Top Banner
Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling
144

Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Jan 17, 2020

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Investigation of Binary Solid Phases

by Calorimetry and Kinetic Modelling

Page 2: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Cover designer: Vladan Glišovi� �

ISBN 978-90-9021838-0 �

Page 3: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Investigation of Binary Solid Phases

by Calorimetry and Kinetic Modelling

Onderzoek van binaire vaste fasen met calorimetrie en kinetisch modelleren

(met een samenvatting in het Nederlands)

Proefschrift �

ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof.dr. W.H. Gispen, ingevolge het besluit van het college voor promoties in

het openbaar te verdedigen op woensdag 31 mei 2007 des ochtends te 10.30 uur �

door �

Marija Matovi� �

geboren op 4 januari 1977

te �a�ak, Servië

Page 4: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Promotoren: Prof.dr. H.A.J. Oonk

Prof.dr. E. Vlieg

Co-promotoren: Dr. J.C. van Miltenburg

Dr. J.H. Los �

Dit proefschrift werd (mede) mogelijk gemaakt met financiële steun van STW

(projectnummer NPC.5738).

Page 5: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

i

Contents

1 General Introduction

1.1 Equilibrium and kinetic phase diagrams 1

1.2 Experimental techniques

1.2.1 Adiabatic calorimetry 6

1.2.2 Differential scanning calorimetry 8

1.3 Influence of crystallization conditions – Polymorphism 9

1.4 Choice of materials 15

1.5 Outline of this thesis 15

References 17

2 Kinetic Approach to the Determination of the Phase Diagram of a Solid

Solution

2.1 Introduction 19

2.2 Experimental method 20

2.3 Qualitative interpretation of the measured enthalpy path 24

2.4 Quantitative analysis, kinetic model 25

2.5 Determination of excess parameters 31

2.6 Summary 33

References 34

3 Kinetic Segregation in Crystallization of Mixed Crystals

3.1 Introduction 35

3.2 Experimental set-up 37

3.3 Linear kinetic segregation model 39

Page 6: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

ii

3.4 Mean field kink site kinetic segregation model 41

3.5 Mass and heat diffusion limitations 43

3.6 Estimation of chemo-physical properties 47

3.6.1 Viscosity estimation 48

3.6.2 Liquid density estimation 48

3.6.3 Solid density estimation 49

3.6.4 Mass diffusion coefficient 49

3.6.5 Thermal conductivity and diffusion coefficient 50

3.7 Results and discussion 51

3.8 Summary 58

References 59

4 Thermodynamic Properties of Pure Triacylglycerols: Tristearin

(SSS), Tripalmitin (PPP) and Trielaidin (EEE)

4.1 Introduction 61

4.2 Prediction of thermodynamic properties of TAGs 63

4.3 Thermal behaviour of pure TAGs: DSC and adiabatic

experiments 66

4.3.1 Tristearin (SSS) 66

4.3.2 Tripalmitin (PPP) 82

4.3.3 Trielaidin (EEE) 84

4.4 Purity determination 86

4.5 Summary 92

References 92

Page 7: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

iii

5 Thermal Analysis of Binary TAG Mixtures

5.1 Introduction 95

5.2 Overview of phase diagrams 97

5.3 The role of kinetics 99

5.4 Equilibrium phase diagrams of the β-polymorph 101

5.5 The crystallization of TAG mixtures at high

cooling rates 116

5.6 Summary 123

References 124

Summary of thesis 125

Samenvatting 129

Acknowledgements 133

Curriculum Vitae 135

List of publications 136

Page 8: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

iv

Page 9: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 10: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 11: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 12: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 13: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 14: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 15: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 16: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 17: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 18: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 19: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 20: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 21: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 22: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 23: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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

Page 24: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 25: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 26: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

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.

Page 27: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

19

Chapter 2

Kinetic Approach to the Determination of the Phase

Diagram of a Solid Solution

2.1 Introduction Multi-component systems of substances that have about the same size and shape of

molecules and do not differ too much in chemical nature tend to form mixed crystals or solid

solutions. Prediction of the phase behavior of such a mixture and characterization of the solid

state require the knowledge of the relevant phase diagram. Several studies on the binary

mixture of our interest, 1,4-dichlorobenzene and 1,4-dibromobenzene, demonstrate methods

for determination of the equilibrium phase diagram1-4. Measuring equilibrium solidus and

liquidus lines by melting samples is a laborious work in the sense of preparing mixtures of a

high degree of homogeneity5-7, while there is always uncertainty about whether the system has

reached the equilibrium during measurements. As for the melting and crystallization of the

molecular mixed crystals, the problem arises due to very low diffusion rates in the solid

phase8 that prevent overall equilibrium between the entire amounts of solid and liquid phase.

Therefore, the result of crystallization will be an inhomogeneous state of solid that contains

composition gradients in its bulk. As an example, previous analysis of the mixture of 1,4-

dichlorobenzene and 1,4-dibromobenzene by Raman spectroscopy illustrates the distribution

of the given components along the length and the diameter of the single crystal9.

In this work, we focus on the crystallization of the issued mixture in an adiabatic

calorimeter, whereby we develop a kinetic model describing quantitatively the crystallization

process. The basic assumption of the kinetic model is that at slow cooling rate equilibrium is

established between the surface of the growing solid phase and the existing liquid phase along

the cooling path. Clearly, this is only valid for low enough cooling rates and thus for slow

solidification processes. Nevertheless, no matter how slow the cooling is, due to the very slow

diffusion rate in the solid phase, the final state of solid will not be the equilibrium state within

a reasonable time scale, as we will show here.

Page 28: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

20

For the purpose of modeling the slow crystallization close to equilibrium as performed

in the adiabatic calorimeter, the effects of mass and heat transport in the liquid phase on the

segregation can be excluded, as it was demonstrated in Ref. [10]. Furthermore, it is shown

how the applicability of the proposed kinetic model can be extended to determine the excess

properties of the solid phase. In previous work11, a preliminary description of this kinetic

approach was given. Here, an improved model, that gives both excess enthalpy and excess

entropy of the solid phase, is presented in detail. Typically, a mixture with high miscibility of

components exhibits lower excess energy and will form a solid solution, while for an eutectic

mixture miscibility in the solid phase is limited corresponding to higher excess energy. As

discussed in Ref. [12], the kinetics usually favors mixing. The advantage of proposed kinetic

modeling over the traditional equilibrium approach5 for the determination of excess properties

is that it is not based on the unjustified assumption that the system is in complete equilibrium.

2.2 Experimental method Several mixtures of 1,4-dichlorobenzene and 1,4-dibromobenzene of different

compositions were measured in the adiabatic calorimeter (laboratory design indication CAL

VII)13. Each mixture weighted around 5 g and it was placed in a copper gold-plated vessel that

was mounted in the calorimeter. The mixture was first heated slowly to 380 K in the way that

was previously described14, while the change of the enthalpy of the mixture with temperature

is measured during the performed experimental procedure. Afterwards, the melt was kept

overnight at 380 K and then cooled down to 250 K with a rate of 0.1 K⋅min-1. Cooling in the

adiabatic conditions is performed by setting the temperatures of the shields to relevant values

with respect to the vessel temperature (see Fig. 1). Here by adiabatic conditions we mean that

the applied cooling is the only heat exchange between the system, i.e. vessel and mixture, and

its surroundings, i.e. shields and wire-heater. Thus, the enthalpy change of the system during

cooling, dHsystem, within a time interval dt as consequence of the cooling power or heat flow

dtdQI coolcool /= , with coolQ the withdrawn heat, is given by:

mixvessystem dHdHdH += = ( ) dtIdTcc coolmixpvesp =+ ,, 1

where dHves and dHmix are the enthalpy changes and vespc , and mixpc , are the heat capacities of

the vessel and the mixture, respectively.

Page 29: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

21

Figure 1. The sketch of the sample container in the adiabatic calorimeter.

During the cooling experiment the change of system’s temperature with time, dT / dt,

is measured, while the heat capacity of the empty vessel vespc , as a function of temperature

was measured independently in a calibration experiment. From the measured heat capacities

of the mixtures in the one-phase temperature regions, we found only small fluctuating

deviations from the ideal values, suggesting no significant excess heat capacity. This implies

that the heat capacity of the mixture in the phase P (liquid or solid phase) is given by:

Pp

Pp

Pmixp zcczc 2,1,, )1( +−= 2

where z = z2 is the overall composition of component 2 (1,4-dibromobenzene), while Ppc 1, and

Ppc 2, are the heat capacities of the pure components in the phase P. These heat capacities are

known from pure components’ measurements. Within the temperature range of interest, they

are very well approximated by Taylor series of the second order in the temperature:

22

,2

,,,, )(

21

)()( ref

T

Pip

ref

T

PipP

refipP

ip TTdT

cdTT

dT

dccTc

refref

−��

��

�+−

��

��

�+= 3

OUTER ADIABATIC SHIELD

INNER ADIABATIC SHIELD

VESSEL

WIRE HEATER

Page 30: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

22

where Prefipc ,, is the pure component heat capacity in phase P at some reference temperature

refT , anddT

dc Pip , , 2

,2

dT

cd Pip are the first and the second derivatives at refT .

The cooling power of the calorimeter as a function of temperature within the one-

phase regions can be calculated straightforwardly from:

dtdT

ccI Pmixpvespcool )( ,,

exp += 4

The heat is withdrawn from the system by radiation to the adiabatic shields and by conduction

through the wire heater. In accordance to this, theoretically the cooling power can be

expressed as the sum of a radiation and a conduction term:

)())(( 3244

1 systemsystemsystemconductionradiationtheorcool TqqTTTqIII ++∆−−=+= 5

where ∆T is a temperature difference between the system and the inner adiabatic shield, being

set to 10 K during the cooling experiments.

By fitting the experimental cooling power (Eq. (4)) to the theoretical expression (Eq.

(5)), parameters q1, q2 and q3 are obtained for each measured mixture. Basically, these

parameters should not depend on the content of the vessel. Nevertheless, to rule out the effects

of small fluctuations in the experimental conditions we determined cooling parameters for

each cooling experiment. The obtained values are given in Table 1. The resulting cooling

powers as a function of temperature are shown in Fig. 2, illustrating that, despite the

differences in the parameters, they are quite close as we expected.

Table 1. Cooling power parameters for each mixture cooled in the adiabatic calorimeter.

mixture

x2

q1⋅⋅⋅⋅1011 (J⋅⋅⋅⋅K-1⋅⋅⋅⋅s-1)

q2⋅⋅⋅⋅103

(J⋅⋅⋅⋅s-1) q3⋅⋅⋅⋅105

(J⋅⋅⋅⋅K-1⋅⋅⋅⋅s-1) I 0.2937 1.3372 - 2.1740 2.8007 II 0.4791 1.3938 - 1.4711 2.4425 III 0.5312 1.3878 - 1.9066 2.6865 IV 0.5338 1.2658 - 3.9556 3.5783 V 0.6025 1.3712 - 1.7688 2.6282 VI 0.7976 1.3423 - 2.0353 2.7332

Page 31: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

23

T / K

260 280 300 320 340 360 380

I cool

/ W

0.015

0.020

0.025

0.030

0.035

Once the cooling power is defined as a function of the system’s temperature, the

enthalpy of the mixture can be calculated as a function of temperature for the applied cooling

by:

''

)()()()(1

expexp dTdtdT

ITHTHTHTHT

Tcoolvesrefvesrefmixmix

ref

��

���

�+−+= � 6

where the integration constant, )( refves TH , is chosen such that the enthalpy of the liquid

mixture is equal to zero at refT , which is chosen to be 365 K.

Figure 2. The cooling power of the adiabatic calorimeter (Icool) as a function of temperature,

calculated for each investigated mixture by Eq. (5) using parameters from Table 1.

Page 32: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

24

2.3 Qualitative interpretation of the measured enthalpy path For each composition of the mixture, the enthalpy path during the described cooling is

calculated by Eq. (6) and plotted in the Fig. 3. Note that all enthalpy curves contain two kinks,

which are typical for the crystallization process as opposed to the presented melting enthalpy

curves.

Upon continuous cooling of the liquid phase, the first kink in the enthalpy curves

appears at point 0 and marks the onset of crystallization by nucleation. Once the

crystallization process starts, the solid phase evolves rapidly as is evident from the registered

temperature increase between points 0 and 1. In all cases the nucleation temperature T0 is

found to be about 8 K lower than the corresponding liquidus temperature of the mixture of the

given composition z. At T0 the liquid phase is highly supersaturated. This means that during

the initial crystallization the composition of growing solid phase is not necessarily lying on

the solidus line of the equilibrium phase diagram, but its determination would require the

knowledge of the so-called kinetic phase diagram, that follows from a non-equilibrium

approach, such as the one given in Ref. [15]. Thus, for the amount of solid phase formed

along the path 0-1 the segregation may deviate from the equilibrium segregation. In contrast,

once being at T1 we assume that equilibrium is established between the remaining liquid phase

and the surface of the solid phase. This assumption is supported by the fact that further growth

of the solid phase from T1 downward is accompanied by gradual temperature decrease, which

points to significantly slower growth of the solid phase than in the initial stage. Accordingly,

the solidification front, i.e. the solid phase that is growing at the surface, is assumed to be in

(near) equilibrium with the remaining liquid phase.

Such a description of solidification implies that the solid phase grows in layers of

different compositions, where each layer is in equilibrium with the remaining liquid phase at

the given temperature during cooling. This approach considers a definite time span at the

given temperature as opposed to the equilibrium, which assumes an infinite time span and

thus allows the equilibrium between completely homogeneous phases. This picture of the

crystallization of a solid solution is known in the literature as the shell model16.

The above qualitative description and assumptions regarding the experimental cooling

enthalpy curves give base to our quantitative modeling of the crystallization process.

Page 33: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

25

-90

-60

-30

0

310 320 330 340 350 360

T / K

Hm

ix /

J/m

ol

I

III II

IV V VI

1

0

1

0

Figure 3. The enthalpy change of mixtures of different compositions (H mix) with temperature

(T), obtained from heating (dashed line) and cooling (solid line) the mixtures in the adiabatic

calorimeter. Denotations for different compositions are in line with those in Table 1. Points 0

and 1 indicate kinks on the cooling enthalpy paths (shown here for only two mixtures for

clarity of the figure).

2.4 Quantitative analysis, kinetic model According to the previous discussion, the impact of kinetics is significant so that a

successful description of the crystallization process requires both thermodynamic and kinetic

factors to be taken into account. We introduce a kinetic way of modeling the crystallization

process by deriving the expressions for calculating the enthalpy path for a given set of excess

parameters. These parameters determine the excess contribution to the thermodynamic

properties of the mixture, and quantify the degree of miscibility of the components in the

given phase. A given excess quantity for a phase P is commonly expressed as a polynomial

function of the composition. Here we adopt the Redlich-Kister expansion17, reading:

Page 34: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

26

�=

−−=N

n

nexcPn

excP xaxxA0

,, )21()1( 7

where 2xx = is the mole fraction of component 2, which is usually chosen to be the

component with the highest melting temperature, and A stands for the excess enthalpy,

entropy or the free Gibbs energy obeying: excP

mixexcP

mixexcP

mix TSHG ,,, −= 8

During the phase transition in the adiabatic calorimeter, the enthalpy change of the system

consists of a contribution for cooling the mixture, coolmixdH , , and a contribution for the phase

transformation, transdH . So we can write:

dsHdTcdHdHdH fusmixptranscoolmixmix ∆+=+= ,, 9

where fusH∆ is the composition dependent enthalpy of fusion of the mixture and ds refers to

the amount of the solid phase that is formed within the time interval dt corresponding to the

temperature change dT.

Let us turn now to the qualitative analysis of the experimental data, where the two

parts of the crystallization process were distinguished. In the initial part the growth of the

solid phase is fast and segregation may deviate from the equilibrium segregation, while during

the second part the crystallization proceeds slowly and we assume that the solid phase at the

surface is in (near) equilibrium with the existing liquid phase. To start, we first derive

expressions for the evolution of solid fraction and the enthalpy as a function of temperature

during the second part of the crystallization, i.e. starting from T1. Here we refer to Fig. 4,

where the cooling path for the melt of overall composition z is schematically presented in the

phase diagram from Ref. [4]. As diffusion in the solid phase is neglected, the lever rule, which

is based on complete equilibrium, is not valid, but we can still apply a differential form of the

lever rule. This implies that the amount of solid formed between T and T-∆T is given by:

)()(

)()()1(

TTxTTx

TTxTxss

liqeq

soleq

liqeq

liqeq

∆−−∆−∆−−

−=∆ 10

where liqeqx and sol

eqx are the equilibrium mole fractions (of component 2) of the liquid and the

solid phase at the corresponding temperature. For ∆T→0, Eq. (10) becomes:

Page 35: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

27

325

335

345

355

365

0 0.2 0.4 0.6 0.8 1

x

T /

K

T1

T0

xeqliq(T)

xeqsol(T) xeq

liq(T-∆∆∆∆T)

xeqsol(T-∆∆∆∆T)

0

1

∆∆∆∆T

z

'''

'

)()(

/)1( dT

TxTx

dTdxsds

liqeq

soleq

liqeq

−−−= 11

implying:

� � −−=

s

s

T

Tliqeq

soleq

liqeq dT

TxTx

dTdx

sds

0 1

'''

'

)()(

/

1 12

Figure 4. The crystallization path for the mixture of overall composition z, schematically

presented in the liquid-solid phase diagram from Ref. [4]. Upon continuous cooling, the

nucleation of solid phase occurs at point 0 (T0), from where temperature rises until point 1

(T1) and then starts decreasing again. For description of solid fraction evolution below T1 we

use Eq. (10), where during temperature decrease ∆T the liquid phase will change composition

along the liquidus line, from xeqliq(T) to xeq

liq(T-∆T), while the composition of growing solid

moves from xeqsol(T) to xeq

sol(T-∆T).

Page 36: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

28

Finally, by analytic integration of Eq. (12) we find that the solid phase fraction as a function

of temperature from T1 downward is given by:

��

��

−−−= �

'''

'

0

1)()(

/exp)1(1)( dT

TxTx

dTdxsTs

T

Tliqeq

soleq

liqeq 13

Similarly, the theoretical enthalpy path is found by substitution of Eq. (11) into the energy

balance Eq. (9), and integrating, which leads to:

� ∆−

−−=T

Tfusliq

eqsoleq

liqeq

mixptheormix dTH

TxTx

dTdxTscTH

1

'''

'

, ))()(

/))(1(()( 14

where fusH∆ is the composition dependent enthalpy of fusion given by:

)()1( 2,1,soleq

excfus

soleqfus

soleqfus xHHxHxH ∆+∆+∆−=∆ 15

Here 1,fusH∆ and 2,fusH∆ stand for the temperature dependent enthalpies of melting of the

pure components, while excH∆ is the difference between the excess enthalpy in the liquid and

solid phase. Our calculated enthalpy paths from T1 downward are obtained by numerical

integration of Eq. (14), inserting the temperature dependent solid fraction of Eq. (13).

However, in order to perform this integration, the amount of solid being formed between T0

and T1, 0s , and its average composition, solavx , have to be determined.

Although we assume that the surface of the solid at T1 is of equilibrium composition,

the total initial solid phase, s0, is in fact inhomogeneous. However, its average composition solavx = sol

avx ,2 must satisfy the mass balance equation:

zxsxs solav

liqeq =+− 00 )1( 16

As the surface of the solid phase is in equilibrium with the remaining liquid at T1, the

following relation holds for the both components of the binary mixture:

���

����

� ∆−∆−=

1

,1,,,,, exp

RT

STHxx fusifusisol

eqisoleqi

liqeqi

liqeqi γγ 17

for i=1,2, where fusiS ,∆ is the temperature dependent melting entropy of the pure component i.

The excess property of the mixture in phase P=liq,sol is expressed in the terms of activity

coefficient of component i in that phase, Peqi ,γ , which is related to the excess free Gibbs energy

of the phase, excPmixG , , by:

Page 37: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

29

Pi

excPmixP

eqi NG

RT∂

∂=

,

, )ln(γ 18

where PiN is the amount of component i in phase P.

From the experiment the value of the enthalpy of the mixture at T1 is known and

should be equal to:

solliqtheormix HsHsTH 001 )1()( +−= 19

where liqH and solH are the liquid and solid enthalpies at T1, given by:

)()1( ,*,2

*,1

liqeq

excliqliqliqeq

liqliqeq

liq xHHxHxH ++−= 20

)()1( ,*,2

*,1

solav

excsolsolsolav

solsolav

sol xHHxHxH ++−= 21

with PiH *, being the pure component enthalpies at T1 and excPH , the excess enthalpies in phase

P=liq, sol. The liquid phase is not treated as completely ideal, since we adopted the excess

properties of the liquid phase as determined in Ref. [4].

With the coupled equations of form Eq. (17) for the both components, the mass

balance Eq. (16) and the enthalpy balance at T1 (Eqs. (19)-(21)), the four unknowns

( solav

soleq

liqeq xxxs ,,,0 ) can be determined.

In Table 2 the experimental temperatures T0 and T1 are given for each mixture,

together with the calculated initial solid fractions 0s and relevant compositions for the initial

part of the crystallization.

Table 2. Nucleation and equilibrium temperatures T0 and T1 for each investigated

composition of the mixture, and corresponding results from modeling the initial

crystallization: initial solid fraction s0, equilibrium compositions of liquid and solid phase at

T1 (xeq liq and xeq

sol) and average composition of the initial solid (x avsol).

x2

T0 / K

T1 / K

s0

xeq liq

xeq sol

x av

sol

0.2937 325.537 330.932 0.1826 0.2262 0.3529 0.5958 0.4791 332.641 337.808 0.1892 0.4338 0.6498 0.6729 0.5312 334.646 339.998 0.1962 0.4613 0.6772 0.8157 0.5338 335.622 339.435 0.1624 0.4472 0.6670 0.9809 0.6025 332.811 341.751 0.3294 0.5187 0.7322 0.7731 0.7976 344.306 351.329 0.2746 0.7461 0.8898 0.9335

Page 38: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

30

-40000

-35000

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

260 280 300 320 340 360 380

T / K

Hm

ix J

/mol

Hmix cooling

Hmix kinetic model

Hmix equilibrium model

To illustrate the performance of the proposed kinetic model, the calculated cooling

enthalpy path is compared with the experimental enthalpy path for the mixture of composition

z = 0.7976, assuming the excess properties to be as determined by van der Linde4. In Fig. 5 we

also include the enthalpy path as calculated from the model, which assumes complete

equilibrium at any time during the crystallization process. Clearly, much better agreement

between the experimental and kinetic enthalpy curves affirms the validity of the kinetic

modeling, while the equilibrium approach fails in describing properly the crystallization

process.

