieved to a certain extent by means of adiabatic calorimetry, where the
measurements are performed very slowly so that it can be assumed that the system has
reached the equilibrium. However, in heating experiments the starting point should be a solid
material that is completely homogeneous. To prepare such materials, the methods like co-
precipitation from a solution2 or zone levelling3,4 have to be undertaken. Accordingly, the
experimental determination of a phase diagram is a laborious work, while there is always
uncertainty if the equilibrium has been reached on the time scale of the measurements.
The part of uncertainty of the measured phase diagram can be overcome by means of
thermodynamic phase diagram analysis. This analysis enables the derivation of
Chapter 1: General Introduction
3
thermodynamic excess properties from experimental phase diagram data. Moreover, in this
way the measured phase diagram is checked and optimized. One of the computational
thermodynamic methods developed for the phase diagram analysis is LIQFIT. This method
allows the derivation of both liquidus and solidus lines, by using the thermodynamic
properties of the pure substances and a set of experimental liquidus points only4,5. Since
LIQFIT yields the excess Gibbs energy difference function, it requires one of the excess
functions to be known in order to enable the calculation of the complete phase diagram. This
is usually overcome by taking the liquid phase as an ideal mixture. Beside LIQFIT, other
methods for analysis of various types of phase diagrams can be found in the literature6,7. In
Ref. [8], an experimental method for measuring the excess properties in the liquid and solid
phase is demonstrated. In combination with the phase diagram analysis, this method has led to
the thermodynamically correct phase diagram of the mixture 1,4-dichlorobenzene and 1,4-
dibromobenzene. However, in order to measure the excess enthalpy of the solid phase, the
samples were prepared as homogeneous mixed crystals before they were melted in the
adiabatic calorimeter. Thus, in this procedure complete equilibrium during measurements is
assumed and consequently, the obtained excess properties refer to the equilibrium state of the
solid phase.
From the foregoing discussion, it is clear that the traditional approach for the
determination of phase diagram is based on the assumption that the overall equilibrium is
established between phases. For vapour and liquid phases this approach may be adequate
because the relatively high diffusion rate in these phases ensures homogeneity within short
times. However, in the case of mixed solid phases, the equilibrium state is often not reached
due to the very low diffusion rates in solid phases9, 10, especially for molecular systems.
Namely, during the crystallization of a liquid mixture, the composition of the liquid phase will
change and the composition of the growing solid will differ from that of the previously
formed crystallites. In this way, composition gradients are built up in the solid phase, resulting
in a non-equilibrium or metastable state of the solid. Thus, on a relevant time scale overall
equilibrium will hardly be reached between the entire amounts of the solid and liquid phases.
One step beyond the equilibrium approach would be to assume that the equilibrium is
established between the surface of the solid phase and the existing liquid phase. As shown in
Chapter 2, this is valid only for transition processes that take place at near equilibrium
Chapter 1: General Introduction
4
conditions. In the case of slow crystallization as performed in the adiabatic calorimeter, the
liquid phase is in equilibrium only with the growing solid phase, while the compositions of
the liquid phase and the surface layer of the solid change along the equilibrium liquidus and
solidus lines, respectively. Consequently, the segregation follows the equilibrium phase
diagram. Based on this qualitative picture, we propose a procedure for the determination of
the excess properties, which uses experimental cooling curves and does not assume complete
equilibrium (see Chapter 2). However, this method is appropriate only for slow crystallization
and it cannot be applied for conditions away from near-equilibrium, which often occur in the
practice.
Generally, the crystallization takes place at a certain degree of undercooling, i.e. at
conditions well away from equilibrium. For the description of crystal growth at conditions far
from equilibrium, the non-equilibrium or kinetic segregation has to be defined. The
compositions of the growing solid, i.e. the kinetic segregation, may considerably deviate from
that predicted by the equilibrium phase diagram. The kinetic segregation can be represented
by kinetic phase diagrams11, which give the growth composition of the solid phase as a
function of the liquid composition and of the undercooling at the solidification front. These
phase diagrams can be calculated by the means of analytical models that describe the kinetic
segregation at the given undercooling. In Chapter 3, we experimentally evaluate the
performances of two kinetic models, which have been introduced in the recent studies as the
linear kinetic segregation (LKS)9 model and mean field kink site kinetic segregation
(MFKKS)12 model. These models have no limitations with respect to the number of
components in the mixture or to the miscibility of the components in the phases.
