relaxation dynamics in amorphous chalcogenides probed by...

UNIVERSITA DEGLI STUDI DI ROMA ‘SAPIENZA’

FACOLTA DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Ph.D. THESIS IN MATERIALS SCIENCE

Relaxation Dynamics in Amorphous

Chalcogenides probed by InfraRed Photon

Correlation Spectroscopy

Tutors: Candidate:

Prof. Giancarlo Ruocco Stefano CazzatoDr. Tullio Scopigno

Ph.D. Coordinator:

Prof. Ruggero Caminiti

Contemplator enim, cum solis lumina cumque

inserti fundunt radii per opaca domorum:

multa minuta modis multis per inane videbis

corpora miscari radiorum lumine in ipso

et velut aeterno certamine proelia pugnas

edere turmatim certantia nec dare pausam,

conciliis et discidiis exercita crebis;

conicere ut possis ex hoc, primordia rerum

quale sit in magno iactari semper inani.

dumtaxat rerum magnarum parva potest res

exemplare dare et vestigia notitiai

hoc etiam magnis haec animum te advertere par est

corpora quae in solis radiis turbare videntur

quod tales turbae motus quoque materiai

significant clandestinos caecosque subesse.

De Rerum Natura

Titus Lucretius Carus (c. 99 - c. 55 BCE)

——

Do but observe:Whenever beams enter and pourThe sunlight through the dark chambers of a house,You will perceive many minute bodies mingling,In a multiplicity of ways within those rays of lightThroughout the entire space, and as it wereIn a never ending conflict of battleCombating and contending troop with troopWithout pause, maintained in motion by perpetualEncounters and separations; so that thisShould assist you to imagine what it signifiesWhen primordial particles of matterAre always meandering in a great void.To this extent a small thing may suggestA picture of great things, and point the wayTo new concepts. There is another reasonWhy you should give attention to those bodiesWhich are seen wavering confusedlyIn the rays of the sun: such waverings indicateThat beneath appearance there must beMotions of matter secret and unseen.

3

Contents

1 Introduction 7

I Glassy dynamics and amorphous chalcogenides 11

2 Dynamics of supercooled liquids 13

2.1 Phenomenology of the glass transition . . . . . . . . . . . . . . 132.1.1 Interplay of different time scales . . . . . . . . . . . . . 142.1.2 Free energy landscape . . . . . . . . . . . . . . . . . . 182.1.3 Correlation functions . . . . . . . . . . . . . . . . . . . 182.1.4 The calorimetric glass transition . . . . . . . . . . . . . 202.1.5 Viscosity, structural relaxation time and fragility . . . 23

3 Amorphous chalcogenides 27

3.1 The chalcogens . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.1 Sulfur (S) . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 Selenium (Se) . . . . . . . . . . . . . . . . . . . . . . . 323.1.3 Tellurium (Te) . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Binary As-Se and As-S chalcogenides . . . . . . . . . . . . . . 353.2.1 Structural properties . . . . . . . . . . . . . . . . . . . 363.2.2 Thermal properties . . . . . . . . . . . . . . . . . . . . 393.2.3 Viscosity and structural relaxation . . . . . . . . . . . 423.2.4 The onset of an intermediate phase in binary chalco-

genide glasses . . . . . . . . . . . . . . . . . . . . . . . 443.2.5 Optical properties . . . . . . . . . . . . . . . . . . . . . 50

II Materials and methods 53

4 Dynamics investigation through light scattering 55

4.1 Light scattering theory . . . . . . . . . . . . . . . . . . . . . . 564.1.1 Light scattered from an isotropic medium . . . . . . . 564.1.2 Dynamic Light Scattering . . . . . . . . . . . . . . . . 59

4.2 Homodyne Photon Correlation Spectroscopy . . . . . . . . . . 614.2.1 The Gaussiam approximation . . . . . . . . . . . . . . 624.2.2 Discrete scatterers . . . . . . . . . . . . . . . . . . . . 66

4.3 The heterodyne correlation function . . . . . . . . . . . . . . . 714.3.1 Heterodyne method . . . . . . . . . . . . . . . . . . . . 724.3.2 Discrete scatterers under flow . . . . . . . . . . . . . . 73

5 InfraRed Photon Correlation Spectroscopy 75

5.1 Setup description . . . . . . . . . . . . . . . . . . . . . . . . . 775.1.1 General sketch of the IRPCS setup . . . . . . . . . . . 77

5

CONTENTS

5.1.2 The sample environment . . . . . . . . . . . . . . . . . 795.1.3 The detectors . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Setup alignment and calibration . . . . . . . . . . . . . . . . . 825.3 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3.1 Effects due to finite intensity . . . . . . . . . . . . . . . 875.3.2 Effects due to finite experiment duration . . . . . . . . 875.3.3 Effects due to unwanted scattered light . . . . . . . . . 88

5.4 The samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

III Results 91

6 Liquid chalcogens 93

6.1 Liquid sulfur . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.1.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . 936.1.2 Data treatment . . . . . . . . . . . . . . . . . . . . . . 956.1.3 Temperature dependence of chain relaxation time . . . 966.1.4 Momentum dependence of chain relaxation time . . . . 1006.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Liquid selenium . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.2.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . 1036.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 Chalcogenide glass formers 107

7.1 As-Se chalcogenides . . . . . . . . . . . . . . . . . . . . . . . . 1077.1.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . 1077.1.2 Data treatment . . . . . . . . . . . . . . . . . . . . . . 1087.1.3 Structural relaxation in the As-Se series . . . . . . . . 1137.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2 As-S chalcogenides . . . . . . . . . . . . . . . . . . . . . . . . 1177.2.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . 1177.2.2 Data treatment . . . . . . . . . . . . . . . . . . . . . . 1187.2.3 Structural relaxation in the As-S series . . . . . . . . . 1197.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Conclusions 125

6

1Introduction

The main subject of this experimental work is the development of a novelspectroscopic technique, namely dynamic light scattering with infrared radi-ation (IRPCS), aiming to investigate the slow dynamics (10−6s up to 100 s)of non transparent inorganic glass formers, belonging to the vast family ofchalcogenide glasses (ChGs), in the supercooled liquid regime above the glasstransition temperature.

ChGs are a group of inorganic glassy materials which contain one or moreof the chalcogen elements: sulphur, selenium and tellurium (but not oxygen),in conjunction with more electropositive elements, most commonly arsenicand germanium, but also phosphorus, antimony, silicon, tin and other. Theiruse in a wide range of optical, electronic and memory applications [1] hasstimulated a wide interest for a more thorough understanding of such ma-terials properties and of variations of such properties with composition. Inparticular, one of the interests for the ChGs came from the attempt to ex-tend the IR transparency region of glasses past the 8 µm wavelength region,a limit which remains implicit in the use of oxide glasses and heavy oxidematerials. Moreover, ChGs have been shown to exhibit a dazzling varietyof structural modifications (photo-structural changes) when exposed to light[2]. This property of ChGs, i.e. their sensitivity to light illumination, ren-ders them ideally suitable media for many important applications (opticalgratings, microlenses, waveguides, optical memories, holographic media, etc.)[3, 4, 5, 6, 7, 8, 9, 10].

Chalcogen elements are also interesting systems from the viewpoint of thedynamics in the warm and supercooled liquid phases. Both Se and S are wellknown to be easily obtainable in a glassy state and among the reasons thathave stimulated an intense interest in investigating such systems there is alsothe notion that they constitute one of the simplest polymers possible. In par-ticular, vitreous selenium became of interest for the scientific community atthe beginning of the 20th century when Wood [11] and Meier [12] reported thefirst research on the subject. Later on, in 1979, amorphous selenium has beenreferred to as “one of the best studied of all substances” [13]. Liquid S and Sehave been recognized to be assemblies of living polymers [14], i.e. polymerswhose lengths fluctuate and attain an equilibrium length distribution at anygiven temperature T.

7

CHAPTER 1. INTRODUCTION

The first works on ChGs were attributed to Frerichs in the early 50’s onthe As2S3 glass [15, 16]; As2Se3 was firstly investigated by Fraser and Dewulf[17]. Other important studies followed in the same years [18, 19, 20]. In 1969Abrikosov et al. [21] reported in their monograph the phase diagram for theAs-S and As-Se systems. As-S alloys can be formed with an As content up to46% while in As-Se this maximum content can be raised to almost 60%. Theseglass formulations can contain a non-stoichiometric amount of chalcogen andexcess S and Se atoms can form chains, probably responsible of changes incooperativity of structural relaxation with composition. The stability againstcrystallization in bulk As-S and As-Se alloys and their optical properties hasstimulated activity in these materials, as well as an increasing interest in theability to tailor from bulk glasses films and fibers for a range of applications[4, 22, 23, 24, 25].

A part of this work will focus on the investigation of supercooled liquiddynamics of arsenic selenide and sulfide ChGs on approaching the glass tran-sition temperature Tg from above. This view from the liquid side allows theexperimentalist to follow the dynamical processes triggering structural arrestat Tg. Indeed, still lowering the temperature, the characteristic time-scaleof structural relaxation continues to increase, eventually falling outside anyexperimentally accessible window. The resulting glass phase can be viewed,in this respect, as a liquid phase in which molecules are so tightly packed thatthe relaxation time needed to reach thermodynamic equilibrium becomes ex-perimentally infinite, while, macroscopically, its viscosity diverges.

Despite the apparent simplicity of this qualitative description, the meta-stable glassy and supercooled states of matter have been, and still are, thesubject of numerous investigations, both theoretically and experimentally,and the formulation of a coherent microscopic picture of the glass transitionstill remains a challenge for the scientific community [26, 27, 28, 29]. However,on a phenomenological ground, the rapid - often non-Arrhenius - increase ofthe shear viscosity η on decreasing the temperature of a supercooled liquid,approaching the glass transition temperature, has long ago been recognizedas one of the salient features of the dynamical glass transition effect. Thistemperature behavior of viscosity has stood as the basis for the strong andfragile classification of supercooled liquids [30].

As suggested by many theoretical approaches, a good variable capable ofdescribing the dynamics of supercooled liquids is the density time correla-tion function, which encodes information about the microscopic relaxationtimescales in the material. Typically, a two step decay is observed: a fastrelaxation accounts for local rearrangements of the particles, while a slowdecay is related to the highly cooperative structural relaxation process. Thetimescales of these long-time collective rearrangements represents the micro-scopic counterpart of the viscosity.

Light scattering techniques are a valuable tool to follow the dynamics ofsupercooled glass-forming liquids, as the scattered intensity signal provides

8

direct information on the timescales characterizing the decay of the den-sity fluctuations. In particular, homodyne photon correlation techniques areapplicable to the measurement of dynamics evolving on timescales rangingfrom the µs to 100 s, corresponding to what, on a molecular scale, are long-distance, long-time phenomena, including the typical timescales of structuralrelaxation in glass formers near the glass transition temperature Tg.

A Photon Correlation Spectroscopy (PCS) setup, using as probe a near-Infrared laser (λ=1064 nm) radiation, has been implemented with the aim ofreaching the best experimental conditions for the investigation of the density-density time correlation function in systems offering a transmission maximumin the near IR. Through this technique we are allowed, for the first time, toinvestigate the nature of homodyne correlation function in supercooled liquidChGs as well as in liquid chalcogen elements S and Se. The strong variation,over several decades, of the typical relaxation time as a function of tempera-ture is studied for these systems. Moreover, also the exchanged momentum qdependence of relaxation function can be accessed with the IRPCS setup, byperforming acquisitions at five different values of the scattering angle at onetime. This was made possible by the implementation of five independent ac-quisition channels, collecting the scattered signal from the sample at differentscattering angles.

The thesis is divided into three parts:

1. The first part, split into two chapters, introduces the reader to thescientific problem. In the first chapter, the complex phenomenologyrelated to the dynamical glass transition is introduced. In the secondchapter structural, thermal, rheological and linear optical properties ofChGs belonging to the As-Se and As-S series, as well as of amorphouschalcogens, will be examined.

2. The second part, also divided into two chapters, presents the techniqueand the materials used in the experiments. In the first chapter, the basictheories for dynamic light scattering are presented. The second chapteris dedicated to the description of the experimental IRPCS techniqueand of samples preparation.

3. In the last part, divided into two chapters, the experimental resultsare presented and discussed in the light of the background provided inthe first part. In particular, results on liquid chalcogens dynamics arereported in the first chapter, while the second chapter is dedicated toresults on the structural relaxation dynamics in arsenic selenide andsulfide ChGs.

9

Part I

Glassy dynamics and

amorphous chalcogenides

11

Even the definition of glass is arbitrary:basically a rate of flow so slow that itis too boring and time-consuming towatch.

Kenneth ChangThe New York Times, July 29, 2008 2

Dynamics of supercooled liquids

2.1 Phenomenology of the glass transition

Consider the use of the glass transition made by glass makers. At high tem-perature the glass melt is a viscous liquid. The viscosity increases with fallingtemperatures and a ‘forming interval’ is passed, where the glass makers canwork. This interval can be characterized by a viscosity range or by a cor-responding temperature range. Further lowering the temperature, the glassmelt becomes a solid, i.e., glass. This continuous transition from a liquid toa glass is called the glass transition and is the subject of the present chapter.

The glass transition is different from solidification by crystallization. Thelatter is a (first order) phase transition with a well defined thermodynamictransition temperature, the crystallization temperature. From a dynamicpoint of view the matter is not so simple. Crystal formation needs time tocrystal nuclei of the new phase to form, and time for crystals to grow.

Crystallization is also relevant for the glass industry. The aim is usuallyto avoid crystallization in order to get a clear optical glass. Other industries,dealing with ceramics or enamels, for instance, are interested in a manageableinterplay between the glass and phase transitions.

The glass transition is of general interest not only for manufacturing andindustrial applications. It is also of general interest in natural science, ma-terials science, and the life sciences. As an example, the continuous natureof the glass transition, as opposed to the discontinuous one characterizingthermodynamic phase transitions, lays at the basis of the cryopreservation ofbiological tissues in nature. Many animals in winter survive freezing weatherby producing glucose, or other natural sugars (e.g., trehalose), within theirbodies to act as an antifreeze [31, 32]. The most commonly cited exampleis the wood frog who freezes solid in winter but returns to life during thespring thaw [33]. At the onset of cold temperatures, glucose produced by theliver circulates throughout the body. In addition to stabilizing proteins [34],the glass-forming properties of glucose reduces the likelihood that ice willform inside cells or capillaries. Should this crystallization occur, the rapidexpansion would rupture the cell membrane or otherwise destroy the delicateinternal components [33, 35].