Figure 5. The experimental enthalpy cooling path of the mixture (Hmix cooling, solid line) of

overall composition z = 0.7976, compared to those reproduced from the kinetic (Hmix kinetic

model, solid bold line) and equilibrium model (Hmix equilibrium model, dashed line). All three

enthalpy paths are calculated by using the excess properties of the mixture as determined in

Ref. [4].

Page 39: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

31

2.5 Determination of excess parameters As it has been demonstrated, the kinetic model successfully describes the

crystallization process when the excess properties of the phases are known. On the other hand,

the kinetic model can be used to determine the excess properties by fitting the theoretical

enthalpy path (Eq. (14)) to the experimental one as measured in the adiabatic calorimeter

during cooling. In principle, the difference between the theoretical and experimental value of

the enthalpy at the given temperature Ti, starting from T1 downward, is minimized:

� −= 2exp,, )( imix

theorimix HHF 22

The result of proposed procedure is a set of the excess parameters that define dimensionless

excess properties, written as follows:

excEliq

sol hxxxhxhxxRTH ~

)1())1()(1( 1221

,

−=+−−=∆

23

excEliq

sol sxxxsxsxxR

S ~)1())1()(1( 1221

,

−=+−−=∆

24

excEliq

sol gxxxgxgxxRTG ~)1())1()(1( 1221

,

−=+−−=∆

25

Note that the relevant quantity for the phase behaviour is the difference in the excess free

energy for the liquid and solid phase. During the calculation, we restrict ourselves to fitting

with three parameters, two for the excess enthalpy (h12 ≠ h21) and one for the excess entropy

(s12 = s21), while the parameters that describe the excess Gibbs energy follow from the above

relation Eq. (8). As we observe no excess heat capacity, the excess properties are not

dependent on temperature, but only on the composition of the mixture. For each mixture of

different composition that has been measured in the adiabatic calorimeter, the set of the excess

parameters is obtained and given in Table 3. The values of the fitting parameters, h12, h21 and

s12 are not exactly the same for different compositions. However, the final coefficients in

Redlich-Kister expressions for the excess properties, obtained from the below described

procedure, are not dependent on composition. Calculated excess functions of composition

exch~

, excs~ and excg~ are shown in Fig. 6. Their values can be fitted by a polynomial of an

optional order, which will imply different number of the relevant Redlich-Kister excess

parameters. The proposed procedure provides both excess enthalpy and entropy as functions

of composition. These two quantities give the excess free Gibbs energy, enabling the

Page 40: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

32

y = 0.0172x + 0.6987

y = 0.0607x + 0.1674

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x

exce

ss fu

nctio

n

calculation of the phase diagram. Accordingly, the excess enthalpy and entropy curves

presented in Fig. 6 are fitted in the linear functions of composition, so that finally two

Redlich-Kister parameters are obtained for these excess properties, being: EliqsolH ,

0,∆ = 2146.64

J⋅mol-1, EliqsolH ,

1,∆ = - 26.1 J⋅mol-1; EliqsolS ,

0,∆ = 1.14 J⋅(K⋅mol)-1, EliqsolS ,

1,∆ = - 0.252 J⋅(K⋅mol)-1.

Table 3. Excess parameters that determine dimensionless excess functions (excess enthalpy,

excess entropy and excess free Gibbs energy).

x2

h12

h21

s12

g12

g21

hexc

gexc

0.2937 0.611 0.741 0.213 0.399 0.529 0.597 0.699 0.4791 0.688 0.604 0.073 0.615 0.531 0.616 0.574 0.5312 0.757 0.755 0.270 0.487 0.485 0.715 0.489 0.5338 0.723 0.799 0.264 0.459 0.535 0.574 0.531 0.6025 0.753 0.569 0.151 0.602 0.419 0.649 0.531 0.7976 0.753 0.524 0.230 0.523 0.294 0.625 0.488

Figure 6. Dimensionless excess quantities as functions of overall composition:

� – excess enthalpy exch~

; � – excess entropy excs~ ; � – excess Gibbs energy excg~ .

Page 41: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

33

0.00 0.20 0.40 0.60 0.80 1.00

x

325

335

345

355

365

T / K

Using these values the phase diagram is calculated and compared to the phase diagram as

determined in Ref. [4] (Fig. 7). Similarity between the presented phase diagrams illustrates

that the crystallization follows closely the equilibrium phase diagram, as expected since the

crystallization is performed very slowly.

Figure 7. The phase diagrams calculated using the excess properties from equilibrium model

in Ref. [4] (dashed line) and from the kinetic model (solid line).

2.6 Summary The validity of the introduced kinetic model is demonstrated by successful

reproduction of the enthalpy path of the mixture during cooling for the known excess

properties. The applicability of the model is extended so that it yields the excess properties

when the cooling path of the mixture is at disposal. By fitting the measured enthalpy of the

mixture to the theoretical expression, the parameters that define the excess enthalpy and

entropy of the solid phase as the functions of overall composition, are calculated. In this way,

Page 42: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 2: Kinetic Approach to the Determination of the Phase Diagram

34

the excess quantities can be obtained by using a relatively simple method, which basically

requires only the knowledge of the cooling curve of the mixture. Finally, the phase diagram is

achieved having the advantage over traditionally determined phase diagrams, in the sense that

both excess enthalpy and entropy are derived without adopting the approach of complete

equilibrium between totally homogeneous phases.

References: [1] A.N. Campbell, L.A. Prodan, J. Amer. Chem. Soc. 70 (1948) 553.

[2] M.T. Calvet, M.A. Cuevas-Diarte, Y. Haget, P.R. van der Linde, H.A.J. Oonk, Calphad 15

(1991) 225.

[3] R. Stosch, S. Bauerecker, H.K. Cammenga, Z. Phys. Chem. 194 (1996) 231.

[4] P.R. van der Linde, Molecular mixed crystals from thermodynamic point of view, Ph.D.

Thesis, Utrecht University (1992).

[5] J.A. Bouwstra, Thermodynamic and structural investigations of binary systems, Ph.D.

Thesis, Utrecht University (1985).

[6] A.C.G. van Genderen, C.G. de Kruif, H.A.J. Oonk, Z. Phys. Chem. Neue Folge, 107

(1977) 167.

[7] Z. Zbio�ski, J. Crys. Growth 58 (1982) 335.

[8] P.A. Reynolds, Molecular Physics 29 (1975) 519.

[9] M.A. Korshunov, Crystallography Reports 48 (2003).

[10] J.H. Los, M. Matovic, J. Phys. Chem. B 109, (2005), 14632.

[11] M. Matovic, J.C. van Miltenburg, J.H. Los, J. Cryst. Growth 275 (2005), e211-e217.

[12] J.H. Los, W.J.P. van Enckevort, E. Vlieg, E. Flöter, J. Phys. Chem. B 106 (2002), 7321.

[13] J.C. van Miltenburg; H.A.J. Oonk, J. Chem. Eng. Data 46 (2001) 90.

[14] J.C. van Miltenburg, A.C.G. van Genderen, G.J.K. van den Berg, Thermochim. Acta 383

(1998) 13.

[15] Z. Chvoj, J. Šesták, A. T�iska, Kinetic Phase Diagrams, Elsevier: Amsterdam (1991).

[16] R.E. Timms. Prog. Lipid Res. 23 (1984) 1.

[17] O. Redlich, A.T. Kister, Ind. Eng. Chem. 40 (1948) 345.

Page 43: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 35

Chapter 3

Kinetic Segregation in Crystallization of Mixed

Crystals

3.1 Introduction Crystallization is a kinetic process. Therefore, the composition of a multi-component

solid phase formed at the surface may deviate considerably from the composition according to

the equilibrium phase diagram. Due to the change in composition of the liquid phase, the

composition of growing solid will also change during the crystallization, leading to

composition gradients in the solid phase. These gradients remain for very long periods due to

very low diffusion rate in the solid phase, especially in molecular solid phases. Thus, in such

mixed systems, the equilibrium state will hardly ever be reached and the system will stay in a

metastable state.

A prediction of this non-equilibrium state of the solid phase requires a kinetic

modeling of the crystallization process. The kinetic modeling should contain a description of

the kinetic segregation at the solidification front as a function of the interfacial undercooling

and composition of the liquid phase. Moreover, especially at large supersaturation, the

segregation will induce concentration and temperature gradients in the liquid phase near the

growth front. This results in different properties of the liquid phase at the interface and in the

bulk. To define the actual temperature and composition of the liquid phase at the interface,

both mass and heat transport limitations have to be considered.

So far in literature the kinetic segregation was studied mainly for atomic systems, such

as metal alloys and semiconductors1,2. As compared to these systems where segregation is

mainly affected by transport limitations, in the crystallization of molecular systems the effect

of undercooling is more dominant. It was shown that the segregation of the molecular systems

with relatively large melting entropy was significantly reduced already at modest

undercoolings3.

Page 44: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

36

In a previous study of a binary molecular system4, we showed that even at very low

cooling rate the final state of mixed crystals would not be the equilibrium state. Here, the

crystallization conditions were close to equilibrium and the undercooling was not remarkable.

Still, a kinetic approach was needed for successful description of the crystallization. Since in

this case the crystallization was very slow, we could assume that the surface of the growing

solid phase and the existing liquid phase were in near equilibrium. Accordingly, the

segregation was following the equilibrium phase diagram.

At large undercoolings the impact of the kinetics on the segregation is expected to be

much stronger. In the present work we performed crystallization experiments on binary

mixture of 1,4 – dichlorobenzene and 1,4 – dibromobenzene, where the melt was solidified at

the conditions well away from equilibrium. After crystallization, the composition of the solid

phase was measured. The experimental results were compared to theoretical calculations

based on two kinetic segregation models, being the linear kinetic segregation model (LKS)5

and mean field kink site kinetic segregation model (MFKKS)6.

The results of proposed kinetic models have been compared to Monte Carlo (MC)

simulations elsewhere6. When the bond energies between alike particles, φ11 and φ22, are equal

and the excess bond energy is zero, i.e. φ12=(φ11+φ22)/2, the results of the LKS and MFKKS

models are exactly the same. In the case of different bond energies between particles, i.e.

φ11≠φ22, the LKS model gives a good description for small undercooling, i.e., near

equilibrium. At large undercooling the MFKKS model shows better agreement with the MC

results. In this study, we will use the combined LKS-MFKKS model, which tends to the LKS

model for small undercoolings and switches smoothly to the MFKKS model for increasing

undercooling. In the limit of zero undercooling, the combined kinetic model is consistent with

the thermodynamic equilibrium phase diagram. The other model, MFKKS, was introduced as

more kinetic than the LKS model, since it includes more details on the growth kinetics at the

surface.

In previous work3, the LKS model was coupled with the mass and heat transport

limitations. It was shown that a coupled description of the interfacial segregation and the mass

and heat transport effects is particularly relevant for molecular systems. This extended model

allows for the construction of so-called effective kinetic phase diagrams, where the

composition of the solid phase that is growing at a given undercooling of the bulk liquid phase

Page 45: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 37

is given as a function of the bulk liquid composition and temperature. In this study, we will

experimentally validate the results of the combined LKS-MFKKS model coupled with the

mass and heat transport limitations.

3.2 Experimental set-up The crystallization experiments were performed in the apparatus, schematically

presented in Figure 1. A glass tube containing the sample is placed in a metal block that is

equipped by a heater. In this work, the samples are the mixtures of 1,4 – dichlorobenzene and

1,4 – dibromobenzene. An insert into the glass tube contains a cooling block and two

thermometers. One thermometer is placed in the cooling block, the other in the tube itself to

monitor the temperature of the liquid sample. A magnetic stirrer is placed at the bottom of the

glass tube and it can be driven at different speeds. The cooling block is connected to a

controlled flow of water by a double walled tube; the cooling water enters the block through

the inner tube and flows out through the outer tube. The crystallization of the mixture on the

cooling block can be observed through a glass window in the heating block.

Figure 1. The sketch of the experimental set-up; 1 – glass tube, 2 – metal block equipped by

heater, 3 – cooling block with thermometer inside, 4 – thermometer measuring the liquid

temperature, 5 – controlled water flow.

FC

1

2

3

4

5

Page 46: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

38

The solid compounds were placed into the glass tube and melted. Before cooling, a

small sample of the melt was taken for measuring the overall composition of the mixture by a

gas chromatograph (GC). The stirring rate was adjusted to a certain speed and the cooling

started by letting the flow of cold water through the cooling block. The temperature-time

profiles during cooling in different experiments are given in Figure 2. Clearly, the cooling

profiles are not the same in all experiments, since the water flow could not be regulated very

accurately in our set-up.

At a certain moment, the crystals appeared on the surface of the cooling block. The

solid layer was left growing under continued cooling, where the total cooling time was 180

seconds in all experiments. After this period, the cooling block with the solid layer on it was

taken out from the remaining melt. The grown solid phase was scratched from the block and

dissolved in diethyl ether for analysing its composition by the GC. The composition of the

remaining liquid phase was also measured.

The described experimental procedure yields the composition of the solid phase

formed at the conditions far from equilibrium. The obtained results are compared to the solid

compositions predicted by equilibrium and by kinetic models, which are described further in

the text.

Page 47: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 39

0 30 60 90 120 150 180

t / s

300

305

310

315

320

325

330

335

340

345

T / K

no stir 50 100 200 300 400

Figure 2. The change of the cooling block temperature in different experiments performed

under the stirring rates as indicated in the figure.

3.3 Linear kinetic segregation (LKS) model During crystal growth, atoms or molecules attach and detach from the interface, i.e.,

they transform from the liquid to the solid phase and vice versa. According to non-equilibrium

thermodynamics, the flux of a component i from the liquid to the solid phase, +iJ , and the

reverse flux, −iJ , are related by:

���

����

� ∆=−

+

surf

surfi

i

i

RTJJ µ

exp 1

where Tsurf is the temperature at the surface and surfiµ∆ is the difference between the chemical

potentials of component i in the liquid at the surface, surfli,µ , and that of the growing solid

phase, grsi

,µ .

Page 48: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

40

In accordance with chemical reaction theory, the attachment flux can be written as: surfl

iii aKJ ,++ = 2

where +iK is a kinetic constant and surfl

ia , is the activity of component i in the liquid phase at

the surface.

The net flux or growth rate Ri for component i is derived as follows:

surfii

grleqi

surfliisurf

surfisurfl

iiiii KaaKRT

aKJJR σµ +++−+ =−=��

��

����

����

� ∆−=−= )(exp1 ,

,,, 3

where surfiσ is the absolute supersaturation for component i at the surface.

The activity of component i in phase P is a function of the composition of that phase, being

related as Pi

Pi

Pi xa γ= , where P

ix is the mole fraction and Piγ the activity coefficient of

component i in the phase P. The latter quantity is also composition dependent and is related to

the excess Gibbs free energy in phase P, excPG , , by:

Pi

excPPi N

GRT

∂∂=

,

)ln(γ 4

where PiN is the amount of component i in phase P.

According to standard thermodynamics, the equilibrium activity of component i in the liquid

phase with respect to a growing solid phase of composition grsix , , is given by:

��

��

����

����

� ∆+��

���

�∆−

∆∆==

surf

surfi

surfiip

surfi

surfiigrs

igrs

igrleqi

grleqi

grleqi T

TT

TR

c

TRT

THxxa lnexp ,0,,,,

,,,

,, γγ 5

where iT and 0,iH∆ are the pure component melting temperature and melting enthalpy at iT ,

respectively, ipc ,∆ is the difference between the heat capacities of the liquid and the solid

phase, and isurfsurf

i TTT −=∆ .

The increase of the amount of component i in the solid phase, siN , is a function of the growth

composition:

)( ,grsii

si xR

dtdN

= 6

For a steady state it holds that sgrsi

si NxN ,= , with �=

isi

s NN . Combined with Eq. 6, this

leads to the following set of coupled first order differential equations for grsix , :

Page 49: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 41

)(1 ,

,

RxRNdt

dx grsiis

grsi −= 7

where �==i i

s RdtdNR / is the total growth rate. In Ref. [5] it was shown that the steady

state and physically relevant solution of this first order differential equation has to fulfill the

common stability criterion for a stable steady state solution, being:

0)( ,, <− RxR

dxd grs

iigrsi

8

In addition, the individual growth rates have to be positive, i.e., 0≥iR .

For a binary mixture, the steady state solutions of Eq. 7 are equivalent with the solutions of

the following equation:

)(

)(,,2

,,2

,2

,22

,,1

,,1

,1

,11

22

11,

2

,1

grleq

grleq

surflsurfl

grleq

grleq

surflsurfl

surf

surf

grs

grs

xxK

xxK

KK

xx

γγγγ

σσ

−−

== +

+

+

+

9

which can be solved numerically for the composition of growing solid phase grsix , for the

given surflix , and surfT . Equation 9 can have more stable steady state solutions, yielding the

simultaneous growth of two solid phases with different compositions. Typically, this can

occur for eutectic or peritectic systems. Apart from this compositional kinetic phase

separation, it is possible that different polymorphs grow simultaneously. In that case, Eq. 9

should be solved for each of the polymorphic forms.

3.4 Mean field kink site kinetic segregation (MFKKS) model The mean field kink site kinetic segregation model has been introduced in Ref. [6],

where its performances have been compared to the results of LKS model and Monte Carlo

simulations based on the Kossel model. It includes the important role of kink sites in the

growth process. The model allows the composition at the kink site to be different from that of

the bulk solid phase. Here, we will give a brief overview of the MFKKS model.

The net incorporation of component i, or growth rate, can be approximated as follows:

)~

( skii

liiki xKxKNR −+ −= 10

where kN is the average number of kink sites per unit surface area, skix is the average mole

fraction of component i at kink sites and where −iK

~ is the average detachment rate of a

Page 50: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

42

particle i at a kink site. Analogously to the LKS model, the growth composition can be

calculated from:

skl

skl

s

s

xKxK

xKxKRR

xx

2222

1111

2

1

2

1~~

−+

−+

−−

== 11

For a kink particle, having three bonds, the MFKKS model assumes a mean field distribution

of neighbors, implying:

�=

−− =2

1,,

~

lkjijkl

sl

sk

sji KxxxK 12

where the compositions of three neighbor sites are assumed to be equal to the bulk

composition, while the time average composition at the kink site itself, skix , is permitted to be

different. −ijklK is the detachment rate of a particle i from a kink site with neighbors of type j, k

and l along the step, on the terrace and down in the bulk respectively, that equals to:

��

��

� ∆−∆+∆+∆−= +−

Tk

STKK

B

eviilikijiijkl

,0,

~exp

φφφ 13

where +0,iK is a kinetic constant for particle i, ijφ∆ , ikφ∆ , ilφ∆ are the corresponding liquid

and solid bond energy changes and Bk is the Boltzmann constant. The change in vibrational

entropy for a selected event, eviS ,~∆ , is defined in the Monte Carlo model6 and it is taken to be

the pure component dissolution entropy, iS∆ , reading:

i

ii

i

iievi TT

HSS

φ∆=

∆=∆=∆

3~, 14

where Ti is the melting temperature and iH∆ is the melting energy of the pure component i.

Considering that the removal of a kink site particle along a straight step generates another

kink site particle, the expression for the variation of skx1 in time is as follows:

sskssksklsklsk

xxKxxKxxKxxKdt

dx21121221122211

1 ~~ −−++ −+−= 15

where −ijK

~ (i � j) is the average detachment probability for a kink particle of type i with a

neighbor of type j along the step, which is assumed to become a kink particle itself after

Page 51: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 43

removal of its neighbor at the kink site. Similar to Eq. (15) holds for component 2, but it is not

independent. For steady state growth, i.e., 0/ =dtdx ski , it follows that:

ssll

slsk

xKxKxKxK

xKxKx

2121212211

121111 ~~

~

−−++

−+

++++

= 16

while sksk xx 12 1−= . The latter equation expresses the kink composition as a function of the

bulk composition of the growing solid phase. By combining equations 11, 12 and 16, and

using ss xx 21 1−= , Eq. 11 becomes an equation with only one variable, sx2 , and can be solved

numerically.

For the combined kinetic model the net incorporation of a component i is expressed as

a linear combination of those according to the LKS and MFKKS model:

)~

)1(()1( ,skii

leqii

liik

MFKKSi

LKSii xKsxKsxKNsRRsR −++ −−−=+−= 17

Here, s is a smooth switching function depending on the total growth rate, R, and

undercooling, ∆θ, taken as:

εθθ+∆

∆=R

Rs 18

where ε is a small number introduced as a parameter that determines how fast the switch will

be. Accordingly, when there is no undercooling the value of s is 0 and the combined model

becomes equal to the LKS model. For large undercooling, s-value approaches 1, and the

MFKKS model becomes dominant. In fact, the combined model switches to the MFKKS

model already at rather small undercooling, where it quite well predicts the MC results6. For

convenience, no temperature dependence is included in the above description. However, in the

calculations the bond energies and all relevant thermodynamic properties are taken as

temperature dependent.

3.5 Mass and heat diffusion limitations The proposed kinetic segregation models describe the segregation at the solidification

front, as a function of the interfacial undercooling and composition of the liquid phase. To

determine the effective segregation, being the actual segregation at the interface, we have to

take into account that the properties of the liquid phase at the interface and in the bulk are not

the same. During crystal growth, mass and heat transport boundary layers develop. They

Page 52: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

44

contain concentration and temperature gradients and their widths are related to the width of

the convective boundary layer. The widths of the mass and heat boundary layers are the

important parameters in hydrodynamic relations, which correlate the properties of the liquid at

the interface with those in the bulk. In practice, this connection is quite important for the

interpretation of the experimental data, since usually only the bulk properties are

experimentally accessible.

The width of the convective layer depends on the geometry and scale of the system,

and hydrodynamic properties of the liquid phase. Besides, it is also determined by the regime

of convective transport in the liquid phase. The convection caused by stirring is called forced

convection and at no stirring conditions one speaks of free (natural) convection.

