Ph.D. Thesis - COnnecting REpositories · 2017. 3. 22. · For 3 months I have been working in Physics and Astronomy laboratory in ... Ikram and Souad Benrahal, Souad Haouari, Wafa

Faculty of Mathematical, Physical and Natural Science

Department of Physics

2004-2006

Ph.D. Thesis

Investigations of the molecular

dynamics in binary glass-forming

systems and polymers by means of

broadband dielectric spectroscopy

Candidate Supervisors

Khadra Kessairi Prof. Pierangelo RollaDr. Simone Capaccioli

Durant toutes ces annees de these, tu as toujours ete la pour me soutenir

et m’encourager, sans toi je n’aurais certainement pas pu atteindre

mon but. Cette these ne peux etre dediee qu’a toi cher mari, car elle

represente le denouement d’une souffrance et d’une esperance partagee.

Tu es vraiment un mari exceptionnel...............................................,

Je n’oublierai pas de remercier mes parents et mes freres, Samir,

Yacine et Zaki, qui ont ete toujours present a cote de moi quand j’en

avais besoin, merci de m’avoir permis d’aller aussi loin dans mes etudes.

Un remerciement speciale a mes beaux parents pour leurs support.

Acknowledgment

Since a long time, i had a dream that i always wanted to fulfill; studying for a

Phd degree. In that time, it seems unreachable but my father learn me an im-

portant thing, which is when we want, we can. Several people encouraged

and supported me to complete this dream. It is a pleasant aspect that i have

now the opportunity to express my gratitude for all of them.

I want to thank first Prof. Pierangelo Rolla for giving me permission to

commence this thesis in the first instance, to do the necessary research work

in the Polylab (Laboratory for Industrial Application of Polymers) and which

confirmed this permission and encouraged me to go ahead with my thesis.

A special thank’s to Dr. Simone Capaccioli which i am deeply indebted,

for his constant support, stimulating suggestions and encouragement in all the

time of research. His wide knowledge and his logical way of thinking have been

of great value for me.

I wish to express my warm and sincere thanks to Prof. K. L. Ngai from

Naval Research Laboratory, Washington, for his detailed and constructive com-

ments, and for his important support and important guidance throughout this

work. Furthermore, a special thank’s for taking time to assess my manuscript.

For 3 months I have been working in Physics and Astronomy laboratory in

Leuven University, Belgium. Working in this place had great advantages. I

appreciate so much Prof. Michael Wubbenhorst for his continuous help

any time i needed, and his valuable assistance in the research work.

I furthermore warmly thank Simone Napolitano, for his valuable advice

and friendly help.

I want to thanks the members of the laboratory for their help, support, inter-

vi Acknowledgment

est and valuable hints, Prof. Mauro Luchessi, The extrasolution groupe:

Gabriele, Alessandro and Ugo, Maria Laura, Mario, Mauro, Marco,

Soheil, Agnese.

I warmly thanks Dr. Daniele Prevosto and his wife Irene for their help

and their support any time i needed.

During the last year I enjoyed very much the numerous non-scientific dis-

cussions I had with Flavia, Cinzia, Michela and Elisa (Secretary of the

Polylab), I appreciate their good advice and the continuous support they offered

to me in the difficult times.

I gratefully thanks Dr. Monica Bertoldo(Polylab) for providing DSC mea-

surement.

I am very happy and proud of making wonderful friends in Pisa who made me

feel in a big family, they make the life much easier, and the days shorter and

happier. I am speaking about, Mohamed Al Shourbagy, Awad Ibrahim,

Masoud Amirkhani, Said AlSalmi, Bachir Bouhadef, Abeer Tem-

raz, Sandra Piconni, Randa, Anice Bousbia Salah, Annissa. I will

always cherish the wonderful time spent together with them. I want also to

thank’s my algerian freinds, Ikram and Souad Benrahal, Souad Haouari,

Wafa Benstaali, Dr. Imad Bensafa who support me along all this years.

The financial support of the Galileo school is gratefully acknowledged.

The chain of my gratitude would be definitely incomplete if I would forget

to thank the first cause of this chain, Prof. S. Hamzaoui and Prof. K.

Bousbahi.

Khadra Kessairi

Pisa, April 2007

Abstract

Binary mixtures of rigid polar molecules dissolved in apolar solvents are ex-

cellent model systems to deepen the understanding of the relaxation dynamics

in glass forming systems. The first part of this thesis provides an investiga-

tion, by means of dielectric spectroscopy and thermal analysis at ambient and

elevated pressures, of the molecular dynamics of polar rigid molecules (het-

erocyclic compounds: tert-butylpyridine, Quinaldine) dissolved in an apolar

solvent (tristyrene). By changing in a systematic way the relative concentra-

tion of the components it was observed a transition from a relaxation scenario

with a structural α- process and an excess wing (for neat polar systems), to

that with a structural process and a well resolved β- secondary process. Our

result show that the excess wing is nothing but a Johari Goldstein secondary

relaxation, so close in the time scale to the structural relaxation that its low

frequency side is hidden beneath the structural peak. The time scale separa-

tion between α- and β- process increases with increasing the relative amount

of the slower apolar component (tristyrene). Among the numbers of new ex-

perimental findings coming form this work, two results can be mentioned, of

particular interest for the current debate on the literature on dynamics of

glass-forming systems. First is the invariance of the relaxation scenario un-

der different thermodynamic conditions: for different combinations of pressure

and temperature that maintain the α- relaxation time constant, the frequency

dispersion of the α- relaxation is unchanged but remarkably also the JG β-

relaxation time remains constant, allowing the construction of a master-curve.

The second result of particular interest is that, when it has been possible to

clearly reveal the JG β- relaxation from the glassy state up to well above Tg

, a strong deviation from the glassy state trend has been found for both the

viii Acknowledgment

temperature and pressure dependence, with an apparent kink in the relaxation

map always occurring at the crossing of the glass transition. All these results

support the idea that the Johari-Goldstein relaxation acts as a precursor of

the structural relaxation and therefore of the glass transition phenomenon. All

these evidences have been successfully compared to the predictions of the Cou-

pling Model, providing a rationale to explain the mutual correlation between (

α-) cooperative and local ( β-) relaxation. Taking into account the experience

gained from the work on binary mixtures in the first part, the second part of

the thesis was devoted to the study of large rigid polar molecules, dissolved

in apolar polymeric systems. The main idea was to investigate, by means of

dielectric spectroscopy, the orientational dynamics of the polar molecule, and

from that to get information on the molecular dynamics of the polymeric ma-

trix, to which it should be strongly coupled. In order to widen the possible

applications of this technique, polymeric systems of high industrial interest

were chosen, such as atactic poly(styrene) PS and poly(propylene) PP. For

poly(styrene), a systematic study was done, by varying the molecular weight

for 8 different systems from tristyrene (370 g/mol) to very high molecular

weight (160 kg/mol). As dielectric probe, the rigid polar molecule (4,4-(N,N-

dibutylamino)-(E)-nitrostilbene (DBANS)) was used. The results show that,

at low concentration, the probe molecule motion is coupled to the coopera-

tive α- molecular dynamics of the polymer matrix, while it is not affected by

additional processes, like intermolecular defect motions (due to helices in PS)

or β- relaxation (for PP). The variations of the glass transition and of the

steepness index with molecular weight revealed by the dielectric probe match

that reported in literature for the undoped polymeric matrix. This evidence

confirms that the dielectric probe technique can be an useful tool for the study

of polymer dynamics in the case of apolar or weakly polar systems.

Keywords: glass transition; pressure; binary mixtures; dispersion; structural

α- relaxation; Johari Goldstein β- relaxation; excess wing; apolar polymer;

polystyrene; polypropylene; dielectric probe

Contents

Acknowledgment v

General Introduction 1

1 Theoretical Background 5

1.1 Glass Transition and Relaxation phenomena in Glass systems . 5

1.1.1 Glass transition . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Relaxation phenomena in glass systems . . . . . . . . . . 8

1.2 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 The Free Volume approach . . . . . . . . . . . . . . . . . 18

1.2.2 Adam-Gibbs Theory approach . . . . . . . . . . . . . . . 20

1.2.3 Coupling model . . . . . . . . . . . . . . . . . . . . . . . 22

2 Experimental Techniques 27

2.1 Dielectric Relaxation Spectroscopy . . . . . . . . . . . . . . . . 28

2.1.1 Linear response theory . . . . . . . . . . . . . . . . . . . 28

2.1.2 Molecular dynamics and correlation function . . . . . . . 31

2.1.3 The Experimental Setup . . . . . . . . . . . . . . . . . . 34

2.2 Differential scanning calorimetry (DSC) . . . . . . . . . . . . . . 39

2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Conclusions 43

4 List of publications 45

References 47

General Introduction

Glasses belong to the oldest materials used by mankind. Already in prehis-

toric times, our early ancestors used obsidian, a volcanic glass, to manufactures

knives and arrows tips. Man-made glass is believed to have first emerged in

the Near East, some thousand of years BC in the form of glass beads. The first

glass vessels showed up in Egypt about 1500 BC and the important discovery

of glassblowing was made presumably in Phoenicia in the 1st millenium BC

[1].