The main problem, in attempting a physical description of the glass tran-

13

CHAPTER 2. DYNAMICS OF SUPERCOOLED LIQUIDS

sition phenomenon, is that no generally accepted microscopic theory is so faravailable. Hence all pictures remain vague to a certain degree, and we mustuse general principles for the description. Such description is based on a viewfrom the liquid side, in order to embrace the dynamic glass transition farabove Tg. Apart from some picosecond/teraherz processes, the material is ina state that cannot be understood from the solid standpoint. Moreover, thephenomena that will be described are typical for all liquids and disorderedmaterials. In a certain respect, the molecular dynamics in liquids is synony-mous with the dynamic glass transition. In the present exposition references[28] and [29] will be mainly followed.

2.1.1 Interplay of different time scales

A glass can be viewed as a liquid in which a huge slowing down of the diffusivemotion of the particles has destroyed its ability to flow on experimental time-scales. The slowing down is expressed through the relaxation time τeq, that is,generally speaking, the characteristic time on which the slowest measurableprocesses relax to equilibrium.

Cooling down from the liquid phase the slow degrees of freedom of theglass former are no longer accessible and the viscosity of the under-cooled meltgrows several orders of magnitude in a relatively small temperature interval.As a result, in the cooling process, from some point on, the time effectivelyspent at a certain temperature is not enough to attain equilibrium.

The preparation, indeed, plays a fundamental role to get a glass out of aliquid, thus avoiding the crystallization of the substance. Depending on thematerial, the ways of obtaining a glass are very diverse, and consist not onlyin the cooling of a liquid but also include compression, irradiation of crystalswith heavy particles, chemical reactions, polymerization, evaporation of sol-vents, deposition of chemical vapors, etc. Many kinds of materials present aglass phase at given external conditions if prepared in the proper way. Foran exhaustive literature, the reader can refer to [26, 36, 37, 38].

The crystal state is always at lower energy than the glassy state, but theprobability of germinating a crystal instead of a glass during the vitrificationprocess is negligible when cooling fast enough: the nucleation of the crystalphase is practically inhibited. In a nucleation event, a small but criticalnumber of unit cells of the stable crystal combine on a given characteristictime-scale, the nucleation time τnuc.

In order to clarify this point, a simplified picture of a nucleation event canbe given. Consider a system which, at a certain time, is found in a metastablestate M (see Fig. 2.1.a), and in which thermal fluctuations has brought tothe formation of a bubble of the stable state S. Let l represent the linear sizeof the bubble, and fM , fS be the free energies per unit volume pertaining tothe metastable and stable states respectively; thus fM > fS. The energeticcost to be payed for the nucleation of the stable state has a negative volumecontribution −(fM − fS)l3 = −δfl3 playing in favor of nucleation, and a

14

2.1. PHENOMENOLOGY OF THE GLASS TRANSITION

Figure 2.1: (a) Schematic representation of the nucleation of a stable state bubbleS within a metastable phase M. (b) The energetic cost for the formation of a nucleusof size l of the stable S phase: Cost(l) = −δfl3+σl2. This function has a maximumfor l∗ = 2σ/3δf .

surface contribution accounting for the two phases interaction, which can bedescribed in terms of a surface tension σ. The latter contribution will bepositive, thus playing against the stable state nucleation. The net energeticcost will read

Cost(l) = −δfl3 + σl2 ,

which, as function of the nucleus size l > 0, attains its maximum for l∗ =2σ/3δf (see Fig. 2.1.b). If l > l∗ the nucleus will tend to grow. In theopposite case, this will tend to implode. Thus there is a critical size forcrystal nuclei, generated by spontaneous thermal fluctuations, to give riseto stable configurations capable of accreting. The nucleation time τnuc is thetime necessary for thermal fluctuations to give rise to a stable crystal nucleus.

In a good glass former, the number of molecules involved in the nucleationmust be much larger than the number of molecules cooperating in the struc-tural relaxation of the glass phase, composing what is called a ‘cooperativerearranging region’, yielding, in this way, a nucleation time much longer thanthe structural relaxation time τeq.

Many processes are involved at the glass transition, or better, around it,since the transition region depends on the way it is reached on an experiment,and the time scales of the process play an essential role for the properties andthe behavior of the glass former.

Imagine to follow a liquid glass former during a cooling procedure, startingfrom a high temperature (look at Fig. 2.2 as a guide, starting from the leftside). Already in the warm liquid, different processes occur at different timescales. At a given temperature, that we will simply denote as Tcage [29], thethermal movement of particles is slow enough for the diffusion to be hinderedby the formation of cages. A cage is, in this sense, a dynamic concept relativeto each particle in the liquid, whose motion is constrained to occur next to

15


other particles around it, with which it collides. This is at difference with thepurely collisional motion taking place in the warm liquid. Cooling further, afirst bifurcation of time scales takes place between the relative fast rattlingtime and the relaxation time τeq for the system. The relaxation time is, here,the characteristic time scale of the process of diffusion from the cage, thatbecomes longer and longer as the temperature decreases. This process iscalled in many different ways in the literature. Staying close to the notationof Ref. [28, 29], we chose the name αβ. The reason will become clear in ashort while.

Figure 2.2: Diagram for the relaxation time τ of a glass former (in seconds) vs.the inverse temperature. From left (high temperature) to right (low temperature)a first bifurcation of the warm liquid characteristic times occurs at a temperatureTcage. Then, at Td, a second bifurcation occurs, more important for the onsetof the glass formation, in the dynamic crossover region where the dynamic glasstransition takes place. Eventually, at Tg, the material freezes and becomes a glass,since the structural relaxation time becomes longer than the experimental time.Figure from Ref. [29]

In summary, the relaxation time in this temperature region is the charac-teristic time needed to have one long distance diffusion process of a particlewhile it is rattling with a high frequency among its neighbor particles forminga cage around it.

Cooling further, in the so called crossover region (always refer to Fig. 2.2),a second bifurcation of time scales takes place between processes involving aglobal rearrangement of the system, tanks to a large cooperativeness of theparticles (we will call them α processes), and processes that involve only alimited number of molecules in a local, microscopically small, rearrangement,thus not contributing to the structural relaxation of the glass former. Thelatter are usually called β processes. We reserve the label α for the slowestprocesses, needing a huge cooperativeness to occur below the crossover region,stressing that, in general, different molecular mechanisms may be responsiblefor the lower temperature α process and the higher temperature αβ processes.

16


In the crossover region, thus, αβ processes bifurcate in α processes – with timescale the structural relaxation time – and β processes with a much shortercharacteristic time: this way we are in the presence of a separation of timescales, that becomes more and more enhanced as temperature is lowered.

In the literature the temperature at which the crossover takes place isnamed either as Tc (crossover or critical) [28], as Td (dynamic) [39] or asTmc (mode coupling) [40]. Traces from dielectric susceptibility measurementson two distinct polymer systems are reported on an Arrhenius plot – i.e. aplot of the typical relaxation frequency (the inverse of relaxation time) vs theinverse temperature – in Fig. 2.3.

Figure 2.3: Dielectric traces for dynamic transition in the Arrhenius diagramfor two glass forming liquids. (a) Temperature dependence of the dielectric peakfrequency for bis-methoxy-phenyl-cyclohexane (BMPC) [41]. The open symbolsrefer to the dielectric results for the α- (◦) and β-process (△). The dashed linesare a Vogel-Fulcher-Tammann (VFT, this model function will be introduced laterin Section 2.1.5) fit to the α-relaxation for T < Tβ , and an Arrhenius fit to theβ-relaxation data. Tβ ≈ 270K marks the bifurcation temperature regarding the α-and β-relaxation. (b) Temperature dependence of the dielectric peak frequencyfor ortho-terphenyl (OTP). The open symbols refer to the dielectric results for theα- (◦, [41]) and β-process (△, [42]). The dashed lines are a VFT fit to the α-relaxation for T < Tβ , and an Arrhenius fit to the β-relaxation data. The solid linerepresents scaled viscosity data of OTP. The diamonds refer to photon correlationspectroscopy (PCS) data and the triangles represent Brillouin light-scattering data(see Ref. [41] and references cited therein). Tβ ≈ 290K marks the bifurcationtemperature regarding the α- and β-processes. Figure from Ref. [41].

The descriptive stile used so far in the present section, should not inducethe reader to believe that all the points raised here have been brought to astage of full understanding [29, 28]. They should serve instead as a necessarystarting point on approaching a convenient description of such a problematictopic as the relaxation dynamics of glass forming systems.

17


2.1.2 Free energy landscape

In terms of a free energy landscape description of the phase space, wheremetastable states are represented by local minima and stable states by globalminima, a two level structure appears: some minima of the free energy areseparated by very small barriers and between them β processes take place;group of those minima are contained in larger basins separated by barriersrequiring a much bigger free energy variation to be crossed. To make thesystem go from a configuration in one of these basins to another configurationin another basin, i.e., to have an α process, a longer time is needed. Indeed,the typical crossing time τeq is related to the free energy barrier ∆F separatingthe valley where the system is currently located from the rest of the landscape,τeq ∼ exp βδF . The time scale on which these processes are occurring is,however, at Td and below (but well above the temperature of the formationof the solid glass) still very short in comparison with the observation time.Below Td the system is, thus, still at thermodynamic equilibrium. The phaseis disordered but the number of minima of the free energy increases, and somelocal minima become deeper. The dynamics of the slowest process (αβ forT > Td, or α for T < Td) displays a huge slowing down, but the temperature isnevertheless high enough for the system to attain equilibrium on experimentaltime scales [29, 28].

2.1.3 Correlation functions

We consider in the present section the time behavior of relaxation functionsin supercooled liquids. Our system comprises N elementary units, i.e. atoms,ions, molecules, with positions ri(t) and momenta pi(t), varying with timeaccording to the classical mechanics equations related to the Hamiltonian

H =1

2

∑

i

p2i

m+ VN(r1, ..., rN ) .

Consider a dynamical variable A(pN , rN), as function of the 6N coordi-nates rN and momenta pN . A central role in our description will be played bytime correlation functions of dynamical variables, and among all such possiblevariables there are certain which play a major role in the theoretical as wellas experimental description of relaxation phenomena in liquid systems. Weconsider for the present purpose the number density variable ρ(r, t), definedas

ρ(r, t).=

1√N

∑

i

δ(r − ri(t)) ,�

�

�

�2.1

We can also define its spatial Fourier transform

ρ(q, t) =1√N

∑

i

eiq·ri(t) ,�

�

�

�2.2

18


which will turn to be useful since, generally, scattering experiments provideinformation on the reciprocal q space to the real space (see Chapter 4). Thenumber density time autocorrelation function is defined as (we now specializeon samples which are isotropic on average, so that the average quantitiesdepend only on the modulus q of the scattering vector)

F (q, t) = 〈ρ(q, t)ρ∗(q, 0)〉 ,�

�

�

�2.3

also called intermediate scattering function. It is experimentally accessiblewith inelastic (or dynamic) scattering techniques and provides dynamicalinformation on the system under study. In Eq. (2.3), the symbol 〈· · · 〉denotes the statistical ensemble average

〈· · · 〉 =

∫

Γ

dpNdrN(· · · )P(pNrN)

being P(pNrN) the system probability distribution over the entire phase spaceΓ. Its initial value is the so-called static structure factor

S(q) =⟨

|ρ(q, 0)|2⟩

,�

�

�

�2.4

being the space Fourier transform of the spatial autocorrelation function ofthe density. It is experimentally accessible with static scattering techniquesand provides structural information on the system under study.

The normalized time autocorrelation of the density

Φ(q, t).=

〈ρ(q, t)ρ∗(q, 0)〉⟨

|ρ(q, 0)|2⟩ =

F (q, t)

S(q)

�

�

�

�2.5

is generally called the density correlator. Its initial value is one and it is anon increasing function of time.

The simplest relaxation model assumes stochastic dynamics. It was usedby Maxwell in its viscoelastic theory for the shear relaxation. In the presentcontext it is usually named after Debye, since he used this model for thediscussion of dipole relaxation in liquids:

Φ(q, t) = exp

[

− t

τ(q)

]

,�

�

�

�2.6

being τ(q) the characteristic decay time of density fluctuations. Structuralrelaxation (α) processes in supercooled liquids, are often associated with afairly broad spectrum of relaxation times. A simple expression that describesthe spectrum reasonably well over a wide range of time is the ‘stretchedexponential’, or Kolraush-Williams-Watts (KWW) expression:

Φ(q, t) = exp

[

−(

t

τ(q)

)β]

.�

�

�

�2.7

19


Here τ is again a characteristic relaxation time, and β ≤ 1 is an exponentthat can vary with temperature. β = 1 implies monoexponential relaxation(i.e. a spectrum with a single relaxation time), while a smaller β correspondsto a wider spectrum.

The stretching phenomenon was observed already in the 18th century byKohlraush [43]. For the description of his relaxation data for the electricalpolarization of glassy materials he modified the stochastic law Eq. (2.6) by astretching exponent. The stretched exponential function entered the modernliterature on glassy dynamics as an empirical fit formula for α relaxation ofmechanical [44], electrical [45], or other variables (compare Ref. [46] for acompilation of examples).

Such a stretched behavior of relaxations has been sometimes interpretedas arising from the distribution of relaxation times associated with differentexponentially relaxing regions or domains of different size or structure (theso-called cooperative rearranging regions). Anyway there is lack of a generalconsensus about the origin of nonexponential relaxations and several modelshave been proposed [27].

The KWW expression is convenient, but not sacrosanct; other expressionscan be used to approximate the spectrum. However stretched exponentialrelaxations are predicted by Mode Coupling theory [40].

It is usually observed experimentally, mainly with dielectric relaxation orlight scattering techniques [27, 47], that the relaxation evolves from a Debyeexponential at T > Td to a two step process at lower temperature, that ismore and more enhanced as T → Tg. This corresponds to the bifurcation atTd between structural, α, and fast, β, processes underlined in Section 2.1.1.Also the β process follows a stretched exponential behavior. In terms ofcorrelation functions, this means that they first decay rather quickly to aplateau and, on a longer time scale, start to decay again towards equilibrium;see Fig. 2.4 for a pictorial example holding for T < Td.

2.1.4 The calorimetric glass transition

If we cool down a glass forming liquid with a given cooling rate T = dT/dt,at the glass temperature Tg thermal fluctuations become too slow to estab-lish their contribution to thermodynamic variables. As a consequence, ther-modynamic quantities show a signature of the glass transition. A samplephenomenology is illustrated below.

• We start with the volume versus temperature path. As a typical ex-ample in Fig. 2.5.a, the specific volume of glucose is reported. Considera solid which has been heated to well above the melting point. Whensuch a melt is gradually cooled its volume decreases continuously downto its freezing point Tm. At Tm the volume generally decreases abruptlydue to crystallization. Upon further cooling the volume again decreasescontinuously, but with a reduced slope, which is characteristic of the

20


Figure 2.4: Typical correlation function for undercooled liquids. A first decayoccurs on the time scale of β-relaxation, and a further decay on the time scale ofα-processes leads to equilibrium. At decreasing temperatures the structural (α-process) relaxation time strongly increases near Tg (the curves · · ·, −−− and —–correspond to temperatures respectively decreasing). Figure from Ref. [29].

crystalline solid. On the contrary, if the melt is cooled very fast, so asto bypass crystallization, the volume below Tm continues to decreaseat the same rate as above Tm. At low enough temperature, a changeoccurs in slope of variation of the volume, and the now-rather-viscousmelt solidifies. The expansivity of this amorphous solid is similar tothat of the crystalline solid.