In previous work3, the mass transport effect was treated for the case of 1-dimensional

geometry with a flat surface, leading to the following expression for the liquid composition at

the surface with respect to the liquid bulk composition:

( ) ( )surfm

grsi

grssurfld

bulkli

bulklsurfl

grsi

grssurfld

surfli qxggxgxggx σ~exp,,

,,,

,,,

,, −+= 19

where surfσ~ is a weighted supersaturation at the surface, defined as:

�=

=2

1

,

ˆ~

ii

grsisurf Rv

Vσ 20

with grsiV , the volume per particle of component i in the growing solid phase and v̂ is an

average crystal growth velocity constant. The quantity dg is a factor that corrects for

differences in molar densities between the liquid and the solid phase, defined as:

( )( )1)~exp(

1~exp,,

,,

−−

=surf

mgrs

surfl

surfm

bulklsurfl

d qg

qgg

σσ

21

where the density ratios are defined in the terms of the liquid and solid phase molar

concentrations, being: surflgrsgrssurfl ccg ,,,

, /≡ and surflbulklbulklsurfl ccg ,,,

, /≡ with �=i

grsi

grs cc ,, ,

�=i

surfsi

surfs cc ,, and �=i

bulkli

bulkl cc ,, . The parameter mq in Eq. (19), defined as:

m

mm D

vq

∆=

ˆ 22

Page 53: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 45

is the crucial parameter for mass transport limitation, where m∆ is the width of mass transport

boundary layer and mD is the mass diffusion coefficient. In case of equal molar densities,

equation 19 becomes:

( ) ( )surfm

grsi

bulkli

grsi

surfli qxxxx σ~exp,,,, −+= 23

In the non-diffusion limited case, i.e. for small values of mq , the composition of the liquid at

the surface becomes equal to that of the liquid bulk, which means that there is no mass

transport limitation. This will be the case for slow crystal growth or a thin mass transport

boundary layer. High values of mq imply large exponential term, which needs to be

compensated by a small number coming from the difference grsi

bulkli xx ,, − . This means that the

composition of the growing solid phase will approach the composition of the liquid bulk,

giving the reduced segregation.

In contrast to mass transport, heat can be transported into both liquid and solid phase.

Correspondingly, there are two parameters for heat transport limitation, being:

lT

lTl

T Dv

q∆

and sT

sTs

T Dv

q∆

24

where superscripts l and s refer to the liquid and solid phase, while the quantities are

analogously defined as those for the mass transport. In this work we derived the expression

for the temperature of the interface, similar as in Ref. [3], but for the case that the

temperatures of the liquid and solid bulk are not equal:

)1()1(

)1)(1()1()1(~~

~~,,,~,~

surflT

surfsT

surfsT

surflT

surflT

surfsT

qsp

sqlp

l

qqgrsgrsbulksqsp

sbulklqlp

lsurf

eccecc

eeHcTeccTeccT

σσ

σσσσ

−−

−+−

−−∆+−+−= 25

where lc and sc are the concentrations of the liquid and solid bulks, respectively, lpc and s

pc are

their heat capacities and grsH ,∆ is the composition and temperature dependent melting

enthalpy.

Furthermore, we also included the fact that in the beginning of the crystallization there

is no steady state. In that sense, the mass boundary layer actually builds up with time until it

reaches a final steady state width. At each moment, the mass conservation balance has to be

fulfilled and by that the width of the mass boundary layer will be determined. Namely, the

total concentration of component i in the layer of grown solid phase and in the boundary layer

Page 54: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

46

near interface, has to be equal to its concentration in the liquid bulk. For the given cylindrical

geometry, the mass conservation reads:

� � −=+)( )(

)(

,20

22

1

0

2

1

)(22tr

r

tr

tr

bulkliqi

liqi

soli xrrdrrxdrrx πππ 26

where 0r is a diameter of the cooling block; )()( 01 tdrtr += is the coordinate representing the

position of the solid-liquid interface with d(t) the time dependent width of the solid layer; and

)()()( 12 ttrtr m∆+= with )(tm∆ the width of the mass boundary layer.

The concentration profile of component i within the boundary layer3 to be substituted

in Eq. (26), reads:

))exp(1()exp(1

)(,,

, rD

D

xxxrx

m

surfliqi

bulkliqisurfliq

iliqi

νν −−

����

����

∆−−

−+= 27

In this way, the width of boundary layer is introduced as time dependent parameter till

the steady state is reached. The final steady state width of the boundary layers is determined

by the regime of convection. In case of stirring, the width of the convective boundary layer

( c∆ ) for forced convection is calculated by using the following equation, while taking into

account the given geometry7: 2/1

4.2 ��

���

�=∆ων

c 28

where ω is the angular velocity and ν is the kinematic viscosity. Now, the steady state widths

of the mass and heat transport boundary layers in the liquid phase, m∆ and lT∆ , can be

calculated from the relations:

cm Sc∆�

���

�=∆3/11

and clT ∆�

���

�=∆3/1

Pr1

29

where mDSc /ν= is the Schmidt number and lTD/Pr ν= is the Prandtl number. mD

represents the mass diffusion coefficient, while lTD refers to the thermal diffusion coefficient

in the liquid phase.

When there is no stirring, the final widths of mass and heat boundary layers are determined by

the natural convection. In this case, the motion of liquid occurs due to the temperature and

Page 55: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 47

composition gradients in the liquid phase in combination with gravity. To calculate the widths

of mass and heat boundary layers, the so-called Grashof number is required, expressed as:

DLg

Grνβ 3

= 30

where g is the acceleration due to gravity, L is a characteristic length (in our case the length of

the cooling block), D is taken as the mass or thermal diffusion coefficient in the liquid phase,

depending on to which of these two transport phenomena the Grashof number is applied. β is

a volumetric thermal expansion coefficient, defined as bulkliq

surfliq

bulkliq ρρρβ /)( −= . Finally, the

steady state widths of mass and heat boundary layers for free convection are calculated as

follows:

4/156.02 mm Gr

L+

=∆ and 4/156.02 T

lT Gr

L+

=∆ 31

For the purpose of our modeling, the combined segregation model is coupled with the

relations for mass and heat transport limitation, where everything is solved for time dependent

boundary conditions. As a result we obtain the composition of the growing solid and that of

the liquid at the interface, as well as the temperature at the surface. These are calculated for

the experimental set of data, consisting of the measured liquid bulk composition and the

measured liquid bulk and the cooling block temperatures. In this way, we can determine the

effective undercooling as the difference between the equilibrium liquid temperature for the

given composition of the liquid at the interface and the actual temperature of the interface. In

addition, we can follow the development of the mass and heat boundary layers during the

crystallization and make an estimation of time needed to reach the steady state.

3.6 Estimation of chemo-physical properties As previously discussed, in order to include the mass and heat transport limitations,

the knowledge of chemical and physical properties including transport properties of the two

components, 1,4 – dichlorobenzene and 1,4 – dibromobenzene, is required. Here, we will give

a brief overview of estimation methods we used. A good estimation method should be easy to

use, requiring a minimum amount of easily accessible input data, and it should be reasonably

accurate, which we have checked.

Page 56: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

48

3.6.1 Viscosity estimation

The following expression enables the estimation of the liquid dynamic viscosity of a

compound at any temperature8:

���

����

�−+=

bLbL TT

B11

lnln 4ηη 32

where Lη [cp] is the dynamic viscosity at given temperature T [K] and Lbη is the viscosity at

the normal boiling point, Tb. The parameter B4 is calculated from:

( )[ ]RTTRTKn

B bbF −+= ln75.81

4 33

where the values of n are given for different types of compounds and KF is a constant related

to the structure of a compound8. Here, R is the gas constant, being 1.987 cal⋅mol-1⋅K-1. The

average deviation of the proposed method from the available experimental values is 19.0%.

Thus, the viscosities of 1,4 – dichlorobenzene and 1,4 – dibromobenzene can be fairly well

estimated at any temperature, just by knowing their normal boiling points and molecular

structures.

3.6.2 Liquid density estimation An estimation of the liquid density of a compound at any temperature T is given by8:

m

bLbL T

TM

��

����

����

�−= 23ρρ 34

where the input data are the molar density at the normal boiling point (ρLb), the molar mass

(M) and the value of the exponent m. The latter parameter varies, depending on the chemical

class of the compound8. For 1,4 – dibromobenzene, the value of the liquid density at 328.15 K

from the literature9 is 1.248 g⋅cm-3. By using the proposed method, the calculated value is

1.262 g⋅cm-3, having an error of 1.1%. This indicates quite good accuracy of the proposed

method for the estimation of the liquid density.

Page 57: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 49

3.6.3 Solid density estimation The solid density of a compound can be estimated using the following relation8:

Ss V

M660.1=ρ 35

where the unity for the solid density is in g⋅cm-3.

Here, VS is the crystal volume for a single molecule, calculated by:

�=i

iiS vmV 36

where mi is the relative stoichiometric multiplicities and vi is a unit volume of the atomic

element i, where each atom, ion or structure in the molecule has its own specified volume

contribution. This method for solid density estimation was developed from a database of 500

organic crystalline compounds ranging in molecular weight from 50 to 1000 g⋅mol-1. An

average absolute error between experimental and estimated data is 2.0%. The solid densities

of 1,4 – dichlorobenzene and 1,4 – dibromobenzene are taken as independent of temperature.

3.6.4 Mass diffusion coefficient For the description of mass transport limitation, the determination of the mass diffusion

coefficients for the both components in the liquid phase is required. These are estimated by an

equation that expresses the diffusion of an organic solute A, diluted into organic solvent B10,

being:

3/10

ABAB V

KTD

η= 37

where 0ABD is a mutual diffusion coefficient of solute A in solvent B [cm2⋅s-1], Bη is the

viscosity of solvent B [cp] and VA is the molar volume of solute A at its normal boiling

temperature [cm3⋅mol-1]. The dimensionless parameter K is defined as follows:

��

���

����

�+⋅= −

3/2

8 31102.8

A

B

VV

K 38

In principle, the diffusion coefficients estimated in this way represent the infinitely dilute

diffusion coefficients. We calculated the binary diffusion coefficient for non-diluted solutions

Page 58: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

50

simply by the linear interpolation of the two diluted solution coefficients, 0ABD and 0

BAD ,

according the given composition of the melt.

To check the reliability of the described estimation method, we performed a simple

experiment that enables the determination of binary diffusion coefficient in the liquid. A 2 cm

high layer of the solid 1,4 – dibromobenzene was placed in a glass tube. Another layer of 2

cm of the solid 1,4 – dichlorobenzene was put on top of the first one. According to the

measured masses of the components, the overall composition was x2=0.571. The glass tube

was vertically placed in the oven at 100 oC, where the components stayed for 18 hours in the

melt. Thereafter, the tube was taken outside the oven where its contents immediately

solidified. The samples of the solid mixture were taken at different heights and their

compositions were analysed by the gas chromatograph. In this way, we obtained the

distribution profile of the components in the tube, as the consequence of the diffusion in the

liquid phase. The experimentally estimated value of diffusion coefficient is 2.66⋅10-5 cm2⋅s-1.

The corresponding value calculated using the above described estimation method (Eq. 37 and

38) is 3.34⋅10-5 cm2⋅s-1. This gives an absolute error of 25 %. To avoid this inaccuracy, we

scaled the calculated diluted solution coefficients of both components by factor 0.8, so that the

estimated value for the mixture matches the experimental one at 100 oC.

3.6.5 Thermal conductivity and diffusion coefficient

The width of thermal boundary layer in the liquid phase ( lT∆ ) is related to the width of

laminar layer due to the convection ( c∆ ) by:

c

lTl

T

D∆��

����

�=∆

3/1

ν 39

where lTD [cm2⋅s-1] is the thermal diffusivity in the liquid phase and ν [cm2⋅s-1] is the

kinematic viscosity.

The thermal diffusivity coefficient in the liquid phase is calculated as follows:

lp

l

llT cc

kD = 40

Page 59: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 51

where cl is the molar concentration of the liquid phase [mol⋅cm-3], cpl is the heat capacity

[J⋅(mol⋅K)-1], and kl is the thermal conductivity in the liquid phase [J⋅(cm⋅s⋅K)-1]. The latter

property can be estimated for the given compound from the relation8:

��

−+−+⋅=

3/2,

3/2,

5.0

3

)1(203

)1(2031064.2

irb

irli T

T

Mk 41

where Tr,i is the reduced temperature and Trb,i refers to the reduced boiling point. The reduced

properties of a compound are obtained by dividing actual properties by the corresponding

critical values. The results of Eq. (41) compared to the experimental values for thermal

conductivity in the liquid phase, give the average absolute error of 7.1%.

The presented methods are used for the estimation of chemo-physical properties of the

pure components. For the binary mixture of given composition, the relevant properties are

calculated as a linear interpolation of the pure components’ properties. Additionally, the

characteristics of the liquid phase within the boundary layer are determined for the

composition taken as the average between the liquid composition at the interface and that in

the bulk.

3.7 Results and discussion

The binary mixtures of 1,4-dichlorobenzene and 1,4-dibromobenzene of two different

overall compositions, expressed as mole fraction of 1,4-dibromobenzene (see Table 1), were

crystallized in the described experimental set-up. The sample 1 was crystallized under three

different stirring rates of the melt and the sample 2 under six stirring rates, as indicated in

Table 1. Before each experiment, the overall compositions were measured. However, they did

not remarkably change, as well as, the compositions of the melt before and after

crystallization, because relatively small amounts of the solid were grown.

As mentioned above, our kinetic model by means of Eq. (25) allows us to calculate the

change of interface temperature for the given experimental conditions, owing to the registered

temperature changes of the cooling block and the melt (see Fig. 2). For clarity, we present in

Figure 3 the results from one experiment done under the stirring rate of 300 rpm. At the very

beginning of crystal growth, the temperature of the interface is equal to the temperature of the

cooling block. As the solid growth evolved, the interface temperature rises slightly towards

Page 60: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

52

the temperature of the liquid bulk. The increasing trend of the interface temperature, caused

by the motion of the interface towards the warmer melt, was observed in all experiments.

Anyway, the interface temperature did not rise significantly, so that it could be considered as

almost constant during the most of the crystallization time.

Table 1. The experimental, kinetic and equilibrium compositions of the solid phase are given

together with the corresponding overall compositions of the melt and the applied stirring

rates. The last two columns present the difference between the solid compositions as

measured by GC and those determined from the kinetic model and equilibrium.

x overall

Stirring rate / min-1

x sol experiment

x sol kinetic

x sol equilibrium

dx exp-kinetic

dx exp-equil

SAMPLE 1

0.578 0 0.626 0.625 0.704 0.001 -0.078 0.584 50 0.612 0.631 0.713 -0.019 -0.10 0.569 200 0.610 0.653 0.730 -0.043 -0.12

SAMPLE 2 0.323 0 0.335 0.363 0.418 -0.028 -0.083 0.325 50 0.367 0.356 0.417 0.011 -0.050 0.323 100 0.400 0.378 0.443 0.022 -0.043 0.329 200 0.398 0.379 0.446 0.019 -0.048 0.330 300 0.396 0.397 0.469 -0.001 -0.073 0.329 400 0.423 0.389 0.462 0.034 -0.039

In Table 1, the average solid phase compositions calculated from the described kinetic

model are given together with the experimental and equilibrium values. The kinetic and

experimental results show very good agreement, although there are fluctuations in the

differences. The average equilibrium solid compositions in Table 1 are obtained by integrating

the equilibrium solid compositions determined with respect to the calculated time dependent

liquid composition and temperature at the interface. These equilibrium results, as compared to

the experimental solid compositions, show much stronger segregation. Thus, the kinetic

model reproduces the experimental results much better than the equilibrium approximation.

Page 61: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 53

0 20 40 60 80 100 120 140 160 180

t / s

300

310

320

330

340

350

360T

/ K

Tsolid bulk Tliquid bulk Tinterface

Figure 3. The monitored temperature changes of the metal block (solid bulk) and the liquid

bulk during the experiment done under stirring rate of 300 rpm, together with the calculated

interface temperature.

To illustrate further the performance of the kinetic model, we present in Figure 4 the

change of the liquid composition at the interface during crystallization together with the

corresponding kinetic, equilibrium and measured solid compositions. Here, we show the

results from one experiment, done under the stirring rate of 300 rpm, where the solid

composition calculated from the kinetic model agrees perfectly with the experimental value.

As can be seen, the equilibrium model predicts too strong segregation. The graph also

demonstrates clearly the effect of mass diffusion limitation. Namely, although the overall

liquid bulk composition is x2 = 0.33, the liquid composition at the interface is somewhat less.

Since 1,4-dibromobenzene has a higher affinity to build in the solid phase, it will be depleted

in the liquid phase near the solidification front. Consequently, a concentration gradient occurs

in the liquid phase, with less 1,4-dibromobenzene in the liquid at the interface than in the

bulk.

Page 62: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

54

0 20 40 60 80 100 120 140 160 180

t / s

0.25

0.30

0.35

0.40

0.45

0.50

0.55x pd

ibro

mob

enze

ne

xliquid surface xsolid kinetic xsolid equilibrium xsolid experiment

Figure 4. The change of the solid composition as calculated from the kinetic model, from the

equilibrium model and as measured by GC, together with the change of the liquid

composition at the interface. The presented results are for the experiment done under the

stirring rate of 300 rpm.

In Figure 5, we show how the mass transport boundary layer builds up with time under

different stirring rates. As mentioned before, in the beginning of crystallization there is no

steady state. However, from the calculations it appears that the steady state is reached quite

fast. Under low stirring rates it takes more time until the layer reaches its final width. The

thickest boundary layer is formed in the case of no stirring, where the width of the layer is

determined by the natural convection. As the stirring rate increases, the final width of the

boundary layer becomes smaller, and it takes less time to reach the steady state.

Page 63: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 55

Figure 5. The evolution of the mass transport boundary layer during crystallization

experiments performed with the sample 2 under different stirring rates.

The heat is transferred through both liquid and solid phase during crystallization,

causing the temperature gradients in the grown solid as well as in the liquid phase near the

interface. The temperature profiles in the solid and liquid phase during the crystal growth are

given in Figure 6 at different time periods, together with the temperature levels of the metal

block and the liquid bulk at the given times. The dots indicate the temperature at the interface,

which after it has initially decreased, remained almost constant during the rest of

crystallization process. The bars represent the distance from the metal tube interface at which

the liquid phase has the liquid bulk temperature. Thus, the horizontal distance between the

dots and the bars indicates the width of the thermal boundary layer at the given time.

However, the linear trend of the temperature gradient from the metal interface to the liquid

bulk suggests that the heat transport is quite fast.

0

0.005

0.01

0.015

0.02

0.025

0 20 40 60 80 100 120 140 160 180time / s

thic

knes

s of m

ass b

ound

ary

laye

r / c

m no stirring

50 min-1

100 min-1

200 min-1

300 min-1

400 min-1

Page 64: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

56

Figure 6. The temperature profiles in the liquid and solid phase at different times during the

crystallization: 0, 45, 90, 135 and 180 seconds.

Figures 7a and 7b give an illustration of the concentration profiles in the liquid (bold

lines) and solid phase (thin lines) after several time periods of crystallization. The vertical

dotted lines indicate the position of the interface at the given time. The bars in Figures 7a and

7b represent the distance at which the liquid phase is of the liquid bulk composition.

Analogously to the thermal boundary layer (Figure 6), the horizontal distance between the

interface line and the corresponding bar indicates the width of mass boundary layer. In the

beginning of the crystallization, the mass boundary layer evolves quite rapidly until it reaches

the final width. This is in the agreement with the observations based on Figure 5.

After 1 second of crystallization, a thin layer of solid is formed, containing a

composition gradient as shown in Figure 7a. The concentration profile in the solid phase

shows a decreasing trend also after 10 seconds of crystallization. However, after approx. 30

seconds, the composition of the solid phase has already reached a value, which remains

almost constant over the rest of crystallization time (Fig. 7b). This composition gradient in the

solid phase is both due to the change of liquid composition and temperature at the interface.

305

315

325

335

345

355

-0.05 0 0.05 0.1 0.15 0.2

T /

K

distance / cm

0 s

45 s

90 s 135 s

180 s

Page 65: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 57

Figure 7. The concentration profiles in the solid phase (thin lines) and in the mass transport

boundary layer in the liquid phase (bold lines) at several time periods of crystallization: 1,

10, 60 and 100 seconds. Note the difference in scale on the horizontal axis.

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.005 0.01 0.015 0.02 0.025

distance /cm

com

posi

tion

p-di

brom

oben

zene

1s

10s

1s 10s

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1distance / cm

com

posi

tion

p-di

brom

oben

zene

60s100s

60s 100s

b)

a)

Page 66: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

58

3.8 Summary The crystallization of a molecular mixture of 1,4-dichlorobenzene and 1,4-

dibromobenzene was performed under non-equilibrium conditions, where the composition of

the grown solid phase was measured by the gas chromatography. The experimental results

were used for the validation of a kinetic model that describes the segregation. Additionally,

we consider that the segregation induces composition and temperature gradients in the liquid

phase, so that the properties of liquid phase near solidification front differ from those in the

bulk. Thus, for the description of crystallization process we included mass and heat transport

limitations, which enabled the determination of so-called effective segregation. The

experimental results agree very well with the solid compositions calculated from the proposed

kinetic model. On the other hand, the solid compositions predicted by equilibrium suggest

remarkably stronger segregation and deviate significantly from the measured data. In this

way, we have justified the utilization of the kinetic model coupled with mass and heat

transport limitations, showing a better performance in predicting the state of the solid phase

than the equilibrium approach. In fact, these results show the importance of interfacial

undercooling for the segregation during growth in mixed molecular system. For the presented

model system, this effect is still moderate. However, for molecular systems with higher

melting entropy, such as fats, this effect of the interfacial undercooling will be considerably

larger.

Page 67: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals 59

References: [1] K.M. Beatty, K.A. Jackson, J. Cryst. Growth 174 (1997) 28.

[2] K.A. Jackson, K.M. Beatty, K.A. Gudgel, J. Cryst. Growth 271 (2004) 481.

[3] J.H. Los, M. Matovic, J. Phys. Chem. B 109 (2005), 14632.

[4] M. Matovic, J.C. van Miltenburg, H.A.J. Oonk, J.H. Los, Calphad 30 (2006), 209.

[5] J.H. Los, W.J.P. van Enckevort, E. Vlieg, E. Flöter, J. Phys. Chem. B 106 (2002), 7321.

[6] 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.

[7] F. Rosenberg, Fundamentals of Crystal Growth I, Springer: Berlin, (1979).

[8] W.J. Lyman, W.F. Reehl, D.H. Rosenblatt, Handbook of Chemical Property Estimation

Methods: Environmental Behaviour of Organic Compounds, Am. Chem. Soc. (1990).

[9] D.R. Lide, Handbook of Organic Solvents, CRC Press (1995).

[10] R.H. Perry, Perry’s Chemical Engineers’ Handbook, McGraw-Hill 6thedition (1984),

286-287.

Page 68: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 3: Kinetic Segregation in Crystallization of Mixed Crystals

60

Page 69: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 61

Chapter 4

Thermodynamic Properties of Pure Triacylglycerols

Tristearin (SSS), Tripalmitin (PPP) and Trielaidin (EEE)

4.1 Introduction It has been known for over a century that natural oils and fats are complex multi-

component mixtures of different triacylglycerols (TAG), containing generally 96 to 99 % of

TAG. TAGs are made by an esterification between a glycerol molecule and three fatty acids,

where the three fatty acids in a TAG can be identical or different. Nearly all TAGs found in

nature have an even number of carbon atoms in each of the three acid chains. Many important

properties of oils and fats are controlled by phase transitions of their component TAGs. These

include solid-liquid (melting), liquid-solid (crystallization) and solid-solid (aging or

polymorphism) transitions. Phase behaviour of oils and fats is often complicated by mutual

interactions among the component TAGs.