To describe properly the crystallization process, one should realize that the segregation
during crystal growth induces concentration and temperature gradients in the liquid phase near
the solidification front. This will result in different properties of the liquid phase at the
interface and in the bulk. To determine the actual segregation at the interface, so-called
effective segregation, the composition and temperature of the liquid at the interface are
required. These can be calculated from hydrodynamic relations, which correlate the liquid
properties at the interface with those in the bulk. The crucial parameters in the hydrodynamic
relations, so-called q-parameters, are defined in Chapter 3 in the equations 22 and 24. They
are expressed as a kinetic growth velocity constant times the width of the boundary layer
Chapter 1: General Introduction
5
divided by the corresponding diffusion coefficient. In principle, the q-parameters determine
the extent of the mass and heat transport limitations for given crystallization conditions.
The LKS model was coupled with the mass and heat transport limitations elsewhere13,
leading to the construction of so-called effective kinetic phase diagrams. To illustrate the
performances of this extended LKS model, the effective kinetic phase diagrams of an arbitrary
model system that forms solid solution are compared to its equilibrium phase diagram (Fig.
2). The given kinetic phase diagrams are calculated for three situations: a) without including
transport limitations (qm = qlT = 0), b) including only mass transport limitation (qm = 5; ql
T =
0), and c) including both mass and heat transport limitations (qm = 5; qlT = 0.1). In the
presented diagrams, the dimensionless temperature θ is defined as the actual temperature
divided by the melting temperature of a pure component with the highest melting temperature.
Accordingly, the relative liquid bulk undercooling ∆θ is calculated as a ratio of the difference
between the equilibrium liquidus temperature and the actual temperature of the liquid bulk for
the given mixture, and the melting temperature of the pure component with the highest
melting temperature. In all cases, the relative value of the applied bulk undercooling is ∆θ =
0.05. The kinetic liquidus lines are simply constructed by a downward shift of the equilibrium
liquidus over the given value of ∆θ.
For an easier interpretation of the presented graphs, the effective segregation is marked
for one point on the kinetic liquidus, corresponding to a mole fraction of 0.3. The horizontal
distance between the dot on the kinetic liquidus and the arrow end pointing at the growth
composition on the kinetic solidus indicates the magnitude of the effective segregation. Figure
2a clearly demonstrates the effect of bulk undercooling, which significantly reduces
segregation with respect to that according to the equilibrium phase diagram. When mass
transport limitation is included, the effective segregation decreases further (Fig. 2b). The
dashed line in Fig. 2b is the surface kinetic liquidus that gives the composition of the liquid
phase at the surface. For the case with both mass and heat transport limitations, an additional
help line is required giving the temperature at the interface (dashed-dotted line in Fig. 2c) and
the graph is read as follows: the growth composition is found by moving vertically upward
from a point on the kinetic liquidus until this help line is crossed and then moving horizontally
toward the kinetic solidus.
Chapter 1: General Introduction
6
Additionally, by moving horizontally to the surface kinetic liquidus (dashed line), the
corresponding surface liquid composition is found. The heat transport limitation reduces the
effective undercooling at the interface and thus leads to an increase of the effective
segregation.
1.2 Experimental techniques 1.2.1 Adiabatic calorimetry
Adiabatic calorimetry is an experimental method for the investigation of the
thermodynamic properties of solids and liquids, suitable for measuring heat capacity of solids
and liquids, heats of solution and formation, and heat effects associated with structural
changes14. The main objective of the adiabatic measurements is the determination of the heat
capacity cp. The measuring procedure, described in detail earlier15, consists of repeating
stabilization and input periods. During input period a known amount of energy is supplied to a
sample. In the stabilization period, the temperature of the sample container is measured as a
function of time (temperature drift), while the sample is under adiabatic conditions. The
recorded temperature rise, ∆T, is the result of dissipating the given amount of energy ∆q into
the sample container. The heat capacity for the time averaged temperature over the increment
∆T is calculated as follows:
Tq
c p ∆∆= 1
Corrections for the heat exchange with the surroundings are made using the data from the
stabilization period. During the stabilization periods, the thermal behaviour of the sample can
be followed, because any temperature drift during stabilization is an indication for a change of
thermodynamic state. Thus, the adiabatic calorimetry is a powerful tool for measuring very
small heat effects or slow dynamics of phase transitions.