The temperature of the change of slope is, by definition, the calori-metric glass temperature Tg. But this Tg (Fig. 2.5.a) is not a uniquetemperature and it depends on the rate of cooling; the slower the cool-ing, the lower the Tg. From Fig. 2.5.a one can also see that the volumeof the glass is slightly higher that that of the parent crystal and this isalmost always the case. The regime of temperatures between Tm andTg is referred to as ‘supercooled region’.

• Let us examine the specific heat plot (CP vs T , Fig. 2.5.b) for the sametypical case as in Fig. 2.5.a. On cooling the melt, its CP decreases verylittle till when at Tm it drops abruptly to the CP value of the crystal.But when so cooled as to bypass crystallization, the supercooled meltcontinues to follow the same heat capacity behavior of the melt aboveTm. The supercooled melt, therefore, always has a higher heat capacitythan the crystal. On cooling further, however, the supercooled meltexhibits an almost abrupt decrease in CP at Tg where it solidifies intoa glass. The glassy state heat capacity is only slightly higher than thatof the crystal.

• Let us turn to entropy. Upon cooling the melt, if crystallization occurs,entropy drops discontinuously at Tm, in the same way as the specificvolume does (Fig. 2.5.a), to the value characteristic of the crystal.When crystallization is bypassed, entropy decreases down to Tg where

21


it is closed to, but slightly higher than, the entropy of the crystal. If thecooling rate is slow, the slope changes at temperatures still closer to theentropy curve of the crystalline solid. But it never crosses the entropycurve of the crystal itself because that would be a thermodynamic ab-surdity, whereby a supercooled melt would possess lower entropy thanthe crystalline solid itself. This is often referred to in the literature asthe ‘Kautzmann paradox’ [48].

Figure 2.5: Thermodynamic properties of glucose, as an example of the thermalglass transition phenomenology. (a) The specific volume V vs. temperature, and(b) specific heat CP vs. temperature. When the liquid is cooled from T > Tm,the melting point, to T < Tm, either crystallization may occur (solid lines withdiscontinuity at Tm) or the liquid can be supercooled (solid line continuous at Tm).The thermodynamic glass transition (region) corresponds to the change of slopein V or to the jump in CP . Note that lower cooling rates bring to lower glasstemperatures (dashed curves). Figure is taken from Ref. [48].

The glass state below the glass temperature Tg is often referred to as the‘thermodynamic’ state of a vitrified substance. It is true that, for not toolong observation times and/or well above Tg, parameters practically do notshow any time dependence and the amorphous solid seems, therefore, to bein a properly defined thermodynamic state. However, even in this case, theglass and the liquid phase cannot be connected by any path in the timeindependent parameter space, nor adiabatically slow state change can everbring from the liquid phase to the glass phase below Tg [29]. Time will alwaysplay a fundamental role in the formation and description of the glass, thefundamental reason simply being that it is not an equilibrium state. Glassysubstances that look like a solid on experimental time-scales, second or years,may look like a liquid on geological time-scales.

The glass temperature Tg rather marks the transition from ergodic to(practically) non-ergodic behavior. Below Tg, the system degrees of freedomleading the structural relaxation, like the diffusion processes that make thematerial flow, are frozen. This implies, for example, that in a given time tw

22


(the time waited after the quench to the glass state), only a limited region ofthe configuration space, connected to the initial configuration, can be visitedin the evolution of the system. The system cannot explore all energeticallyaccessible parts of the configuration space in this time. As a consequence,ensemble average and time average are no longer equivalent, at least on thetime windows accessed by the experimentalist.

As a consequence, the location of the glass transition temperature Tg

depends, as already observed, on the cooling rate, the pressure and the com-position, but also on the experimental conventions adopted for its operativedetermination. Indeed, a convention has to be established to fix the propercooling and heating rates since the glass transition is kinetic in origin. Finallya further convention identifies which point of the smeared vitrification step,in specific heat, one has to adopt, because no true cusp, strict discontinuityor divergence occurs in the curves.

2.1.5 Viscosity, structural relaxation time and fragility

As we already pointed out in section 2.1.1, in all substances with a certainmolecular disorder the velocity of thermal fluctuations dramatically slowsdown with falling temperatures. Assuming that no phase transition interventsthat would kill or change the disorder, we then have, for the present, no typicaltimes. Instead the fluctuation time τeq increases continuously from 10−12 s(picoseconds) to 102 s (minutes), i.e., about fifteen orders of magnitude (seethe relaxation time vs. temperature path reported in Fig. 2.2). This variationis out of all proportion with the concomitant changes in density or structure.If the fluctuation time τeq arrives at a typical experimental time, the substancevitrifies when cooling is continued.

A glass is then an amorphous solid formed from the melt by cooling torigidity without crystallization. Rigidity means zero or weak steady responseto a permanent shear stress. In the liquid we observe such a response: theviscosity η. To a certain approximation, it is connected to the fluctuationtime τeq by the mechanical Maxwell equation,

η ≈ G∞τeq ,�

�

�

�2.8

where G∞ ia a glass shear modulus of order 109 to 1012 Pa. The increase influctuation time is reflected by an increase in viscosity.

Without crystallization, the solidification of glasses is gradual, with no dis-continuity. For ordinary glasses, we pass a pouring interval (101 − 103Pa · s,where 0.1Pa · s = 1Poise ), a forming or working interval, important for glassmakers (105 − 108Pa s), and other intervals. From a viscosimetry experiment,the glass temperature is conventionally defined as the temperature at whichthe viscosity attains 1013 poise, which is an immense value if compared toviscosities of normal, warm, liquids. In normal liquids the diffusive dynamicstakes place on time scales of some percent of a picosecond, thus, considering

23


a typical value of the glass modulus G∞ ∼ 1011, Eq. (2.8) yields η ≈ 10−3

poise. For instance, in normal conditions the viscosity of water is 10−2 Poise.A widely used three-parameter equation capable of reproducing exper-

imental viscosity data on a wide temperature range is the Vogel-Fulcher-Tammann (VFT) equation [49, 50, 28, 29],

logη

η∞=

B

T − T0

.�

�

�

�2.9

The meaning of the parameters is still a mystery, and an example of a long-running issue. Two of them are asymptotes. For T → ∞ we have formallyη → η∞, and for η → ∞ we have formally T → T0 > 0, the Vogel temperature.Both asymptotes are far away from the range of application: the viscosityin the working interval is many orders obove η∞, and Tg is several tens onkelvins above T0. Eq. (2.9) can be considered as a phenomenological equation,although theoretical results predict an analogous relation [51].

Important for glass makers is the temperature interval ∆T correspondingto the working interval of viscosity. For a given cooling rate T = dT/dt, theyhave a lot of time for glasses with large ∆T . These are the ‘long’ glasses, andthe others the ‘short glasses’. This property of a glass-forming melt, can becharacterized by one parameter [52] that may be determined from propertiesnear Tg. This parameter was called fragility by Angell [53, 54]. There ismuch misunderstanding outside the glass transition community because thisconcept has, in general, nothing to do with fragility of glasses in the commonsense, i.e., breaking easily, or being brittle. Instead, Angell’s pristine inter-pretation was related to molecular pictures for dynamics or thermodynamicsin the liquid.

Fragility is usually characterized [36] by a dimensionless ‘steepness index’related to T = Tg by

m = limT→Tg

d log10 η

dTg/T.

�

�

�

�2.10

An Arrhenius plot of viscosity versus inverse temperature for a variety ofglass forming liquids is reported in Fig. 2.6. For most of the liquids theviscosity seems to extrapolate to 10−4 poise for T → ∞. In this plot, thetypical Arrhenius behavior of an activated process would be represented by astraight line. For the majority of glass formers this is actually not the case, infact they can be conveniently represented with a VFT behavior. The fragilityEq. (2.10) can be estimated for a VFT, Eq. (2.9), thus leading:

mV FT =BTg

(Tg − T0)2= log

(

η(Tg)

η∞

)

Tg

Tg − T0

,�

�

�

�2.11

but from our previous considerations about the orders of magnitude of vis-cosity in warm liquids and at the glass temperature, log η(Tg)/η∞ ≈ 16, and

mV FT ≈ 16Tg

Tg − T0

.�

�

�

�2.12

24


Figure 2.6: Viscosity vs. inverse temperature for glass-forming liquids, showingbehavior classified as ‘strong’, typified by open tetrahedral networks, to ‘fragile’,typical of ionic and molecular liquids. Here Tg is defined by the criterion thatη(Tg) = 1013 poise. For most of the liquids, the viscosities seem to extrapolateto a common value of around 10−4 poise at high temperatures, corresponding toa fundamental molecular vibrational frequency of around 1013 s−1 . Figure fromRef. [30].

Glass formers can thus be classified according to their fragility parameter, asfollows [29]:

• Certain materials exhibit an Arrhenius behavior of viscosity for temper-ature above Tg. In such cases, Eq. (2.9) holds with T0 = 0. These showa high resistance to structural changes, usually small jumps of specificheat (with the exception of cases where hydrogen bond plays a majorrole), their vibrational spectra and radial distribution functions showlittle reorganization in a wide range of temperature and the free energyhyper-surface (or landscape) has few minima and high barriers, see Fig.2.7. Their fragility is m ∼ 16 (for very strong glasses), as can be seenfrom our rough estimate Eq. (2.12). They are called strong liquids.Examples of strong glass formers are silica, germanate dioxide (GeO2)and open network liquids such as boron trioxide (B2O3) (see Fig. 2.6).

• In other materials, instead, the viscosity temperature dependence presentsa large deviation from the Arrhenius law and the viscosity pattern isphenomenologically reproduced by the VFT law Eq. (2.9). These ma-terials are referred to as fragile liquids. In fragile glass formers the mi-croscopic amorphous structure at Tg can be easily made collapsing and,

25


with little thermal excitation, it is able to reorganize itself in structureswith different particle orientations and coordination states. In terms ofthe free energy landscape the fragile glass presents much more degen-erate minima, separated by sensibly smaller barriers than the strongglass, see Fig. 2.7. Their fragility varies in a wide spectrum, and canbe as high as 200 for very fragile glasses. Some examples of fragileglasses are K+Ca2+NO3−, K+Bi3+Cl−, orto-terphenyl (OTP), toluene,chlorobenzene, but also selenium and sulfur. In general these are liquidscharacterized by simple, non-directional Coulomb interactions or, else,van der Waals interactions [55]. The most fragile substances known arepolymeric [36].

Figure 2.7: One dimensional pictures of the free energy landscape F in fragile andstrong glass formers. The horizontal axis represents the one dimensional projectionof the configurational coordinates of the degrees of freedom {r}. The crystal globalminimum is omitted. Figure from Ref. [29].

Other phenomenological relations than the VFT are also used to fit experi-mental data for the viscosity pattern of glass forming liquids, one example isthe so-called generalized Vogel-Fulcher-Tammann law

η = η∞ exp

(

B

T − T0

)γ

.�

�

�

�2.13

The exponent γ is usually set equal to 1, and an argument for setting γ = 1was originally given by Adam and Gibbs [51]. An alternative explanationfor this choice is provided in the framework of the so-called Random FirstOrder Transition (RFOT) theory [56]. However these studies do not excludeexponents γ > 1, always compatible with data, merely affecting the width ofthe fitting interval. On the contrary, analitic approaches [57] yield γ = 2 inthree dimensions. The same broadening can be implemented for strong glassformers, for which a generalized Arrhenius relaxation law, i.e. Eq. (2.13)with T0 = 0, can be used to properly fit the data of supercooled liquids [29].

A review of theoretical models for the glass transition would be far beyondthe scope of the present exposition. Further insight can be found in references[28] and [29].

26

Giallo e un color che si chiama orpi-mento (...) ed e propio tosco. Ede di color piu vago giallo resimiglianteall’oro, che color che sia.

Cennino CenniniIl libro dell’arte (c. 1390) 3

Amorphous chalcogenides

In the last part of this work, experimental investigations by means of Dy-namic Light Scattering (DLS) with infrared radiation will be reported onarsenic selenide and sulfide ChGs (see Chapters 6 and 7), as well as on amor-phous chalcogen elements S and Se. These compounds properties will besummarized in the present Chapter, without any intent of completeness. Foreach system, the main findings will be highlighted concerning structural, ther-mal, rheological as well as the basic optical properties. Light induced struc-tural changes, in which a great deal of research is currently involved, will beleft apart, mainly because this topic is not of direct interest for our presentachievements. For an extensive account on this subject the book by Popescu[1] can be consulted.

3.1 The chalcogens

The chalcogen elements belong to the VI-A subgroup of the periodic table.These elements are: sulfur, selenium and tellurium (the VI-A subgroup con-tains also the oxygen and polonium). The chalcogenides are compounds ofsulfur, selenium and tellurium with electropositive elements or with organicradicals. The name chalcogenide originates from Greek: χαλκoζ=copper,γενναω=born and ειδoζ=type being given initially to the chalcogenide min-erals that contain copper in combination with sulfur, selenium and tellurium.

3.1.1 Sulfur (S)

Sulfur has the atomic number Z = 16 and its atomic mass is 32.064. Thevalence electronic shell has 6 electrons with the disposal 3s23p4. Sulfur canbe found in the following oxidation states: -2, 0, +2, +3, +4, +5, +6. Thiselement exhibits several crystalline and non-crystalline forms. In order toexplain the atomic scale structure in the solid and liquid phase we must takeinto account some peculiarities of the chemical bonding.

Sulfur forms di-covalent bonds. It has two unpaired p electrons and canform σ-type bonds. The p orbitals are oriented reciprocally at 90◦ angles.The angle between sulfur bonds is 105◦. This value is very near to the char-

27

CHAPTER 3. AMORPHOUS CHALCOGENIDES

acteristic angle for the sp3 hybridization. Starting from these bonds it ispossible to define two distinct positions in the series of five bonded atoms:the cis or eclipses position and the trans or staggered position (Fig. 3.1).

Figure 3.1: The eclipsed (cis) and staggered (trans) configuration of the chalcogenbonds. Figure from Ref. [1].