The complexity of the thermal behaviour of fats is due to the great variety of TAGs,

which may differ by the length of hydrocarbon chains, by their unsaturation degree and by

their position on the glycerol backbone. Regarding the unsaturated TAGs, only cis type

unsaturated fatty acids are found in crude fats (oleic, linoleic, etc.), while trans double bonds

occur rarely in nature. During the commercial process of hydrogenation, in order to harden

vegetable oils, cis double bonds are often converted to the trans form. The presence of trans

fatty acids in the refined fractions modifies their technological properties – polymorphism,

kinetics of polymorphism and intersolubility. From a nutritional point of view, the trans

isomers may induce modifications of biochemical processes. Thus, the variation of fatty acids

in the chain-length and degree of unsaturation can dramatically affect the bioavailability and

digestibility of oils and fats.

Analysis of the TAG composition of an oil or fat requires methods of separating their

complex mixtures into individual components or at least into simpler mixtures that contain

only a few TAG each1. The complex mixtures of TAGs from oils and fats have usually been

Page 70: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

62

analyzed by reversed-phase high-pressure liquid chromatography (HPLC) and gas-liquid

chromatography GLC. The advantage of HPLC analysis over GLC composition analysis is

that HPLC results contain structural information, for example the position of the fatty acid on

the glycerol backbone. However, complete determination of the TAG profile can be achieved

only by several successive procedures that are tedious and time-consuming, which is less

practical for industrial application.

In order to meet more practical requirements, thermo-analytical techniques have been

used in oil and fat characterization to determine melting and crystallization profiles, heats of

transition, phase diagrams and solid fat content. Such techniques provide a wealth of

information that need to be correctly interpreted in thermodynamic terms. DSC is the most

widely used thermo-analytical technique in oil and fat research. DSC promises to offer a

sensitive, rapid and reproducible fingerprint method for the identification of oils and fats.

Results in most of the scientific literature show that one critical limitation of using DSC is the

dependence of the thermal transition on the scanning rate 2-5.

Polymorphism is also one of the aspects, which relates the molecular structures of oils

and fats to their macroscopic physical properties. In practical applications the polymorphic

properties of oils and fats are determined by multiple factors, particularly by fatty acid

compositions and their positions at the glycerol backbone. For example, the fats with simple

and symmetric fatty acids tend to exhibit typical α, β’ and β forms, whereas those with

asymmetric mixed-acid moieties often make the β’-form the most stable6.

Apparently, the prediction of phase behaviour of an oil or fat is a complex task, and its

complete understanding is still not achieved although a lot of research has been done on this

issue. Therefore, the study of their main constituents – TAGs, and further, of binary mixtures

of TAGs, is one step closer to a better insight into the phase behavior of oils and fats. In this

chapter, we will focus on the thermal analysis of three monoacid TAGs: tristearin (SSS),

tripalmitin (PPP) and trielaidin (EEE). Moreover, we will determine their purities based on

the data from melting experiments performed in the adiabatic calorimeter. The knowledge of

thermodynamic properties and purity of the TAGs is of great importance for further studying

of their binary mixtures (Chapter 5).

Page 71: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 63

4.2 Prediction of Thermodynamic Properties of TAGs Information about the enthalpy of fusion and the melting point of TAGs is important

to the prediction of solid-liquid equilibrium of TAG mixtures. Consequently, it is of great

interest for the production of fat-containing products like chocolates, salad dressings,

margarines or cooking oils. With regard to fat-containing products, it is important to know

and control the melting or the solidification properties of the edible fats. In the literature, most

of the experimental values for the enthalpy of fusion or the melting point are given for

saturated TAGs with an even number of carbon atoms in each chain7- 9. However, these pure

component properties have not been measured for all TAGs. Instead, correlations between

structural characteristics and thermal properties have been developed.

The first expressions that appeared in the literature concerned the prediction of melting

points of different polymorphs for saturated monoacid TAGs as a function of chain length10. It

was noticed that there is an alternation of melting points for even and odd homologues of the

β-form (Fig. 1, Ref. 11), which is probably due to the difference in methyl group packing that

normally occurs with tilted chains. However, the melting points of α and β’ forms as a

function of the chain length show smooth curves, indicating the similar methyl group packing

in these polymorphs for odd and even TAGs. In a later study12, the correlations were extended

to unsaturated TAGs, predicting both the melting points and the enthalpies of fusion for the

most stable, β-form. The enthalpy and entropy of the β-to-liquid and α-to-liquid transitions

for even-numbered carbon TAGs are given in Figure 2 (Ref. 11). The slopes of the enthalpy

versus the number of carbons in the acyl chain are rather linear, although there is a tendency

toward nonlinearity for the TAGs with fewer that 12 carbons in the chain. The entropy shows

a similar trend. The precise reason for nonlinearity in lower homologues is not clear.

In the work of Ollivion13, the enthalpies and entropies of fusion of unstable α and β’

polymorphs of saturated monoacid TAGs are reported as linear functions of the carbon

number of acyl chains. A more advanced model was proposed by Westdorp14 for different

polymorphs of saturated TAGs, where the enthalpy of fusion and melting points are not given

only as functions of the total number of carbon atoms in the three chains, but also the position

and chain length of the three acyl groups in TAG is taken into account. The group

contribution method15 considers also the position and the individual size of the acyl groups in

TAGs and it can be applied to any of the polymorphic forms of the saturated TAG.

Page 72: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

64

Figure 1. Melting points of pure TAGs as a function of the number of carbons in each acyl

chain, n, for three polymorphic crystalline states. Open symbols: odd TAGs; filled symbols:

even TAGs. (�) α-form; (�) β’-form; (�)β-form.

However, the models for the prediction of thermal properties of TAGs require

experimental data to proof their reliability. Therefore, we focused on studying the thermal

properties of SSS by adiabatic calorimetry and DSC, while detailed investigations of thermal

behaviour of PPP and EEE were previously reported16, 17.

n 10

15 20

-60

-40

-20

0

20

40

60

80

m.p. 0C

� �'

5

Page 73: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 65

Figure 2. Enthalpy (a) and entropy (b) of the melting transition from the α-form (�)

and β-form (�) of simple TAGs with an even number of carbons in the acyl chains as

a function of the number of carbons in each acyl chain (n).

Page 74: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

66

4.3 Thermal Behaviour of Pure TAGs: DSC and adiabatic

experiments

4.3.1 Tristearin (SSS) Tristearin, for short often indicated as SSS, is one of the saturated monoacid TAGs,

where three fatty acids of 18 carbon atoms are attached to the glycerol backbone. Despite the

fact that SSS is probably the most commonly investigated TAG concerning thermal

behaviour, we undertook this research as we feel that the combination of adiabatic calorimetry

and DSC can give more precise data and a better insight in the crystallization processes.

Moreover, the use of adiabatic calorimetry enables the measurements at such a low

temperature that the absolute entropy of the stable phase could be calculated. SSS exhibits

four polymorphic forms (α, β2’, β1’ and β), which have been identified by different

techniques, like X-ray studies18-26, DSC22-32, Raman spectroscopy25, calorimetry7, 13, 33 and

microscopy34. In this work we observed the existence of two β’ forms by DSC, but the small

temperature range in which the β2’-phase occurred, prevented detailed measurements on this

form. Here, we will discuss the possibilities for the preparation of the β’-form and investigate

its stability through the sets of DSC experiments. Furthermore, the heat capacities of the α-

and β-form, as measured in the adiabatic calorimeter, will be presented and compared to the

literature values. In addition, the values of other thermodynamic properties (enthalpy, entropy

and relative Gibbs energy) will be given for these two polymorphic forms of SSS.

DSC experiments

First melting. SSS, received in a solid state, was heated at a rate of 5 K·min-1 from room

temperature to 363 K. One endothermic peak was observed, indicating the melting of the

stable �-phase. The onset of melting was at 346 K and the enthalpy of fusion was found to be

219.6 J·g-1.

Isothermal crystallization of SSS into the �’-phase. Before the �1’ and the �2’ forms of SSS

were identified and measured as separate phase entities there was a discussion in the literature

about the different stabilities of �’ polymorphs for odd and even TAGs10. These studies

Page 75: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 67

indicated that the �’-form of even triglycerides were decidedly more fleeting and thus are

more difficult to characterize.

Typically, the melting of the �-phase is followed by the re-crystallization of the �’-

form, but the melting peak of the �’-form could not be detected due to the interference of the

�’� � re-crystallization. Previous studies on fat crystallization14 suggested that the pure �’

polymorph could be formed by rapid cooling of the melt to (1 or 2) K above the �-melting

point and stabilizing at that temperature for 30 minutes to 1 hour. Therefore, we cooled the

melt using a cooling rate of - 10 K·min-1 to 329 K and left the sample at this temperature for

40 minutes. After cooling the obtained solid to 273 K, the heating was performed under a

scanning rate of 5 K·min-1. The recorded scanning pattern corresponded to that of the �2’-form

reported by Simpson25, showing two endothermic peaks that indicated the melting of the �2’-

phase at 334 K and of the �1’-phase at 338 K. The energy level of the stable �2’-form,

prepared in the described way, falls in between the levels of the �- and the �-phase, as it is

illustrated in Figure 3. The value for the enthalpy of fusion of the �2’ polymorph (see Table 6)

is calculated as the difference between the extrapolated enthalpy levels of the liquid and the

solid phase at the melting point of the �2’- form.

However, when the �-form was heated to the same temperature of 329 K and its melt

left there for 40 minutes, then the preparation of the �’-form failed. On scanning the obtained

solid, only one endothermic peak occurred at 345 K, demonstrating the melting of the �-form.

For comparison between the �-melt induced crystallization and the crystallization from the

undercooled melt, the heat flows registered in both cases during the isothermal crystallization

at 329 K are given in Figure 4. Accordingly, the crystallization of the �’-form requires no

induction time when generated in the �-melt, but it converts readily to the �-form as can be

seen from the second heat effect in the relevant curve. On the other hand, the crystallization of

the �’-phase from the undercooled melt has a delay, there is a time interval during which �’-

nuclei are formed. The �’ polymorph, obtained in such a way, does not transform so easily to

the �-form. The higher stability of the �’-form crystallized directly from the melt compared to

the one induced from the �-melt was already reported.35

Page 76: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

68

305 315 325 335 345 355

T / K

0

50

100

150

200

250

300

350

h / J

·g-1

0 300 600 900 1200 1500 1800

t / s

-1.50

-1.10

-0.70

-0.30

0.10

heat

flow

/ m

W

Figure 3. Pseudo-enthalpy paths of the three polymorphs of SSS measured upon constant

heating at 5 K⋅min-1: �, α-form; �, β2’-form; �, β-form.

Figure 4. Isothermal crystallization of SSS at 329 K from the α-melt (solid line) and from the

undercooled melt (dashed line).

Page 77: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 69

Phase stability of the �’ polymorph. To get more insight into the fleeting existence of the �’-

phase the following set of isothermal crystallization experiments were done at respectively

(328.15, 329.15 and 330.15) K. First the sample was heated to 373.15 K in order to melt it

completely and to remove possible crystallization grains. Then it was cooled at a rate of - 10

K⋅min-1 to the stabilization temperature and kept at that temperature for respectively 30 min,

1 h, 2 h and 4 h. After the stabilization period the sample was cooled to 273.15 K and was

subsequently scanned to 373.15 K at a heating rate of 5 K.min-1. In the Figures (5(a, b) to 7(a,

b)) the results are given, where the observed exothermic heat effects during the stabilization

are given in the (a) figures and the heating scans are given in the (b) figures. The stabilization

temperatures were chosen to be above the melting/crystallization temperature of the α-phase

and below the temperature where � crystallization was observed in the adiabatic experiments.

At the chosen stabilization temperatures, there is a considerable undercooling of the � as well

as of the �’-phase; however in all cases the �’-phase did crystallize first.

Typically, the crystallization of the �’-phase from the undercooled melt was

appreciably faster at lower temperatures. Indication that solidification was not complete

during the isothermal period is the melting peak of the α-phase that occurs during the final

heating scan. For instance, no α-phase was observed in the solids formed at 328.15 K (Figure

5b), while after 30 min at 329.15 K an amount of liquid remained that solidified in the α-

phase upon cooling to 273 K (Figure 6b). Independently of the investigated temperatures,

once the �’-phase was formed it converted to the �-phase. However, the rate of �’��

transition depends on the temperature, being slower at higher temperatures. The sample kept

for 4 hours at 328.15 K totally transformed to the �-phase, while when it was kept at 330.15 K

for the same time some amount of the �’-phase remained.

The stable �-phase can be prepared by very slow cooling of the liquid, as will be

demonstrated in the adiabatic calorimeter experiment below, or by first forming the �’-phase

and allowing the formed solid sufficient time to transform to the �-phase. In the light of very

slow nucleation rate for the β-form, these observations suggest that it is more efficient to

prepare the most stable phase via the intermediate β’-phase than by direct crystallization from

the undercooled melt.

Page 78: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

70

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

t / h

-0.35

-0.25

-0.15

-0.05

0.05

heat

flow

/ m

W

310 320 330 340 350 360 370

T / K

-10

0

10

20

heat

flow

/ m

W

b)

a)

Figure 5. Isothermal crystallization of SSS at 328.15 K: �, 30 min; —, 1h; – – –, 2h;

·······, 4h.

a) exothermic effects registered for different stabilization period

b) corresponding heating scans of the solids formed as indicated in a)

Page 79: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 71

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

t / h

-0.25

-0.15

-0.05

0.05

heat

flow

/ m

W

310 320 330 340 350 360 370

T / K

-5

0

5

10

15

20

heat

flow

/ m

W

a)

b)

Figure 6. Isothermal crystallization of SSS at 329.15 K: �, 30 min; —, 1h; – – –, 2h;

·······, 4h.

a) exothermic effects registered for different stabilization periods

b) corresponding heating scans of the solids formed as indicated in a)

Page 80: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

72

Figure 7. Isothermal crystallization of SSS at 330.15 K: �, 30 min; —, 1h; – – –, 2h;

·······, 4h.

a) exothermic effects registered for different stabilization periods

b) corresponding heating scans of the solids formed as indicated in a)

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

t / h

0.20

0.30

0.40

0.50

heat

flow

/ m

W

a)

b)

310 320 330 340 350 360 370

T / K

-10

0

10

20

heat

flow

/ m

W

Page 81: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 73

Adiabatic Calorimetry Experiments

The ββββ phase. From the DSC experiment, we knew that the sample as received was in the β-

form. To allow optimal relaxation of this crystal form, the sample was first heated till about 2

K below the melting point of the stable β polymorph. At this temperature the sample was

stabilized for 24 h before cooling to liquid helium temperature. In Table 1 the subsequent

measurements from (10 K to 370 K) are given. The heat capacity curves of this phase and of

the other polymorphs discussed below are shown in Figure 8. Around 130 K a small,

unexplained hump in the data is observed, we repeated the measurements in this temperature

region twice with the same result. The absolute entropy sabs and melting enthalpy of the β

polymorph were calculated by numerical integration. The starting values of this integration

were calculated at 10 K by assuming that below this temperature the Debye low temperature

limit for the heat capacity could be applied. The thermodynamic functions are given in Table

2 at selected temperatures. The enthalpy of fusion of the β-phase was measured twice; the

mean value was (221.6 ± 1) J⋅g-1 and the triple point value was found to be (345.94 ± 0.01) K.

In Table 3, the results of the two melting experiments are given, together with the linear fits of

the heat capacity of the solid and the liquid phase around the melting temperature, which were

used in the calculation.

Table 1. Experimental heat capacities of the stable β phase of SSS from 10 K to 370.

T

cp

T

cp

T

cp

T

cp

K J⋅K-1⋅g-1 K J⋅K-1⋅g-1 K J⋅K-1⋅g-1 K J⋅K-1⋅g-1

Series 1 30.62 0.176 143.02 0.830 283.16 1.520 297.32 1.609 Series 4 145.92 0.852 286.11 1.538 298.40 1.611 30.93 0.168 148.83 0.863 289.07 1.556 299.84 1.632 33.29 0.207 151.73 0.873 292.03 1.584 301.82 1.653 35.42 0.230 154.64 0.887 294.99 1.602 303.79 1.674 37.51 0.242 157.55 0.890 297.96 1.625 305.75 1.692 39.73 0.252 160.46 0.918 300.94 1.645 307.70 1.709 41.97 0.269 163.37 0.927 Series 6 309.65 1.723 44.25 0.295 166.29 0.940 304.05 1.670 311.60 1.741 46.55 0.307 169.20 0.952 306.72 1.699 313.55 1.761 48.96 0.330 172.12 0.961 308.94 1.716 315.49 1.772 51.41 0.350 175.04 0.976 311.17 1.732 317.43 1.793 53.88 0.373 177.95 0.985 313.39 1.751

Page 82: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

74

319.37 1.826 56.39 0.398 180.87 0.999 315.61 1.768 321.32 1.842 58.93 0.418 183.79 1.011 317.83 1.793 323.27 1.862 61.51 0.442 186.70 1.034 320.05 1.822 325.23 1.887 64.13 0.466 189.62 1.053 322.28 1.846 327.20 1.907 66.78 0.487 192.54 1.074 324.50 1.869 329.17 1.931 69.45 0.507 195.45 1.091 326.72 1.899 331.15 1.960 72.14 0.525 198.37 1.095 328.95 1.922 333.14 1.988 74.85 0.543 201.29 1.119 331.17 1.953 335.14 2.030 77.59 0.562 204.21 1.131 333.40 1.992 337.14 2.068 80.34 0.577 207.13 1.138 335.63 2.031 Series 2 83.10 0.595 210.05 1.148 337.85 2.073

9.65 0.009 85.88 0.612 212.96 1.155 340.08 2.142 9.71 0.014 88.67 0.626 215.88 1.164 342.28 2.395

10.85 0.025 91.47 0.642 218.80 1.172 344.14 8.560 12.51 0.030 Series 5 221.72 1.176 345.13 54.85 13.96 0.037 85.71 0.606 224.64 1.186 345.47 144.88 15.56 0.049 88.61 0.622 227.55 1.197 345.62 278.92 17.30 0.070 91.44 0.635 230.47 1.212 345.71 439.05 19.33 0.078 94.27 0.649 233.39 1.228 345.76 998.91 21.58 0.092 97.11 0.657 236.30 1.242 345.79 1233.21 23.86 0.113 99.95 0.670 239.22 1.258 345.86 285.47 26.19 0.131 102.80 0.685 242.14 1.272 346.95 3.418 28.56 0.154 105.65 0.687 245.06 1.289 349.12 2.211 30.99 0.178 108.50 0.691 247.98 1.307 351.37 2.211

Series 3 111.36 0.704 250.90 1.321 353.63 2.216 9.60 0.012 114.22 0.718 253.82 1.338 355.89 2.223

11.58 0.025 117.09 0.731 256.74 1.355 358.16 2.228 13.59 0.034 119.95 0.742 259.67 1.371 360.45 2.235 15.41 0.044 122.82 0.753 262.60 1.389 362.74 2.241 17.41 0.057 125.70 0.763 265.53 1.407 365.05 2.249 19.45 0.074 128.58 0.775 268.46 1.426 367.37 2.252 21.54 0.089 131.46 0.818 271.39 1.442 369.71 2.261 23.71 0.110 134.35 0.818 274.33 1.460 25.95 0.129 137.23 0.804 277.27 1.489 28.25 0.152 140.13 0.817 280.21 1.502

Page 83: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 75

100 170 240 310 380

T / K

0.50

1.00

1.50

2.00

2.50c p /

J. K

-1. g

-1

Figure 8. Comparing the experimental heat capacity curves of SSS with literature data:

αααα-phase: �, this work; �, Simpson31; ∇, Hampson et al.32; �, Charbonnet7;

ββββ-phase: �, this work; �, Simpson31; +, Hampson et al.32; �, Charbonnet7;

ββββ’-phase: -·-·-, this work; �, (β1’) Simpson31; �, (β2’) Simpson31;

, Hampson et al.32

liquid phase: —, this work; , Morad et al.29; ♦, Phillips30; +, Charbonnet7.

Table 2. Derived thermodynamic properties of the β-form of SSS at selected temperatures.

T

cp

sabs

h-h(0)

K J⋅K-1⋅g-1 J⋅K-1⋅g-1 J⋅g-1 10 0.019 0.006 0.05 20 0.068 0.036 0.50 30 0.170 0.084 1.72 40 0.254 0.146 3.91 50 0.338 0.212 6.88 60 0.428 0.282 10.71 70 0.510 0.354 15.43 80 0.575 0.427 20.87 90 0.628 0.498 26.90

Page 84: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

76

100 0.671 0.566 33.39 110 0.697 0.631 40.25 120 0.743 0.694 47.46 130 0.796 0.755 55.08 140 0.817 0.815 63.21 150 0.867 0.873 71.64 160 0.915 0.931 80.50 170 0.955 0.987 89.84 180 0.994 1.043 99.58 190 1.056 1.098 109.8 200 1.109 1.154 120.7 210 1.148 1.209 132.0 220 1.174 1.263 143.6 230 1.210 1.316 155.5 240 1.262 1.368 167.8 250 1.317 1.421 180.7 260 1.373 1.474 194.2 270 1.434 1.527 208.2 280 1.501 1.580 222.9 290 1.564 1.634 238.2

298.15 1.626 1.678 251.2 300 1.639 1.688 254.2 310 1.724 1.743 271.0 320 1.821 1.799 288.7 330 1.935 1.857 307.5 340 2.140 1.917 327.7 350 2.211 2.612 567.7 360 2.233 2.675 589.9

Table 3. Melting of the β-phase of SSS.

Used linear functions of the heat capacity __________________________________

J⋅K-1⋅g-1

Temperature interval _________________

K-K

∆∆∆∆ hfus _____________

J⋅g-1

cp(solid, 300 K-317 K)=-0.664+0.00768 T/K

cp(liquid, 350 K-365 K)=1.264+0.00267 T/K

317 to 349

220.9 (first melt)

222.4 (second melt)

Page 85: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 77

The αααα phase. Cooling the sample from the liquid phase at a rate of 16 K⋅h-1 or larger resulted

in the formation of the α-phase. On heating the obtained solid from 175 K, the heat capacity

of the α-phase is measured and the transition of α�β�liquid is observed. The experimental

heat capacity data are given in Table 4. As in the measured temperature range exothermic

effects occurred due to re-crystallization, we did include the relative enthalpy values in this

table. The enthalpy values of the measuring set starting in the α-phase were shifted, so that

they coincide in the liquid phase with the enthalpy values of the liquid phase obtained from

the measurement of the stable β-phase.