In the last half century, the appearance of numerous new types of glasses as

metal alloys, polymers, simple liquids, etc. has made such materials ubiqui-

tous, and of inestimable importance to material engineering and science. But

from a physical point of view the theory of the nature of glass and the glass

transition is still the deepest and most interesting unsolved problem in solid

state theory [P. W. Anderson, Science 1995][2].

However, during the recent years significant theoretical and experimental ad-

vances have led to an increasing interest in physics of glasses, giving rise to

the hope that a breakthrough in the understanding of the glassy state may

be within reach in the near future [3]. Glassy systems exhibits some very in-

triguing relaxations processes: the main or the structural α-relaxation, usually

interpreted as due to the cooperativity motions of molecules which become ki-

netically frozen upon cooling the liquid to form a glass, and the secondary

β- relaxation, which is a faster process and its relaxation time changes with

temperature more slowly than that of the structural relaxation one, and it can

be monitored in the glassy state over a wide range of temperatures. For this

reason, the β- process is a unique source of information on relaxation prop-

erties of a glassy state and consequently it is considered by many researchers

2 Introduction

in the field as a fundamental characteristic of glasses. In some systems, the

secondary relaxation originates from the motion of some molecular subgroup

dynamically decoupled from the rest of the molecule(intramolecular origin),

as for example the side group in polymers [4]. In this case, the intramolecu-

lar secondary relaxation is system dependent and consequently the universal

aspect of the glass transition is hardly recognized. In some other systems,

the secondary relaxation observed was supposed to be related to the local

motion of the whole molecule (intermolecular origin). The secondary process

originating by this mechanism is usually recognizable as the Johari-Goldstein,

JG, relaxation, and is considered to reflect an universal relaxation mechanism

of glass formers systems. Therefore, a strong connection with the structural

α- relaxation is expected [5]. However, the microscopic process behind this

kind of β- relaxations is still controversially discussed. Johari and Goldstein

themselves ascribed the β- process dynamics to so-called islands of mobility,

i.e local and highly mobile regions in the glass. On the other hand, there

were experiments who support the idea that all the molecules take part in the

secondary relaxation process [6],[7],[8],[9] thus openly contradicting the idea of

islands of mobility or any other defect model. This thesis is mainly focused on

acquiring new experimental evidences together with a discussion of the phe-

nomenology of secondary relaxation to demonstrate the existence of a genuine

Johari-Goldstein relaxation which is strongly correlated with the structural α-

relaxation. We based our study of binary glass-forming mixtures of polar rigid

molecules in apolar solvent. Our choice was supported by several reasons:

Firstly, the dynamics of each component in the mixture can be investigated by

changing the concentration of the rigid molecules. For example, in the limit of

low concentrations of the polar rigid molecules, the question can be addressed

in how far the dynamics of the probe reflects the behavior of the host sys-

tem. The binary mixtures of glass-formers also serve as model systems for the

study of the technically important polymer-plasticizer problem. Instead, at

high concentration of the polar rigid molecules the dynamics of neat system

can be studied, and so even simple molecules can be suitably supercooled and

the crystallization is avoided.

Second, dielectric relaxation analysis can widely contribute significantly to the

Introduction 3

understanding of molecular dynamics of glass-forming binary mixtures, be-

cause it can provides information over a wide time-scale on the orientational

correlation function of dipoles in molecular environment. Thus, it can be used

to investigate both the structural and the secondary relaxation time over a

wide interval of time scales. Consequently, binary mixtures are excellent sys-

tems to deepen the understanding of the molecular slowing down characteristic

for the glass transition and to find out the origin of the secondary relaxation

process and it’s relation with the structural relaxation.

The thesis is organized as follows: Chapter 1 will introduce the general top-

ics and phenomenology of glass forming liquids. The experimental setup and

practical procedures are described in Chapter 2. Chapter 3 presents the results

divided in two parts.

In the first part, the effect of temperature and pressure on the structural and

secondary relaxation in binary mixture of polar rigid molecules in apolar sol-

vent are discussed. In particular, the importance of combining temperature-

and pressure conditions to investigate the relation between the structural and

secondary relaxation is reported.

In the second part, the results are from experiments where the polar rigid

molecules act as a dielectric probe in apolar polymer host with the purpose of

investigating the molecular dynamics of the polymer host.

Chapter 1

Theoretical Background

1.1 Glass Transition and Relaxation phenom-

ena in Glass systems

Understanding the glass transition and glassy dynamics is one of the greatest

challenges in theoretical physics. This knowledge is not only relevant for the

comprehension of this particular phenomenon but also could be a good test

for the extension of the equilibrium thermodynamics to the non-equilibrium

regime. This condensed matter background is thought as a initiation of the

topics which are the focused areas of research in this thesis. So first, we will

explain how we can get a glass from a liquid and then we will explain the

different dynamic behaviors found in a typical glass system.

1.1.1 Glass transition

If we quench a liquid from a high temperature well above its melting point,

at a fast enough rate, we avoid crystallization and the resulting material is a

glass. On cooling through the melting temperature Tm, the liquid enters in

a metastable equilibrium phase called the supercooled phase. In this region,

the dynamics of the supercooled liquid slows down and the system needs more

and more time to equilibrate. This slowing down can be also revealed by the

increase of the viscosity. At some characteristic temperature Tg , called the

glass transition temperature, the characteristic equilibration timescale of a su-

6 Theoretical Background

Figure 1.1: Temperature dependence of the volume (Enthalpy) of a glass former. Tm is

the melting temperature. Tg is defined as the temperature at which the liquid and the glass

V(T) curves intersect.

percooled liquid exceeds the experimental timescale and the system falls out of

its thermodynamic equilibrium. The obtained glass system has the structure

similar to that of liquid, i.e. lacks of long range structural order, but their

macroscopic behaviour is more similar to a solid [10]. The glass transition is

accompanied by changes of different thermodynamic and dynamic quantities,

such as heat capacity, expansivity, viscosity, and structural relaxation time, i.e

the time necessary to the system to recover equilibrium after an external per-

turbation [10],[11]. For example, in the Fig.( 1.1) a typical curve representing

the volume of a supercooled liquid as function of temperature is reported. At

the glass transition the value of the volume change gradually with temperature,

while the thermal expansion coefficient changes slope. The latter fact occurs

in a narrow temperature interval, usually identified with a single value, i.e. the

glass transition Tg , determined conventionally by the intersection of the liquid

and the glassy curve Fig. (1.1). The glass transition temperature is also de-

fined as a temperature where the viscosity η is approximately 1013 poise [Pa.s],

or can be referred as the temperature range where the structural relaxation

motions occurs on the time scale of the order of 102 s. The glass transition is

1.1 Glass Transition and Relaxation phenomena in Glass systems 7

a complex phenomenon, whose fundamental nature is yet unknown [11],[12].

In relatively recent years high-pressure studies have grown in the literature

about the liquid-glass transition, showing that several features of the temper-

ature dependent response of glass-forming materials can also be recognized

in the manner in which they respond to a pressure variation. The effect on

the molecular motions of an isothermal compression is indeed similar to that

produced by an isobaric cooling and the use of pressure as an additional ther-

modynamic variable can represent an importance tool for testing of theories

that claim to explain the glass formation.

There has been some debate on whether the glass transition is a kinetic or a

thermodynamic phenomenon. According to the kinetic theory, the glass tran-

sition is a transformation of liquid to solid when the characteristic time of the

molecular motion becomes of the same order or greater than the experimen-

tal characteristic time scale, this is the strong basis that glass transition is

purely kinetic phenomenon. On the other hand, in the thermodynamic theory

the glass transition is a second order phase transition which implies that the

transition is continuous over enthalpy, entropy or volume but discontinuous

at specific heat capacity, compressibility and coefficient of thermal expansion

which are second derivatives of Gibbs free energy or volume. But a true second

order phase transition must verified the Ehrenfest relations [13].

dTg

dP= TgV

∆αp(Tg)

∆Cp(Tg)(1.1)

dTg

dP=

∆KT (Tg)

∆αp(Tg)(1.2)

Where αp is the thermal expansion coefficient at constant pressure, Cp specific

heat capacity, KT Isothermal Compressibility Coefficient, V and P are the

volume and pressure of the system, respectively. For any parameter ∆ indicates

the difference between the value measured in the liquid and the value measured

in the glass. Equations (1.1) and (1.2) are obtained imposing the continuity

of the entropy and the volume from the liquid to the glass, respectively. In a

true phase transition the rapport π (The Prigogine-Defay ratio) between Eq:


(1.1) and Eq: (1.2) must be equal to 1.

Π =∆KT ∆Cp)

TgV ∆α2p

(1.3)

For polymeric glasses the ratio π is generally between 2 and 5 [13], therefore,

the glass transition is not a true second phase transition. On the other hand,

the kinetic nature of the glass transition is also supported by the observed

dependence of Tg on the cooling rate. From the Fig.(1.1), we can see that the

slower a liquid is cooled, the longer the time available at each temperature for

changing its structure, and hence the liquid cooled with high cooling rate fall

out the equilibrium before. Consequently, Tg increases with the cooling rate.

The properties of a glass, therefore, depend on its cooling history. In practise,

the dependence of Tg on the cooling rate is weak, and the transformation range

is narrow, so that Tg is an important material characteristic.