Basic to accurate adiabatic measurements is the design of calorimeter in order to
minimize heat exchange between the sample container and the surroundings. As illustrated in
Figure 1 (Chapter 2), the vessel is surrounded by regulated shields, where the inner shield is
kept at the same temperature as the vessel and the outer shield is regulated at about 10 K
below the temperature of the inner shield. The wire-heater, situated between the shields, is
also kept at the vessel temperature.
Chapter 1: General Introduction
7
Figure 2. Kinetic phase diagrams (bold lines, including auxiliary lines) compared to
the equilibrium phase diagram (dotted lines) for an arbitrary binary mixture that
forms solid solution.
Chapter 1: General Introduction
8
In addition, high vacuum is maintained inside the calorimeter to reduce the heat exchange by
conduction and radiation between the adiabatic shields and the vessel. The described
regulation is typically applied in the heating mode of the calorimeter, which is a standard
procedure for the determination of the heat capacity. The estimated accuracy of the
calorimeter in the heat capacity measurements is within 0.5 % and that in latent heat effects,
such as melting, is about 0.2 %. In Chapter 2 we demonstrate the usage of the calorimeter in
the cooling mode, which enables derivation of the enthalpy change of the mixture during
crystallization.
1.2.2 Differential Scanning Calorimetry (DSC) Differential scanning calorimetry is a technique for thermal analysis of materials,
which found numerous applications in different research fields. For the purpose of our
measurements in Chapters 4 and 5, a Mettler Toledo DSC 821e, equipped with an intracooler,
has been used. This is a heat flux DSC, where the sample and reference (usually an empty
pan) are placed in the same furnace (Fig. 3). The basic principle of this technique is that the
difference in heat flow into the sample and the reference is measured as the sample
temperature is increased or decreased linearly. By observing this difference in heat flow, the
DSC is able to measure the amount of heat absorbed or released during phase transitions. The
result of this difference between the two heat flows is a peak in the DSC curve. The
monitoring of the heat flow difference can be done at a constant temperature (isothermally) or
during temperature changing at a constant rate (heating or cooling).
The advantage of DSC is that relatively small samples (few milligrams) can be
measured within a short time. On the other hand, a disadvantage of the DSC is relatively high
scanning rate that may prevent thermodynamic equilibrium to be reached. Moreover, some
transitions display very weak energy effects that can hardly be detected. One of the main
problems in the analysis of DSC-curves is the occurrence of a thermal lag. Namely, the
sample might contain temperature gradient, so that during scanning the temperature of the
inner part of the sample lags behind that of the apparatus. The thermal lag can disturb the
shape of the DSC-curve for too large sample size and scan rates. At very low scanning rate or
sample size thermal lag is negligible, but the DSC signal becomes very weak, leading to noisy
Chapter 1: General Introduction
9
DSC-curves. To prevent the thermal lag during experiments, the right balance between
thermal lag and instrument sensitivity must be found.
Figure 3. Schematic representation of the DSC furnace. S – sample pan, R – reference pan.
1.3 Influence of crystallization conditions - Polymorphism The amount and composition of the phases are basically determined by the
thermodynamic equilibrium, but in practical situations, the crystallization may lead to
significant deviations from equilibrium. Although the knowledge of the phase diagram of the
system is very important for controlling crystallization, the effects of crystallization
conditions on crystallization kinetics must also be known. To control the formation of the
correct number, size, shape and polymorph of crystals, the kinetics of nucleation and crystal
growth must be described.
When a liquid mixture is cooled, the crystallization generally does not start at the
corresponding point on equilibrium liquidus, but at a lower temperature. At this temperature,
the liquid phase is undercooled and it is thermodynamically unstable. Such a state of the
P
F
V
V V
S R
Chapter 1: General Introduction
10
liquid, where a certain degree of supersaturation is achieved, is needed to trigger the
crystallization. The energy supplied by supersaturation is consumed to overcome the
nucleation energy barrier for the formation of a new surface. Generally, for small
supersaturations the nucleation rate is practically zero, while with a relatively moderate
increase of the supersaturation, the nucleation rates increase dramatically by many orders of
magnitude.
The degree of undercooling and further crystallization path, are determined by the
crystallization conditions. In practice, the properties of the solid phase are often tuned by
adjusting the processing factors, like cooling and stirring rate. In melt systems, a high rate of
cooling generally leads to crystallization at a lower temperature than in the case of slow
cooling. As for stirring, it is well known that the increase of the stirring rate decreases the
extent of metastable zone. The proper regulation of the crystallization process is especially
important in food and pharmaceutical industries. Namely, fats, oils and most of the
pharmaceutical compounds exhibit polymorphism. After crystallization of these compounds,
the solid phase may consist of a number of different coexisting phases, which are not of the
same polymorphic form. To obtain a desirable polymorph, an adequate control of the
crystallization process is required.