As can be seen in Figure 8, the heat capacity of the α-phase increases rather sharply

before the melting point. In the adiabatic calorimetry experiments the transition of the α-phase

to the β-phase or the β’-phase started already before the melting point of the α-phase was

reached. At the melting temperature of the α-phase, being 326.6 K according to the DSC

experiments, the formed liquid is directly transformed to the β or the β’ phase. In Figure 9 the

relative enthalpy curves of the different phases are given. From this picture it is clear that

within the experimental time used, the transition to the β-phase was not completed and it is

only at the very end of the melting process of the β-phase that the curves coincide.

Table 4. Experimental heat capacities and enthalpy increment of the α-phase of SSS.

T cp

h(T)-h(0) T

cp

h(T)-h(0)

K J⋅K-1⋅g-1 J⋅g-1 K J⋅K-1⋅g-1 J⋅g-1

178.91 1.010 134.09 283.59 1.828 279.67 181.60 1.030 136.87 285.80 1.848 283.72 184.26 1.057 139.63 288.00 1.874 287.82 186.88 1.093 142.40 290.20 1.895 291.96 189.48 1.106 145.24 292.39 1.925 296.15 192.06 1.117 148.11 294.59 1.944 300.40 194.63 1.120 150.99 296.78 1.978 304.70 197.18 1.128 153.86 298.97 2.005 309.06 199.71 1.155 156.75 301.16 2.037 313.47 202.23 1.182 159.69 303.34 2.069 317.97 204.74 1.205 162.68 305.53 2.106 322.53 207.23 1.222 165.70 307.71 2.142 327.16 209.71 1.215 168.72 309.89 2.179 331.88 212.17 1.224 171.72 312.07 2.217 336.66

Page 86: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

78

214.62 1.246 174.75 314.26 2.259 341.54 217.06 1.265 177.81 316.44 2.307 346.52 219.48 1.287 180.90 318.63 2.316 351.60 221.90 1.298 184.02 320.84 2.169 356.55 224.30 1.313 187.15 323.13 1.384 360.60 226.69 1.330 190.31 326.73 -5.225 349.59 229.06 1.347 193.49 330.77 -2.092 333.57 231.43 1.366 196.70 333.70 0.373 330.65 233.79 1.389 199.94 336.23 0.983 332.35 236.13 1.409 203.22 338.66 1.437 335.29 238.46 1.426 206.53 340.96 2.653 339.93 240.79 1.445 209.86 342.89 7.391 348.96 243.10 1.467 213.23 344.13 26.47 366.17 245.41 1.489 216.64 344.75 66.64 390.88 247.71 1.503 220.08 345.07 126.87 419.16 250.00 1.524 223.54 345.26 210.24 449.01 252.28 1.544 227.04 345.38 312.73 479.63 254.56 1.568 230.58 345.47 400.99 510.65 256.83 1.585 234.16 345.60 167.56 541.20 259.09 1.600 237.76 346.77 3.134 559.55 261.34 1.620 241.39 349.01 2.220 565.50 263.59 1.639 245.05 351.31 2.208 570.59 265.83 1.659 248.75 353.61 2.212 575.66 268.07 1.677 252.47 355.92 2.218 580.78 270.30 1.700 256.24 358.24 2.223 585.93 272.53 1.719 260.05 360.57 2.230 591.12 274.75 1.742 263.89 362.91 2.241 596.35 276.96 1.770 267.78 365.27 2.244 601.64 279.18 1.784 271.70 367.64 2.250 606.95 281.39 1.806 275.67 370.02 2.257 612.32

The ββββ’ phase. The results from the isothermal experiments in the DSC explain why in the

adiabatic calorimeter experiments the �’-phase could not be observed. These adiabatic

measurements are so slow that the sample has transformed to the �-phase before the melting

point of the �’-phase was reached. To measure the β’-phase in the adiabatic calorimeter, we

removed the vessel from the calorimeter, immersed it in hot water to melt the contents and

transferred it to a temperature regulated block. The vessel was kept at 328 K for 4 hours, then

cooled to room temperature and replaced in the calorimeter.

Hereby we point to the essential influence of the mass of SSS on the time needed to

obtain the �’-form. For instance, the sample of 1.55 mg crystallized in the β2’-form within 40

Page 87: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 79

300 310 320 330 340 350 360

T / K

120

205

290

375

460

h re

l./ J. g-1

min at 328 K in the DSC. However, the same mass of SSS transformed completely to the β-

phase while staying at the given temperature for 4 hours. The β2’-phase, which heat capacity

curve is shown in Figure 8 31, was formed in a capillary tube by annealing the melt at 328 K

for 15-25 min. To conclude, the optimal annealing time for obtaining the pure β’-form

depends on a sample size.

The heat capacity of the solid formed outside the calorimeter is presented in Figure 8.

The values are intermediate to the curves for the α- and the β-phase, but we cannot be sure

whether it is due to the formation of the pure β’-phase or else the measured solid was actually

a mixture of the α-phase and one or two of the other polymorphs. We believe that the α-phase

is present as the re-crystallization starts at the melting temperature of the α-phase (see Fig. 8).

More detailed characterization of the solid structure is not possible by using only thermal

analysis, thus we cannot give reliable heat capacity values for the β’-phase.

Figure 9. Experimental relative enthalpy curves: �, α-phase; �, β-phase; �, the solid

formed by annealing the melt at 328 K for 4 h. The curves are shifted so that they coincide in

the liquid phase.

Page 88: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

80

Enthalpy and entropy values of the αααα- and the ββββ- phase at 298.15 K. The enthalpy and

entropy values of the different polymorphic phases can be calculated by constructing a

reversible path connected to the liquid phase. First the enthalpy of the liquid phase was

extrapolated to the melting point of the α-phase by assuming that the heat capacity could be

extrapolated linearly. The enthalpy of fusion of the α-phase was taken as the difference

between the extrapolated enthalpy values of the liquid and the relevant solid at the melting

point. Then the same procedure was repeated for the entropy of fusion, calculated as the

enthalpy of fusion divided by the temperature of fusion. In Table 5 the calculated enthalpy

and entropy values of the α- and the β-phase at 298.15 K are given, together with the

calculated enthalpies of fusion.

Table 5. Enthalpy, entropy and relative Gibbs energy values at 298.15 K for the two

polymorphic forms of SSS and the temperatures of fusion and the enthalpy of fusion.

phase

Tfus

_________

K

∆∆∆∆hfus

__________

J⋅g-1

h(298.15)-h(0) _____________

J⋅g-1

sabs

_________

J⋅K-1⋅g-1

g ________

J⋅g-1

ββββ 345.9 221.6 251.2 1.678 -249.1

αααα 327.3 144.8 307.4 1.816 -234.0

Comparing the results. In Figure 8 the literature data for the heat capacities of the different

phases are plotted. Generally the correspondence is good, especially for the liquid phase.

However, the results of Hampson et al.32 significantly deviate from our data, which could be

due to the relatively high heating rate of 10 K⋅min-1 that they used for scanning solids in the

DSC. In Table 6 we compare the temperatures and enthalpies of fusion of the different phases

with the literature values. Here, the correspondence is also within the error margins, except for

reference 35. We think that in that case the authors integrated the DSC melting curve in the

melt of the β-phase without taking into account the re-crystallization, which took place in the

same run.

Page 89: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 81

Table 6. Melting temperatures and enthalpies of fusion of the three polymorphic phases of

SSS and comparison with the literature.

β β2’ α References

Tfus ∆hfus Tfus ∆hfus Tfus ∆hfus

K J⋅g-1 K J⋅g-1 K J⋅g-1

345.9

346.0

346.6

344.6

346.1

345.6

346.7

346.3

343.1

345.7

345.5

346.3

345.7

221.6

219.6

214.7

220.7

70.4

228.0

213.0

211.4

216.7

336.7

336.4

337.1

337.6

336.1

336.4

337.1

336.7

337.5

154.2

162.7

178.4

168.9

175.2*

327.3

327.3

328.1

327.6

327.3

327.6

327.8

327.1

327.1

326.0

328.3

327.9

144.8

128.0

162.9

162.8

153.6

122.6

126.9

Adiabatic (this work)

DSC (this work)

18

19

20

9

22

26

28

29

35

7

33

13

14

*) not indicated in the reference for which β’ polymorph the value is given.

Page 90: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

82

4.3.2 Tripalmitin (PPP) Tripalmitin (further notated as PPP) is a TAG composed of three identical fatty acids

attached to the glycerol backbone, each chain containing 16 carbon atoms (palmitin acid). PPP

belongs to the group of monoacid, saturated TAGs and according to the long spacing

measurements done by X-ray diffraction it has an L-2 structure in the solid state36.

By using different experimental techniques, such as: DSC17, 36-38, X-ray diffraction36-38,

NMR36, optical microscopy37, 39, it is known that PPP has basically the three polymorphs: α,

β’ and β form. As for the other TAGs, the least stable polymorph of PPP, the α-form,

crystallizes upon fast cooling of the melt without appreciable undercooling. On heating the α-

form, it melts at 318.15 K, but transforms readily into the β’-form. It has been reported36 that

the β’-phase crystallized from the melt is more stable than that obtained from the α-form, in

the sense that it does not convert that easily into the β-form. That the less fleeting β’-phase

can only be obtained from the isotropic melt, we have also observed in the case of SSS.

However, some authors have found two β’-forms of PPP38, β’1 and β’2-form, where the β’1-

form is of higher stability, as reflected by a higher melting temperature and melting enthalpy.

Still, the differences in melting behaviour and crystal structure of these two β’-forms are very

slight and can be explained in terms of crystal perfection. Accordingly, there is no sufficient

evidence to regard both β’ forms as independent phases. Regarding the most stable β-form, it

can be obtained by very slow cooling of the melt or by isothermal crystallization above

330.15 K37. However, the β-form for which the crystallization was β’-initiated is

characterized by a large volume expansion due to the inclusion of air (blooming effect), while

the β-form crystallized from the melt above 327.15 K, does not exhibit such a large volume

expansions38.

Experimental Section PPP was purchased from Larodan, with a reported purity > 99 wt%. The compound was

received as a finely divided powder, probably crystallized from a solution. In previous study,

molar heat capacities of two forms of PPP, the α- and β-form, were measured between 10 K

and 350 K in an adiabatic calorimeter16. The enthalpies of fusion of these two forms were also

derived from both DSC and adiabatic measurements. In this work, we measured only the most

stable β-form in the adiabatic calorimeter and DSC. This was done in order to check and

Page 91: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 83

115 135 155 175 195 215 235 255 275 295 315 335 355

T / K

0

1

2

3

4

5

c p / J

·K-1

·g-1

compare the thermal properties of PPP, since it has been used further in investigations of

binary mixtures of TAGs. Moreover, we were interested in the determination of the purity of

the given PPP sample, since it might have significant influence on the thermal behaviour of

both pure PPP and its mixtures with other TAGs.

An amount of 0.328 g of PPP was introduced into a vessel that was mounted in the

adiabatic calorimeter. Before melting, PPP was first cooled down to 115 K and then slowly

heated to 350 K, where stabilization periods of 800 s were used. In Figure 10, the heat

capacity obtained from this measurement is presented as a function of temperature. No

exothermic effect was registered during the heating, meaning that PPP was in the most stable

phase, the β-form. This measurement was used for the determination of purity, as it will be

demonstrated below in the text.

Figure 10. The heat capacity in the function of temperature for the β-form of PPP.

Page 92: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

84

4.3.3 Trielaidin (EEE) Trielaidin (further notated as EEE) is a TAG consisted of three identical unsaturated

fatty acids (elaidic acid) attached to the glycerol backbone, each containing 18 carbon atoms

with a double bond on position C9. Moreover, it is a trans-unsaturated TAG - the elaidic acid

is the trans homolog of the cis-unsaturated oleic acid. In general, cis-unsaturated acyl chains

are abundant in naturally occurring TAGs, whereas the TAGs containing trans-unsaturated

acyl chains have been found in a limited number of organisms. However, during a

hydrogenation process the fraction of trans-unsaturated acyl chains sometimes amounts to

over 15 % in industrial products, due to positional isomerization reactions. Therefore, it is

important to clarify how trans-unsaturation influences the polymorphism of TAG blends.

As the other TAGs, EEE also shows polymorphic behaviour, where it appears in three

crystalline forms: sub-α, α and β. Their existence was confirmed by: DSC, X-ray powder

diffraction and NMR40. The structure of the β-form has been analyzed at atomic level, while

the structure of metastable phases, sub-α and α, could not be resolved by the usual X-ray

analysis, due to the wealth of disorder present. To obtain more details on metastable forms,

and particularly the α-form in which hydrocarbon chains are oscillating, other experimental

techniques have to be used, such as vibrational spectroscopy, IR and Raman spectroscopy41.

Accordingly, the β-form of EEE is almost the same as that of saturated monoacid TAGs, the

so-called chair shape structure. The α-form exhibites spherulite texture and by measuring

polarized micro IR spectra the molecular orientation has been clarified. It corresponds to the

hexagonal subcell.

In general, by quenching the melt EEE crystallizes in the sub-α-form that is observed

in the temperature range 233-243 K, while from 243 K to 287 K the α-form is present, which

at 287 K transforms into the β-form 17, 40. There is no evidence found for the existence of the

β’-form of EEE. Furthermore, EEE shows the double-chain-length structure in all forms, just

like SSS, PPP and other C16-C18 saturated mixed TAGs. Such a behaviour is unlike for other

unsaturated TAGs, which commonly show triple-chain-length structure.

Page 93: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 85

Experimental Section EEE was purchased from Larodan, with a reported purity > 99 wt%. The thermal behaviour of

EEE has already been investigated by DSC and adiabatic calorimetry elsewhere17. The molar

heat capacities of the α- and the β-form were measured between 10 K and 360 K in the

adiabatic calorimeter, while the thermal behaviour of EEE under different cooling and heating

rates was analysed in the DSC. Both techniques enabled the calculation of the enthalpy of

fusion of two polymorphs. The existence of amorphous sub α phase was also discussed, while

no evidence of the β’-form was found. Although the thermal properties of EEE are already

known, we measured the β-form in the adiabatic calorimeter, in order to determine the amount

of impurity in the compound.

The vessel was filled with 0.515 g of EEE and placed in the adiabatic calorimeter.

First heating was done from the room temperature till 360 K and the measured heat capacity is

presented in Figure 11. Afterwards, the melt was cooled down at a rate of 0.24 K/min, where

the crystallization occurred at 291.1 K, but on further cooling one more exothermic effect was

registered at 288.5 K. Considering that the melting points of the α- and the β-form are 288 K

and 314.8 K, respectively, this suggests that one part of EEE crystallized in the β-form and the

other in the α-form. That is also confirmed by an exothermic effect occurring before the

melting point of EEE, as can be seen from the measured heat capacity curve (see Figure 11).

To obtain the most stable polymorph, the sample was first cooled down slowly and then re-

heated to 312 K, where it was stabilized for 4 days. The measured heat capacity is also given

in Figure 11, coinciding with the values from the first melting experiment. For the

determination of the purity of EEE we used the data from the first melting and those of the

stabilized sample, both referring to the β-form.

Page 94: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

86

160 180 200 220 240 260 280 300 320 340 360

T / K

0

1

2

3

4

5c p /

J·K

-1·g

-1

Figure 11. The heat capacity of EEE in the function of temperature: the first melting (bold

line), heating of EEE after slow cooling (solid line), heating after stabilizing the compound at

312 K for 4 days (dashed line).

4.4 Purity determination Thermal analysis of TAGs can be used as their fingerprint, the way to identify the

polymorphic form. However, the phase behaviour of a TAG may be disturbed by presence of

a certain amount of impurities, which would lead to different values of thermal properties, as

well as to different kinetics of crystallization and polymorphism. For instance, one of the

deviations caused by impurities is a pre-melt effect, where the heat capacity of the solid phase

starts deviating from a baseline far below the melting point. In that sense, the compound

begins to melt at unexpectedly low temperature, which complicates the correct choice of the

solid heat capacity baseline, resulting in different values for the heat of fusion. For example, it

is hard to estimate at which temperature the pre-melt effect actually begins. Such a behaviour

was observed during the thermal investigation of PPP16. However, beside the impurities this

deviation might be also caused by the increase of the mobility of the fatty acid chains as

Page 95: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 87

temperature rises. In investigations of binary mixtures of TAGs, the purity of TAGs has to be

considered, because of its significant impact on the phase behaviour of the mixtures.

Here, we performed a purity determination of SSS, PPP and EEE by adiabatic

calorimetry. Measurement of the equilibrium temperatures during fractional melting in the

adiabatic calorimeter enables calculation of the purity of a compound. The results of this

method will be compared to a model for purity determination based on the effect of pre-

melting 42.

When the main compound forms a eutectic phase diagram with the impurity, the function of

the equilibrium temperature against the reciprocal of the melted fraction is linear:

F

x

HRT

TT imp

fus

measeq ∆

−=2

00 1

where measeqT is the measured equilibrium temperature at the melted fraction F, ∆Hfus is the

enthalpy of fusion of the main component and T0 is the melting point of the pure compound,

ximp being the mole fraction of the impurity.

The measured equilibrium temperatures in function of the reciprocal of the melted

fraction are presented in Figure 12, from two melting experiments of EEE. Figure 13 shows

the results from two meltings of SSS, whereby SSS was stabilized at 343 K before the second

melting. As can be seen from Figures 12 and 13, for both TAGs the curves derived from the

first and second melting have different slopes, where those from the second melting are

obviously steeper. This suggests the increase of impurity in the second melting. However,

extrapolation of the curves to a value 1/F = 0 give the same melting point for the pure

compounds. In Figure 14, the equilibrium temperatures measured during the melting of PPP

are plotted in the function of the reciprocal of the melted fraction. In case of PPP, the function

exhibits linear trend, but we did not perform the second melting of this compound. The purity

values of the three TAGs, calculated according to Eq. 1 and by using the corresponding linear

fits presented in Figures 12, 13 and 14, are given in Table 7. Beside purity, we report the

melting points of EEE, SSS and PPP as estimated from two melting experiments.

Page 96: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

88

0 2 4 6 8 10

1/F

344.80

345.04

345.28

345.52

345.76

346.00

T equi

l / K

0 2 4 6 8 10

1/F

314.00

314.20

314.40

314.60

314.80

315.00

T equi

l / K

Figure 12. The equilibrium temperature in the melt against the reciprocal melted fraction of

EEE - (�) the first melting; (�) the second melting.

Figure 13. The equilibrium temperature in the melt against the reciprocal melted fraction of

SSS - (�) the first melting; (�) the second melting.

Page 97: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 89

0 2 4 6 8 10

1/F

338.80

338.90

339.00

339.10

339.20

339.30

339.40

T equi

l / K

Figure 14. The equilibrium temperature in the melt against the reciprocal melted fraction of

PPP - (�) the first melting.

Table 7. Triple points and purities calculated by using the linear function of measured

equilibrium temperatures against the reciprocal of melted fraction.

First melting Second melting

Compound

T0 / K

Purity / mol %

T0 / K

Purity / mol %

EEE 314.92 99.3 314.89 98.6 SSS 345.89 98.4 345.92 97.6 PPP 339.28 99.1 - -

Page 98: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

90

Between two melting experiments the compounds could not get in contact with any

contamination. In that sense, the reason for the observed impurity increase could be that the β-

form of TAGs that crystallized during slow cooling in the adiabatic calorimeter had certain

defects in the crystal lattice. Moreover, the given plots do not show exactly a linear trend,

which can be caused by the wrong choice of the extrapolation of the heat capacity of the solid

through the melting region. Apparently, there is a considerable margin or error in this way of

calculation, depending on which points are included in the linear fit. One of the explanations

for non-linearity could be that the main compound and impurity do not form a eutectic

mixture, but a solid solution.

Near the melting transition the heat capacity of the compound consists of two parts,

being: the “normal” heat capacity of the solid and an increasing contribution due to the

formation of a small fraction of liquid (F). For the calculation of the enthalpy of fusion, one

needs to know the “normal” heat capacity function in the melting region. Often the heat

capacity outside the melting region is linearly extrapolated, but one has to be sure that the

extrapolation is done with a data set outside the pre-melt region and a linear extrapolation

does not need to be the optimal solution. A wrong choice of the heat capacity function used to

calculate the melted fraction can lead to losing of a small part of the fusion, which influences

mostly the melted fraction at low temperature values. The pre-melt heat capacity contribution

can be used to estimate a base line for the heat capacity of the solid, assuming again that the

main compound and impurity form a eutectic system. Marti42 derived the specific heat

contribution of the pre-melt effect, starting from the equation for the solubility equilibrium at

low values of one component (impurity):

���

����

�−

∆==

0

11TTR

H

F

xx fusimpliq

imp 2

where liqimpx is the mole fraction of the impurity in the liquid phase and the rest of quantities are

already defined in Eq. 1.

The pre-melt heat capacity contribution or excess specific heat function can be written as:

dTdF

dFdH

dTdH

ex

⋅=��

���

� 3

Page 99: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 91

The derivative dF / dT is calculated from Eq. 2, being:

20

20

)( TTT

H

Rx

dTdF

fus

imp

−⋅

∆= 4

The other derivative can be formed using the following relation, which holds because of the

limitation to eutectic systems and the restriction to ideal liquid mixture:

fusHdFdH ∆= 5

Inserting the derivatives dH / dF and dF / dT into equation 3, leads to the heat capacity

function that depends solely on the amount of impurity ximp:

20

20

, )( TTRT

xdTdH

c impex

exp −=�

���

�= 6

Starting with a quadratic heat capacity base line, we calculated the excess heat capacities. By

using the coefficients of the quadratic function and the mole fraction of the impurity as

variables, we minimized the differences between the experimental excess heat capacity values

and those calculated with equation 6. This optimization procedure was applied to the three

TAGs, resulting in the purity values given in Table 8. Using the solid base lines determined

by the proposed method, we calculated the enthalpies of fusion (see Table 8). If we compare

the purity values from two methods, we find good agreement for EEE and PPP, while for SSS

the discrepancy is quite larger. For higher accuracy of the purity measurement, the knowledge

of the phase diagram of the main component and the impurity is of great importance. Both

methods are applicable only for eutectic systems, and moreover are limited to a high purity

region.

Table 8. Enthalpies of fusion and purities calculated by using the pre-melt method.