1.1.2 Relaxation phenomena in glass systems

In physics, the relaxation phenomenon means the return of the macroscopic

system to the thermodynamic equilibrium after removal of an external pertur-

bation. For example, a system can be perturbed from its equilibrium state by a

variety of mechanical, as well as electrical and magnetic stresses, etc. whereby

the energy from the external source can be dissipated in the system in char-

acteristic time-regions which are determined by the molecular structure of the

glassy system. These are the relaxation processes in which configurational re-

arrangement of atomic or molecular constituents of the glass are induced by the

external stress which must be small compared with for example the electrical

forces acting between the constituent particles (more details can be found in

the section 2.1.1). These relaxation processes may be monitored by different

ways, some of which the volume or enthalpy of the system are determined,

while others the conductivity or optical density at frequencies characteristic

of specific subunits of the glass structure are determined. From the correla-

tion and contrasts of these many different type of experiments, much can be

inferred about the dynamic relationship amongst the different elements of the

glass structure and among the different energy and volume-changing processes


which occur within that structure. Moreover, the characteristic relaxations are

of particular interest in material science because, in contrast with resonance

frequencies, they are highly temperature-dependent, varying over 13-15 orders

of magnitude between the glass temperature and the normal melting temper-

ature. Generally, they are named successively by the order of appearance at

fixed frequency on cooling the system, by the Greek letters α, β, γ, δ and can

be classified according to their time scale as the following order:

(i) At the longest time scale structural changes are allowed by collective, co-

operative motions that involve the rearrangement of groups of molecules or

the segmental motions in polymer chains. In this case the dynamic is con-

trolled by the α relaxation, named also the structural relaxation which is the

main universal dynamical processes related to the glass transition (Fig. (1.2))

depicts the situation near Tg with the α- peak situated at a rather low fre-

quency. With increasing temperature it will rapidly shift to higher frequencies.

(ii) At shorter times, the molecular motions is non-cooperative. This process

is present even in the viscous liquid state or in the glassy state and is generally

more localized and activated and because of the disordered structure of the

glass, a large distribution of the barrier energy can be present.

Generally various dynamic features can be observed in the glass- forming sys-

tems as shown by Fig. (1.2), in some systems, an excess wing shows up in

the high frequency power law flank of the α- peak, and in other systems, a

slow β- peak is shows up. It should be noted that the amplitudes of all these

processes beyond the α- process are rather small compared to the dominating

α- peak. Nevertheless they have attracted considerable interest in recent years

to understand their microscopic origin.

(iii) At some THz a second loss peak shows up that can be identified with

the called boson peak (bp) known from neutron and light scattering [3]. The

boson peak is a universal feature of a glass systems materials and corresponds

the vibrational density of state.

The most significant parameter of a single relaxation processes is the charac-

teristic relaxation time τ which present the most probable time, or the average


Figure 1.2: Schematic view of the different relaxation loss spectra in glass-forming system.

two typical cases are shown, namely the response of: a glass showing an excess wing; and a

glass formers with well resolved β- relaxation peak.

time, that is the area below the peak curve of the different relaxation spectra.

In the following sections, we will introduce and discuss the main feature of

the structural α- relaxation and secondary β- relaxations. Moreover, we will

introduce the properties of the different secondary relaxations present in the

glassy systems, from those related to intramolecular degrees of freedom and

that which are more intermolecular and therefore universal in nature. Indeed,

the properties of the latter are of special interest, since they may provide useful

hints concerning what features have to be reproduced by a full theory of the

glass transition.

The structural relaxation

The structural relaxation in supercooled liquids is the process that slows down

more rapidly on approaching the glass transition, and it the more affected by

the vitrification.


Consequently, the glass transition is referred to the temperature range where

the structural relaxation motions occur on a time scale of the order of 102

s. An other property that is intimately related to the characteristic time for

molecular rearrangement is the viscosity η. One can write η = G∞τ , where

G is the shear modulus and τ is the relaxation time. Since G is only weakly

temperature dependent, η therefore gives a direct measure of a typical relax-

ation timescale τ in supercooled liquids. The point where η reaches the value

1013 Poise (1012 Pa.s) is often used to define the glass transition temperature

Tg . Operationally; given typical values of G, this corresponds to relaxation

timescales τ of the order of hundreds of seconds or more. The most promi-

nent features of the α-relaxation resulting from structural disorder are non-

exponential relaxation patterns and a non-Arrhenius temperature dependence

of the characteristic timescale [13], [14].

a) Non-exponentially and dynamic heterogeneity:

Differently to a simple Debye relaxation phenomena which describe the di-

electric response of a system of noninteracting molecular dipoles rotating in

a non-polar viscous environment by a simple exponential function(see section

2.1.2). The experimentally observed response of glass-forming substances, as

well as that of a variety of solid dielectric materials in a remarkably wide range

of time, have been found to exhibit stretched exponential response. In fact, the

structural relaxation as function of time is generally described by the empirical

stretched exponential [15] or the Kohlrausch-Williams-Watts type decay for a

very broad class of materials [13].

φ(t) = φ(0) exp[−(t

τKWW

)βKWW ] (1.4)

The quantities φ(t) and φ(0) are the dynamic response at time t and 0, respec-

tively, after a perturbation, while τKWW is a characteristic time, the exponent

β is the stretching parameter and is generally in the range between 0 and 1.

For simple Debye relaxation behavior, the response is an exponential function

with β=1. The deviations from simple exponential to stretched exponential of

the time dependence of the relaxation functions when a liquid is supercooled

could be related to two reasons:


• The cooperativity effect which become more pronounced as the density

increases at lower temperature. Thus, the long time dependence of the

relaxation functions should have a larger temperature dependence on ap-

proaching the glass transition and the motion will be more and more co-

operative as the molecular packing increases which explains the stretch-

ing of the structural α- relaxation. A general characteristic of the the

structural α- relaxation is the tendency of the relaxation function to

become exponential as the temperature increases, as expressed by the

increase βKWW , which is some cases reach the limiting value [1]. This ef-

fect is usually interpreted considering that at relative high temperatures

the cooperativity effects on the dynamics become less important, and

structural relaxation reflects a nearly independent molecular motions.

• Dynamic heterogeneity which is now recognized as a fundamental fea-

ture of the slow dynamics of supercooled fluids and glasses in condensed

matter. Heterogeneity regarding the dynamics refers to the picture in

which fast and slow relaxing modes exist. The sizes of these dynamically

distinct domains are assumed to be of the order of several nm [16],[17].

Taking a snapshot of a glass-forming system, in the same environment

some molecules are characterized by fast motions, while others are much

slower, acting as cages for the faster. This picture is dynamic, in the sense

that, few moments later (usually on the time scale of the structural re-

laxation τα) due to the mutual interactions, the fast molecules could be

become slow and vice versa. Fast and slow molecules are usually clustered

in domains. One could be tempted to explain the stretched exponential-

ity character of the relaxation in glass-forming systems, simply assuming

that, in each spatial domain, relaxation occurs exponentially but with a

different relaxation time in each domain: averaging over the ensembles,

this should lead to a distribution of relaxation times. Following this idea,

Ediger and al [18] proposed a spatially heterogenous model of transla-

tional and rotational diffusion in supercooled liquids, i.e. the different

temperature behaviour of the two quantities reported in the region Tg

<T<1.2Tg . In fact translational diffusion was found less temperature

dependent than the rotational one. According to Ediger the decoupling


could be due to the different importance that fast and slow motions have

on translational and rotational diffusion respectively: therefore, if spa-

tial heterogeneity reflects into a given distribution of relaxation times

for the α- relaxation, wider is the distribution stronger is the decoupling

between translational and rotational diffusion. Some experimental evi-

dences for organic glass-formers (for instance ααβ-tris-naphthylbenzene

(TNB) [19], [20], where the width of the relaxation peak (and so the dis-

tribution of relaxation times) was found temperature independent over

a broad temperature range, while in the same range a decoupling be-

tween translational and rotational diffusion was found. An alternative

explanation was proposed by the coupling model(CM)(see section 1.2.3)

according to which decoupling of translational and rotational diffusion in

glass-forming liquids is a special cases of a more general phenomenon, i.e.

that different dynamic observables weigh the many-body relaxation dif-

ferently and have different coupling parameters n (i.e., different degrees

of intermolecular cooperativity) that enter into the stretch exponents

βKWW of their Kohlrausch correlation functions but also in the temper-

ature dependent of the considered observable [21],[22].

b) Arrhenius and Non-Arrhenius temperature dependence

Another characteristic feature of the α- relaxation of glass-forming systems

is the deviation of its temperature dependence from that expected for ther-

mally activated processes, namely the Arrhenius law in which it is assumed

the existence of unique energy barrier E which controls the transition from two

different states.