The bonding in the configuration cis leads to the formation of ring molecules(S6, S8, ...) and the bonding in the configuration trans leads to the forma-tion of chain-like molecules. The angles between the planes defined by theatoms of a given configuration are called dihedral angles. These angles aresituated in the interval 90 − 100◦. If only σ-bonds would exist then a ran-dom rotation angle around a common bond should be expected. The specialsituation of two types of configurations appears due to the contribution ofthe π-bonds between the p-electron pairs on neighboring atoms. The ringmolecules S8 give the most stable structural configuration in the solid state.Other molecules as e.g. S6, S4 and also long chains of atoms (Sµ) can bepacked in the solid state of sulfur. The study of elemental sulfur offers anattractive challenge because of the unique diversity of stable molecules thatit can form in the gaseous, liquid and solid state and because the chemicalconversion of molecular species occurs at moderate temperature conditions[58, 59, 60, 61].

The stable crystalline modification at room temperature is orthorhombicsulfur (Sα). It is believed to consist of S8 rings as structural units; see Fig.3.2. After an orthorhombic to monoclinic transition at 96◦C, solid sulfurmelts at about 119◦C, forming a light yellow liquid of relatively low viscosity.According to an interpretation of IR and Raman experiments [62, 63], thisliquid is made up mainly of S8 ring molecules, too.

Liquid sulfur exhibits an anomalous dependency on temperature of theviscosity [64]; see Fig. 3.3.a. By heating the liquid, its viscosity firstly de-creases (down to about 0.1 poise at 155◦C, a viscosity near that of linseed oilin n.c.), following the behavior of most other liquids. Then, still continuing toraise the temperature the viscosity increases abruptly at Tλ = 159◦C and theliquid color becomes dark-brown (see Fig. 3.4). At about 187◦C the sulfurviscosity reaches its maximum (about 930 poise, comparable to the viscosityof peanut butter) and, thereafter, gradually decreases so that at 300◦C it

28

3.1. THE CHALCOGENS

Figure 3.2: The unit cell of orthorhombic sulfur. (a) The ring packing. (b), (c)Front and side view of the S8 ring. Figure from Ref. [1]

becomes again a soft, fluid mass. The shape of the viscosity change in thistemperature range, has led this transition to be called a ‘λ-transition’ (Fig.3.3.a). At the transition, other physical properties, such as the refractiveindex, density, thermal expansion and specific heat show anomalous behavior[61, 62, 63, 65, 66, 67, 68, 69].

These changes are known to be caused by a polymerization process involv-ing (i) the opening of sulfur rings to form diradical S∗

8 chains (initiation step)and (ii) the concatenation of S∗

8 to long chains Sµ (propagation step). Thetransition is thermoreversible and hence S8 rings form again upon cooling be-low Tλ. These macromolecule chains do not appear at low temperatures dueto the sulfur tendency to the bond in cis (eclipsed) configuration. Neverthe-less, when the temperature is raised, the probability for the bonding in transconfiguration increases. This leads to an equilibrium for the concentration ofS8 rings and chains of variable length, a processes which is usually termed‘equilibrium polymerization’ [75].

The experimental determination of the degree of polymerization Φ(T ), i.e.the relative fraction of polymer to monomer content, above the polymeriza-tion transition is still a matter of debate due to the inherent experimental dif-ficulties. The first available reliable set of experimental data is that providedby Koh and Klement [71] where Φ(T ) was estimated by rapidly quenchingliquid sulfur to room temperature. Equilibration temperatures where cho-sen from the interval 135-300◦C and quenching rates were estimated to be∼ 105 Ks−1. Solution of the quenched product in CS2 gave the weight frac-tion of Sµ as the insoluble portions.

Attempt to calculate Φ(T ) from Raman spectra on liquid sulfur across the

29


Figure 3.3: (a) Temperature dependence of liquid sulfur viscosity from Ref. [64].The inset shows viscosity values for the range up to 160◦ C. (b) Temperaturedependence of the extent of polymerization Φ(T ) from different experimental aswell as theoretical works. Full circles: data from [70]; crossed squares: data takenfrom Koh and Klement [71]; open circles: data taken from Kozhevnikov et al. [72].The thick solid line represents the prediction of a theoretical model from Wheeleret al. [14] while the dashed line the prediction of the Tobolsky-Eisenberg model[73]. Figure taken from [70].

λ-transition were made by Ward and Myers [76] who were able to demonstratethe increasing trend of this quantity but failed in providing a quantitativeand accurate description of this parameter. More recently [70], a new Ramaninvestigation provided a more quantitative insight in the degree of polymer-ization of liquid sulfur. A summary of experimental results is reported in Fig.3.3.b. These experimental findings suggest a rapid, monotonic increase in Sµ

content just above Tλ, followed by a less steep increase for T > 250◦C.

According to the Maxwell relation Eq. (2.8), the abrupt increase in vis-cosity at the polymerization transition should trigger an equivalent increaseof the structural relaxation time of liquid sulfur. Considering the ideal be-havior of a temperature-independent shear modulus G∞, at the λ-transitionone should expect the relaxation time to rise from ∼ 1 ps to ∼ 10 ns. Sincethe frequency position of the resonance is ruled by the condition ωτ ∼ 1, thisshould imply the detection of relaxation effects in the GHz frequency domainby means of those techniques, like Brillouin Light Scattering (BLS), that mea-sure the longitudinal modulus M = K + 4/3G (K being the bulk modulus).None of the correlated effects has ever been detected in BLS experiments inliquid sulfur [77, 78]. However, a recent investigation with InfraRed Photon

30

3.1. THE CHALCOGENS

Figure 3.4: Liquid sulfur at temperatures below (left side) and above (right side)the λ-transition. Figure from Ref. [74].

Correlation Spectroscopy [79] allowed to overcome photoinduced and absorp-tion effects, to reveal a so-called chain relaxation process with characteristictime in the ms range. The temperature trend of the average chain relaxationtime is found to abruptly increase just below Tλ, reaching its maximum valueat the transition. The present findings will be discussed more thoroughlyin Section 6.1, where a further study of the chain relaxation process, as afunction of both temperature and exchanged momentum, will be reported.

Theoretical efforts made to relate the dynamics of liquid sulfur to theunderlying polymerization phenomenon are mainly related to thermochemi-cal models [80, 81, 82], and provide hypothesis [82] on physical mechanismsaffecting this fluid behavior. A simple equilibrium polymerization theory ofliquid sulfur has been proposed by Tobolsky and Eisemberg [73], who definedtwo equilibrium constants associated to chemical reactions of polymerizationinitiation and propagation:

S8 ⇄ · S8·�

�

�

�3.1

· Sn8 · + S8 ⇄ · S(n+1)8·�

�

�

�3.2

Eq. 3.1 represents the opening of a S8 ring triggered by thermal effects, re-sulting in a ·S8· diradical. Eq. 3.2 describes the growth process of interactionbetween a diradical activated chain with an inactivated ring.

Wheeler and co-workers have shown [14, 83] that polymerization tran-sitions can be treated as second-order phase transitions described by non-classical exponents, and that the Tobolsky-Eisemberg model represents themean-field limit of their theory. Despite these theoretical efforts, an accuratequantitative description of the observed viscosity and degree of polymeriza-tion is still lacking. Predictions for Φ(T ) from the Tobolsky-Eisemberg andWheeler models are reported in Fig. 3.3.b.

31


3.1.2 Selenium (Se)

Selenium is the element with Z = 34 in the Periodic Table, and its atomicmass is 78.96. The configuration of the valence electrons is 4s24p4. Theoxidation states of selenium are -2, 0, +2, +4, +6. The sp3 hybridization isless stable than in sulfur.

The particular features of the crystalline states of selenium are based onthe tendency of the selenium atoms to have the trans configuration moreexpressed than sulfur atom. The most stable crystalline form of selenium isthe hexagonal (also known as grey or metallic Se). The lattice is made fromparallel helical chains (see Fig. 3.5). Every atom has two neighbors in itsown chain and four neighbors situated in neighboring chains. The radius ofthe helix is 0.98 A. Within the selenium chains the atoms are bonded bycovalent bonds and between chains act molecular forces of the Van der Wallstype [84]. The hexagonal unit cell contains three atoms. The bond angleis about 103◦ and the torsion (dihedral) angle is about 100◦. However, aweak covalentlike attraction between chains arising from the partial overlapof the unoccupied lone-pair orbital in one Se atom with the unoccupied p-like antibonding orbital in a neighboring atom is believed to exist since theintermolecular distances are appreciably smaller than the distances dictatedby the Van der Walls interactions . Monoclinic and rhombohedral Se are thecorresponding molecular crystals where the structural building blocks are theeight-membered Se8 and the six-membered Se6 rings, respectively [85, 1].

Figure 3.5: Structure of hexagonal selenium. (a) Chain configuration in the unitcell. (b) The atom chain (view along the c-axis). Figure from Ref. [1].

Amorphous selenium (a-Se) is a dark-grey solid, presumably the onlymonatomic glass known, at ambient temperature and pressure conditions.It was suggested that the amorphous phase would be built from disorderedchains and rings of di-covalent atoms. The covalent distance, the valence an-gle and the dihedral angle in non-crystalline selenium seem to be very similar

32

3.1. THE CHALCOGENS

to those from the hexagonal selenium crystal. The same Van der Walls forcesact between the chains. The density deficit of ∼ 10% in the amorphous phasesuggests that the packing of the structural units is far from close packing [1].

Diffraction data on this system could not answer the question concerningthe relative population of ring-like and polymer-like conformations, an issuethat is closely related to the nature of the medium-range structural order.This arises from the fact that by using either rings or chains one is ableto model equally well the obtained radial distribution functions [86]. Thesame argument has also been noticed by Ref. [87] where it was shown thatthe static structure factor reflects mainly the properties of the short-rangestructural order that is dominated by two- and three-body correlations; itwas quite insensitive, however, to a medium-range structural order that isrelated to more than four-body correlations.

Thus, in order to firmly establish the constitution of amorphous seleniumchemical methods for analysis are needed. Brieglieb [88] dissolved vitreousselenium in CS2 and demonstrated on the basis of selective dissolution thatthis type of selenium is made of a mixture of chains and rings. The proportionfound between chains and rings are closely dependent on the preparationconditions. Different experimental techniques have been used to estimate thechain length as a function of temperature, including viscosity measurements[89, 90, 91, 92], viscoelastic measurements, recoverable shear creep compliance[93], NMR [94, 95, 96]. Reported chain-lengths have been as small as 50 atomsto as large as 4 · 105 atoms in the vicinity of the melting point [89, 90, 91, 92,94, 95].

In a-Se the structural relaxation has been investigated by a number ofmeans including calorimetric [97], dilatometric [98], ultrasonic [99, 100, 101],and dielectric [102] techniques. Bohmer and Angell [103] reported a study ofthe stress relaxation modulus in a-Se. The mechanical correlation functionsof stabilized supercooled liquid Se have been found to exhibit two distinctdecay channels, attributed by the authors to the existence of a cooperativetransition, near 300 K, between ringlike and chain elements. The authors donot exclude a different interpretation, representing the step in the correlationfunction as the entanglement plateau, which is a well-known feature in themechanical spectra of high molecular weight polymers [104].

More recently, Raman spectroscopy has been employed to study the effectof temperature on the medium-range structural order of a-Se trough its glasstransition [84]. In this study the authors assert that a-Se is found to undergoa structural transition – partly related to monomer ↔ polymer equilibrium– even at temperatures as low as Tg.

From the viewpoint of numerical and theoretical investigations, the short-range structure of a-Se has been investigated by means of recently developedtight-binding [106, 107, 108] and ab initio [109, 110, 111, 112] moleculardynamics simulations, while a theoretical analysis taking into account themedium-range structure of this system is presented by Nakamura and Ikawa

33


Figure 3.6: Transmission of amorphous and liquid Selenium. Figure from Ref.[105].

[87].

Fig. 3.6 shows the absorption edge of selenium as a function of wavelength[105]. The refractive index has a maximum value of n=3.13 at about 500 nm.In the infrared range n=2.46. Optical properties associated with photoin-duced structural changes such as the photoconductivity and photodarkeningphenomena [113, 114, 112] have received much attention in semiconductortechnologies.

3.1.3 Tellurium (Te)

Tellurium has the atomic number Z = 52, the atomic mass 127.60 and theconfiguration of the valence shell is 5s25p4. It is a hard solid with metallicaspect. The oxidation states in compounds are +2, +4 and +6. Because thecis configuration is not favored in tellurium, it exists only one crystalline stateat normal pressure. This form is called α-tellurium, it exhibits hexagonalsymmetry and is analogous to hexagonal (grey) selenium.

Tellurium cannot be obtained in glassy state by melt quenching. The vis-cosity of the tellurium melt is, on the basis of the partial delocalization of thevolume electrons, so much reduced that the transition in the glassy state canbe reached only trough cooling rates as high as 1010 K/s [115]. The amor-phous state is obtained by evaporation and deposition on solid substratesmaintained at very low temperatures. Stuke [116] suggested that amorphoustellurium should have a distorted chain structure where the interchain bond-ing is weaker than in the hexagonal tellurium but the bonds within the chainsare longer and nearer to the covalent bond.

34

3.2. BINARY AS-SE AND AS-S CHALCOGENIDES

3.2 Binary As-Se and As-S chalcogenides

The present section focuses on properties of amorphous ChGs in the As-Sand As-Se binary series. The properties relative to the binary and ternarysystems formed by these latest are extensively described by Popescu [1].

The best studied are amorphous As2Ch3 systems, where Ch denotes achalcogen element. These systems show high stability against aggressive me-dia; they are stable in humid atmosphere, but less stable in alkali solutions.The stability of the glasses depends on the character of the chemical bonds,and increases in the series S → Se → Te. In the crystalline state these sys-tems are isostructural with monoclinic lattice. In particular, As2S3 is knownas a mineral under the name of orpiment; a sample is shown in Fig. 3.7.a.

In the systems As-S and As-Se several compounds in the crystalline stateare known besides As2S3 and As2Se3. The compounds As4Ch4 and As4Chwere described, but no reliable structural data are available up to today [1].As4S4 is known as a mineral under the name of realgar (Fig. 3.7.b), and itcrystallizes in the monoclinic system. The compound As4S is known as amineral under the name of duranusite, and it exhibits a layered crystallinestructure related to that of As. Its elementary cell is orthorhombic.

Figure 3.7: Arsenic sulfide minerals. (a) An orpiment sample (As2S3). (b) Arealgar sample (As4S4).

For binary series AsxSe100−x and AsxS100−x the glass-forming ability stronglyvaries with composition. Alloys can form glasses for 0 < x < 65 in the caseof AsxSe100−x, and for 0 < x < 45 in the case of AsxS100−x. It is remarkablethat, although As4S4 and As4Se4 are isostructural, As4Se4 is very stable inthe glassy state, while As4S4 cannot be obtained in the glassy state even byvery rapid quenching; only rapid quenching accompanied by high pressurecan determine the formation of glass.