First melting Second melting

Compound

∆∆∆∆Hfus / J⋅⋅⋅⋅g-1

Purity / mol %

∆∆∆∆Hfus / J⋅⋅⋅⋅g-1

Purity / mol %

EEE 172.79 99.3 172.63 99.0 SSS 220.92 99.6 216.81 98.7 PPP 212.67 99.2 - -

Page 100: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

92

4.5 Summary Specific heat capacities of the α- and β-polymorphic forms of SSS are measured in the

adiabatic calorimeter. The enthalpies of fusion and melting temperatures for each polymorph

are calculated from both the adiabatic and the DSC experimental data and compared to the

literature values. Additionally, we examine the fleeting existence of the β’-polymorph of SSS

and its transformation to the β-phase under isothermal conditions in the DSC. The adiabatic

data from melting of the β-polymorphs of SSS, EEE and PPP are used for the determination

of their purities. The purities are calculated by two methods, both based on the assumption

that the compound and impurity form a eutectic mixture. One method is using the linear

function of measured equilibrium temperatures against the reciprocal of melted fraction, while

the other includes the pre-melt heat capacity contribution for the estimation of a base line for

the heat capacity of the solid. The results of both methods, as compared for the first and the

second melting of the compounds, show some discrepancies due to crystal imperfections in

the β-polymorph formed after the first melting. Nevertheless, the reported purities are quite

satisfactory and the amounts of impurities present in the considered TAGs will not disturb the

investigation of the phase behaviour of their binary mixtures.

References: [1] C.P. Tan, Y.B. Che Man, JAOCS 77 (2000).

[2] M.L. Herrera, J.A. Segura, G.J. Rivarola, M.L. Anon, JAOCS 69 (1992), 898.

[3] E. ten Grotenhuis, G.A. van Aken, K.F. van Malssen, H. Schenk, JAOCS 76 (1999),

1031.

[4] Y.B. Che Man, C.P. Tan, Phytochem. Anal. 13 (2002), 142.

[5] W.K. Busfield, P.N. Proschogo, JAOCS 67 (1990), 171.

[6] P. Elisabettini, G. Lognay, A. Desmedt, C. Cultot, N. Istasse, E. Deffense, F. Durant,

JAOCS 75 (1998), 285.

[7] G.H. Charbonnet, W.S. Singleton, JAOCS (1947), 140.

[8] J.W. Hampson, H.L. Rothbart, JAOCS 46 (1969), 143.

[9] F. Lavigne, C. Bourgaux, M. Ollivon, J. Phys. IV Proc. 3, (1993), 137.

Page 101: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs 93

[10] E.S. Lutton, A.J. Fehl, Lipids 5 (1970), 90.

[11] M.D. Small, The Physical Chemistry of Lipids, Plenum Press, New York, (1986).

[12] R.E. Timms, Chemistry and Physics of Lipids 21 (1978), 113.

[13] M. Ollivon, R. Perron, Thermochim. Acta 53 (1982), 183.

[14] L.H. Wesdorp, Liquid-multiple Solid-phase Equilibria in Fats. Ph.D. Thesis,

Technische Universiteit Delft, (1990).

[15] C.K. Zeberg-Mikkelsen, E.H. Stenby, Fluid Phase Equilibria 162 (1999), 7.

[16] J.C. van Miltenburg, E. ten Grotenhuis, J. Chem. Eng. Data 44 (1999), 721.

[17] J.C. van Miltenburg, P.J. van Ekeren, F.G. Gandolfo, E. Flöter, J. Chem. Eng. Data

48 (2003), 1245.

[18] R.D. Kodali, D. Atkinson, T.G. Redgrave, D.M. Small, J. of Lipid Research 28

(1987), 403.

[19] C.E. Clarkson, T. Malkin, J. Chem. Soc. (1934), 666.

[20] E.S. Lutton, JAOCS 32 (1955), 49.

[21] D.J. Cebula, P.R. Smith, JAOCS 67 (1990), 811.

[22] J.H. Whittam, H.L. Rosano, JAOCS (1975), 128.

[23] J.W. Hagemann, J.A. Rothfus, JAOCS 60 (1983), 1123.

[24] J.H. Oh, A.R. McCurdy, S. Clark, B.G. Swanson, Journal of Food Science 67 (2002),

2911.

[25] T.D. Simpson, J.W. Hagemann, JAOCS 59 (1982), 169.

[26] M. Kellens, H. Reynaers, Fat Science Techn. 94 (1992), 286.

[27] W.W. Walker, JAOCS 5 (1987), 754.

[28] R.R. Perron, Revue Francaise des corps gras (1984), 171.

[29] N.A. Morad, M. Idrees, A.A. Hasan, J. Therm. Anal. 45 (1995), 1449.

[30] J.C. Phillips, M.M. Mattamal, J. Chem. Eng. Data 21 (1976), 228.

[31] T.D. Simpson, JAOCS 61 (1984), 883.

[32] J.W. Hampson, H.L. Rothbart, JAOCS 60 (1983), 1102.

[33] I.T. Norton, C.D. Lee-Tuffnell, S. Ablett, S.M. Bociek, JAOCS 62 (1985), 1237.

[34] M. Okada, Journal of Electron Microscopy (1964), 180.

[35] A. Desmedt, C. Culot, C. Deroanne, F. Durant, V. Gibon, JAOCS 67 (1990), 658.

[36] V. Gibon, F. Durant, Cl. Deroanne, JAOCS 63 (1986), 1047.

Page 102: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 4: Thermodynamic Properties of Pure TAGs

94

[37] K. Sato, T. Kuroda, JAOCS 64 (1987), 124.

[38] M. Kellens, W. Meeussen, H. Reynaers, Chemistry and Physics of Lipids 55 (1990),

163.

[39] M. Kellens, W. Meeussen, H. Reynaers, JAOCS 69 (1992), 906.

[40] C. Culot, B. Norberg, G. Evrard, F. Durant, Acta Cryst. B56 (2000), 317.

[41] K. Dohi, F. Kaneko, T. Kawaguchi, J. Cryst. Growth 237-239 (2002), 2227.

[42] E.E. Marti, Thermochim. Acta 5 (1972), 173.

Page 103: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 95

Chapter 5

Thermal Analysis of Binary TAG Mixtures

5.1 Introduction Fats, multi-component mixtures of TAGs, are main components in many food

products. The physical properties of fats firmly influence the processing of fats in food

materials, the final characteristics of food products (hardness and texture) and the storage time

of these products1. For instance, the composition of fat mixture is a crucial factor that

determines the properties like rheology and crystallization behaviour2. In order to achieve

desirable characteristics of the products during manufacturing, the adequate control of fat

solidification is an important step. The crystallization conditions, such as cooling rate and

thermal history, have significant effects on the crystallization kinetics and physical properties

of the crystallized products3. To reveal valuable information about crystallization kinetics and

the type of polymorphs formed, recent studies deal with modeling of fat crystallization under

isothermal4 and non-isothermal5 conditions. Besides the chemical composition of fats and the

crystallization conditions, one of the most important pieces of information required to control

crystallization is the equilibrium phase diagram of the system. These diagrams are quite

complex for real fats due to various interactions between different types of TAG molecules6.

Besides, the crystallization kinetics of fats is highly affected by operating conditions. That is

reflected in the occurrence of the crystalline phases and phase transformations between them,

which do not involve only thermodynamic phase diagram, but also the kinetics of these

processes.

Most investigations of the mixtures of TAGs refer to binary blends. Studying of their

phase behaviour, which is not that complex as for the real fats, provides useful information

about the interactions of TAG molecules in the mixtures. Monitoring of the phase transitions

of binary mixtures of TAGs is usually performed by means of the DSC and X-ray diffraction

(XRD) techniques7-12. Some authors use the combination of the mentioned techniques with

NMR, which enables studying the organization of the fatty acid chains in the most stable

Page 104: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

96

polymorphic forms13, 14. In other studies, the synchrotron radiation XRD (SR XRD) has been

introduced as a powerful tool for the construction of binary phase diagrams and distinct

characterization of the metastable polymorphs15, 16. All these investigations reveal the various

polymorphic forms occurring in the binary mixtures and give insight in the mixing properties

of different solid phases. Regarding the miscibility in the solid phase, the effect of TAG

interactions in mixtures may result in the formation of solid solutions (mixed crystals),

eutectic mixtures and chemical compounds8.

The primary factors determining the phase behaviour of TAG mixtures are the

differences in the chain lengths of the TAGs, the degree of saturation and the position of the

fatty acid chains, and polymorphic forms involved. If the difference in the chain lengths of

monoacid saturated TAGs is two carbon atoms or more, the components typically exhibit

limited miscibility in the most stable polymorph (example PPP-SSS)10, 13. The bigger the

difference in the number of carbon atoms, the regions where the mixture forms solid solutions

become smaller. Poor miscibility in the β-form also occurs for the monoacid saturated and cis-

monoacid unsaturated TAGs7,8,13, while trans-unsaturated molecules exhibit higher solubility

in the saturated ones11. Furthermore, for sets of TAG mixtures containing mixed chains of

saturated and unsaturated acids (SOS-SSO, PPO-POP), the formation of a molecular

compound with molar ratio 1:1 is quite common16. Regarding the less stable forms (α and β’),

they often exhibit higher miscibility than the most stable polymorph, forming solid solution

phases in all proportions of the components. However, increasing the difference between the

properties of two TAGs may result in limited miscibility for all polymorphic forms (example

PPP-POP, POP-PPO)12, 17.

In this work, we investigate the phase behaviour of the three binary TAG mixtures,

being EEE-SSS, EEE-PPP and PPP-SSS, by DSC and adiabatic calorimetry. The phase

diagrams, constructed from the DSC measurements of the β-polymorph, are presented and

discussed. To reveal more information about the mixing properties in the solid phase, the DSC

results are supplemented by several adiabatic measurements of the given TAG mixtures.

Page 105: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 97

5.2 Overview of phase diagrams The phase diagram allows the determination of the amount and the composition of the

solid phase fraction in a crystallized fat, if equilibrium conditions are attained. However, it is

important to point out that fats are usually not in equilibrium, and the phase diagram does not

necessarily tell us what will happen in complex food systems. Kinetic constrains may have

significant impact on the crystallized product and may lead to the formation of different

phases than those predicted by the phase diagram. Nevertheless, the equilibrium phase

diagrams of fats provide a useful guideline for proper control of crystallization during

processing of food products. However, they are not well known in many cases and are usually

difficult to obtain.

Here, we give a classification of main types of phase diagram that are commonly

observed in binary mixtures of TAGs (Fig. 1), as introduced by Timms18:

• Monotectic continuous solid solutions, which are formed when the TAGs are

very similar in melting temperature, molecular volume and polymorphism.

• Eutectic systems, which are often found in TAG mixtures, tend to occur when

the components differ in molecular volume, shape and polymorph, but not too

much in melting temperature.

• Monotectic partial solid solutions form in preference to a eutectic system if the

difference in melting temperature of the TAG components is increased.

• Peritectic systems have been found to occur only in mixed

saturated/unsaturated systems where at least one TAG has two unsaturated

acids.

For the description of the solid-liquid phase equilibrium, beside the pure component

properties, the excess Gibbs free energy properties in the coexisting phases must be known.

Since the fats exhibit polymorphism, the excess Gibbs energy of all possible solid phases is

required. In principal, there is relatively little information about the excess properties of TAG

mixtures. Generally, the deviation from ideal mixing of TAGs in the liquid phase becomes

noticeable if the difference in chain length exceeds 15 carbon atoms. As for the mixing

properties in the solid phase, we refer to the work of Wesdorp9, who developed a

thermodynamic model to describe the phase behaviour of TAG mixtures. In this work, a

review of experimental binary phase diagrams of TAGs is provided. These diagrams were

Page 106: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

98

used to determine the interaction parameters that occur in an excess Gibbs energy model.

Using a fitting procedure the interaction parameters were adjusted until the calculated and

measured phase diagrams agreed. By extracting binary interaction coefficients between

TAGs, it is possible to extrapolate to ternary and more complex mixtures. Based on these

results, the phase behaviour of multi-component TAG mixtures in any polymorphic form was

obtained. In general, the α-polymorphs behave as ideal mixtures, while β’- and β-forms

exhibit significantly non-ideal behaviour.

Figure 1. Phase diagrams in binary TAG mixtures: (A) Monotectic continuous solid solution;

(B) Eutectic; (C) Monotectic partial solid solution; and (D) Peritectic.

Page 107: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 99

However, the unreliability of the interaction parameters obtained from the measured

phase diagrams is considerable. Even relatively small experimental errors in the position of

the liquidus and solidus lines lead to large uncertainty in the interaction parameters.

Moreover, it is not possible to measure reliable phase diagrams of unstable polymorphs. Study

of the phase behaviour of a TAG mixture in a metastable solid state (α and β’) can only be

established under certain conditions, which definitely do not respect the rule of a true

thermodynamic equilibrium.

Namely, the accuracy of the measured phase diagram is disturbed by several

experimental parameters, such as the purity of materials and the stabilization procedure used

for producing the most stable phase9. The usage of the DSC also introduces certain

experimental limitations. Sometimes small heat effects are difficult to detect using the DSC

technique due to too large sample size or scanning rate. In some cases the melting peaks are

not sharp, so that it is not possible to determine exact starting and end temperatures of the

transitions. These issues, concerning the usage of the DSC, may result in a set of false

liquidus and solidus points. Usually, the measured solidus line is unreliable due to the kinetics

effects. Therefore, revealing accurate phase diagrams for fat mixtures remains a challenging

task to be done mostly.

5.3 The role of kinetics The strong influence of kinetics on the crystallization of fats is observed in the

occurrence of different polymorphs. Preferably, fats crystallize in the thermodynamically least

stable α-polymorph. Although the final energy state of the lower-stability polymorph is

higher, the energy barrier that has to be overcome for the nucleation is less (see Fig. 2). On

heating or during storage, the α-polymorph eventually transforms or re-crystallizes to more

stable forms (β’- or β). From experiments it turns out that no significant undercooling is

needed for the nucleation and crystallization of the α-polymorph. The reason for that is the

ordering of molecules into lamellae that takes place in the liquid phase before the

crystallization16. Typically, at the crystallization temperature of the α-polymorph the

undercooling for β’- and β-polymorphs is very high, but these phases will not crystallize due

to a high nucleation barrier and the slow growth kinetics. The β- polymorph tends to

Page 108: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

100

crystallize from the melt under conditions where little or no undercooling of the less stable

form is present.

Figure 2. Schematic diagram presenting the Gibbs free energy change during nucleation of

different polymorphs.

Another influence of the kinetics is reflected in the existence of composition gradients

within crystals, so that the crystallized material is not homogeneous. By slow cooling of the

melt, the composition of the growing solid that is in contact with the liquid phase might reach

the equilibrium. However, the composition of inner regions of the crystal does not change due

to the low diffusion rate in the solid phase. Additionally, the effect of the undercooling at the

interface is quite strong and it will induce segregation that might deviate significantly from

that predicted by the equilibrium phase diagram. In case of TAG mixtures, it is shown that

increased undercoolings result in high reduction of segregation19. Finally, for an adequate

description of the crystallization process, the segregation at the given undercooling should be

Page 109: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 101

coupled with mass and heat transport limitations, as discussed in Chapter 3. It was shown that

during the crystallization of TAG mixtures, the heat transport limitation could significantly

enlarge the segregation compensating (partly) the effect of undercooling19.

Rapid crystallization of TAG mixtures may result in poorly packed crystals, which

thermal properties may deviate significantly from those of well-packed ones. For example, the

melting temperature of such imperfect crystals is typically lower than the melting point

predicted by thermodynamics. These imperfections in the crystals may persist for years if the

liquid phase is not present20. However, by re-crystallization the badly packed crystals can

easily rearrange into well-packed crystals.

Clearly, the kinetics plays an important role in the fat crystallization. For prediction of

the amount and composition of the solid phase, the knowledge of equilibrium

thermodynamics is indispensable, but not sufficient. Kinetic factors are as important as

thermodynamic ones in determining which polymorph will crystallize from the melt and the

amount, composition and properties of the crystalline phase.

5.4 Equilibrium phase diagrams of ββββ-polymorph In this section, the phase diagrams of three TAG binary mixtures measured by DSC

are presented. Generally, the diagrams are constructed by using the melting temperatures

derived from the heating scans of the measured mixtures. The diagrams presented here also

contain the corresponding crystallization temperatures obtained from the cooling scans. When

the mixture forms a solid solution upon crystallization, the DSC heating scan contains one

endothermic effect and the temperature of this endotherm lies between the melting

temperatures of the pure components. In case that a mixture exhibits limited miscibility in the

solid phase, in principle two endothermic effects are present on the DSC heating scan.

However, these two endothermic peaks usually overlap on the experimental scans, so that it is

difficult to determine their melting onsets21. Therefore, we constructed the phase diagrams by

using the values of melting peak temperatures. The exothermic effects on cooling DSC scans,

registered at low cooling rates, do not show the overlap. Thus, we were able to derive the

onset temperatures of the exothermic peaks, which were plotted as a function of composition

in the presented phase diagrams.

Page 110: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

102

280 290 300 310 320 330 340 350 360 370

T / K

-15

-10

-5

0

5

heat

flow

/ m

W Beside the DSC results, we present the adiabatic measurements of the binary TAG

mixtures. As it will be discussed further in the text, the diluted TAG mixtures might exhibit

different phase behaviour when measured in the adiabatic calorimeter as compared to the DSC

results.

EEE-SSS mixture Several mixtures of EEE-SSS were measured in the DSC. Their melts were cooled to

273 K at a rate of 0.2 K⋅min-1 and the obtained solids were melted with the rate of 1 K⋅min-1.

No exothermic re-crystallization effects were observed on the heating scans, meaning that the

solid phases crystallized in the most stable polymorphic form. In Figure 3, the heating and

cooling scans are shown for the mixture xSSS = 0.558. Both on cooling and heating scans two

peaks are clearly distinguished, implying that the mixture exhibits phase separation in the

solid phase.

Figure 3. Cooling and heating DSC scans of the EEE-SSS mixture of composition xSSS=0.558,

registered at the cooling rate of 0.2 K⋅min-1 and the heating rate of 1 K⋅min-1.

Page 111: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 103

280 290 300 310 320 330 340 350 360

T / K

-4

-3

-2

-1

0

1

2

3

4

heat

flow

/ m

W

310 320 330 340 350 360 370

T / K

-20

-10

0

10

20

heat

flow

/ m

W

xSSS=0.05

xSSS=0.98

On continuous cooling of the melt, the SSS-rich crystalline phase solidifies first, followed by

the crystallization of another solid phase, rich in EEE. According to our measurements, the

mixture of EEE-SSS shows the described behaviour in most of composition range.

Figure 4. Cooling (bold lines) and heating (thin lines) DSC scans of two diluted EEE-SSS

mixtures of compositions xSSS=0.05 and xSSS=0.98, registered at the cooling rate of

0.2 K⋅min-1 and the heating rate of 1 K⋅min-1.

Page 112: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

104

Still, we observed complete solubility of EEE in SSS for the composition range xSSS >

0.9. In another study of the EEE-SSS mixture11, a broader region of solubility was reported,

occurring for xSSS > 0.65. To illustrate the solubility in the EEE-SSS mixture, we present the

scans recorded for two diluted mixtures, being of compositions xSSS = 0.05 and xSSS = 0.98

(see Fig. 4). On both cooling and heating scans of the SSS-rich mixture, one peak was

observed, indicating that the mixture formed a solid solution. For the EEE-rich mixture,

although one exothermic peak appears on the cooling scan, two endothermic effects can be

distinguished on the heating scan. This suggests that there is still some separation of the

components in the solid phase. These results support already observed mixing trend in the

mixtures of saturated-unsaturated TAGs; the saturated TAGs exhibit less or no solubility in

the unsaturated ones13.

Based on the transition temperatures extracted from the heating and cooling scans of

the measured mixtures, the diagram shown in Figure 5 is constructed. The line that represents

the onset temperatures of the first crystallization lies about 20 K below the liquidus line

derived from the corresponding melting temperatures. This temperature difference indicates

the undercooling needed to induce crystallization at the cooling rate of 0.2 K⋅min-1.

Accordingly, the space between these two lines represents a metastable zone, where

nucleation does not take place. The width of the metastable zone depends on the

crystallization conditions, particularly on the cooling rate. When higher cooling rates were

applied, being 5 and 10 K⋅min-1, more undercooling was needed to initiate crystallization. The

lines showing the crystallization onsets at different high cooling rates are very close to each

other (Fig. 5). This suggests that there is a maximal value of undercooling that can be

achieved for the given mixture. However, the solid phases obtained at higher cooling rates

were not in the most stable polymorphic form.

The line presenting the onset temperatures of the second crystallization under the rate

of 0.2 K⋅min-1 lies about 4 K below the line of the corresponding melting temperatures (Fig.

5). Thus, less undercooling is needed for the solidification of the second crystalline phase,

since it nucleates on already existing crystals of the first solid phase, which is easier. The

values of the crystallization and melting temperatures of the second solid phase as plotted for

different compositions of the mixture, form lines at approximately constant temperature. The

Page 113: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 105

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

x (SSS)

280

290

300

310

320

330

340

350

T / K

line showing the melting temperatures of the second solid phase is very close to the melting

point of pure EEE.

Figure 5. The phase diagram of the EEE-SSS mixture. Bold lines with circles present the

melting temperatures derived from the heating scans registered at the rate of 1 K⋅min-1; thin

lines with triangles present the crystallization temperatures upon slow cooling of the melt at

the rate of 0.2 K⋅min-1; dashed lines correspond to the crystallization temperatures during

cooling of the melt at the rates of 5 and 10 K⋅min-1.

In the DSC experiments the mass of the sample is in the order of a few milligrams. For

the experiments in the adiabatic calorimeter, about 1000 time higher amounts of sample can

be used, which might cause different phase behaviour of the mixture. To compare with the

DSC results, we performed experiments in the adiabatic calorimeter with an EEE-SSS

mixture of composition xSSS = 0.499. Before the sample was mounted in the calorimeter, the

mixture was melted in an oven, then shaken to ensure complete mixing in the liquid phase,

Page 114: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

106

280 290 300 310 320 330 340 350 360

T / K

500

1000

1500

2000

Hve

ssel

+mix /

J

heating enthalpy cooling enthalpy

and immersed into ice. Inside the calorimeter the solid phase was again melted, and the melt

was cooled to 250 K at a rate of 0.1 K⋅min-1. The obtained solid phase was heated to 370 K

and no re-crystallization exothermic effects were observed, meaning that the mixture

crystallized in the β-polymorph. The enthalpy curves measured during cooling and heating of

the mixture are given in Figure 6, where each curve shows two transitions. This illustrates the

limited miscibility of the components in the solid phase. It was also noticed that the first

crystallization took place at somewhat lower undercooling in the adiabatic calorimeter than in

the DSC. The reason for that is the larger amount of sample as used in the adiabatic

measurements.