In Chapter 4, the thermal behaviour of different polymorphs of the three pure TAGs,
being tristearin (SSS), tripalmitin (PPP) and trielaidin (EEE), is studied. Three main crystal
forms - α, β’ and β - are generally accepted in order to describe polymorphism of TAGs and
their mixtures. These polymorphs are based on the unit cell structures, as determined by the
cross-sectional packing modes of the zigzag aliphatic chain16 (Fig. 4). They can be
characterized as follows:
- α - the least stable form, loosely packed system in which the chains form a
hexagonal lattice
- β’ - intermediate form, an orthorhombic perpendicular unit cell
- β - the most stable form, triclinic unit cell.
Chapter 1: General Introduction
11
Figure 4. The unit cell structures of the three most common polymorphs in TAGs
viewed along chains of the TAG molecules.
Basically, the rate of crystallization differs remarkably between the polymorphic
forms, being highest for the α-form and lowest for the β-form. The thermal history and the
rate of cooling and heating a TAG sample can cause major differences in the appearance of
the calorimetric traces. Extremely rapid quenching can solidify materials in a glass – a solid
state quite similar to the liquid state in molecular orientation (sub-α form). Under fast cooling
most of TAGs crystallize in the α-form, in which the specific chain-chain packing, typical for
the stable forms (β’ or β), has not been allowed to occur. Moreover, since the molecular
motion is greatly restricted in the solid state, the reorganization into more specific crystalline
forms may occur only very slowly. However, once the α-form has partly melted,
rearrangement easily takes place and other forms, β’ or β, may crystallize. Thus, the α-form is
thermodynamically unstable, but kinetically favorable.
To illustrate the impact of the cooling rate on the occurrence of different polymorphs
of SSS, we will discuss the results of several DSC experiments. The sample was heated with
the rate of 5 K·min-1 to (10 to 15) K above the melting temperature of the �-phase;
subsequently in each experimental set the melt was cooled at a different rate, being (-10, - 5, -
1 and -0.1) K·min-1. The scanning patterns of the solid phases formed under these various
cooling rates were recorded from 273 K at a heating rate of 5 K·min-1, and were transformed
into pseudo-enthalpy curves, presented in Figure 5. These curves are obtained by integration
Chapter 1: General Introduction
12
of the registered heat-flow signals over time and matching the enthalpy levels in the liquid
phase. At high cooling rates, SSS crystallizes in the �-polymorph, which melting behaviour is
represented by the group of curves 1 in Fig. 5. The first endothermic effect at 327 K indicates
the melting of the �-form, followed by a broad exothermic peak due to the crystallization of
the �’-phase from the � melt. The melting of the �’-phase is not observed, since it converts
readily into the �-phase that melts at the temperature lower than that according to the melting
experiment with the sample as delivered. As already noticed17, the melting temperature of the
�-phase is dependent on the manner of preparation. Rapidly prepared �-phase exhibits a lower
melting point, likely due to defects built in the crystals.
Although crystallized at different cooling rates, the α-phase always crystallized in the
vicinity of 326.6 K. The onset of melting of the α-phase (327 K) is very close to this value,
meaning that no significant undercooling is needed for the nucleation of the α-form. The
absence of undercooling for the α-phase has already been discussed18, 19, while it is also
observed during measurements in the adiabatic calorimeter20.
The curves 2 and 3 in Fig. 5 represent the scanning patterns of the solid forms that
crystallized during cooling of the melt with the rate of -0.1 K·min-1. Despite the fact that the
same cooling rate was used, the crystallization occurred at different temperatures, resulting in
different solid states. The solid phase that solidifies at 334 K exhibits only one endothermic
peak on the scanning pattern, indicating that SSS crystallized in the �-form upon slow cooling
of the melt. However, in the other experiment, the crystallization occurred at 329 K under the
same cooling rate and the formed solid consisted of more than one polymorph. The scanning
pattern of this solid phase (curve 3, Fig. 5) shows three endothermic effects, where the newly
identified peak at 337.5 K corresponds to the melting of the �’-polymorph. We suppose that
part of the melt crystallizes in the �’-form, but under constant cooling of -0.1 K·min-1, the
crystallization of the �’ polymorph does not come to completion and the remaining liquid
crystallizes in the �-form.