For increasing concentration of arsenic in the system As-S, the stabilityagainst crystallization increases. The glass with the content of 6 at.% Ascrystallizes at room temperature in a day with the formation of rhombic

35


sulfur, while the glass of composition As2S5 (x=28.6) cannot be crystallizedby thermal annealing. The glasses situated in the range 5÷16 wt.% As usuallycrystallizes by long time annealing at 60◦C with the formation of rhombicsulfur. As2S3 (x=40) crystallizes completely by annealing at 280◦ in 30 days.The glasses with high sulfur content crystallizes under light irradiation.

Crystalline As2Se3 was prepared by annealing glassy materials at 688 Kfor ten days [117]. In the system As-Se the glasses crystallize under pressureand high temperatures in all the composition range.

3.2.1 Structural properties

The local structure of binary ChGs, considering both the bulk and thin filmforms, was extensively studied using neutrons and x-rays in the 1960s and1970s. The vast majority of studies have been made on As2S3 and As2Se3,which have been primarily the only alloy used in the As-S-Se system untilrecently.

Goriunova and Kolomiets pointed out the importance of covalent bondingin ChGs as the most important property to guarantee stability of these glasses[118]. Vaipolin and Porai-Koshits reported x-ray studies in the beginning ofthe 1960’s [119, 120] of vitreous As2S3 and As2Se3 and a number of binaryglass compositions based on these two compounds. These glasses were shownto contain corrugated layers, which were deformed with increasing size of thechalcogens. The character of the bonds was also shown to become more ionicwhen at equimolar compositions. Nevertheless, resolution in both reciprocaland real space was very often not sufficient to obtain details of the glassnetwork organization.

The emergence, during the last decade, of third generation synchrotronas well as spallation neutron sources allowed new possibilities to be explored.On one hand, x-ray absorption near edge structure (XANES) and extendedx-ray absorption fine structure (EXAFS) have exposed researchers to a morethorough understanding of the electronic and structural properties of theseamorphous materials, and especially in As2S3 [121, 122, 123]. These two pre-vious techniques are element-specific and allow one to investigate the short tomedium range structure around the absorbing atom [124, 125]. On the otherhand high resolution neutron diffraction measurements over a wide range ofexchanged momenta, Q, with or without isotopic substitution, anomalousx-ray scattering, and high energy x-ray diffraction experiments were carriedout, mostly for stoichiometric glass compositions [126].

Byrchkov et al. [127] conducted a high energy x-ray diffraction investi-gation on arsenic selenide glasses at compositions covering the entire glassformation region. These glasses are confirmed to remain homogeneous onmesoscopic scale over the entire vitreous domain. Moreover, the lowest-Q diffraction peak of the static structure factor S(Q) exhibits remarkablechanges with composition which reflect changes in the glass network on boththe short- and intermediate-range scales.

36


More involved is the investigation of short- and medium-range structurein AsxS100−x systems, mainly because the S-rich binaries are believed to phaseseparate. Kawamoto and Tsuchihashi [128] were the first who observed twotypes of sulfur in bynary GexS100−x glasses, on the S-rich side. They per-formed solubility experiments in liquid CS2 and found (i) an insoluble species,as usual attributed to sulfur chains, existing above x ≈ 20, and (ii) a solublespecies, interpreted as molecular sulfur rings, appearing at x < 20. The solu-ble species was actually recognized as being constituted of S8 rings after laterRaman spectroscopy investigations [129, 62]. A simple model for sulfur-richstructure of AsxS100−x glasses was proposed to account for these results [130].Let us consider the stoichiometric glass structure, in which AsS3/2 pyramids

Figure 3.8: Schematic representation of the simple structural model for sulfur-richAsxS100−x glasses: (a) A fragment of the stoichiometric As2S3 glass with two CS-AsS3/2 pyramidal units. (b) Transformation of bridging sulfur into sulfur dimerin the domain 25 < x < 40. (c) A fragment of the glass structure in the domainx< 25 with four isolated ISO-AsS3/2 pyramids, separated by S2 dimers, and a S8

ring. Figure is taken from Ref. [130].

share their corner S atoms (‘Corner Sharing’ or CS-AsS3/2 structure); seeFig. 3.8.a. Excess S atoms are then added to the stoichiometric composition,which transform bridging S into S dimers (Fig. 3.8.b). Then it is supposedthat a ‘saturated’ composition exists, below which all the AsS3/2 pyramidsbecome isolated, i.e., they do not share any corners and are separated bythree S dimers. This saturated composition will be x = 25 (i.e. independentAsS3 pyramids linked by S-S bonds). For x < 25 S8 rings start forming in theglass network (Fig. 3.8.c).

Phase separation in As-S glasses is the subject of a recent experimentalstudy again by Byrchkov et al. [130], in which high energy x-ray diffraction,small angle neutron scattering (SANS), Raman spectroscopy (RS), and dif-ferential scanning calorimetry (DSC) techniques are employed. RS results arefound not to support the simple structural model previously introduced. Inparticular, for what concerns the ‘saturated’ composition, the characteristicvibrations of sulfur rings are found to appear at x . 28; Raman spectra arereported in Fig. 3.9.a. Nevertheless, the existence of the two suggested com-

37


Figure 3.9: (a) Raman spectra for selected sulfur-rich AsxS100−x glasses. Char-acteristic vibrations of sulfur rings at 152 and 220 cm−1 appear at x ≤ 28.5, i.e.,below the ‘saturated’ composition in the simple model. (b) Amplitude of the4.5 A peak in the radial distribution function g(r) (obtained from neutron andhigh-energy x-ray diffraction techniques), reflecting S-S third neighbor in the sul-fur rings, as a function of the As content in the binary AsxS100−x glasses. Sulfurrings concentration in the network is proportional to this amplitude. Figure fromRef. [130].

position domains seems to be evident. From neutron and x-ray diffractionit is found that the pair correlation function g(r) develops characteristic fea-tures in the sulfur-rich region, similar to those observed in liquid sulfur belowthe polymerization transition, where it is believed to consist of S8 rings. Theamplitude of these features, plotted against the As content, reflects the con-centration of sulfur rings. The trend obtained by Byrchkov et al. from one ofthe addressed peaks is reported in Fig. 3.9.b. The sulfur-rich region is alsoaccompanied by characteristic an systematic changes in the SANS intensitybetween 0.01 and 0.1 A−1 exchanged momentum. Thus the typical size of theglassy phase formed upon S addition is estimated between 20 and 50 A. Theonset of phase separation for x . 28 is also confirmed by DSC measurements,but this topic will be further stressed in Section 3.2.2.

A characteristic common to ChGs is their electronic structure, originat-ing in the p-like lone pair electrons of the chalcogen. The sulfur atom, forexample, has four outer p electrons. The two remaining p electrons form aso-called ‘lone pair’ of electrons. Note that there exist no essential differencein the band structure between crystalline and glassy As2Ch3. While the twofilled (σ) p-states lie at the bottom of the valence band , the filled lone pair(LP) p-states at the top of the valence band are separated from the empty σ∗

states of the conduction band by the bandgap [131]. These LP states which

38


can be excited by light irradiation to the bottom of the conduction bandwith the creation of an electron-hole pair, are at the bases of many modelsproposed in the years for accounting photo-structural changes in ChGs [1].

3.2.2 Thermal properties

Differential scanning calorimetry (DSC) has been used to establish glass tran-sition temperatures, Tg, for the past 50 years. Although the glass transitionitself is quite wide (> 20◦C), one can usually localize the inflection point ofthe heat flow endotherm to within ±2◦C, and thus define Tg. DSC investiga-tions are documented for AsxS100−x [132, 133] and AsxSe100−x systems [134]at varying compositions.

However, as already pointed out in section 2.1.4, it is well known that notonly does the shape of the heat flow endotherm depend on the baseline of theinstrument, but also on the thermal history of the sample and the scan rateemployed. Several of these limitations have been overcome in a recent variantof DSC, known as modulated DSC (MDSC). The new instruments are widelyused in industry for quality control, especially of plastics, but are enteringresearch practice only slowly [135]. In MDSC, the programmed heating rate[136] includes a sinusoidal temperature modulation superimposed on a lineartemperature ramp used to scan through the glass transition. Because ofincreased sensitivity, an order of magnitude reduction in scan rates (1 −3◦C/min) can be used in MDSC in relation to those (10− 20◦C/min) used inDSC. Furthermore, in MDSC it is possible to deconvolute the total heat-flowrate into a part that tracks the temperature modulation and is known as thereversible heat-flow rate, leaving a part that does not track the temperaturemodulation, which is known as the non-reversing heat-flow rate.

Experience on a wide variety of glass systems shows that the non-reversingheat-flow rate displays a Gaussian-like peak as a precursor to the glass tran-sition [137]. The latter is observed as a smooth step-like shift of the reversingheat-flow rate as illustrated in Fig. 3.10. The area under the non-reversingheat-flow rate will be denoted henceforth as ∆Hnr, the non-reversing heat-flow. It is found to saturate in time typically after 100 hours to a fully relaxedglass. Hence, ∆Hnr can be conveniently used to denote the saturated value,and it measures the latent heat between the relaxed solid glass and its melt.The sigmoidal jump in the reversing heat-flow observed in these experimentsis independent of the baseline of the instrument. It establishes the thermo-dynamic jump ∆CP in the specific heat between the glass and its melt andits inflection point can be taken to define Tg.

MDSC data are available for both series AsxSe100−x [139, 138] and AsxS100−x

[140] of our interest, and values for reversing and non-reversing heat-flows arereported for these systems in Fig. 3.11. In particular the AsxS100−x series isquite peculiar as far as its thermal behavior is concerned.

We already pointed out that the onset of phase separation has been pro-posed for this series in the sulfur-rich region (Section 3.2.1), and thus a signa-

39


Figure 3.10: MDSC scan of As45Se55 glass showing deconvolution of the total heatflow Ht into reversing, Hr and non-reversing, Hnr, components. Figure is takenfrom Ref. [138].

ture of phase separation is somehow expected also from calorimetry measure-ments. T. Wagner et al. [140] showed that below x=25 two jumps ∆Cp1(x)and ∆Cp2(x) of the specific heat path are found at two well separated tem-

peratures T(1)g (x) and T

(2)g (x). The situation, which had never been reported

previously with conventional DSC investigations, is illustrated in Fig. 3.11.d;thus two glass temperatures are obtained for x < 25.

While the lower glass temperature T(1)g depends on the As content, the

higher glass temperature T(2)g remains almost constant, and very close to

the typical Tg of As28.5S71.5 (As2S5). These findings are confirmed by recentconventional DSC results [130]. The authors suggest the presence of two glass

phases in the sulfur-rich region and assume that the nearly constant T(2)g close

to the Tg of the As2S5 sample is an indication of the similarity of the structureof the As2S5 and the high temperature phase in the AsxS100−x samples withx < 25.

40


Figure 3.11: Summary of MDSC results for AsxSe100−x (a, b, c from Ref. [138])and AsxS100−x (d, e from Ref. [140]) glasses. (a) Tg(x) variation for As-Se. (b)∆Hnr(x) variation for As-Se. (c) ∆CP (x) variation for As-Se. In panels a, b and cthe (•) data points are taken from Ref. [138], the (△) data points are taken fromRef. [139] and the (◦) data points are taken from Ref. [134]. (d) Tg(x) (�), Tg1(x)(N), Tg2(x) (•) and non-reversing heat-flows ∆H(x) (�), ∆H1(x) (�), ∆H2(x) (◦)variation for As-S glasses [140]. (e) Specific heat capacity vs. composition forAs-S glasses CP (x) (�), CP1(x) (◦), CP2(x) (△). Values are taken at Tr = 0.9Tg.Change in the specific heat capacities ∆CP (x) (♦), ∆CP1(x) (�), ∆CP2(x) (◦) atthe glass transition [140]. Lines are drawn as guides for the eye.

41


3.2.3 Viscosity and structural relaxation

As-Se

Viscosity and recoverable shear creep compliance measurements of amorphousAsxSe100−x binary solutions have been made below and above the glass tran-sition temperature by Bernatz et al. [93]. The solutions all show a VFTtemperature dependence of the viscosity, except for the 30at.% As, which isnearly Arrhenius in nature. The thermal gradient of viscosity near Tg appearsto decrease as the arsenic concentration x increases in the explored composi-tion range (5 ≤ x ≤ 30); viscosity data are reported in Fig. 3.12.a. Estimates

1.8 2.1 2.4 2.7 3.0 3.30

5

10

15

1.5 1.8 2.1 2.4 2.7 3.0 3.3

5

10

15

x=40

x=30

x=20

x=12

.5

x=40 x=

30

x=20

x=10 x=

5(a)

AsxSe

100-x

1000/T [K-1]

Lo

g10 (η/p

oise)

Lo

g 10(η

/po

ise)

1000/T [K-1]

AsxS

100-x

(b)

a-Se

Figure 3.12: (a) Arrhenius plot of the viscosity for AsxSe100−x alloys. The viscos-ity of amorphous selenium (a-Se) is also reported. Data are taken from Ref. [93],apart from the open diamonds for a-Se which are taken from Ref. [141] and theopen circles for x=40 which are taken from Ref. [142]. VFT (- - -) and Arrhenius(—) models, obtained from best fit procedures, are also reported. (b) Arrheniusplot of the viscosity for AsxS100−x alloys. Data are taken from Ref. [143]. VFT (-- -) models are also reported.

of the fragility parameter for these samples indicate a significant decrease forcompositions x=20 and 30. Data are summarized in Tab. 3.13. Moreover,the recoverable shear creep compliance plotted versus time is found to exhibita plateau whose magnitude is temperature dependent for all but the 30% Assample. In more detail, as the As concentration increases, the overall height ofthe plateau decreases, and this implies the inability to reduce the compliancecurves corresponding to different temperatures to a single reference curve (forx< 30). Such phenomenology is known as thermorheological complexity.

More recently Malek and Shanelova [142] reported on a viscosimetry inves-tigation of the stoichiometric mixture As2S3 (x=40). Viscosity dependence ontemperature has been described with an Arrhenius relation (data are includedin Fig. 3.12.a). Non-exponential relaxations obtained by length dilatometry[142] are described with a KWW function, providing a stretching parameterβ ∼ 0.7.

42


AsxSe100−x AsxS100−x

As at. % Tg [◦C] m T0 [◦C] Tg [◦C] m T0 [◦C]

0 32.0 5, 44 7 64 5, 87 11 -45 5 -30 9 116 10 –5 45.0 5 64 5 -32 5 – – –10 61.0 5 70 5 -12 5 33 2 – –

12.5 – – – 43 1 50 1 -65 1

15 97 3 – – 55 2 – –20 89.0 5 40 5 -95 5 94 2, 78 1 44 1 -30 1

25 119 3 – – 137 2 – –30 112.0 5, 137.0 3 31 5 -273 5 148 2, 125 1 40 1 -41 1

35 160 3 – – 178 2 – –40 187 3, 166.8 4 38 4 -273 4 212 2, 178 1, 37 1 -98 1

185 8

42.7 – – – 161 1 – –45 194 3,188 6 – – – – –

Figure 3.13: Summary on composition dependence of glass temperature Tg,fragility m and Vogel temperature T0 for arsenic selenide and sulfide solutions.These parameters are reported for comparison also in the case of pure chalcogens(0 As at.%). Estimates are derived from: (1) viscosimetry [143]; (2) MDSC [140];(3) MDSC [139, 138]; (4) viscosimetry [142]; (5) viscosimetry [93]; (6) conventionalDSC [133]; (7) conventional DSC [134]; (8) viscosimetry [144]; (9) thermodynamicmeasurements [145]; (10) inelastic x-ray scattering [146]; (11) mechanical relaxationexperiments [36].