τ = τ∞ exp(∆E

KBT) (1.5)

The extent of this deviation is very different for different glass-forming ma-

terials. Oldekop and later Laughlin and Uhlmann [23] were the first to use

Tg as a corresponding state parameter for liquid viscosity η to compare the

flow behavior of different liquids. Greet and Magill compared η and η normal-

ized by its value at the melting temperature, Tm, of glass-formers in a plot

against Tm/T, and find variations [23]. By comparing log η in a plot against

Tg/T of three different classes of materials, (i) SiO2 and GeO2, (ii) B2O3 and


the alkali silicate glasses and (iii) the organic small molecule liquids including

orthoterphenyl (OTP), salol, α-phenyl-o-cresol, and 1,3- bis(1-naphthyl)-5-(2-

naphthyl)benzene (TNB) as well as a Ca(NO3)2KNO3 melt, Laughlin and

Uhlmann found liquids within each class have nearly the same Tg - scaled

temperature dependence. Furthermore, the Tg /T dependencies of log η of

the three classes have very different curvatures. Angell [24] recognized the

importance of the Laughlin-Uhlmann plot as a means to classify the transport

properties of glass-formers. He included more non-polymeric materials leading

to a categorization [25], [26] into strong glasses following equation Eq: (1.5)

and fragile ones which show a bent curve in the normalised activation plot log

τ versus Tg /T, which means that the apparent activation energy is increasing

on decreasing T (Fig. (1.3)). A similar pattern has been found for polymers

by plotting log τα against Tg /T , where τα is the local segmental relaxation

time[27], [28],[23]. Moreover, the viscosity of polymers depends sensitively on

molecular weight and polydispersity, and the plot of log τ against Tg /T should

not be used. Nowadays, the words strong and fragile are often used to con-

vey the curvature or location of either log η or log τ of the glass-former in the

Oldekop-Laughlin-Uhlmann-Angell (OLUA) plot. In experiments with fragile

glasses, temperature dependence of the α- relaxation time can be described

empirically with the Vogel-Tamman-Fulcher (VTF) law [29].

τ(T ) = τα0 exp(

DT0

T − T0

) (1.6)

where T0 is the Vogel Fulcher temperature which indicates the divergence of

the relaxation time at infinite viscosity, corresponding to the complete blocking

of the structural relaxation, τα0 is the high temperature limit of relaxation

time, and D is the strength parameter whose value is related to the degree

of deviation of the τα(T ) curve from the Arrhenius equation [30]. Typically,

for a fragile system D is about 3 and for a strong one D is about 100. An

other parameter m (called the fragility or steepness index) is associated to this

classification and defined as the slope of the relaxation time curve vs. Tg /T at

the glass transition temperature Tg in the Oldekop-Laughlin-Uhlmann-Angell

(OLUA) plot.

m =d log(τ)

d(Tg

T)|T=Tg (1.7)


Figure 1.3: Angell plot of the viscosity against Tg /T in log-scale. In this representation,

strong glass formers such as SiO2 with their Arrhenius dependence of timescales on tem-

perature gives straight lines, while the super-Arrhenius (such as VTF) divergence in fragile

glasses (for example OTP) leads to curved plots.([31],[32])

The classical value of m are between 50 and 150 for fragile systems, (for in-

stance m=140 for cis-Polyisoprene), and m are between 16 and 20 for strong

systems, for instance, m=20 for GeO2 and SiO2. Of course, there are a continu-

ity of systems with intermediate fragility. All experimental evidences indicate

the complex dependence of the structural relaxation time on the temperature.

Actually, there are several experimental and numerical simulation results that

reported that the slowing down of the dynamics on approaching Tg and the

increasing of the apparent activation energy could be related to the onset of

cooperative motions, involving molecules on an increasing correlation length

[18],[14] [33]. The size of the cooperative rearranging region has been found

to be related to the fragility and to the complementary as well as to βKWW

parameter [34], [16],[35],[17], leading to the increase of the activation energy

and so of the fragility.

The secondary relaxation

It is well known that glass-forming liquids, in addition to the α- relaxation,

show secondary β- relaxations actives also in the glassy state, which in di-


electric loss spectra appears as a broad peak, more often symmetrical slowly

moving at lower frequency on decreasing temperature([15],[1]). Microscopi-

cally, two different mechanism can contribute to the β process : firstly, motions

of molecular subunits that can relax independently from the whole molecules

(intramolecular origin). Besides, in some other systems, the observed sec-

ondary relaxation reflects the motion of the whole molecule (intermolecular

secondary relaxation). The first contribution can be well detected in polymers

with side-chain dipolar groups and in systems with complex molecular struc-

ture, where the motion of a side group or a small molecular unit such as the

methyl group can originate relaxation processes independent from the struc-

ture one [4]. Moreover, there is also other polymers in which the secondary

relaxation involves also some small scale motion of part of the repeat unit

on the main chain as poly(n-methyl mathacrylates)(PMMA) and poly(n-ethyl

methacrylates)(PEMA) in which the secondary relaxation involve not only a

180° flip of the O-(C=O)- plane of the side group but also the rocking motion

of the part of the repeat unit on the main chain [36],[37]. The second contribu-

tion was singled out by G. Johari and co-workers [38], which found a secondary

relaxation process in simple glass forming system, having no internal degree

of freedom. Two approaches have been proposed to explain the origin of the

intermolecular β- relaxation (called Johari Goldstain (JG)relaxation). Johari

considered the concept of the islands of mobility which are responsible for

the β- process. The islands of mobility are isolated regions of loosely packed

molecules caged in the glassy matrix. Consequently, only molecules within

these regions possess enough space to perform, in the glassy state, relatively

fast and independent motions which appear as the β- relaxation. These defects

are due to freezing-in of local density fluctuations during the vitrification pro-

cess. According to the Johari’s model, the β- relaxation is a non-homogenous

process, which originates only from molecules within the islands of mobil-

ity [39],[40]. A completely different interpretation was later put forward by

Williams and Watts [41]. They postulated that the β- process could be at-

tributed to a faster, small angle (thermally-activated) reorientational motion

of all molecules. Thus, it has a homogenous nature, in contrast to Johari’s

concept. On the other hand, Wagner and Richert investigated the solvation


dynamics in d-sorbitol and concluded, after comparing these results with data

from conventional dielectric measurements, that all molecules take part in the

secondary relaxation process [9]. This view was strongly supported by NMR

measurements: Vogel and Rossler [7],[8] investigated the β-process in toluene

and polybutadiene by means of 1D- and 2D- NMR and were able to show that

below Tg all molecules in the sample participate in highly restricted small angle

reorientation, thus openly contradicting the idea of islands of mobility or any

other defect model. Similar results were obtained by Bohmer et al. analyzing

the spin lattice relaxation times in toluene [6]. However, recently, Johari pro-

posed arguments to show that the picture of a spatially homogeneous mobility

is inconsistent with the properties of the secondary processes usually observed

in the dielectric spectra [40]. Although Johari’s concept was proposed over 30

years ago it still remains a subject of very an ongoing debate in the literature.

The debate results from the fact that there is no unambiguous experimental

evidence for the existence of the islands of mobility. Consequently, the mi-

croscopic mechanism at the origin of the β- relaxation still remains an open

question. There is currently strong effort by many researchers to explain its

real nature. It is worth noting that in various glass-formers no secondary re-

laxation processes is observed in the loss spectra, but an excess contribution

to the high- frequency tail of the α- peak, called excess wing. Some researcher

proposed that the excess wing and β-relaxations are different phenomena [42],

and classified the glass formers according two classes: type A with an excess

wing and type B with a β- process [43]. However, based on experimental obser-

vations it seems natural to explain Johari-Goldstein β- processes and excess

wing phenomena on the same footing. Indeed it was recently confirmed by

a broadband dielectric spectroscopy study that in a binary mixture of polar

rigid molecules in apolar solvent, the so-called excess wing, characteristic of

the spectra of neat simple and quite rigid molecular glass formers, evolves in a

well resolved β-relaxation, when a mixture with a more viscous apolar solvent

is carried out [44]. Besides, the Polyalcohols with small chain length present

an excess wing but in the sample with longer chain length the excess wing

reveals as secondary relaxation peak ([45],[46],[47]). Schneider and co-workers

observed the EW of glycerol and propylene carbonate transforming into a sec-


ondary peak after aging at temperatures below Tg [42] due to the different

dependence on aging of the structural and secondary peaks. Similarly, dielec-

tric spectra of the trimer of propylene glycol showed an EW that, becomes

a distinct relaxation peak at high pressure [48]. Besides, the application of

very high pressure transform the EW of threithol and glycerol into a partially

merged secondary peak [49].

At temperature below the glass transition temperature when the structural

relaxation moved out the frequency interval of the measurement, the tem-

perature dependence of the secondary β-relaxations is usually the Arrhenius

equation

τβ = τβ∞ exp(

∆Eβ

RT) (1.8)

where R is the gas constant (R = 8.314 J mol−1K−1), τβ0 the relaxation time

at high temperature and ∆Eβ is the activation energy. Typical value for ∆Eβ

are 20-50 KJ/mol. Other broadband studies have shown a more complicated

behaviour of the temperature dependence of the secondary relaxation time

around and above Tg .

1.2 Theoretical models

1.2.1 The Free Volume approach

The concept of the free volume was proposed in 1951 by Doolittle [50]. The

starting point of this theory was that the internal mobility of the molecu-

lar systems expressed as viscosity is related the free volume υf , available for

translational molecular motion. The subsequent derivation of Doolitle’s fluid-

ity equation within the free volume model is based on the following:

Consider a glass, Let the volume per molecule be v, and let v o denote the vol-

ume per molecule excluded to all other molecules. Upon heating, the excess

volume, v - v o, increases. At very low temperatures, packing constraints im-

pose a severe energy penalty upon any nonuniform redistribution of the excess

volume.