Chapter 1: General Introduction
13
Figure 5. Pseudo-enthalpy heating curves of the solid states of SSS formed under
various cooling rates.
According to Figure 6, the crystallization process that took place at 334 K was
appreciably faster than the one occurring at 329 K under the same cooling rate. The nucleation
rate of the �-form is very low, but once the �-nuclei are formed, instant crystallization
follows. On the other hand, due to the slow nucleation kinetics of the �-form, the melt can
easily be undercooled below the onset of the �-form crystallization, leading to the nucleation
of the less stable polymorphs. In this case the onset of crystallization occurred at lower
temperature (329 K), while the shape of recorded exothermic peak points to two-step
crystallization. Eventually, the final solid state consists of both the �- and the �’- polymorphs.
Clearly, the influence of the applied cooling rate is evident in the final energy level of
the compound. Faster cooling provides high-enthalpy solid forms (curves 1, Fig. 5), while
slower cooling results in a lower enthalpy state of the final solid (curves 2 and 3). These
observations point to the necessity of very slow cooling of the melt in order to obtain nuclei of
310 320 330 340 350 360
T / K
-20
60
140
220
300
h / J
·g-1
1
2
3
Chapter 1: General Introduction
14
320 325 330 335 340
T / K
-0.40
-0.30
-0.20
-0.10
0.00
0.10
heat
flow
/ m
Wthe �-form. They also illustrate the impact of kinetics on the crystallization, which is typical
for the solidification of TAGs. The discussed influence of the cooling rate shows that
principally the �-form crystallizes when the melt is cooled fast enough, while only for very
low cooling rates the crystallization of the �-form is to be expected. In between, we are left
with a range of cooling rates where there is a possibility that solidification in more than one
polymorphic form takes place.
Figure 6. Crystallization of SSS, starting at 334 K (dashed line) and at 329 K (solid
line) during the cooling of the melt at a rate of 0.1 K⋅min-1.
Chapter 1: General Introduction
15
1.4 Choice of materials In Chapters 2 and 3 we examined the kinetics of crystallization for mixed molecular
systems, where the mixture of 1,4-dichlorobenzene and 1,4-dibromobenzene is used as a
model mixture. This mixture has been chosen, since its phase diagram is very well known.
The thermodynamic properties of the pure components were reported in Ref. [8], as well as
the thermodynamic excess mixing properties of the liquid and the solid phase.
Solid 1,4-dichlorobenzene occurs in three different crystalline forms: γ, α and β. The
low-temperature stable form is γ, which at 275 K transforms into the α-form that is stable at
room temperature. At 306 K the α-form converts into the high-temperature β-form, which
melts at 326.24 K. The reported melting temperature of 1,4-dibromobenzene is 360.48 K and
no polymorphism occurs for this compound. The crystal structure of 1,4-dibromobenzene is
the same as that of the α-form of 1,4-dichlorobenzene. The two components co-crystallize in
the α-form, forming solid solutions over virtually the whole composition range8.
The thermal behaviour of three pure TAGs, being SSS, EEE and PPP, is studied in
Chapter 4. The TAGs were purchased from Larodan, with stated mass fraction purity of
> 99%. Their thermodynamic properties are reported in Chapter 4 and the purities are
determined by means of adiabatic calorimetry. These compounds are important for being main
constituents of edible fats and oils. Understanding of their polymorphism and phase behaviour
leads to a better insight into the complex crystallization behaviour of fats in food products.
Furthermore, in literature there is little information about the miscibility of TAGs in the solid
phase of their mixtures. To reveal more information about the mixing properties, we
investigated the binary mixtures of the mentioned TAGs in Chapter 5.
1.5 Outline of this thesis As it has been discussed in Section 1.1, it is not justified to assume that the system is
in complete equilibrium on a relevant time scale during phase transition processes. One part of
our investigation, to be presented in this thesis, deals with a kinetic description of the
crystallization of the mixture of 1,4-dichlorobenzene and 1,4-dibromobenzene. In Chapter 2,
we propose a kinetic model that very successfully reproduces the enthalpy curve of the
mixture measured during slow cooling in the adiabatic calorimeter. The performance of the
Chapter 1: General Introduction
16
kinetic model is extended to the determination of the excess enthalpy and entropy of the solid
phase. In this way, we developed a method for the determination of the phase diagram without
adopting the assumption of complete equilibrium between the phases.