As-S

Viscosimetry investigation of supercooled AsxS100−x for a broad range of com-positions (12.5 ≤ x ≤ 42.7) have been reported in 1964 by Nemilov [143], andthis remains the most accurate characterization of rheological properties ofthis series up to date. An estimate of the fragility m for these systems canthus be obtained, which is reported in Tab. 3.13. Viscosity data are reportedin Fig. 3.12.b for a selection of compositions. The non-Arrhenius characterof the temperature dependence of viscosity versus temperature paths is moreand more pronounced as the stoichiometric concentration (x=40) is aban-doned towards lower arsenic content. The glass temperature estimated froma viscosity attaining 1013 poise, T13, is compared with the calorimetric glasstemperature Tg in Fig. 3.14. We note that, even if T13 systematically under-estimates Tg, its behavior follows the Tg path at decreasing x up to x=25,where two glass temperatures are found by MDSC (see Section 3.2.2). Then

it follows the lower glass temperature T(1)g .

More recently Malek [144] reported viscosity data for the supercooledAs2S3 melt, confirming the Arrhenius nature of the temperature dependenceof viscosity in this system over the range of four decades. Nevertheless, theauthor points out that his data are slightly shifted to higher temperaturein comparison with Nemilov’s data, resulting in higher estimation of viscosi-

43


Figure 3.14: Comparison between calorimetric glass transition Tg, obtained fromMDSC data (•, [140]) and ‘viscosimetric’ glass transition T13 (N) obtained as thetemperature at which viscosity attains 1013 poise [143], for AsxS100−x alloys. Forx < 25 two glass temperatures are obtained by MDSC (see Section 3.2.2), whileT13 follows the lower Tg path. The substantial offset between the two estimatesof the glass temperature may depend, according to an objection rased by Malek[144], on a calibration offset in the viscosity measurements by Nemilov.

metric glass temperature T13. This is ascribed by Malek to the calibrationprocedure, unavoidable if penetration technique with a cylindrical indenter isused, as in the work by Nemilov.

The structural relaxation of the stoichiometric arsenic sulfide glass wasalso studied by length dilatometry [144]. Non-exponential relaxations are de-scribed with a KWW function, and provide an almost temperature-independentstretching parameter β ∼ 0.8, corresponding to a relatively narrow distribu-tion of relaxation times.

3.2.4 The onset of an intermediate phase in binary chalco-genide glasses

The present section focuses on several aspects of some recent data [138, 147,148] on chalcogenide network glasses, which bears on a long-standing issueconcerning the existence of a topological transition proposed by Tanaka [149];a picture which was subsequently extended by Boolchand et al., who consid-ered the occurrence of a further topological transition. These issues providea tentative account for variation of amorphous chalcogenide materials prop-erties with composition, thus disclosing interesting perspectives for a moresystematic study of these systems.

In two recent reviews [150, 135], Boolchand et al. consider the elastic(Raman mode frequencies) and thermal (MDSC paths) response of chalco-genide glasses examined as a function of cross-linking, or mean-coordinationnumber, r. The authors describe how the highlighted experimental results

44


lead to the existence of three distinct glass phases, the so-called floppy, in-termediate and rigid phases that are characterized by distinct elastic powerlaws as a function of r. In the following we will stress the basic theoretical aswell as experimental findings underlying the onset of an intermediate phasein binary ChGs, mainly following Ref. [150].

In 1979 Phillips introduced ideas based on mechanical constraint-countingalgorithms to explain glass forming tendencies in network forming systems[151]. He reasoned that for a liquid to form a glass composed of a networkpossessing well-defined local structures, interatomic forces must form a hi-erarchical order. The strongest covalent forces between nearest neighborsserve as Lagrangian (mechanical) constraints defining the elements of localstructure (building-blocks). Constraints associated with the weaker forces ofmore distant neighbors must be intrinsically broken leading to the absence oflong-range order. He speculated that the glass forming tendency is optimizedwhen the number of Lagrangian local-bonding constraints per atom, nc, justequals the number of degrees of freedom. For a 3d network

nc = 3 .�

�

�

�3.3

In covalent solids there are two types of near-neighbor bonding forces; bondstretching (α-forces) and bond-bending (β-forces). The number of Lagrangianbond-stretching constraints per atom is nα = r/2, and of bond-bending con-straints nβ = 2r − 3, with r being the valence of the atom. For the casewhen all α- and β-constraints are intact and no dangling ends (i.e. one-foldcoordinated atoms) exist in the network, Eq. (3.3) implies that the optimummean coordination number is r = 2.40. Highly overcoordinated or undercoor-dinated structures are not conducting to glass formation and, upon cooling,lead to crystalline solids.

In 1983, Thorpe pointed out [152] that undercoordinated networks wouldpossess, in the absence of the weaker longer range forces, a finite fraction ofzero-frequency normal vibrational modes, the floppy modes. In fact, he foundfrom simulations on random networks that the number of floppy modes peratom, f , is rather accurately described by the mean-field constraint countaccording to the relation

f = 3 − nc .�

�

�

�3.4

This led to the realization that a glass network will become spontaneouslyrigid when f → 0, defining a floppy to rigid phase transition.

In glasses one rarely observes the solitary rigidity glass transition rc = 2.40[153]. In part this is because glasses soften at a finite temperature Tg, incontrast to the constraint and harmonic elasticity theory predictions whichare based on T = 0K calculations. Since α-forces exceed β-forces by a factorof three or more, in glass systems possessing a high Tg and/or weak β-forces,constraints associated with β-forces may be intrinsically broken [154, 155].For a network with a finite fraction mr/N of r-fold (r ≥ 2) coordinatedatoms that have their β-constraints broken, the mean-field rigidity transition

45


will be upshifted [153] in r according to

rc = 2.40 +2

5

(m2

N

)

+6

4

(m3

N

)

+ 2(m4

N

)

.�

�

�

�3.5

Even with the aforementioned correction, the proposed picture of therigidity transition in covalent glass forming networks would not change on aqualitative ground. To understand the occurrence of an intermediate phasecovering a finite composition range, a distinction between the onset of rigidityand the onset of stress should be done.

A network can be isostatically rigid, i.e. with no stress present. Stressesare said to exist when already present bond length and/or angles must changeon the addition of more atoms to the structure. Thus, the imposition ofadditional cross links or constraints can lead to redundant constraints andto the accumulation of stress. An example can serve to illustrate the idea;with reference to Fig. 3.15. A square is not a rigid structure because it

Figure 3.15: Three examples of elastic network skeletons in two dimensions withnearest-neighbor forces only. A quadrilateral is floppy (isostatic, stressed) with no(one, two) cross braces. Note that there are eight degrees of freedom for the fouratoms, but three of these are associated with two translational and one rotationalrigid modes. Each spring, here, is an independent constraint. Figure from Ref.[135]

can be sheared into a rhombus. A crossbar attached to opposite vertices ofthe square will result in an isostatically rigid structure. A second crossbarattached across the remaining two vertices provides a redundant constraintand will result in an accumulation of stress in the structure. Isostatically rigidand stressed rigid random structures are expected to be physically differentin behavior. This results in the onset of two transitions, a floppy to anisostatically rigid state, followed by an isostatically rigid to a stressed rigidstate upon varying the system composition. We will address to these twotransitions as r1 and r2, respectively, and will consider in the following theexample of a binary ChG alloy as, for example, systems like AsxSe100−x andSixSe100−x.

Thus, we distinguish among three average coordination regions, and un-derline the experimental evidences put forward by Boolchand et. al [150, 135]for the peculiar glassy behaviors in each region.

46


• r < r1, or floppy region. This region corresponds to underconstrainedglass compositions. Examples are given by AsxSe100−x for x< 29 orSixSe100−x for x< 20.

As already stated in Section 3.2.2, the non-reversing heat-flow ∆Hnr canbe experimentally probed by MDSC. The floppy region is associatedwith a non-vanishing ∆Hnr. This quantity is found to decrease onincreasing r, and to become almost zero at the floppy to rigid transitionr1. An example is given by MDSC data on AsxSe100−x shown in Fig.3.16.e (white circles in the floppy region) [135]. As further examplesnon-reversing heat-flows are reported for for GexSe100−x and SixSe100−x

glasses in Fig. 3.16.b and 3.16.d, respectively.

Molar densities increase in this region on increasing r towards r1. Inother words, the results of the space filling-condition is an increase indensity when chains are cross linked.

Substantial ageing phenomenology characterizes this window.

In many cases, when vibrational bands associated with a specific net-work building block – as could be the As(Se1/2)3 pyramid in AsxSe100−x

– can be resolved in the Raman lineshape, it is possible to quantitativelyfollow mode-frequency changes with glass composition. A plot of themode frequency νCS in systems like GexSe100−x or SixSe100−x glasses asa function of composition x, shows a kink [150] at r1. Below this value apeculiar linear behavior is suggested for νCS as a function of r by exper-imental investigations on binary ChGs [150], and measurements at stilllower average coordinations [150] show that νCS becomes independenton sample composition. Raman modes frequency vs. composition plotsare reported for GexSe100−x and SixSe100−x glasses in Fig. 3.16.a and3.16.c, respectively.

The viscosity depends on temperature for these glass compositions ina non-Arrhenius fashion, and these glasses can be classified as fragile(see Section 2.1.5); see also Fig. 3.16.e (black dots).

• r1 < r < r2, or isostatically rigid region. This region corresponds tooptimally constrained networks, and this phase can be said to be in anisostatic mechanical equilibrium. Examples are given by AsxSe100−x for29 <x< 37 or SixSe100−x for 20 <x< 26.

∆Hnr varies extremely rapidly near the transition r1, dropping in fa-vorable cases nearly to zero in this intermediate region, thus forminga so-called ‘reversibility window ’. This suggests that glass- and liquid-structures in the window compositions are closely similar to each otherand that both are stress free.

In this strain-free intermediate phase, across a narrow range of composi-tions the density is nearly constant as percolative strain-free backbonesare added to the network.

47


There is almost no ageing in this region [156].

Raman mode frequency as a function of composition can be fitted toan underlying elastic power law by plotting ν2

CS against r − r2 on aLogLog scale. Sub-linear power laws with exponent p < 1 are generallyobtained in this region [150].

The viscosity depends on temperature for these glass compositions inan Arrhenius fashion, and these glasses can be classified as strong.

• r > r2, or stressed rigid region. This region corresponds to overcon-strained networks.

This region is again associated with a non-vanishing ∆Hnr, which startto rapidly increase as the isostatically rigid region is abandoned.

Molar densities again start to decrease as the network is overconstrained.

Again substantial ageing is found for these glass compositions [156].

By plotting squared Raman mode frequency ν2CS against r − r2 on a

LogLog scale, one obtains a power law of p which is predicted to be inthe rigid region p = 1.40− 1.50. It has been confirmed by experimentalfindings [150].

Non-Arrhenius temperature dependence of viscosity: these are fragileglasses.

48


Figure 3.16: (a) Raman mode frequency variation of corner-sharing tetrahedra inGexSe100−x plotted as a function of x [150]. (b) Non-reversing heat variation ∆Hnr

in GexSe100−x glasses [150]. (c) Raman mode frequency variation of corner-sharingtetrahedra in SixSe100−x plotted as a function of x [157]. (d) Non-reversing heatvariation ∆Hnr in GexSe100−x glasses [157]. (e) Non-reversing heat variation (◦),∆Hnr, and activation energy (•) for viscosity dLogη/dT at Tg, in AsxSe100−x glassesas a function of As content. Note both observable track each other suggesting acorrelation between glassy network structure and liquid’s dynamics [150]. (f) Non-reversing heat variation ∆Hnr as a function of mean coordination number r for fourdifferent glass systems. Ternary systems are also included for comparison. The Ge-As-Se ternary shows the widest window [150] while the Ge-S-I the narrowest [150].

49


From our viewpoint, a relevant aspect related to the onset of a reversibilitywindow in ChGs, is that concerning the compositional trend of glass fragility:a quantity which is accessible by studying the structural relaxation dynamicsin these systems.

In fact, the fragility behavior with composition is found to mimic thenon-reversing heat-flow, with a sharp decrease in the reversibility window, asshown for the case of AsxSe100−x in Fig. 3.16.e. However, it is worth nothingpointing out that the fragility parameter is found to decrease roughly by afactor two in the intermediate region, while ∆Hnr decreases by a factor tenor more in the same range. The correlation observed in Fig. 3.16.e shows aconnection between the glass structure and liquid dynamics. Thus, conceptsof global connectivity developed to describe glass networks would extend wellinto the liquid state and, conversely, the underlying dynamics of structuralarrest of a liquid on solidification would be related to the global connectivityof the liquid structure. Glass compositions in the intermediate phase giverise to strong liquids, while both floppy and rigid glasses give rise to fragileliquids. Fragility thus emerges from the proposed picture as a multivaluedconcept, and a new question arises on what manner does the temperaturedependence of viscosity of floppy liquids differ from that of rigid liquids.

However, if by one hand the fragility classification has the advantage ofproviding a one-parameter description for the glass transition, on the otherhand it is well known that this classification is by no means a naive represen-tation of a more subtle phenomenology.

In perspective, a deeper connection between liquid dynamics and glassynetwork rigidity than that emerging from the comparison proposed in Fig.3.16.e, would serve to shed more light on the picture presented so far. Inother words, two questions are brought to our attention:

Can one recognize a signature of the reversibility window not onlyin the way the structural relaxation time behaves with temperature(Arrhenius-strong behavior inside the window, non-Arrhenius-fragilebehavior outside), but also in the shape of relaxation functions?

If different behaviors are found, is the transition sharp enough to berelated to distinct dynamical regimes?

An answer to these questions could be provided by a careful analysis of longtime dynamics in supercooled ChGs probed by means of photon correlationtechniques, at varying compositions of well defined systems.

3.2.5 Optical properties

Sulfides usually are transparent in the high wavelength range of the visibleregion where selenides or tellurides are completely opaque (see also Fig. 5.10in Section 5.4). However them all are highly transmissive in the near-infrared

50


and mid-infrared regions, specifically 12 µm, 15 µm and 20 µm for sulfides,selenides and tellurides respectively [158].