Consequently,thermal expansion results mainly in increased anharmonicity in

1.2 Theoretical models 19

the vibrational motion of molecules about essentially fixed value, and the ex-

cess volume is uniformly distributed. Furthermore, the entropy increases since

the volume change is not accompanied by randomization of molecular motion.

Eventually, v -v o reaches a critical value, beyond which any additional volume

can be redistributed at random without an energy penalty. That part of the

excess volume that can be distributed randomly without energy cost is referred

to as the free volume vf ,

v − v0 = vf + ∆vc (1.9)

where ∆vc is that part of the thermal expansion that cannot be randomly

distributed. The randomization of molecular motion arising from the redis-

tribution of free volume contributes significantly to the entropy. The ideal

glass transition occurs when v f=0 and therefore v − v0=∆vc. For laboratory

transition, vf > 0.

Molecules in a dense liquid are confined to a cage formed by their imme-

diate neighbors. spontaneous density fluctuations occasionally result in the

formation of holes within a cage that are large enough to allow an appreciable

displacement of the trapped central molecules. This can give rise to diffusive

motion only if another molecule occupies the hole before the first one can re-

turn to its original position. We give here the main result of Cohein-Turnbull

theory, that is the relationship between translational diffusion coefficient and

free volume. Denoting the free volume associated with a given molecules as

v ′, the translational diffusion coefficient is written as a sum over all possible

contributions from molecules having different available free volumes:

D =

∫ ∞

υ∗D(υ)P (υ′)dυ′ = gu

∫ ∞

υ∗a(υ′)p(υ′)dυ′ (1.10)

where D(υ′)is the contribution to the translational diffusion coefficient due to

molecules having free volume υ′ and υ′+ dυ′; a(υ′)is the length scale indicative

of the cage diameter corresponding to a molecule having free volume υ′, u is the

thermal velocity [u≈ (kBTm

)12 , with m the molecular mass]; g is a geometrical

factor; and υ∗ is the minimum free volume capable of accommodating another

molecule after the original displacement in the cage. The function P(υ) is


calculated as

P (υ′) =γ

υf

exp(−γυ′

υf

) (1.11)

Since D(υ′)varies more slowly than P(υ′), and in general υ∗ À υf , we can write

D =

∫ ∞

υ∗D(υ)P (υ′)dυ′ ≈ D(υ∗)

∫ ∞

υ∗P (υ)P (υ′)dυ′ = ga∗u exp(−γυ′

υf

)

(1.12)

Where, consistently with the interpretation given to υ, we take a∗ to be of the

order of the molecular diameter.The free volume expression predicts that D,

and hence the possibility for translational motion, will vanish when the liquid

has no more free volume available for random redistribution (υf=0).

Eq: (1.12) is related to the empirical Doolittle equation, proposed originally

to describe fluidity, η−1,

η−1 = A exp(−bυ0

υf

) (1.13)

where A and b are constants, υ0 and υf are the occupied and free volume per

molecule, respectively; and the implied inverse relationship between transla-

tional diffusion and viscosity follows from Stokes-Einstein equation, D=KBT6πaη

,

in which a is the molecular diameter.

1.2.2 Adam-Gibbs Theory approach

In 1965, Adam and Gibbs suggested a theory of the glass transition phenom-

ena while supposing that the glass system can be divided into independent,

distinguishable regions containing molecules which move cooperatively in dy-

namically cooperative regions (cooperative rearranging regions CRR) and that

the structural α- relaxation ordinates from the rearrangement of the molecules

in this later. When decreasing the temperature toward the glass transition tem-

perature, the cooperativity increases, leading to an increase of the size of the

CRR’s , and consequently to reduction of the number of configurations avail-

able (reduction of the configurational entropy Sc). The Adam-Gibbs theory

relates the growth in molecular relaxation time to the configurational entropy


Sc which decrease with the growth of the cooperative rearranging regions.

At constant T, we can consider that the system is divided into N regions each

composed by z molecules, moreover, n of these regions are supposed in a ther-

modynamic state that allows a molecular cooperative rearrangement and can

transform to an other configuration under the effect of the energy fluctua-

tion( Enthalpy), independently on its environment. The probability transition

W(T) of a cooperative region region with z molecules can be expressed :

W (T ) = Aexp(−z∆µ

kT) (1.14)

Where A is a normalizing factor, and ∆µ is the difference between the chem-

ical potential of molecules in region where rearrangement is permitted and

where the rearrangement is not permitted. The majority of the configura-

tional transitions occurs when the size of a single CRR, exceed a critical size

z∗, consequently, the transition probability can be written;

W (T ) = Aexp(−z∗∆µ

kT) (1.15)

A is an average proportionality, independent on the temperature. z∗ depend

on the thermodynamic parameters, and in particular on Sc. In fact, con-

sidering the CRR’s as weakly interacting, the configurational entropy of the

macroscopic system can be defined as the summation of the configuration of

each CRR: Sc= N sc. Moreover, if we consider the system composed by a mole

of molecules, and each CRR composed by z molecules, the number N of CRR

in the system is Na/z, where Na is the Avogadro’s number, consequently z =

Na sc/Sc. For the smallest size z∗ that permit a configurational transition is

associated a critical value of the configurational entropy s∗c , which can not be

less than k ln2, since two distinct configurations are the minimum required

for the rearrangement to occur. Therefore z∗ = Na s∗c/Sc, inserting into Eq:

(1.16), yields

W (T ) = Aexp(−−∆µs∗ckBTSc

) = A exp(−CAG

TSc

) (1.16)


Since the characteristic molecular time τ is inversely proportional to W , the

Adams-Gibbs leads to the important result

τ = τ0 exp(−CAG

TSc

) (1.17)

This equation relates the growth in molecular relaxation to the configurational

entropy. This configurational entropy is not an experimental available quantity

and for this reason a direct test of the model cannot be performed. However,

in the hypothesis that the vibrational contribution to the entropy of the su-

percooled liquid, Smelt and the of the crystal Scrist or S(glass)are the same, Sc

can be replaced by the excess of entropy of the supercooled liquid with respect

to the crystal (or the glass) Sexc=Smelt-Scryst(Sglass=Smelt-Scryst) [51].

1.2.3 Coupling model

Glass-forming substances are made of units having nontrivial mutual interac-

tions. Therefore, structural relaxation of glass-formers is a many-body relax-

ation problem, which unfortunately is still an unsolved problem in statistical

mechanics and hence glass transition is unsolved problem too. Conventional

theories and models of glass transition bypass solution of this problem, and of-

ten take into account the effect of the mutual interaction according to a mean

field approach, i.e., by averaging the effect of the surrounding molecules on the

relaxing one. There are plenty of experimental evidences for the many-body

relaxation dynamics. The Coupling Model is not a full solution of the many-

body relaxation problem either. Nevertheless, its predictions can explain the

properties of structural relaxation originating from many-body relaxation and

their relations to its precursor, namely the Johari-Goldstein secondary relax-

ation [52].

In the framework of the coupling model, exists a temperature independent

time tc separating the non- cooperative dynamics characterized by a single

exponential to a cooperative one characterized by stretched exponential [53].

A schematic description of the dynamic evolution can be described as follow-

ing: at very short time t<tc , molecules units relax independently with the


correlation function

φ(t) = exp(− t

τ0

) (1.18)

characterized by the independent (primitive or non-cooperative) relaxation

time, τ0, which present the inverse of the rate W0=τ−10 at which the molecule

star to relax, but the many-molecule dynamics, starting at tc , prevent all

attempts of molecules to be simultaneously successful, resulting in faster and

slower relaxing molecules or heterogeneous dynamics. However, when aver-

aged, the effect is equivalent to the slowing down of τ−10 by another multiplica-

tive factor, which is time dependent. The time-dependent rate W(t) has the

product form, τ−10 , where f(t) is a decreasing function with values less than

unity. In particular, the slowing down factor f(t)−1 was found depending on

time according to a sublinear power law, and hence W(t)∝ t−nτ−10 , where n

is the coupling parameter of the CM and 0<n<1. Higher is the extent of

the dynamic heterogeneity, stronger is the slowing effect of the many-molecule

dynamics, larger is the coupling parameter n and vice versa. Therefore, the

corresponding correlation function of the model is the Kohlrausch stretched

exponential function (Eq: (1.4)), that holds only for t>>tc .

φ(t) = exp[−(t

τα

)1−n] (1.19)

and the cross- over at tc leads to a relation between τα and τ0 given by

τα = [t−nc τ0]

11−n (1.20)

There are experimental evidence [54], [55],[56] and numerical simulations

[57],[58],[59] that the cross-over from the primitive relaxation to Kohlrausch

relaxation is at tc ≈ 2ps for molecular systems. The measurements was per-

formed at temperatures high above Tg where τ0 becomes short and of the

order of ten picoseconds or less. The correspondence between the independent

(primitive) relaxation time of the coupling model (CM),τ0, and the the (JG)

β- relaxation time, τβ , was predicted and demonstrated by K.L Ngai [60].