Figure 6. The enthalpies of the vessel and the EEE-SSS mixture (xSSS=0.499) measured

during cooling and heating in the adiabatic calorimeter. Both cooling and heating rates were

0.1 K⋅min-1.

Page 115: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 107

From this analysis, we conclude that the β-polymorph of EEE-SSS mixtures show

phase separation in the most of composition range, while it forms solid solution in the SSS-

rich mixtures for xSSS > 0.9. Such a type of phase diagram belongs to the group of monotectic

partial solid solution phase diagrams and it is typical for TAG mixtures where the difference

in the melting temperatures of components is high. In case of EEE and SSS, this difference is

quite significant, being about 30 K. However, according to Wesdorp9, EEE and SSS should

form completely miscible β-form for the whole composition range. The shape of the liquidus

line could point to such a conclusion, but our results clearly show that phase separation occurs

in the solid phase.

EEE-PPP mixture Mixtures of EEE-PPP were measured in the DSC under the same experimental

conditions as the mixture of EEE and SSS. The derived crystallization and melting

temperatures of the β-polymorph are plotted for different compositions of the mixture,

resulting in a diagram given in Figure 7. The line presenting the first crystallization

temperatures upon cooling of 0.2 K⋅min-1 lies about 15 K below the corresponding melting

temperature line. For the second crystallization much less undercooling is needed, about 2 K.

Besides, we plot the crystallization temperatures registered during fast cooling. As for the

EEE-SSS mixture, it appears to be a maximal achievable undercooling for increasing cooling

rate.

When the mixture becomes richer in EEE, the crystallization at the cooling rate of 0.2

K⋅min-1 starts at temperatures that exceed the undercooling of 15 K. For the mixtures of

compositions xPPP = 0.019 and xPPP = 0.039, the crystallization onset was 26 K lower than the

corresponding liquidus point. In these cases, the crystallization temperatures are quite close to

the melting point of the EEE α-polymorph, being 288 K. On the cooling scans of these

mixtures, broad crystallization peaks appear, while the heating scans contain the exothermic

re-crystallization effects just before the melting (Fig. 8). Clearly, in the case of these EEE-rich

mixtures the β-polymorph was not obtained during cooling of the melt at the rate of 0.2

K⋅min-1. Therefore, only the melting points are presented for these two diluted mixtures in

Figure 7. In contrast, EEE-rich samples of the EEE-SSS mixtures solidify in the most stable

Page 116: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

108

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

x (PPP)

280

290

300

310

320

330

340

350

T / K

form upon slow cooling. Probably, that is so because SSS molecules are longer than PPP ones

and thereby allow for easier formation of the most stable form.

Figure 7. The phase diagram of the EEE-PPP mixture. Bold lines with circles present the

melting temperatures derived from the heating scans registered at the rate of 1 K⋅min-1; thin

lines with triangles present the crystallization temperatures upon slow cooling of the melt at

the rate of 0.2 K⋅min-1; dashed lines correspond to the crystallization temperatures during

cooling of the melt at the rates of 5 and 10 K⋅min-1.

Page 117: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 109

-10

0

10

20

30

40

50

270 280 290 300 310 320 330 340 350 360

T / K

heat

flow

/ m

W

xPPP = 0.019

-10

0

10

20

30

40

50

270 280 290 300 310 320 330 340 350 360

T / K

heat

flow

/ m

W

xPPP = 0.039

Figure 8. Cooling and heating DSC scans of two EEE-rich mixtures of EEE and PPP, having

the compositions of xPPP=0.019 and xPPP=0.039. The cooling rate was 0.2 K⋅min-1 and the

heating rate was 1 K⋅min-1.

Page 118: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

110

-60

-40

-20

0

20

40

60

290 300 310 320 330 340 350 360

T / K

heat

flow

/ m

W

xPPP = 0.95

On the other hand, the PPP-rich mixtures crystallize in the β-polymorph under slow

cooling. The cooling scan of an EEE-PPP mixture of composition xPPP = 0.95 shows one

sharp exothermic peak (Fig. 9). On the heating scan two endothermic effects can be

distinguished, suggesting that the components do not mix ideally in the solid phase. Note the

difference in shape of the crystallization peaks for the EEE-rich and PPP-rich mixtures. The

broad peaks by the crystallization of the EEE-rich mixtures point to the crystallization of more

unstable polymorphs, while the PPP-rich mixture solidifies instantly in the β-polymorph.

Figure 9. Cooling and heating DSC scans of the PPP-rich mixture of EEE and PPP, being of

the composition xPPP=0.95. The cooling rate was 0.2 K⋅min-1 and the heating rate was

1 K⋅min-1.

Page 119: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 111

280 290 300 310 320 330 340

T / K

1

2

3

4

5

c p / J

(gK

)-1

solid after slow cooling of melt solid stabilized at 313 K

To get a better insight into the behaviour of the EEE-rich mixtures, an EEE-PPP

mixture of composition xPPP = 0.017 was measured in the adiabatic calorimeter. The mixture

was prepared for adiabatic measurements in the same way as for EEE-SSS mixture. After

slow cooling of the melt under the rate of 0.1 K⋅min-1, no β-polymorph was formed. The heat

capacity curve, measured upon slow heating of the obtained solid, contains an exothermic

kink just before melting (see Fig. 10). Thus, in both adiabatic calorimeter and DSC we could

not obtain the most stable polymorph for an EEE-rich mixture just by slow cooling of the

melt. The heat capacity of the same EEE-PPP mixture stabilized at 313 K for 24 hours is

given in Figure 10. This heat capacity curve corresponds to the most stable polymorph, which

seems to have the property of a solid solution. Apparently, the β-polymorph exhibits enhanced

miscibility when re-crystallized from the less stable and basically more miscible α-

polymorph.

Figure 10. The heat capacities of the EEE-PPP mixture of the composition xPPP=0.017

measured in the adiabatic calorimeter at the heating rate of 0.1 K⋅min-1.

Page 120: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

112

The above-presented results suggest that the components EEE and PPP are highly

immiscible in the β-polymorph. We found no evidence of the solid solution formation during

slow cooling of the melt. Such behaviour originates from significant differences between their

molecules; the chain lengths differ in two carbon atoms and EEE is trans-unsaturated.

However, accurate determination of mixing properties in EEE-rich mixtures remains difficult,

since these mixtures do not tend to crystallize in the β-polymorph upon slow cooling.

PPP- SSS mixture The phase diagram of the PPP-SSS mixture, as measured by DSC22, is shown in

Figure 11. The experimental procedure consisted of cooling the melt at a rate of 0.1 K⋅min-1

and afterwards heating of the obtained solid phase at a rate of 1 K⋅min-1. The diagram of the

β-polymorph is constructed by using the corresponding melting temperatures registered

during the experiments. Additionally, we present the crystallization temperatures of the α-

polymorph, derived from rapid cooling of the melts. The phase diagram of the β-form of the

PPP-SSS mixture clearly shows eutectic character. The components exhibit limited

immiscibility in most of the composition range, having the regions of complete mixing close

to the pure component sides. According to the experimental data, the mixed crystals are

expected to form in the following composition ranges: at PPP side for xSSS < 0.046 and at SSS

side for xSSS > 0.88. The phase diagram presented here is in agreement with that determined in

Ref. [9].

To illustrate the phase behaviour of the diluted PPP-SSS mixtures, we present cooling

and heating scans of the SSS-rich mixture (xSSS = 0.95) as registered during the described

DSC experiments (Fig. 12). At a cooling rate of 0.1 K⋅min-1 the mixture crystallized at 327.6

K, which is close to the melting point of the α-form of SSS, being 327.3 K. According to the

heating scan, which contains one exothermic re-crystallization effect at 327.4 K, we conclude

that the mixture did not crystallize completely in the most stable polymorph. However, when

the mixture of the same composition was cooled under the same rate in the adiabatic

calorimeter, the β-polymorph crystallized at 331.7 K, 4 K above the crystallization

temperature in the DSC. This discrepancy illustrates the effect of the difference in mass of the

adiabatic and the DSC samples. Surprisingly, the β-polymorph obtained in the adiabatic

Page 121: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 113

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

x (SSS)

310

320

330

340

350

T / K

calorimeter shows limited miscibility of the components. This is in contrast with the phase

diagram measured by DSC (Fig. 11), which suggests the formation of mixed crystals at the

given composition of the PPP-SSS mixture. The DSC results point to higher miscibility in the

solid phase, since the β-polymorphs of the diluted PPP-SSS mixtures were formed by the re-

crystallization of the less stable α-polymorph. In conclusion, regarding the mixing status in

the most stable β-polymorph, one should consider the manner in which it was formed.

Figure 11. The phase diagram of the PPP-SSS mixture. Bold lines with circles present the

melting temperatures derived from the heating scans at the rate of 1 K⋅min-1 and dashed line

with squares corresponds to the crystallization temperatures during rapid cooling of the melt

at the rate of 10 K⋅min-1.

Page 122: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

114

Figure 12. Cooling and heating DSC scans of the SSS-rich mixture of PPP and SSS, being of

the composition xSSS=0.95. The cooling rate was 0.1 K⋅min-1 and the heating rate was

1 K⋅min-1.

To investigate the mixing properties of SSS-rich mixtures, we measured a mixture of

the composition xSSS = 0.97 in the adiabatic calorimeter under the same conditions as the

previous mixture. The β-polymorph formed during slow cooling of the melt shows phase

separation, as can be seen from the measured heat capacity curve (Fig. 13). When the melt

was cooled rapidly, the α-polymorph solidified. Subsequently, on heating it re-crystallized at

332 K (Fig. 13). In the next experiment, the thermodynamically unstable α-form was heated

to 332 K and left at that temperature to stabilize under adiabatic conditions for 24 hours.

Afterwards, the sample was cooled slowly to 250 K and the heat capacity of the obtained solid

phase was measured (Fig. 13). The β-polymorph obtained by re-crystallization is apparently

in the state of a solid solution. Thus, the β-polymorph tends to higher miscibility when formed

via re-crystallization of the less stable polymorph. Using the measured enthalpies and heat

capacities of two types of the β-polymorphs, one being in solid solution and other in eutectic

form, we calculated their relative Gibbs free energies (see Fig. 14). From thermodynamic

point of view, the solid phase with the lowest value of the Gibbs energy should preferably

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

300 310 320 330 340 350 360

T / K

heat

flow

/ m

W

Page 123: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 115

280 290 300 310 320 330 340 350 360 370 380

T / K

0

1000

2000

3000

4000

5000

c p / J

(mol

K)-1

crystallize from the melt. Since the given Gibbs energy curves are very close to each other,

the difference in driving force for the crystallization of the β-polymorph into separate phases

and a solid solution is very small. This suggests that the solid solution may occur due to

kinetics.

Figure 13. The heat capacities of PPP-SSS mixture xSSS = 0.97 measured in the adiabatic

calorimeter during heating at the rate of 0.1 K⋅min-1; α-form (dashed line), β-form

crystallized during slow cooling of the melt at the rate of 0.1 K⋅min-1 (solid line with circles),

β-form obtained by re-crystallization at 332 K (bold line).

Page 124: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

116

270 280 290 300 310 320 330 340

T / K

-15

-10

-5

0

5

(Tho

usan

ds)

Gre

l / J

mol

-1

solid solution limited miscibility

Figure 14. Relative Gibbs free energies of the PPP-SSS mixture (xSSS = 0.97), for solid

solution formed by re-crystallization at 332 K and for the solid phase with limited miscibility

that crystallizes during slow cooling of the melt at the rate of 0.1 K⋅min-1.

5.5 The crystallization of TAG mixtures at high cooling rates EEE-SSS mixture

Several mixtures of EEE and SSS were melted in the DSC. After keeping the melt at

373 K for 30 minutes the mixtures were cooled at high cooling rates, being 5, 10 and 20

K⋅min-1. After each cooling to 273 K, the heating scans of the obtained solids were taken at a

rate of 5 K⋅min-1. It was expected that the mixtures would crystallize in the least stable

polymorphic form at the mentioned high cooling rates. However, this was not always the case,

since the mixture of composition xSSS=0.229 crystallized in the β-form under all three cooling

rates. Each cooling scan basically contains two exothermic peaks (Fig. 15), where the onset of

Page 125: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 117

270 280 290 300 310 320 330 340 350 360 370

T / K

-80

-70

-60

-50

-40

-30

-20

-10

0

heat

flow

/ m

W

-5 K·min-1 -10 K·min-1 -20 K·min-1

the first crystallization is in the vicinity of 314 K. During cooling of the mixture at a rate of 20

K⋅min-1, one additional exothermic peak appears. On the heating scan of this solid phase, a

small exothermic effect occurs at 288 K. This temperature corresponds to the melting

temperature of EEE α-polymorph, suggesting that part of EEE component solidified in the

least stable form. Anyway, the rest of the mixture crystallized in the β-form characterized by

limited miscibility of the components. The appearance of more exothermic peaks on the

cooling scans suggests that the components EEE and SSS do not mix easily in the solid phase

even under fast cooling.

Figure 15. Cooling scans of the EEE-SSS mixture of composition xSSS=0.229.

On the other hand, an EEE-SSS mixture of composition xSSS=0.883 crystallized

around 323.5 K for all three cooling rates, giving one crystallization peak. On each of the

heating scans one exothermic peak appears at 327.2 K (Fig. 16). This temperature coincides

with the melting temperature of SSS α-polymorph, suggesting that part of the sample, rich in

SSS, solidified in the α-form upon fast cooling. In contrast to the previously discussed EEE-

Page 126: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

118

290 300 310 320 330 340 350 360 370

T / K

-10

0

10

20

heat

flow

/ m

W

SSS mixture, which crystallized in the most stable form at high cooling rates, this mixture

obviously solidified in more polymorphs. Thus, as the mixture becomes richer in SSS, the

probability of the formation of the α-polymorph, apparently rich in SSS, increases.

Figure 16. Heating of the EEE-SSS mixture of composition xSSS=0.883. The solid phase is

obtained using a cooling rate of 10 K⋅min-1.

EEE-PPP mixture The phase behaviour of the EEE-PPP mixture at fast cooling was investigated in the

same experimental conditions as described for the EEE-SSS mixture. The cooling scans of the

mixture xPPP=0.657 show two exothermic peaks in case of all three cooling rates. The onset of

the first peaks is always in the vicinity of 311 K, while the position of the second

crystallization peak shifts toward a lower temperature for increasing cooling rate (Fig. 17 a, b,

c). Only during the cooling with the rate of 5 K⋅min-1, both solid phases are in the most stable

form (fig. 17a).

Page 127: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 119

270 280 290 300 310 320 330 340 350 360 370

T / K

-25

-20

-15

-10

-5

0

5

10

heat

flow

/ m

W

270 280 290 300 310 320 330 340 350 360 370

T / K

-15

-10

-5

0

5

10

heat

flow

/ m

W

270 280 290 300 310 320 330 340 350 360 370

T / K

-10

-5

0

5

10

heat

flow

/ m

W

a)

b)

c)

Figure 17. Cooling and heating scans of the EEE-PPP mixture of composition xPPP=0.657,

for different cooling rates: a) 5 K⋅min-1, b) 10 K⋅min-1 and c) 20 K⋅min-1.

Page 128: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

120

For the other two cooling rates, being 10 and 20 K⋅min-1, the solid phases formed during the

second crystallization are clearly in the less stable form, since they re-crystallize just before

the melting (Fig. 17 b, c). In these cases, the final solid phase contains more than one

polymorph - one stable that crystallizes at 311 K, and the other unstable, of which the

properties depend on the intensity of cooling. Accordingly, in the heating scan of the solid

phase formed at the cooling rate of 20 K⋅min-1, an additional melting peak occurs at 288 K,

followed by a re-crystallization effect (Fig. 17c). This temperature corresponds to the melting

temperature of the α-polymorph of EEE, suggesting that a certain portion of EEE crystallized

in the less stable polymorph due to the very fast cooling.

The PPP-rich mixture, xPPP=0.946, exhibits one exothermic effect in case of all applied

cooling rates, while the onset of crystallization is at 314.7 K. On heating the obtained solid

phases, one exothermic re-crystallization effect occurs at the temperature close to 318 K (Fig.

18). It seems that the most of the sample crystallizes as a completely mixed stable form, while

the remaining part solidifies in the α-polymorph on further fast cooling. The effect of the

crystallization of this α-phase is not visible on the cooling scan, probably due to the fact that

two polymorphs solidify simultaneously within the temperature interval of the crystallization

peak. According to the results presented here, the fast cooling of the EEE-PPP mixtures leads

to solid phases that consist of more polymorphic forms. Considering the results for the

mixture of composition xPPP=0.657, the appearance of two crystallization peaks implies that

the components do not mix easily during fast cooling. However, in the case of the PPP-rich

mixture, the miscibility of the components is enhanced to a certain extent.

Page 129: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 121

280 290 300 310 320 330 340 350 360 370

T / K

-40

-30

-20

-10

0

10

20

heat

flow

/ m

W

Figure 18. Cooling and heating scans of the EEE-PPP mixture of composition xPPP=0.946.

The applied cooling rate was 10 K⋅min-1.

PPP-SSS mixture The experimental procedure applied for measuring of the previously discussed TAG

mixtures in the DSC, is used for the investigation of several PPP-SSS mixtures. For all the

measured compositions of the PPP-SSS mixture, one crystallization peak occurs during

cooling with the above-mentioned high cooling rates. This suggests that the components mix

ideally in the α-polymorph.

To illustrate the effect of composition of the PPP-SSS mixture on the DSC trace, we

present the cooling and heating scans of two PPP-SSS mixtures, being of compositions

xSSS=0.522 and xSSS=0.879 (Fig. 19 a, b). For both mixtures, the scans refer to the solid phases

formed at a cooling rate of 10 K⋅min-1. Both heating scans show first the melting of the α-

polymorph, followed by the immediate re-crystallization into the β’-polymorph. This

polymorph melts on further heating and subsequently re-crystallizes into the most stable β-

polymorph. In principle, the phase transitions are the same for both mixtures, but the

Page 130: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

122

280 290 300 310 320 330 340 350 360 370

T / K

-30

-20

-10

0

10

heat

flow

/ m

W

280 290 300 310 320 330 340 350 360 370

T / K

-50

-40

-30

-20

-10

0

10

20

heat

flow

/ m

W

a)

b)

sharpness and the intensity of the transition peaks are clearly different. When the amount of

the components is about the same, the peaks on the cooling and the heating scans are

somewhat broader than in the case where the component SSS dominates. This is not

surprising, in the sense that the purer the sample, the sharper the transition peaks.

Figure 19. Cooling and heating scans of the PPP-SSS mixtures: a) xSSS=0.552, b) xSSS=0.879.

In both cases the cooling rate was 10 K⋅min-1.

Page 131: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures 123

5.6 Summary The thermal analysis of three binary mixtures of TAGs, being EEE-SSS, EEE-PPP and

PPP-SSS, is performed by means of DSC and adiabatic calorimetry. The phase diagrams of

the β-polymorph of the mixtures are constructed using the DSC data obtained from slow

cooling of the melts and subsequent melting of the formed solid phases. Regarding the EEE-

SSS mixture, the components exhibit limited miscibility in the most of the composition range,

while the SSS-rich mixtures form solid solutions. The components EEE and PPP show no

mixing in the β-polymorph for the whole composition range. Still, it remains difficult to

determine the miscibility in the EEE-rich mixtures, since they do not crystallize in the most

stable polymorph during slow cooling. As for the PPP-SSS mixture, the β-polymorph exhibits

typical eutectic phase diagram with the regions of solid solutions close to the pure

components. However, the β-polymorph of SSS-rich mixtures, formed during slow cooling in

the adiabatic calorimeter, shows limited miscibility of the components. That is in contrast to

the DSC results, which imply that these SSS-rich mixtures should be in the state of solid

solutions. Namely, in the DSC experiments, the β-polymorph of diluted PPP-SSS mixtures is

formed by the re-crystallization of a less stable polymorph. In this way, the miscibility of the

components in the solid phase is enhanced. These observations point to the impact of kinetics

on the mixing properties of the β-polymorph.

Finally, the solid states of the mentioned TAG mixtures, formed during fast cooling of

the melts in the DSC, are discussed. The results show a remarkable difference between the

samples containing the unsaturated EEE component and the PPP-SSS mixture. In the latter

sample, the α-polymorph is easily obtained for high cooling rates, whereas for the mixtures

containing EEE only part of the sample crystallizes in the α-form. An explanation of this

behaviour is that the mixing in the α-polymorph is not ideal for the EEE containing samples,

due to the significant structural differences in the molecular shapes of the saturated and the

unsaturated TAG component.

Page 132: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Chapter 5: Thermal Analysis of Binary TAG Mixtures

124

References: [1] R.W. Hartel, Crystallization in Foods, Aspen Publishers Inc., 2001.

[2] A. Bell, M.H. Gordon, W. Jirasubkunakorn, K.W. Smith, Food Chemistry, 101 (2007)

799.

[3] D. Perez-Martinez, C. Alvares-Salas, M. Charo-Alonso, E. Dibildox-Alvarado, J.F. Toro-

Vazquez, Food Research International, 40 (2007) 47.

[4] I. Foubert, K. Dewettinck, G. Janssen, P.A. Vanrolleghem, J. Food Eng., 75 (2006) 551.

[5] A.G. Marangoni, D. Tang, A.P. Singh, Chemical Physics Letters, 419 (2006) 259.

[6] K. Sato, In: N. Widlak, R. Hartel, S. Narine, editors. Crystallization and Solidification

Properties of Lipids, AOCS Press, 2001.

[7] J.E. Hale, F. Schroeder, Lipids, 16 (1981) 805.

[8] W.L. Ng, JAOCS, 66 (1989) 1103.

[9] L.H. Wesdorp, PhD thesis, Delft University of Technology, 1990.

[10] M. Kellens, H. Reynaers, Fat Science Technology, 94 (1992) 286.

[11] A. Desmedt, C. Culot, Cl. Deroanne, F. Durant, V. Gibon, JAOCS, 67 (1990) 653.

[12] K. Sato, A. Minato, S. Ueno, K. Smith, Y. Amemiya, J. Phys. Chem. B, 101 (1997) 3498.

[13] I.T. Norton, C.D. Lee Tuffnell, S. Ablett, S.M. Bociek, JAOCS, 62 (1985) 1237.

[14] V. Gibon, F. Durant, Cl. Deroanne, JAOCS, 63 (1986) 1047.

[15] D.J. Cebula, P.R. Smith, JAOCS, 67 (1990) 811.

[16] K. Sato, Fett/Lipid, 101 (1999) 467.