Because of the lack of long range order in ChGs, their absorption bandis radically changed when comparing the crystalline and amorphous states[159]. The optical absorption edge dependence on temperature was reportedfor both amorphous and crystalline As2S3 by Zakis and Fritzsche [160].

Tauc compared the absorption edge of amorphous- and crystalline-As2S3

[159] and showed that the high absorption (α > 104 cm−1) region was mostlyassociated with transitions from localized valence band states to delocalizedconduction band states, or vice-versa. For 1 < α < 104 cm−1, the absorptionwas declared to be due to the presence of band tail states (the ‘Urbach tail’)that extends into the gap. The third region of the band (α < 1 cm−1) wasfound to be related to the preparation, purity and thermal history of thematerial.

The refractive indices in the systems As-S and As-Se range from 2 to & 3[161]. Sanghera and Aggarwal [162] summarizes many optical constants forAs2S3.

The optical gap of As2Se3 was determined to be ∼ 1.76 eV [163, 164] andthe photoconductive gap was found to be 2.1 eV [165]. Kitao [166] measuredthe refractive index of many AsxSe100−x glasses.

The amorphous As2Se3 and As2S3 vibrational spectra were detailed byZallen et al. using Raman and infrared tools [167]. These authors show thatthe As atom vibrates in opposition to its three neighboring chalcogens. Lu-covsky and Martin then proposed a schematic representation of the molecularorder in a-As2S3 [168]. Later Raman and IR spectra on ChGs films showedobvious similarities between amorphous and crystalline states [169].

The recent advances in nonlinear optics have contributed to the increasein research on the ChGs subject. Nonlinear refractive index determination inAs2S3 fibers have been attempted [170] for switching applications [171]. Otherinvestigation of nonlinear optical properties in As2S3 and As2Se3 followed, alsoincluding Z-scan investigations [172].

51

Part II

Materials and methods

53

There is another reasonWhy you should give attention to those bodiesWhich are seen wavering confusedlyIn the rays of the sun: such waverings indicateThat beneath appearance there must beMotions of matter secret and unseen.

Lucretius in De Rerum Natura 4Dynamics investigation through light

scattering

Dynamic Light Scattering (DLS) techniques are generally divided into twomain classes: those that use ‘time-domain’ techniques to measure the fre-quency distribution of the scattered light, and those that directly measurethe frequency distribution by placing a monochromator (‘filter’) before thedetection photomultiplier.

The main subject of this chapter, i.e. DLS performed with the use of time-domain techniques, may be said to date back to 1960s, when it was shown,in a work by Robert Pecora [173], that the small frequency broadening in thelight scattered from dilute solutions of macromolecules contained informationabout diffusion of the macromolecules. In addition, the application of DLSto the study of critical phenomena was noted.

The experimental realization of these ideas depended not on directly mea-suring the small frequency changes involved by the use of filter analyzers -they are usually much too small to be measured with conventional monochro-mators - but on the use of ‘optical beating’ techniques. With the gradualreplacement of spectrum analyzers by autocorrelators in most experiments,new terms came to describe the field, the most important of which is perhaps‘photon correlation spectroscopy’ (PCS). PCS denotes a digital technique formeasuring intensity fluctuations in which the number of photons arriving ata detector at a set time interval is repeatedly counted and its time autocor-relation function computed.

Photon correlation techniques are usually applicable to the measurementof frequency changes in the approximate range from 10−2 to 106 Hz, i.e. dy-namics evolving on timescales ranging from the µs to 100 s. These timescalescorrespond to what, on a molecular scale, are long-distance, long-time phe-nomena. Thus, these techniques are especially well suited for measuring dy-namic constants associated with macromolecular and particulate systems.These include diffusion in dilute solutions or suspensions, effects of interac-tions between the large species in more concentrated but still relatively dilutesolutions, and interaction effects in semidilute and concentrated solutions.

The investigation by means of PCS of the structural relaxation dynamicsin supercooled liquids on approaching the glass transition has contributed,

55

CHAPTER 4. DYNAMICS INVESTIGATION THROUGH LIGHT SCATTERING

from the 1980s, to shading more light on such a rich phenomenology. Awealth of PCS data on organic glass formers has been collected since then,providing a fascinating but still open picture of the dynamics responsiblefor the structural arrest [174, 175]. Among inorganic glass formers only fewsystems, mainly oxides, have been studied by PCS techniques [176, 47], whilethe family of non-transparent chalcogenide glasses has remained, up to now,untouched.

The present chapter aims at introducing the reader to the theoreticalframework underlying a DLS experiment with a time-domain technique. Sec-tion 4.1 will deal with the general theory of light scattered from an isotropicmedium, and will show how dynamical information can be extracted from thescattered light. Then the homodyne technique will be introduced in Section4.2, and a brief overview of the salient features of an heterodyne experimentwill be given in Sec. 4.3. References [177, 178] will be mainly followed.

4.1 Light scattering theory

4.1.1 Light scattered from an isotropic medium

Fig. 4.1.a shows the top view of a typical scattering experiment. Radiationis incident on a sample (or scattering medium), say a suspension of colloidalparticles or a solution of polymers or surfactants. Some of the radiationpasses through the sample unaffected, some is scattered. A detector is setup at scattering angle θ, and the intensity I(θ, t) of the scattered radiationis measured. Typically, the incident and scattered fields are shaped by aper-tures, slits, the walls of the sample’s container, or by optics such as lenses.The region of sample which is both illuminated by the incident beam and‘seen’ by the detector is called the ‘scattering volume’ V.

As we will see shortly, a (hypothetical) totally homogeneous medium doesnot scatter radiation away from the incident direction. Scattering is caused byspatial fluctuations in the medium. Usually these fluctuations are associatedwith variations in the ‘density of scattering material’ within the medium.Thus, for example, a colloidal particle scatters light when it has a differentindex of refraction from the liquid in which it is dispersed.

Basically, different types of scattering experiments can be performed.First, measuring the dependence on angle of the averaged scattered intensity– frequently called ‘static’ scattering – yields structural information. Thus,in a dilute system we learn about the shapes of the individual particles andthe arrangement of material within them. In a concentrated system, we gaininformation about positional correlations, the average spatial arrangement ofthe particles in the sample. Second, analysis of the time dependence of fluctu-ations in the scattered radiation (or analysis of frequency or energy changes)yields dynamic information. It tells us, for example, how the particles aremoving in Brownian motion and how their shapes, or configurations, fluctuate

56

4.1. LIGHT SCATTERING THEORY

Figure 4.1: (a) Top view of a typical scattering experiment. (b) Expanded viewof the scattering volume, showing rays scattered at the origin O and by a volumeelement dV at position r. Figure from Ref. [177].

in time.In Fig. 4.1.b, a beam of plane-wave, monochromatic light,

EI(r, t) = E0exp [i(kI · r − ωt)]�

�

�

�4.1

with electric vector E0 polarized perpendicular to the scattering plane (theplane of the paper), is incident on a region of scattering medium; kI is thepropagation vector of the incident light, having magnitude kI = |kI | = k =2π/λ where λ is the wavelength of light in the medium; ω is its angularfrequency. Light of the same polarization is scattered at angle θ to a detectorin the far field; thus we consider only polarised scattering, and assume thatthe dielectric behavior of the medium is scalar, rather than tensorial (fora discussion of depolarized scattering and optically anisotropic media, seereference [178]).

We assume that the scattering is weak, so that:

57


i. most photons pass through the sample undeviated, a few are scatteredonce, and the probability of double and higher-order scattering is neg-ligible;

ii. the incident beam is not distorted significantly by the medium. (Thiscorresponds to the first Born approximation or, in the specific contextof light scattering, the so-called Rayleigh-Gans-Debye approximation.)

we also assume that the scattering process is ‘quasielastic’, implying only avery small change in frequency. Thus the magnitudes kS of the propagationvector kS of the scattered light is also 2π/λ.

We apply Maxwell’s equations to the problem of a plane electromagneticwave propagating through a medium described by a local dielectric constantε(r, t): ε(r, t) is the dielectric constant of the medium at position r, relativeto an arbitrary origin O (Fig. 4.1.b), and at time t. Details of this calculationare given, for example, by Berne and Pecora [178]; here we simply quote theresult. The amplitude ES(R, t) of the dielectric field of the radiation scatteredto a point detector at position R in the far field is given by:

ES(R, t) = −k2E0

4π

exp [kR − ωt]

R

∫

V

[

ε(r, t) − ε

ε

]

exp(−iq · r) d3r .�

�

�

�4.2

Here ε is the average dielectric constant of the medium, V is the scatteringvolume, and the ‘scattering vector’ q is defined as the difference between thepropagation vectors of the scattered and incident light (Fig. 4.1.b)

q ≡ kS − kI , q ≡ |q| =4π

λsin

θ

2;

�

�

�

�4.3

and represents the exchanged momentum between the probe and the sample.Insight into the physics underlying Eq. (4.2) can be obtained by rewriting

it as the sum of the amplitudes of the fields dES(R, t) scattered by the volumeelements dV ≡ d3r at positions r:

ES(R, t) =

∫

V

dES(R, t)�

�

�

�4.4

where

dES(R, t) = −k2E0

4π

exp [kR − ωt]

R

[

ε(r, t) − ε

ε

]

dV exp(−iq · r) .�

�

�

�4.5

Eq. (4.5) can be recognized as the formula describing the radiation dueto an oscillating point dipole. The incident electric field, of strength E0

and propagation vector k, induces in the volume element dV at position r adipole moment of strength proportional to E0 [ε(r, t) − ε] dV which oscillatesat angular frequency ω. This elementary dipole reradiates, or scatters, lightin all directions. The second factor in Eq. (4.5) describes a spherical wave of

58

4.1. LIGHT SCATTERING THEORY

scattered radiation emenating from the origin O. The final term, exp(−iq ·r),allows for the fact that the radiation scattered by the volume element atposition r is shifted in space relative to that scattered by an element at theorigin O. Referring to Fig. 4.1.b, simple geometry shows that the extradistance travelled by the light scattered at r, compared to that scattered atO, is (kI · r − kS · r)/k which, with use of Eq. (4.3), gives a phase shift of−q · r radians.

It is immediately apparent from Eq. (4.2) that, as previously noted, ifthe medium is totally homogeneous, so that ε(r, t) = ε, there is no scatter-ing. In other words, scattering of radiation (for q 6= 0) is caused by spatialfluctuations in the dielectric properties of the medium.

Eq. (4.2) embodies the fundamental physics of light scattering. Thescattered light field consists of a spherical wave emanating from the scatteringvolume with an angle- or q-dependent amplitude which is the spatial Fouriertransform of instantaneous variations in the dielectric constant of the sample.The intensity of the scattering, averaged over time, provides information onthe sample’s structure, essentially spatial correlations of the medium. Clearlyany variation in time of the local dielectric constant is directly reflected intemporal variations of the amplitude of the scattered field (and its intensity).Light scattering therefore probes directly the structure and dynamics of asample in reciprocal space (or q-space).

In the next Section we will deal with dynamic light scattering, in whichphysical information is extracted by the fluctuating intensity of the scatteredfield at a point detector in the far field.

4.1.2 Dynamic Light Scattering

Consider a scattering medium, such as a suspension of colloidal particles,illuminated by coherent light. At any instant, the far field pattern of scat-tered light is not, as one may naively expect, a smooth envelope of intensitywhich varies with angle; rather it comprises a grainy random diffraction, or‘speckle’, pattern (Fig.4.2.a). At some points in the far field the phases of thelight scattered by the individual particles (or more generally, by any smallvolume with linear size of the range of spatial correlations, provided thatsuch range is much smaller than the scattering volume) are such that theindividual fields interfere largely constructively to give a large intensity; atother points destructive interference leads to a small intensity. Furthermore,as the scattering medium evolves in time - for example the particles positionschange due to Brownian motion - the phases change, and the speckle patternfluctuates from one random configuration to another. Thus, as sketched inFig. 4.2.b, the intensity I(q, t) scattered to a point in the far field fluctuatesrandomly in time. Clearly, information on the motion of the particles (or onthe dynamics of spatially correlated volumes in the medium) is encoded inthis random signal. At the simplest level, the faster the particles move, themore rapidly the intensity fluctuates.

59


Figure 4.2: (a) Coherent (laser) light scattered by a random medium such asa suspension of colloidal particles gives rise to a random diffraction, or speckle,pattern in the far field. As the particles move in Brownian motion, their positionschange, as do the phases of the light that they scatter, and the speckle patternfluctuates from one random configuration to another. (b) The fluctuating intensityobserved at a detector roughly the size of one speckle. (c) The time autocorrelationfunction of the scattered intensity shown in (b). The time-dependent part of thecorrelation function decays with a time constant TC equal to the typical fluctuationtime of the scattered light. Figure from Ref. [177].

Normally, in dynamic light scattering (DLS), useable information is ex-tracted from the fluctuating intensity by constructing its time correlationfunction, defined as:

〈I(q, 0)I(q, τ)〉 ≡ limT→∞

1

T

∫ T

0

dt I(q, t)I(q, t + τ) .�

�

�

�4.6

As can be seen from its definition, this quantity effectively compares the signalI(q, t) with a delayed version I(q, t + τ) of itself, for all starting times t andfor a range of delay times τ . Typical behavior of the intensity correlationfunction is sketched in Fig. 4.2.c. At zero delay time Eq. (4.6) reduces to

limτ→0

〈I(q, 0)I(q, t)〉 =⟨

I2(q)⟩

.�

�

�

�4.7

For delay times much greater than the typical fluctuation time TC of theintensity, fluctuations in I(q, t) and in I(q, t + τ) are uncorrelated, so that

60

4.2. HOMODYNE PHOTON CORRELATION SPECTROSCOPY

the average in Eq. (4.6) can be separated:

limτ→∞

〈I(q, 0)I(q, t)〉 = 〈I(q, 0)〉〈I(q, τ)〉 = 〈I(q)〉2 .�

�

�

�4.8

Thus, as shown in Fig. 4.2.c, the intensity correlation function decays fromthe mean-square intensity at small delay times, to the square of the mean atlong times; the characteristic time TC of this decay is a measure of the typicalfluctuation time of the intensity (Fig. 4.2.b).

In the next section we consider the properties of the scattered field,speckle, and the associated correlation functions, all of which underlie thetheory of dynamic light scattering. Then as a simple example the problem ofdynamic light scattering from discrete scatterers will be analyzed.

4.2 Homodyne Photon Correlation Spectroscopy

In the so-called ‘homodyne method’, the time correlation function of thescattered intensity I(q, t) = |ES(q, t)|2 is measured. When normalized, it isdefined as

g(2)(q, τ) ≡ 〈I(q, 0)I(q, τ)〉〈I(q)〉2

.�

�

�

�4.9

In the ‘heterodyne method’, a small portion of the unscattered laser light(called local oscillator) is mixed with the scattered light. The sum of thesetwo interfering fields ELO(t)+ES(q, t) (with ELO(t) for the angle-independentlocal oscillator and ES, as usual, for the scattered field) hits the detectorsurface and the time correlation function of the collected intensity Ihe(q, t) =|ELO(t) + ES(q, t)|2 is computed.