Such a connection is expected from the similar characteristics of the two re-

laxation processes, including their local nature, involvement of essentially the

entire molecule, common properties, and the fact that both serve as the pre-

cursor to the α- relaxation τ0(T, P ) ≈ τJG(T, P ) and this is should be valid at

for different temperature and pressure conditions. Remarkably, this relation,

holds for many small molecular and polymeric glass-formers, where τJG is from

experiment and τ0 obtained by calculation [61],[5],[62],[63].

The CM was applied to also other areas including blend dynamics [64],[65],

polymer viscoelasticity and to ionic conductors [53].

It was mentioned that different mechanism was proposed to identify the Johari-

Goldstein secondary relaxation, the first one by Johari which support the idea

of island of mobility and the second is that essentially all molecules contribute

to the JG relaxation and the motions are dynamically and spatially hetero-

geneous. The coupling model description of the JG relaxation may help to

resolve the different points of view on its nature between Johari and others.

It is important to recall that the timescale where JG relaxation takes place usu-

ally exceeds tc, especially in molecular glass-formers, where the many-molecule

dynamics starts at tc=2 ps. So, when tc<<τ0<<τα, the molecules are essen-

tially all caged at short time and then a gradual development of cooperativity

occurs when increasing numbers of molecules are ready to reorient indepen-

dently. In the region tc<<τ0<<τα all the molecules are attempting to make

independent relaxation, but not all of them are successful because of the inter-

action with or the constraints by the surrounding molecules. The most prob-

able relaxation time τJG, related to the few molecules that have the chance

to independently reorient, should be close to τ0, the primitive relaxation time

of the independent reorientation if molecular interactions would be negligible.

Evidently, the primitive and the JG relaxation processes are not identical, but

they are closely related. Therefore the check of a correspondence between JG

relaxation time and the primitive relaxation time τ0 is of paramount impor-

tance to test the prediction of CM in glass-forming systems.

Chapter 2

Experimental Techniques

The relaxational dynamics of soft matter, e.g. polymeric materials and glass-

forming liquids, is characterized by a large frequency range. The dielectric (or

impedance) spectroscopy, the measurement of dielectric properties, comprises

this frequency range which extends over nearly 18 orders in magnitude: from

the µHz to the THz range close to the infrared region and even it can be per-

formed at different temperatures. It is based on the interaction of an external

field with the electric dipole moment of the sample. The fluctuations of local

electrical fields are measured and connected to the dynamics on a molecular

scale.

Basically there are two different possibilities to achieve a dielectric experi-

ment. Experimental working with sinusoidal alternating fields are called mea-

surements in the frequency domain and the complex permittivity ε∗(ω) of the

susceptibility are measured. Moreover the experiments can also be carried out

in the so called time domain where a step like change of E is supplied to the

sample. In that case the dielectric behaviour is discussed in terms of the time

dependent dielectric function ε(t) obtained from the time evolution of the po-

larization. Generally, there are no reason to prefer time domain or frequency

domain measurements, and the choice of technique is based on consideration

of the analysis procedures and on the characteristic of the experimental appa-

ratus, such as accuracy, and the time require for the measure. For example, in

frequency domain technique the signal is filtered by very narrow band filters,

that allows to eliminate the noise out the band, obtaining high accuracy mea-

28 Experimental Techniques

surements. This is probably the most important reason that makes preferable

to use frequency domain techniques instead of time domain techniques. In this

part, we will describe Dielectric Spectroscopy technique and the experimental

apparatus used in our laboratory. Then, the second part will be devoted to

Differential scanning calorimetry DSC which is a powerful technique to study

the phase behaviour of binary glass forming systems. In this technique, the

heat flow is measured as a function of time or temperature to observe physical

transitions like demixing and crystallization.

2.1 Dielectric Relaxation Spectroscopy

2.1.1 Linear response theory

When an electric field is applied to the dielectric sample, the atomic and molec-

ular charges in the dielectric are displaced from their equilibrium positions and

the material is said to be polarized. Different mechanisms can induce polar-

ization in dielectric upon the application of field:

• Electric/atomic polarization: the displacement of binding electrons or

atoms from their equilibrium position on a molecular level (induced

dipoles) until the field force and the oppositely acting elastic force are

equal. This process is very fast and can be regarded as frequency inde-

pendent.

• Orientation polarization: the orientation of permanent dipoles along the

external electric field. Usually, these latters are randomly oriented, but

when an external field is applied an average orientation parallel to the

field direction is preferred.

• Interfacial or Maxwell-Wagner polarization: a macroscopic build-up of

charges at internal interfaces. This occurs, when two materials with dis-

similar conductivities are subjected to an electric field and a double layer

is built up at the interface. This process is relatively slow (relaxation time

>1 s) and can be investigated by low frequency dielectric spectroscopy.

2.1 Dielectric Relaxation Spectroscopy 29

In the case of static electric field, the polarization caused by the electric field

is in equilibrium with the field. However, in the dynamics case when the

electric field strength varies appreciably within a period of the same order as

the characteristic time of the microscopic motions of the molecules, this latter

will not be sufficiently rapid to build up the equilibrium polarization. On the

other hand, if the time-dependent electric field strength E(t) occur slowly as

compared with the motions, there is enough time to build up a time-dependent

polarization P(t) that is in equilibrium with the electric field at any moment.

In this quasi-static case, the polarization P(t) of a linear and isotropic system

is proportional to the electric field E(t)is :

P (t) = χE(t) (2.1)

where χ is the susceptibility. The dielectric displacement D for time dependent

electric field

D(t) = E(t) + 4πP (t) (2.2)

D(t) = εE(t) (2.3)

ε = 4πχ + 1 (2.4)

The polarization can be separated into two parts P(t)=Pin(t)+ Por(t), the sec-

ond one due to the partial orientation of permanent dipoles along the electric

field Por =∆χE, where ∆χ is simply defined as ∆χ=χs -χ∞ and the first one

due to induced dipole moments Pin=χ∞E(t)=E(t) (ε∞−1)4π

, here ε∞ is the di-

electric constant due to the induced polarization. The polarization due to the

orientation polarization is currently studied by dielectric spectroscopy, cover-

ing the range from ps to hundred of Ks [1]. The time-dependent decrease of

orientation of dipoles upon the removal of the electric field is called dielec-

tric relaxation. Commonly the change of polarization in a system due to the

application of a pulse field is:

Por(t) = ∆χ

∫ t

−∞ϕ(t− t′)E(t′)dt′ (2.5)


Here, the integral over the electric field expresses the fact that the polarization

at a certain time t is in general determined by the development of the field

E(t′) at all times of the past t′ < t. The quantity ϕ(t) is called a pulse response

function, i. e. for Dirac δ-shaped pulse in the electric field E(t) = E0δ(t) one

obtains as a response Por =∆χE0ϕ(t), for t>0. The pulse response function

ϕ(t) is normalized as:∫ ∞

0

ϕ(t)dt = 1 (2.6)

so that for a time independent field E(t)= E0 the static limit Por =∆χE0 is

recovered from Eq: (2.5). By means of Kubo’s identity [66]

ϕ(t) = −Φ(t) t ≥ 1 (2.7)

A step response function Φ(t) can be defined. It describes the polarization re-

sponse after switching off a constant electric field at time t=0: Por =∆χE0Φ(t).

Note that in equilibrium the orientational polarization will decay to zero at

long times, so that Φ(t→∞)=0. Inserting Eq: (2.7) into Eq: (2.6) one thereby

obtains a normalization condition for the step response function:Φ(0) =1. Typ-

ically, the step response function is measured in dielectric time domain exper-

iments. The dielectric relaxation process can be performed in the frequency

domain and in the time domain, equivalently according to the linear response

theory [67]. Applying the Laplace transform to the Eq: (2.5) we obtain a pro-

portionality relation between the Laplace transform of the electric field and

that of the polarization

P (ω) = χ(ω)E(ω) (2.8)

where χ∗(ω)=χ′(ω) − iχ′′(ω) is the Laplace transform of the susceptibility.

Analogously to the time domain, it is possible to define the complex dielectric

permittivity function in the frequency domain according to:

ε∗ = 4πχ(ω) + 1 (2.9)

It should be noted that the above linear relationship between the electric field

and the polarization holds only for low electric-field strengths. Generally, the

complex dielectric permittivity is a material property depending on frequency,


temperature, pressure and structure. As intuitive meaning, we can think to

the a real part ε′ which is related to the energy stored by the interaction of

the electric field with the dipoles in a period of the oscillation and the imagi-

nary part ε′′ which is related to the energy dissipated in the material per cycle

by the out-of-phase motion of dipoles with the field. Another useful quantity

is the dielectric loss factor (or dissipation factor) tan(δ)= ε′′(ω)ε′(ω)

. It contains

no more information than ε′ or ε′′, but it is independent of the capacitor ge-

ometry. This is particularly important if the capacitor geometry is not well

defined. Furthermore, tan(δ) is used as a measure of the sensitivity of a dielec-

tric equipment. Recently, dielectric equipment with tan δ ≈10−5 has become

available [68].