[17] C. Himawan, V.M. Starov, A.G.F. Stapley, Advances in Colloid and Interface Science,

122 (2006) 3.

[18] R.E. Timms, Prog. Lipid Res., 23 (1984) 1.

[19] J.H. Los, M. Matovic, J. Phys. Chem. B, 109 (2005) 14632.

[20] J.W. Hagemann, In: N. Garti, K. Sato, editors. Crystallization and Polymorphism of Fats

and Fatty Acids. New York: Marcel Dekker, 1988.

[21] F.G. Gandolfo, A. Bot, E. Flöter, Thermochim. Acta, 404 (2003) 9.

[22] M. van den Heuvel, The growth behaviour of fat mixtures: The binary tristearin-

tripalmitin system, IMM Solid State Chemistry, Radboud University Nijmegen, (2005).

Page 133: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Summary of thesis 125

Summary of thesis

The traditional methods for the determination of liquid-solid phase diagrams of

mixtures are based on the assumption that the overall equilibrium is established between the

phases. Although these methods reveal valuable information about the phase behaviour and

the mixing properties of the phases, one should realize that on a relevant time scale overall

equilibrium will hardly be reached between the entire amounts of the solid and liquid phases.

Typically, the result of the crystallization of a liquid mixture will be a non-equilibrium or

metastable state of the solid. For a proper description of the crystallization process the

equilibrium approach is practically insufficient. In Chapters 2 and 3, it has been shown that a

kinetic approach is required for successful prediction of the state of the solid phase.

In Chapter 2, the slow crystallization of binary mixtures of 1,4-dichlorobenzene and

1,4-dibromobenzene was performed in the adiabatic calorimeter. For slow transition process

that takes place at near equilibrium conditions, we could assume that the liquid phase was in

equilibrium only with the growing solid phase, i.e., the solid phase at the surface. Then, the

compositions of the liquid phase and the surface layer of the solid change along the liquidus

and solidus lines of the equilibrium phase diagram. This assumption was quantitatively

expressed in a kinetic model, which successfully reproduced the experimental enthalpy curve

of the mixture, measured during slow cooling in the adiabatic calorimeter. The applicability of

the introduced model was extended to a new method for the determination of the excess

properties when the cooling path of the mixture was at disposal. In this way, the excess

quantities can be obtained by using a relatively simple method, which basically requires only

the knowledge of the cooling curve of the mixture. Finally, the phase diagram was achieved

having the advantage over traditionally determined phase diagrams, in the sense that both

excess enthalpy and entropy were derived without assuming complete equilibrium between

totally homogeneous phases. 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.

Page 134: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Summary of thesis

126

The crystallization mostly 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, by which the

actual composition of the growing solid phase can be determined for the given composition of

the liquid phase and temperature. The compositions of the growing solid, i.e. the kinetic

segregation, may considerably deviate from that predicted by the equilibrium phase diagram.

To study the crystallization of the mixture of 1,4-dichlorobenzene and 1,4-dibromobenzene at

non-equilibrium conditions we designed an experimental set-up described in Chapter 3. The

composition of the grown solid phase was measured by the gas chromatography and the

results were compared to that calculated from the equilibrium and kinetic segregation model.

Additionally, we took into account that the segregation induced composition gradients, as well

as the temperature gradients in the liquid phase, allowing the properties of liquid phase near

the solidification front to be different from those in the bulk. Thus, for the description of the

crystallization process we included mass and heat transport limitations, which enabled the

determination of so-called effective segregation. The experimental results, showing a reduced

segregation, agreed very well with the solid compositions calculated from the proposed

kinetic model, while the solid compositions predicted by equilibrium show too strong

segregation. In this way, we have verified the performance of the kinetic model coupled with

mass and heat transport limitations. These results show the effect of interfacial undercooling

on the segregation during growth in mixed molecular systems. It appears that for the presented

model system, this effect is still moderate. However, for molecular systems with higher

melting entropy, such as fats, the effect of the interfacial undercooling will be considerably

larger.

Triacylglycerols (TAGs) are the main constituents of edible fats and oils, which make

the knowledge of their thermodynamic properties very important for food industry. These

compounds exhibit polymorphism and thereby introduce complex phase behaviour in the food

products. In Chapter 4, the thermal behaviour of different polymorphs of the three pure TAGs,

being tristearin (SSS), tripalmitin (PPP) and trielaidin (EEE), was studied. A detailed thermal

analysis of SSS and the investigation of its polymorphism were performed by means of

adiabatic and differential scanning calorimetry. Specific heat capacities of the α- and β-

polymorphic forms of SSS, together with the enthalpies of fusion and melting temperatures

Page 135: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Summary of thesis 127

are reported and compared to the literature values. Additionally, we examined the fleeting

existence of the β’-polymorph of SSS and its transformation to the β-phase under isothermal

conditions in the DSC. The adiabatic data from melting of the β-polymorphs of SSS, EEE and

PPP were used for the determination of their purities. The purities were calculated by two

methods, both based on the assumption that the compound and impurity form a eutectic

mixture. The results of these methods, for the first and the second melting of the compounds,

show some discrepancies due to crystal imperfections in the β-polymorph formed after the

first melting. Nevertheless, we concluded that the investigation of the phase behaviour of the

binary mixtures of the mentioned TAGs would not be disturbed by the estimated amounts of

impurities.

In Chapter 5, the thermal analysis of three binary mixtures of TAGs, being EEE-SSS,

EEE-PPP and PPP-SSS, was performed in the DSC and the adiabatic calorimeter. The phase

diagrams of the β-polymorph of the mixtures were constructed using the DSC data obtained

from slow cooling of the melts and subsequent melting of the formed solid phases. In all three

binary mixtures the miscibility of the components is very limited in the most of the

composition range. For the EEE-SSS system, the SSS-rich mixtures form solid solutions. The

components EEE and PPP do not co-crystallize in a solid solution over the whole composition

range. However, it was difficult to draw definite conclusions on the miscibility in the EEE-

rich mixtures, since they did not crystallize in the most stable polymorph during slow cooling.

Regarding the PPP-SSS mixture, the β-polymorph yields a typical eutectic phase diagram

with some mixing in the regions close to the pure components. The β-polymorph of the

diluted mixtures, formed in the DSC by the re-crystallization of a less stable polymorph, was

in the state of solid solutions. On the other hand, the β-polymorph of SSS-rich mixtures,

obtained by slow cooling in the adiabatic calorimeter, showed limited miscibility of the

components. Apparently, the miscibility of the components is enhanced when the β-

polymorph is formed by the re-crystallization. These results demonstrate the impact of the

kinetics on the mixing properties of the β-polymorph.

Finally, the solid states of the mentioned TAG mixtures, formed during fast cooling of

the melts in the DSC, were discussed. The results show a remarkable difference between the

samples containing the unsaturated EEE component and the PPP-SSS mixture. In the latter

sample, the α-polymorph was easily obtained for high cooling rates, whereas for the mixtures

Page 136: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Summary of thesis

128

containing EEE only part of the sample crystallized in the α-form. An explanation of this

behaviour could be that the mixing in the α-polymorph is not ideal for the EEE containing

samples, due to the significant structural differences in the molecular shapes of the saturated

and the unsaturated TAG component.

Page 137: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Samenvatting 129

Samenvatting

De traditionele methoden voor het bepalen van vloeistof-vaste stof

evenwichtsfasendiagrammen voor mengsels zijn gebaseerd op de aanname dat de

verschillende fasen volledig met elkaar in evenwicht zijn. Hoewel deze methoden in het

algemeen een redelijk beeld geven van het fasengedrag en de mengeigenschappen van een

gegeven mengsel, dient men zich toch te realiseren dat bij kristallijne mengsels totaal

evenwicht meestal niet bereikt zal worden op een relevante tijdschaal, onder andere vanwege

de zeer trage diffusie in de vaste fase. In het algemeen zal het kristalliseren van een vloeibaar

mengsel resulteren in metastabiele vaste fase(n). Om dit goed te beschrijven is de methode

gebaseerd op totaal evenwicht feitelijk ontoereikend. In de hoofdstukken 2 en 3 wordt

getoond dat een kinetisch aanpak vereist is om de toestand van de vaste fase direkt na

kristallisatie adequaat te voorspellen.

In hoofdstuk 2 is het langzaam kristalliseren van mengsels van 1,4-dichlorobenzeen en

1,4-dibromobenzeen uitgevoerd in een adiabatische kalorimeter. Voor de aldus optredende

langzame faseoverovegang kan worden aangenomen dat de vloeistof voordurend vrijwel in

evenwicht blijft met de groeiende vaste fase, dat wil zeggen met de vaste fase in een relatief

dunne laag aan het kristaloppervlak. Dit houdt in dat tijdens het kristallisatieproces de

samenstellingen van de vloeistof en de groeiende vaste fase respectievelijk de liquidus- en de

solidus-lijn van het evenwichtsfasendiagram zullen volgen. Deze voorstelling van zaken kan

kwantitatief uitgedrukt worden in een kinetisch model, dat in staat blijkt de gemeten

enthalpiecurve nauwkeurig te reproduceren. De toepasbaarheid van dit model is vervolgens

uitgebreid tot een relatief eenvoudige methode om nauwkeurig de excess-mengeigenschappen

te bepalen uit een gemeten koelcurve. Als eenmaal de mengeigenschappen bepaald zijn ligt

het fasendiagram vast. Op deze manier kan men dus het fasendiagram bepalen via een

methode zonder daarbij aan te hoeven nemen dat er tijdens de kristallisatie totaal evenwicht

bestaat. Echter, deze methode kan alleen gebruikt worden voor langzame kristallisatie en niet

voor kristallisatie bij condities ver uit evenwicht, die toch vaak optreden in de praktijk.

Page 138: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Samenvatting 130

Vaak treedt kristallisatie pas op bij een zekere graad van onderkoeling, dat wil zeggen

niet dicht bij evenwicht. Om de kristallisatie bij dergelijke condities goed te kunnen

voorspellen moeten we een beschrijving geven van de zogenaamde niet-evenwichtssegregatie

ofwel de kinetische segregatie, waarmee de actuele samenstelling van de groeiende vaste fase

bij gegeven vloeistofsamenstelling en temperatuur bepaald kan worden. Deze vaste stof

samenstelling kan namelijk aanzienlijk afwijken van de samenstelling volgens het

evenwichtsfasendiagram. Om de kristallisatie van 1,4-dichlorobenzeen and 1,4-

dibromobenzeen bij condities ver van evenwicht te bestuderen hebben we een experimentele

opstelling gebouwd zoals beschreven in hoodfstuk 3 en geillustreerd in Fig. 3.1. Na

kristallisatie hebben we de samenstelling van de kristallen gemeten door middel van

gaschromatografie en die vervolgens vergeleken met de berekende samenstelling uit een

evenwichts- en een kinetisch segregatiemodel. Daarbij hebben we tevens rekening gehouden

met compositie- en temperatuurgradienten in de vloeistoffase in een grenslaag aan het

oppervlak, zodat de vloeistofeigenschappen aan het oppervlak in principe anders kunnen zijn

dan in de bulk. Kortom, voor de berekende samenstelling volgens het kinetische model

hebben we rekening gehouden met transportlimitering, zodat we uiteindelijk de effectieve

segregatie konden berekenen. De experimentele resultaten, die allemaal een gereduceerde

segregatie ten opzichte van de evenwichtsegregatie aangeven, stemmen goed overeen met de

voorspelling volgens het kinetisch model, terwijl het evenwichtsmodel een te sterke segregatie

voorspelt. Hiermee hebben we het kinetische segregatiemodel, gekoppeld aan massa- en

warmtetransportlimitering gevalideerd en tevens het effect van hoge oververzadiging op de

groeisamenstelling aangetoond. Voor het hier gebruikte systeem is dit effect nog gematigd.

Echter, als we het bovenstaande resultaat extrapoleren, zal het effect voor systemen met een

grotere smeltentropie, zoals vetmengsels, veel groter zijn.

Triacylglycerols (TAGs) vormen het hoofdbestanddeel van consumptievetten en -

oliën, wat maakt dat hun thermodynamische eigenschappen van groot belang zijn voor de

voedselindustrie. De vetbestanddelen in voedselproducten vertonen een complex fasengedrag

mede door het optreden van polymorfie. In hoofdstuk 4 is het thermische gedrag van

verschillende polymorfen van drie zuivere TAGs, te weten tristearin (SSS), tripalmitin (PPP)

en trielaidin (EEE), bestudeerd. Een gedetailleerde thermische analyse en een onderzoek naar

het polymorfe gedrag van SSS zijn uitgevoerd aan de hand adiabatische kalorimetrie en

Page 139: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Samenvatting 131

"Differential Scanning Calorimetrie (DSC)". De resulterende soortelijke warmte curven,

alsmede de smelttemperaturen en smeltwarmten voor de α- en de β-fase zijn gerapporteerd en

vergeleken met data uit de literatuur voorzover beschikbaar. Tevens is de omzetting van de α-

naar de β-fase via de β'-fase geconstateerd en bestudeerd met DSC bij isotherme condities. De

smeltdata uit adiabatiche kalorimetrie voor de β-polymorf van SSS, EEE en PPP zijn gebruikt

voor het bepalen van de zuiverheden van deze stoffen. De zuiverheden zijn berekend met twee

methoden, beiden gebaseerd op de aanname dat de hoofdcomponent en de onzuiverheid een

eutectisch systeem vormen. De resultaten van deze methoden laten enige discrepantie zien,

die zeer waarschijnlijk te wijten is aan roosterfouten die worden ingebouwd tijdens de

kristallisatie nadat het systeem eerst in volledig vloeibare toestand is gebracht. Desalniettemin

is de conclusie dat de zuiverheid dusdanig hoog is dat een onderzoek naar het fasengedrag van

de binaire mengsels van de genoemde stoffen niet vertroebeld wordt door de onzuiverheden.

In hoofdstuk 5 is een thermische analyse gedaan voor de drie binaire mengsel EEE-

SSS, EEE-PPP en PPP-SSS, wederom door middel van adiabatische kalorimetrie en DSC. De

fasediagrammen voor de β-fase van deze drie mengsel zijn bepaald op grond van DSC

experimenten door het langzaam afkoelen van de smelt en het vervolgens smelten van de

gevormde β-fase. Voor alle drie de systemen blijkt de mengbaarheid in de β-fase zeer beperkt

te zijn. Bij het EEE-SSS mengsel wordt enige menging vastgesteld aan de SSS-rijke kant van

het fasediagram. Daarentegen lijken EEE en PPP in het geheel niet te mengen in de β-fase,

hoewel aan de EEE-rijke kant een slag om de arm gehouden moet worden aangezien in dit

geval geen volledige kristallisatie in de meest stabiele polymorf plaatsvindt. Het PPP-SSS

systeem vertoont een typisch eutectisch fasengedrag voor de β-fase met beperkte menging aan

de beide zuivere component uiteinden van het diagram. Opmerkelijk is ook dat de β-fase voor

mengsels die rijk zijn in een van de beide componenten en die gevormd zijn via rekristallisatie

vanuit een minder stabiele polymorf meer gemengd zijn dan de β-fase die gevormd wordt via

directe kristallisatie in de β-fase bij langzaam afkoelen in de adiabatische kalorimeter. In de

laatst genoemde meting blijkt voor een mengsel van gelijke samenstelling als die voor de

DSC-meting fasescheiding op te treden. Dus kennelijk is de route naar de β-fase mede

bepalend voor het mengtoestand van het systeem na kristallisatie, wat duidelijk de rol van de

kinetiek illustreerd.

Page 140: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

Samenvatting 132

Tot slot is de kristallisatie bij het snel koelen van de hierboven genoemde TAG-

mengsels onderzocht en bediscussieerd. De resultaten laten een opmerkelijk verschil zien

tussen de mengsels met de onverzadigde EEE-component en het PPP-SSS mengsel. Voor het

laatst genoemde mengsel kon de instabiele α-fase gemakkelijk worden verkregen bij hoge

koelsnelheden, terwijl de EEE bevattende mengsels slechts gedeeltelijk in de α-polymorf

kristalliseren en de rest in een meer stabiele polymorf. Een mogelijke verklaring hiervoor is

dat de menging in de α-fase niet ideaal is, zoals algemeen wordt aangenomen, vanwege het

aanzienlijke structurele verschil tussen de moleculen van de verzadigde en de onverzadigde

vetcomponent.

Page 141: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

133

Acknowledgments I would like to express my truthful gratitude to the people who made possible the successful

outcome of this work.

To start with, I would like to thank my promoter Prof. Dr. Harry Oonk for giving me

the opportunity to carry out my study in Chemical Thermodynamics Group (CTG) and for his

constant care that I am satisfied and happy with life in general. I also gratefully acknowledge

my co-promoter Dr. Kees van Miltenburg for guiding me through the world of (adiabatic)

calorimetry, for many useful discussions, for the given freedom and faith in my research.

Beside that, I thank you Kees for being so supportive and attentive to me.

Special gratitude I would like to express to my closest collaborator and co-promoter

Dr. Jan Los. Dear Jan, working with you was an instructive and challenging experience for

me, I have learned so much from you. Thank you for the patience and strength shown for all

these years, for creative and enjoyable atmosphere and above all for unfailing enthusiasm in

my work, that would never be so successful without your support. Thank you very much for

your help! After all, I believe we had a very nice time together.

I am deeply grateful to my colleagues from CTG: Aad, Paul, Gerrit, Erik, Koos and

Marjan, for their support and for making these four years so enjoyable. Thank you all for

being so splendid group-mates! For excellent technical assistance, I owe great gratitude to a

wonderful man and our technician, Gerrit van den Berg. Beste Gerrit, jij en Helma zijn heel

aardig tegen mij gewest, dat zal ik nooit vergeten. I am also truly thankful to my office-mate

Erik Bevers for sharing with me the knowledge about Differential Scanning Calorimetry, for

helping me in many situations and for nice times at AIO events.

Although I belong primary to CTG, I also feel as being a part of Solid State Chemistry

group from Nijmegen. That is thank to the wonderful people I have met there, who always

made me feel welcome and comfortable among them. For that, I would like to thank Prof. Dr.

Elias Vlieg, the leader of this project, and to Hugo, Willem, Jan Los, Natalia, Jan van Kessel,

Wim, Cristina, Neda and many others. I would like to mention and thank to Maaike van den

Heuvel for giving a significant contribution to the part of my research results.

For my longest friendship in The Netherlands I would like to thank my dear friend

Aneliya Zdravkova, a strong and successful woman. Her face was the first I saw when I

Page 142: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

134

entered our apartment in Utrecht more than four years ago. Since then we lived together for

two years and we built mutual trust that became very important for both of us. My dear

Anleke, thank you very much for being there for me in good and bad times, your support

means a lot to me! I am also grateful for knowing other nice people from Bulgaria, Petar,

Veselka, Nikoleta, Ivan, with you it was always fun and I hope we will have more nice times

together.

The person who partly experienced making of the thesis on his own skin, by being the

most intensively beside me in my good and bad moods, is my beloved friend Shahin. My dear

Shy, I cannot imagine that these last years could be so nice, funny and exciting without you.

Only one smile from you makes a grey day blue. Thank you from my heart for being a part of

my life!

����� ���� ��, ������ � ���, ����� ��� �� ���� �� ���� ������� � �����, ��

��� ��� �������� ����� ������� ��� � ����������� ����, � ����� ������ ��� ����, ���

����� �� �������� ����� ���������. ������ ��� �� ��������� ���� �� ������������

������ �� �����.

!���� �� �� �������� ��������� �� ���. “"��# �����”: �����, $����� %., $�����

"., &����� � �������, �� ����� ��������, ���� �� ��, ����� ��, ��������� � ����.

'��� �����, ����������� �������, &���, '�������, '���, '�����, "���, ������,

(�����, )��� – ������ ������ �� ������ ����� – ���� ������� �� ��������� ���� ����� �

����� �������� �� ���� ����������� ����, � � ����, �� ���� ����������� ����� � ����

�� ���� ����. ���� ����� �� ���� �� ���� ����� ������ �� ������� �� �� ����� ���

���� ��������. !����� �� ����!

'���� ������� ���������� � ���� ����� '�����, ����� �� �� �������� ��

����� ���� � ���� � ��� ��� �����. (� ���� ��� ����, � ��� ����, � �� �� ���� �����

�� ��������� ���. (��� �� ��� �� �� ������ � ������. '��� ����� ���� � $����, ���� ���

������, ���� ��� � ����� �������. '��� ������ ������, *��� � �����, ��� �����

���������, ������ � �������, ���� �� �� ������ ���� �� ��� ���������.

'�� "����� �����!

Utrecht, April 2007.

Page 143: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

135

Curriculum Vitae

Marija Matovi� was born in �a�ak, a town in Serbia. After primary school, she

attended secondary school, Gimnazium �a�ak, from 1991. to 1995. The same year (1995) she

started her study at Faculty of Technology and Metallurgy, Belgrade University. In May 2001

she graduated at the department of Chemical Engineering. From September 2001 she was

working for one year as an engineer in the factory “1. Maj” �a�ak, Serbia, where she was

responsible for the line for production of isolation materials based on expanded polystyrene.

In February 2003 she started Ph.D. in The Netherlands in the group of Prof. Dr. Harry Oonk –

Chemical Thermodynamics Group, University of Utrecht. From March 2007 until now she is

working by TNO in Apeldoorn, as a process technologist in the department of Process

Modelling and Control.

Page 144: Investigation of Binary Solid Phases by Calorimetry and ...Investigation of Binary Solid Phases by Calorimetry and Kinetic Modelling Onderzoek van binaire vaste fasen met calorimetrie

136

LIST OF PUBLICATIONS: 1. M. Matovic; J. C. van Miltenburg; J. Los. Metastability in Solid Solution Growth, Journal

of Crystal Growth, 275, (2005), 211-217.

2. M. Matovic; J. C. van Miltenburg. Thermal Properties of Tristearin by Adiabatic and

Differential Scanning Calorimetry, Journal of Chemical Engineering Data, 50, (2005),

1624-1630.

3. J. Los; M. Matovic. Effective Kinetic Phase Diagrams, Journal of Physical Chemistry B,

109, (2005), 14632-14641.

4. M. Matovic; J. C. van Miltenburg; J. Los. Kinetic Approach to the Determination of the

Phase Diagram of a Solid Solution, Computer Coupling of Phase Diagrams and

Thermochemistry, 30, (2006), 209-215.

5. J. Los; M. van den Heuvel; W. J. P. van Enckevort; E. Vlieg; H. A. J. Oonk; M. Matovic;

J. C. van Miltenburg. Models for the Determination of Kinetic Phase Diagrams and

Kinetic Phase Separation Domains, Computer Coupling of Phase Diagrams and

Thermochemistry, 30, (2006), 216-224.