Slow crystallization in the adiabatic calorimeter, as applied for the experiments in
Chapter 2, will lead to a situation of near equilibrium between the liquid phase and the surface
of the solid phase. Therefore, we designed an experimental set-up described in Chapter 3,
where the mixture of 1,4-dichlorobenzene and 1,4-dibromobenzene was allowed to crystallize
at conditions well away from equilibrium. To predict the state of the solid phase, we used a
model that describes the kinetic segregation at the interface as a function of the undercooling
and composition of the liquid phase. Moreover, this kinetic model is coupled with the mass
and heat transport limitations. The measured composition of the solid phase, crystallized well
away from equilibrium, is compared to that calculated from the extended kinetic model and
the equilibrium model.
In Chapter 4, we give an overview of thermal properties of three pure TAGs: SSS, PPP
and EEE. A detailed thermal analysis of SSS and the investigation of its polymorphism are
performed by means of adiabatic and differential scanning calorimetry. Furthermore, the
purity of three TAGs is determined from the adiabatic measurements. Purity is an important
parameter, since it has influence on the phase behaviour of TAG mixtures. The thermal
analysis of three binary mixtures of the mentioned TAGs is presented in Chapter 5. We
focused on the measurement of the phase diagrams of the most stable β-form by DSC. The
analysis of the mixing properties in the solid phase is supported by the adiabatic
measurements.
Chapter 1: General Introduction
17
References: [1] H.A.J. Oonk, Phase Theory: the thermodynamics of heterogeneous equilibria, Elsevier Sci.
Pub. Comp., Amsterdam (1981).
[2] Y. Haget, J.R. Housty, A. Maïga, L. Bonpunt, N.B. Chanh, M. Cuevas, E. Estop, J. Chim.
Phys. 81 (1984) 197.
[3] A.C.G. van Genderen, C.G. de Kruif, H.A.J. Oonk, Z. Phys. Chem. Neue Folge 107 (1977)
167.
[4] J.A. Bouwstra, Thermodynamic and structural investigations of binary systems, Ph.D.
Thesis, Utrecht University, (1985).
[5] J.A. Bouwstra, H.A.J. Oonk, Calphad 6 (1982) 11.
[6] N. Brouwer, Thermodynamic investigations of isobaric binary mixtures; simultaneous
derivation of excess enthalpy and excess entropy functions, Ph.D. Thesis, Utrecht University,
(1981).
[7] M.H.G. Jacobs, TXFIT, a computer program for the derivation of excess properties of
two-phase equilibria, Chemical Thermodynamics Group, Utrecht University (1989).
[8] P.R. van der Linde, M. Bolech, R. den Besten, M.L. Verdonk, J.C. van Miltenburg, H.A.J.
Oonk, J. Chem. Thermodynamics 34 (2002), 613.
[9] J.H. Los, W.J.P. van Enckevort, E. Vlieg, E. Flöter, J. Phys. Chem. B 106 (2002), 7321.
[10] J.H. Los, W.J.P. van Enckevort, E. Vlieg, E. Flöter, F.G. Gandolfo, J. Phys. Chem. B 106
(2002), 7331.
[11] Z. Chvoj, J. Šesták, A. Tiska, Kinetic phase diagrams, Elsevier Sci. Pub. Comp.,
Amsterdam (1991).
[12] J.H. Los, M. van den Heuvel, W.J.P. van Enckevort, E. Vlieg, H.A.J. Oonk, M. Matovic,
J.C. van Miltenburg, Calphad 30 (2006), 216.
[13] J.H. Los, M. Matovic, J. Phys. Chem. B 109 (2005), 14632.
[14] A. Cezairliyan et al., Specific heat of solids, Hemisphere Pub. Corporation (1988).
[15] J.C. van Miltenburg, A.C.G. van Genderen, G.J.K. van den Berg, Thermochim. Acta 319
(1998), 151.
[16] K. Sato, In: N. Widlak, R. Hartel, S. Narine, editors: Crystallization and Solidification
Properties of Lipids, AOCS Press (2001) 1.
[17] M. Ollivion, R. Perron, Thermochim. Acta 53 (1982) 183.
Chapter 1: General Introduction
18
[18] L.H. Wesdorp, Liquid-multiple solid –phase equilibria in fats, Ph.D. Thesis, Delft
University, (1990).
[19] R. Perron, J. Petit, A. Mathieu, Chem. Phys. Lipids 3 (1969) 11.
[20] M. Matovic, J.C. van Miltenburg, J.H. Los, F.G. Gandolfo, E. Flöter, J. Chem. Eng. Data
50 (2005) 1624.