2.1.2 Molecular dynamics and correlation function

In contrast to the polarization response due to an external perturbation, ori-

entational polarization ∆P = P −〈P 〉 can also be observed in thermodynamic

equilibrium as a quantity fluctuating around its equilibrium average value 〈P 〉.Such fluctuations can be characterized by means of an autocorrelation function

of the polarization φ(t):

φ(t) =〈∆P (t)∆P (0)〉〈∆P (0)2〉 (2.10)

ε∗ is related to the correlation function φ(t) of the polarization fluctuation by

the phenomenological theory of dielectric relaxation:

ε∗(ω)− ε(∞)

εs − ε∞=

∫ ∞

0

[−dφ(t)

dt] exp(−iωt)dt (2.11)

Where: εs and ε∞ are the permittivities at very high (relaxed permittivity )

and quasistatic (Unrelaxed permittivity ) frequency, respectively. The macro-

scopic Polarization P is related to the dipole density of N permanent micro-

scopic molecular dipoles µi in a volume V,

−→P =

1

V

N∑i=1

µi =N

V〈µ〉 (2.12)


ε′(ω) and ε′′(ω) can be calculated from the correlation function following Eqs:

(2.13) and (2.14)

ε′(ω)− ε∞εs − ε∞

= 1− ω

∫ ∞

0

φ(t) sin(ωt)dt (2.13)

ε′′(ω)

εs − ε∞= ω

∫ ∞

0

φ(t) cos(ωt)dt (2.14)

For a Debye relaxation process with a characterization time τ , φ(t) follows a

single exponential function:

φ(t) = exp(− t

τ) (2.15)

According to the equation this leads to:

ε∗(ω) = ε∞ +εs − ε∞1 + iτω

(2.16)

and:

ε′(ω) = εs +εs − ε∞

1 + (τω)2(2.17)

ε′′(ω) =(εs − ε∞)ωτ

1 + (τω)2(2.18)

The Debye process has a relaxation time distribution which is symmetrical

around fmax=ωmax/2π=1/(2πτ) and with a full-width at half-maximum of

∼1.14 decades in frequency for the dielectric loss [69]. However, the relax-

ation time distribution in polymers is generally much broader than a single

Debye process and generally asymmetric. The reason is obvious: the Debye

model does not include many-molecules interactions; therefore it applies only

to gases and dilute solutions. In condensed matter multiple collisions, cage

effect of the neighbors and dipole-dipole interactions lead to a distribution of

relaxation times around a most probable value. For the presentation and com-

parison of experimental data it is appropriate to modify Debye’s equations and

in such a way it’s possible to determine the different relaxation parameters.


Figure 2.1: Frequency dependence of the real ε′ and the secondary part ε′′ of the complex

dielectric constant ε(ω)=ε′(ω)-iε′′(ω), typical of a relaxation process. The dispersion in ε

correspond to a maximum of absorption (maximum in ε′′)

Various models can be used : for instance, the Havriliak-Negami (HN) empiri-

cal relaxation function which is the most commonly used in the frequency [70].

ε∗(ω) = ε′(ω)− ε′′(ω) = ε∞ − iσ

ε0ω+

∆ε

[1 + (iτHNω)1−α]β(2.19)

in which ∆ε=εs-ε∞ is the dielectric strength see Fig. 2.1, σ0, ε∞, τHN , ∆ε,

1 − α and β are fitting parameters. σ0 is the dc conductivity constant with

units of S/cm.

The exponents 1−α and β [0<1-α, (1-α).β≤1] are shape parameters describ-

ing the width and the asymmetry of the loss peak, respectively [15] and related

to the limiting low and high-frequency slopes of a log (ε′′)∼log(ω) plot:

∂ log ε′′(ω)

∂ log ω= 1− α(ω ¿ 1

τHN

) (2.20)

∂ log ε′′(ω)

∂ log ω= −(1− α)β(ω À 1

τHN

) (2.21)

τHN is related to the peak location τmax by Eq. 2.13 [71]

τmax = τHN [sin π(1−α)β

2(β+1)

sin π(1−α)2(β+1)

]1/m (2.22)


The dielectric relaxation strength, ∆ε , can be determined from either the

real part ε′ (by εs-ε∞) or from the area under the ε′′∼ln(ω) plot:

∆ε =2

π

∫ +∞

−∞d(ln ω)ε′′(ω) (2.23)

It is related to the specific material density and dipole moment by the Kirkwood-

Onsager-Frohlich equation ( [15],[68])

∆ε = (3εs

2εs + ε∞)(

ε∞ + 2

3)2 4πN

3kTgµ2 (2.24)

where µ is the dipole moment, k the Boltzmann constant, and N the number

density of movable dipoles. The correlation factor, g, is a measure of the ex-

tent to which restricted internal rotation and interaction of neighboring dipoles

influence dipole alignment. However, because g is very difficult, if not impos-

sible, to measure, Eq: (2.23) cannot be used to calculate theoretical ∆ε and,

in stead, it is usually used to calculate the correlation factor g from measured

∆ε. If changes in g and N with temperature are negligible, Eq: (2.23) predicts

that ∆ε will decrease with increasing temperature. This is true for a structural

relaxation process, but not is necessarily correct for secondary relaxations.

In summary, a complete description of a relaxation process includes the calcu-

lation of four parameters: relaxation strength, relaxation time and two shape

parameters is possible using the so-called HN analysis which is also particularly

useful even for separating two superimposing relaxation processes as well as for

separating a conductivity term from the low-frequency end of the spectrum.

2.1.3 The Experimental Setup

Low frequency dielectric measurement

In dielectric relaxation spectroscopy studies, the experimental complex dielec-

tric permittivity ε(ω) is measured covering the frequency range from 1 mHz-10

MHz. This technique were performed employing a frequency impedance mea-

surement technique where the complex impedance Z(ω)=Z ′(ω)+iZ ′′(ω) of a

capacitor, with empty capacity C0, filled by the sample material is measured.

The principle of the impedance measurement is as follow:

A sinusoidal voltage U1(t)=U∗exp(iωt) at fixed frequency ω is applied to the


Figure 2.2: The experimental Setup of the dielectric spectrometer

cell and the current IS(t)=I∗exp(iωt+ϕ) is measured across it. Generally the

measured current is shifted with respect to the applied voltage by a phase angle

ϕ, which depend on the sample properties and the frequency. After measuring

the applied voltage and the generated current the value of Z is obtained by the

ratio U∗/I∗ and consequently the complex permittivity ε=ε′-iε′′ is obtained by

impedance measurement through the relation ε= 1iωC0Z(ω)

.

The experimental setup is shown in Fig. (2.2), the device under test is mounted

in a special sample holder Fig. (2.3) for dielectric measurements which is di-

rectly connected to the input terminals of the impedance analyzer.

The sample holder is part of a cryostat which belongs to the QUATRO

cryosystem. The DUT is heated and cooled with controlled dry nitrogen gas.

The ac bridge as well as the cryosystem are fully remote controlled by a per-

sonal computer. This Alpha high resolution dielectric analyzer can be used

to measure the impedance or complex dielectric function in a frequency range

from 3 µHz through 10 MHz. The temperature range is between -170°C and

+400°C, the resolution limit lies at tanδ<10−5.

The measuring circuit is presented in Fig. (2.4) in simplified form, the voltage

signal U1 generated by the digital sine-wave generator is applied to the cell,

then the generated current across is converted to the voltage U2 by current to


Figure 2.3: Low frequency sample holder, the sample cell is constituted generally by two

round plane plates of diameter 3 cm which are spaced by silica fibers ranging from 30 to 100

µm. The temperature is measured using Platinum resistor sensor (PT100) placed in contact

with one of the capacitors plates

voltage converter. This later is amplified, filtered and converted in two digital

data streams, which are analyzed with discrete Fourier transfer technique in

order to gain information about the amplitude and the phase of the voltage

U2. Once the two voltages are knows the value of the sample impedance ZS is

calculated:

ZS =U1

IS

= −U1

U2

Zx (2.25)

Zx=(R−1x + iωCx)

−1 is the feedback impedance of the inverting amplifier used

in the current to voltage converter. In this type of measurement the main

errors are the detection of the phase of U1 And U2 and of the contribution of

Zx. To reduce this errors, the alpha analyzer has the capability of replacing

the sample by a precision low loss reference variable impedance, ZR, since the

value of the reference capacitor is well known, by this measurement, the value

of Zx can be calculated from:

Zx =UR

2

UR1

ZR (2.26)

and finally the value of the sample impedance is:

ZS =UR

2

UR1

.U1

U2

ZR (2.27)


Figure 2.4: Scheme of the circuit of Alpha dielectric analyzer

Dielectric measurements at high pressure were carried out using a Novocon-

trol Alpha-Analyzer (n=10−3-107 Hz). A sample-holder multilayer capacitor

(geometric capacitance 30pF) was filled with the sample under test and iso-

lated from the pressurizing fluid (silicon oil) by a Teflon membrane of thickness

of 50 mm. The dielectric cell was then placed in a Cu-Be alloy high pressure

chamber, provided by UNIPRESS. Pressure from 0.1 to 700 MPa was applied

to silicon oil using a manually operated pump with a pressure intensifier and

measured by a Nowa Swiss tensometric sensor with a resolution of 1 bar. Tem-

perature was varied in the interval 220-355 K and controlled within 0.1 K by

means of a thermally conditioned liquid flow (Julabo thermal bath circula-

tor). Fig. (2.5) shows the experimental set-up. Calibration measurements

(empty cell and short-circuit) were performed before each pressure scan (see

Fig. (2.6)). The extremely low losses obtained for empty cell allow to measure

relaxations with small amplitude (that is the case of secondary relaxation) also

at very high pressures and low temperatures: in fact, the signal coming from

the relaxation process of the material is enough higher than that coming form

the empty cell and the contribution of the latter can be neglected or singled

out and subtracted.


