1.1 Equilibrium and kinetic phase diagrams · Chapter 1: General Introduction 1 Chapter 1 General...

Chapter 1: General Introduction

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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.


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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


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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


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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


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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.


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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.


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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.


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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


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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


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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.


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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


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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.


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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


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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.


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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


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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.


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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.


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[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

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