Figure 2.5: High-pressure dielectric cell using a pressure-transmitting liquid. Hydraulic

connections are denoted by dashed lines and electrical connections by solid lines.

Figure 2.6: Low loss ”empty” capacitor cell with teflon membrane

2.2 Differential scanning calorimetry (DSC) 39

2.2 Differential scanning calorimetry (DSC)

Differential scanning calorimetry (DSC) is used for investigating the phase

behaviour of a binary glass formers systems. Generally, it is believed that a

single Tg , located between the two pure components, indicates miscibility. In

DSC the difference in heat flow to the sample and a reference at the same

temperature, is recorded as function of temperature. The temperature of both

the sample and reference are increased at constant rate. Since DSC is at

constant pressure, heat flow is equivalent to enthalpy changes:

(dq

dt)P =

dH

dt(2.28)

where dH/dt is the heat flow measured in mcal sec−1. The heat flow difference

between the sample and the reference is:

∆dH

dt= (

dH

dt)sample − (

dH

dt)reference (2.29)

The calorimeter consists of a sample holder and a reference holder. Both

are constructed of platinum to allow high temperature operation. Under each

holder is a resistance heater and a temperature sensor. Currents are applied to

the two heaters to increase the temperature at the selected rate. The difference

in the power to the two holders, necessary to maintain the holders at the

same temperature, is used to calculate ∆dH/dt. A schematic diagram of a

DSC is shown in Fig. (2.7). During the heating of a sample, peaks with

positive and negative dH/dt may be recorded; each peak corresponds to heat

effect associated with a specific process, usually, a first order transition such as

melting, crystallization or phase transition, the glass transition (Fig. (2.8)) is

manifested by a drastic change in the base line, that indicates a change in the

heat capacity of the system. The glass transition is commonly determined by

extrapolating the baselines forwards and backwards as shown by dotted lines

in the Fig. (2.8) and take the base line shift when the transition is about 63%

complete. The value of the glass transition can be considered as the midpoint,

the inflection or the onset point. The differential scanning calorimeter used in

our study is Perkin Elmer DSC7.


Figure 2.7: Schematic of a DSC technique

Figure 2.8: Schematic of a DSC technique

2.3 Conclusions 41

2.3 Conclusions

The transition towards the glassy state has influence on different physical prop-

erties of the system and in particular on the relaxation dynamics of several

macroscopic observable, such as the polarization or the density fluctuation.

Dielectric spectroscopy is a sensitive technique to investigate the relaxation

dynamics of the orientation polarization due to the permanent dipole moment

of the polar molecular groups. From dielectric relaxation measurement DRS

one can gather the following information on a relaxation process:

(i) the relaxation time τ at a specific temperature and pressure;

(ii) the relaxation time distribution, which reflects the Dynamic heterogeneity

of the relaxing unit;

(iii) the relaxation strength, which is proportional to the concentration of the

relaxing dipoles and their dipole moment.

Differential scanning calorimetry (DSC) is used for investigating the phase

behaviour of a binary glass formers systems, while determining the glass tran-

sition Tg , the width of the glass transition and ∆Cp

Chapter 3

Conclusions

An important aspect concerning the relaxation dynamics of glass forming ma-

terials is the investigation of the molecular origin of the secondary relax-

ation and its possible connection to the structural relaxation and thus to

the glass transition phenomenon [60],[61],[5],[76]. Binary mixtures of rigid

polar molecules dissolved in apolar solvents are excellent systems to deepen

the understanding of the molecular slowing down and of the origin of sec-

ondary relaxation in glassy systems [38], [90],[92]. In this work, we stud-

ied by means of dielectric spectroscopy binary mixtures, with rigid polar

molecules (tert-butylpyridine and Quinaldine) dissolved in apolar viscous sol-

vents (tristyrene)[168]. Our results show that the excess wing characteristic of

the high frequency side of α- loss peak in neat systems becomes a well resolved

Johari-Goldstein (JG) secondary relaxation on mixing with the apolar solvent.

We prove that the excess wing is nothing but a JG process so close in the time

scale to the α- process that its low frequency side is hidden beneath the α-

peak. In fact, due to increasing the content of the apolar solvent, that in our

case has always higher Tg than the neat polar systems, the α- process is slowed

down so that the time scale separation from JG relaxation increases and a clear

bimodal relaxation scenario is obtained. Analyzing the temperature behaviour

of the two processes for each mixture, a clear correlation has been found be-

tween the time scale separation of the α- and JG relaxation time and the

dispersion of the α- relaxation (i.e. its Kohlrausch β parameter. The results

can be rationalized in the framework of the Coupling Model (CM) [60],[5],[53].

44 Conclusions

Moreover, we observed, by means of dielectric spectroscopy under high pres-

sure, that the JG secondary relaxation in binary mixtures is very sensitive

to volume reduction and that also its pressure dependence can be rationalised

according to CM. In particular we found that, at a fixed value of the structural

relaxation time, the dispersion of α- relaxation is constant as shown also by

[129], independent of thermodynamic conditions (T and P): that is, the shape

of the structural relaxation function depends only on the α- relaxation time

and we found that this quantity determines also the JG secondary relaxation

time. Moreover, a strong deviation from the glassy state behaviour found for

both temperature and pressure dependence of JG relaxation times, with an ap-

parent kink in the relaxation map always occurring at the crossing of the glass

transition. These results support the idea that the Johari-Goldstein relaxation

acts as a precursor of the structural relaxation and therefore of the glass tran-

sition phenomenon [60]. In the second part of this thesis, Broadband dielectric

spectroscopy (DRS) has been used to study the dynamics of polymer systems

which contains weakly polar groups, by introducing a dielectric probes which

is a polar dopant that introduces dipoles in the polymer matrix without any

chemical modification of this later which become dielectrically active. For this

purpose, we choose to study polystyrene and polypropylene doped by a rigid-

rod-type molecule, (4,4-(N,N-dibutylamino)-(E)-nitrostilbene) (DBANS). Our

results show that the DBANS probe is coupled to the dynamics glass transition

of polystyrene and polypropylene. Instead, the secondary relaxation process

observed in atactic polypropylene is not affected by the probe concentration.

In the end, we can conclude that dielectric spectroscopy can be used as a probe

technique to study polymers with low dipole moments as polyolefins.

Chapter 4

List of publications

[1] Genuine Johari-Goldstein β-relaxations in glass-forming binary

mixtures, S. Capaccioli, K. Kessairi, D. Prevosto, M. Lucchesi, K. L. Ngai,

Journal of Non-Crystalline Solids, 352 4643-4648(2006)

[2] Excess wing and Johari-Goldstein relaxation in binary mix-

tures of glass formers, D. Prevosto, K. Kessairi, S. Capaccioli, M. Lucch-

esi, P. A. Rolla, Philosophical Magazine, 87, Issue 3-5, 643-650( 2007)

[3] Effect of pressure on relaxation dynamics at different time

scales in supercooled systems, S. Capaccioli, D. Prevosto, K. Kessairi, M.

Lucchesi, P. A. Rolla, Philosophical Magazine, 87, Issue 3-5, 681-689(2007)

[4] Molecular Dynamics of Atactic Poly(propylene) Investigated

by Broadband Dielectric Spectroscopy, K. Kessairi, S. Napolitano, S.

Capaccioli, P.A. Rolla, and Michael Wu1bbenhorst, Macromolecules, 40,

1786-1788(2007)

[5] Correlation of structural and Johari-Goldstein relaxations in

systems vitrifying along isobaric and isothermal paths, S. Capaccioli,

K. Kessairi, D. Prevosto, M. Lucchesi and P A. Rolla, J. Phys.: Condens.

Matter, 19, 205133(2007)

46 Appendix 4. List of publications

[6] Effect of temperature and pressure in the structural (α-) and

the true Johari-Glodstein (β-) relaxation in binary mixtures, K.

Kessairi , S. Capaccioli, D. Prevosto, S. Sharifi and P.A. Rolla,(In Press in

journal non crystalline solids)

[7] Relation between the dispersion of the α- relaxation and the

time scale of the β- relaxation at the glass transition, S. Capaccioli,

D. Prevosto, M. Lucchesi K. Kessairi and P.A. Rolla (Accepted in journal

of Physics Condensed matter)

[8] Relaxation dynamics of tert-butylpyridine/tristyrene binary mix-

ture investigated by broadband dielectric spectroscopy, K. Kessairi, S.

Capaccioli, D. Prevosto , M. Lucchesi, P. A. Rolla, (Submitted in journal

Chemical Physics)

[9] Structural (α) and Johari-Goldstein (β) dynamics in binary

miscible blends, K. Kessairi, S. Capaccioli, M. Lucchesi, D. Leporini, P A.

Rolla, V. Castelvetro, (to be submitted to J. Phys: Condens. Matter).

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