Below we will need the (normalized) time correlation function of the scat-tered field, defined by

g(1)(q, τ) ≡ 〈ES(q, 0)ES(q, τ)〉〈I(q)〉 .

�

�

�

�4.10

(As in section 4.1.1, we now specialize to the case of samples which areisotropic on average so that the averaged quantities such as g(1)(q, τ) org(2)(q, τ) depend only on the modulus q of the scattering vector.) As wewill show, homodyne and heterodyne techniques yield different informationon the system’s dynamics. We start, in the present section, by focusing ourattention on the homodyne method, which will be the technique exploitedfor the investigation of the supercooled chalcogenide glasses dynamics. Theheterodyne method will be introduced in a later section, and it will turn tobe useful for a better understanding of some of the experimental results (seeSection 4.3).

In certain circumstances the homodyne correlation function g(2)(q, τ) maybe simply expressed in terms of g(1)(q, τ), as we now discuss on a generalground. A more rigorous argument will be presented in Section 4.2.2, re-stricted to the special case of discrete spherical scatterers.

61


4.2.1 The Gaussiam approximation

The scattering volume V can be subdivided into N subregions of volume vj

small compared to the wavelength of light, so that

V =∑

j=1,N

vj .�

�

�

�4.11

Then the scattered field ES(q, t) can be regarded as a superposition of fieldsfrom each of the subregions. Let us describe by position vectors Rj(t) thecenters of mass of the volumes vj at time t, and by rj(t) the position of apoint vector inside the small volume vj, taken with respect to the center ofmass Rj (see Fig. 4.3).

Figure 4.3: Coordinates of statistically correlated volumes vj , relative to an ar-bitrary origin O. Rj(t) is the position of the center of mass of volume vj at timet. rj(t) is the position of the volume element dVj relative to the center of masscoordinate.

We now recall the expression Eq. (4.2) for the total scattered field to apoint detector at position R in the far field. With this in mind, we can definea ‘local density of scattering material’ within the small volume vj, as

∆ρ(rj, t) =k2

4π

[

ε(rj, t) − ε

ε

]

,�

�

�

�4.12

where ε(rj, t) is the local dielectric constant at position r = Rj + rj, ε is theaverage dielectric constant of the medium. From Eq. (4.2) we obtain for thescattered field

ES(R, t) = −E0exp [i(kR − ωt)]

R

∑

j=1,N

[

∫

vj

∆ρ(rj, t) e−iq·rj d3rj

]

e−iq·Rj(t) .

�

�

�

�4.13

We can associate a momentum- and time-dependent ‘scattering length’

bj(q, t) =

∫

vj

∆ρ(rj, t) e−iq·rj d3rj

�

�

�

�4.14

62


to any of the volumes vj. Since DLS deals with normalized quantities we canomit the pre-factors in Eq. (4.13), to get the following representation for thescattered field at the detector:

ES(R, t) =∑

j=1,N

bj(q, t) e−iq·Rj(t) ≡∑

j=1,N

E(j)S (q, t) ,

�

�

�

�4.15

where E(j)S (q, t) is the scattered field from the jth subregion: basically the

light scattered from the volume vj times a phase factor accounting for thevolume position Rj with respect to a reference point O (see Fig. 4.1.b).

As particles move E(j)S (q, t) fluctuates, both because of the resulting di-

electric constant fluctuations and because the volumes vj may change positionin time. The only important requirement for the present argument is that,as is often the case, a choice of the volumes vj is possible, such that:

i. as already stated, the subregions are small if compared to the wave-length of light;

ii. there is a large number N of subregions in the scattering volume;

iii. the subregions are sufficiently large to permit particles motion in onesubregion to be statistically independent of those in another.

There are circumstances in which this assumption may be invalid. For ex-ample, systems in their critical region may have very long correlation lengths,and the dielectric constant fluctuations may be spatially correlated over largevolumes. From the point of view expressed by Eq. (4.15), this would meanthat the random variables bj(q, t) may be correlated for any choice of vj

satisfying the requirements (i) and (ii).Each one of the terms in the sum Eq. (4.15) can be represented as a vector

in the complex plane of module bj(q, t) (we neglect absorption terms, so thatthe dielectric constant can be thought as real), making an angle q ·Rj(t) withthe real axis. The total scattered field ES(q, t) at time t is the (vectorial)sum of the individual vectors, as shown schematically in Fig. 4.4.a. Aftera time τ any of the individual vectors E

(j)S (q, t + τ) has changed, and two

distinct situations are reported in Fig. 4.4. In Fig. 4.4.b the subdivisionof the scattering volume V in small volumes vj has not changed, hence thephase factors q · Rj(t + τ) are the same as those at time t. Anyway the

modules bj(q, t) are independent random variables, and as a result E(j)S (q, t+

τ) becomes totally uncorrelated if τ is not too small. In Fig. 4.4.c thedielectric constant inside each volume vj doesn’t change with time. Whatchanges is the partition of the scattering volume V . Such case is analogousto what happens in the case of discrete scatterers. The dielectric constantof each single scatterer doesn’t change with time, but the volumes vj can bechosen as to circumscribe the particles, thus moving with time as independentrandom variables.

63


Figure 4.4: (a) Representation in the complex plane of the electric field ES(q, t)scattered by six statistically independent volumes vj centered at positions Rj(t);j = 1, 2, ..., 6. The field is the vectorial sum of the fields scattered by the individualvolumes. (b) The field ES(q, t + τ) at a later time. Here the partition of thescattering volume V has not changed with time. What changes is the scatteringfrom single volumes, due to fluctuations in the local dielectric constant in each vj .From Eq. (4.14) this means that the modules bj(q, t) of the vectors change withtime as statistically independent variables. (c) The field ES(q, t + τ) at a latertime. Here the dielectric constant inside each volume vj doesn’t change with time.What changes is the partition of the scattering volume V . Such case is analogousto what happens in the case of discrete scatterers. The dielectric constant of eachsingle scatterer doesn’t change with time, but the volumes vj can be chosen as tocircumscribe the particles, thus moving with time as independent random variables.

Whatever the choice for the partition of the scattering volume, once theprescriptions (ii) and (iii) are fulfilled, the total scattered field can be rep-resented pictorially as a two-dimensional random walk of N vectors. In thelimit N → ∞ ES(q, t) becomes a two-dimensional Gaussian variable: this isthe so-called ‘Gaussian approximation’, meaning that our a priori knowledgeabout the stochastic variable ES(q, t) at an arbitrary fixed time t is that itis, irrespectively of any previous extraction, a Gaussian variable with time-independent average value 〈ES(q)〉 = 0 and variance

⟨

|ES(q)|2⟩

. Anywaythis doesn’t means that the stochastic variables ES(q, t) and ES(q, t + τ) arestatistically independent, and actually we argue that they aren’t, unless τ ismuch greater than a typical correlation time τC .

We now suppose, for the sake of simplicity, that the scattering medium isin thermal equilibrium, so that the time autocorrelation function of ES(q, t)does not depend on t, being a stationary property. Let us define the stochasticvariables

E0 = ES(q, 0) ; Eτ = ES(q, τ)�

�

�

�4.16

and ask what is the joint probability P (E0, Eτ , τ) that, if at zero time ES(q, 0) =

64


E0, than at time τ ES(q, 0) = Eτ . In order to compute P (E0, Eτ , τ) one mustknow the time correlation function of the scattered field, bringing all theunderlying information about the system’s dynamics:

I1(q, t) ≡∫

dE0 dEτ P (E0, Eτ , τ) E0Eτ = 〈E0Eτ 〉 ,�

�

�

�4.17

or rather its normalized version g(1)(q, τ) (see Eq. (6.1).Under the hypothesis that ES(q, t) is a gaussian variable it can be shown

that [179]

P (E0, Eτ , τ) ={

2π⟨

|ES(q)|2⟩

[1 − I1(q, τ)]1/2}−1

exp[−E20/2

⟨

|ES(q)|2⟩

] ·

· exp

{

−[

Eτ − E0g(1)(q, t)

]2

2⟨

|ES(q)|2⟩

[1 − g(1)(q, t)2]

}

.�

�

�

�4.18

Figure 4.5: The distribution P (E0, Eτ , τ) is represented in the Gaussian approxi-mation, for a fixed value of the zero time field E0, at different times τ . (a) In thelimit τ → 0 it is a delta function peaked at E0. (b) For a time τ ∼ τC , the typicalcorrelation time, such that g(1)(q, τ) = 1/2. (c) In the limit τ → ∞ the fields E0

and Eτ are completely uncorrelated, and the distribution of Eτ does not depend onE0.

With respect to Eτ , i.e. once known E0, this is still a Gaussian distribution,PE0

(Eτ , τ), with average value

〈Eτ 〉 = E0g(1)(q, τ) ,

�

�

�

�4.19

and standard deviation

[⟨

(Eτ − 〈Eτ 〉)2⟩]1/2=√

⟨

|ES(q)|2⟩

[1 − g(1)(q, τ)2] .�

�

�

�4.20

Needless to say, g(1)(q, τ) is a continuous, not increasing function of time, withvalues in [0, 1]. Thus in the limit τ → 0 the distribution function of Eτ is adelta function peaked at E0 (Fig. 4.5.a). As τ increases, g(1)(q, τ) decreasesand PE0

(Eτ , τ) drifts towards smaller values of Eτ , becoming more and more

65


spread (Fig. 4.5.b). In the τ → ∞ limit it is centered around Eτ = 0, wile

the standard deviation takes on the value√

⟨

|ES(q)|2⟩

: it is nothing but our

starting point, i.e. the expected distribution for the scattered field withoutany a priori knowledge (Fig. 4.5.c).

A Gaussian distribution, as P (E0, Eτ , τ), is completely characterized byits first- and second-order moments. It follows that all higher moments ofthis distribution are related to the first order-two moments. This happens inparticular for the fourth-order moment

I2(q, t) ≡∫

dE0 dEτ |E0|2 P (E0, Eτ , τ) |Eτ |2 =⟨

|E0|2 |Eτ |2⟩

,�

�

�

�4.21

which can be computed by direct integration, to obtain

I2(q, τ) = |I1(q, 0)|2 + |I1(q, τ)|2 ,�

�

�

�4.22

or, for the normalized counterparts Eq. (4.9) and (6.1)

g(2)(q, τ) = 1 +∣

∣g(1)(q, τ)∣

∣

2.

�

�

�

�4.23

This result is sometimes called the ‘Siegert relation’, and reflects the factor-ization properties of the correlation functions of a complex Gaussian variable.It is applicable to any fluid system in which the range of spatial correlationsis much smaller than the linear dimension of the scattering volume. However,the Siegert relation is obtained by considering the amplitude of the electricfield scattered to a point in the far field. In reality a detector has a non-zeroactive area, and therefore sees different scattered fields at different points onits surface. Then it can be shown that Eq. (4.23) is modified to

g(2)(q, τ) = 1 + β∣

∣g(1)(q, τ)∣

∣

2,

�

�

�

�4.24

where the factor β represents the degree of spatial coherence of the scatteredlight over the detector and is determined by the ratio of the detector area tothe area of one speckle. When this ratio is much less than 1 (essentially a‘point detector’) β → 1, and when it is large, corresponding to the detectionof many, independently fluctuating, speckles, β → 0. In practice, with aclassical two-pinhole setup for collection, the detector aperture is usuallychosen to accept about four speckles, which gives the maximum value forβ as 0.8 [177]. The size of a speckle in the far field diffraction pattern willbe estimated for the particular case of discrete, spherical, non-interactingscatterers, in Section 4.2.2.

4.2.2 Discrete scatterers

We now specialize to the case of a sample containing discrete scattering ob-jects suspended in a liquid. For simplicity we will call this object ‘particles’,

66


a term which could refer to a polymer molecule, a micelle, and so on, as wellas to a colloidal particle.

We consider N particles in the scattering volume V . it is clear that thereis scattering both from particles, dielectric constant εP (r, t), and the liquid,dielectric constant εL(r, t). Thus we can write

ε(r, t) − ε =

{

εP (r, t) − ε r inside any particleεL(r, t) − ε r outside any particle

�

�

�

�4.25

where ε is, as before, the average dielectric ocnstant of the medium. If wedefine εL to be the average dielectric constant in the liquid, Eq. (4.25) canbe written in the more usefull form

ε(r, t) − ε =

{

[εP (r, t) − εL] + (εL − ε) r inside any particle[εL(r, t) − εL] + (εL − ε) r outside any particle

or, by recognizing that a term (εL−ε) is present everywhere in the scatteringvolume,

ε(r, t) − ε =

εP (r, t) − εL r inside any particleεL(r, t) − εL r outside any particle

(εL − ε) r everywhere in V.

�

�

�

�4.26

We now make a convenient choice of a partition Eq. (4.11) of the scatter-ing volume. N of the M > N small volumes vj will circumscribe the parti-cles, such that the corresponding positions Rj (j = 1, 2, ..., N) is the centersof mass of particle j. The other M − N volumes vj (j = N + 1, ...,M) areall in the suspension medium. when substituted into Eq. (4.12), Eq. (4.26)leads to three contributions for the scattering. The first is the contributionof interest, essentially the scattering by the particles due to the differencebetween their dielectric constant (or refractive index) and that of the liquidin which they are suspended. The second corresponds to the scattering fromspontaneous density fluctuations of the liquid itself (or density plus concen-tration fluctuations if the suspension medium is a mixture of liquids). Forsimplicity we will assume the density of this ‘background’ scattering to benegligible (though in practice it can be measured and subtracted from thetotal scattering). The third contribution in Eq. (4.26) does not depend onposition r, so leads to a term proportional to

∫

Vexp[−iq ·r] d3r. This integral

describes diffraction by the hole scattering volume which, if the dimensions ofV are much greater than the wavelength of the radiation, is strongly peakedaround q = 0 and can also be neglected.

In conclusion Eq. (4.14) becomes, for the case of discrete scatterers,

bj(q, t) =k2

4π

∫

vj

[

εP (rj, t) − εL

ε

]

e−iq·rj d3rj ,�

�

�

�4.27

where vj is the volume of particle j.

67


We consider now the simple situation of a dilute suspension of identicalspherical particles, in which case the scattering lengths bj(q, t) have the same,time-independent value b(q), and can therefore be taken outside the sum inEq. (4.15), giving a total scattered field (we neglect, as usual, pre-factors)

ES(q, t) =∑

j=1,N

exp [−iq · Rj(t)] .�

�

�

�4.28

Each of the phase factors exp [−iq · Rj(t)] can be represented by a unit vectorin the complex plane, while the total scattered field is the sum of the indi-vidual vectors. The situation is pretty much like what depicted in Fig. 4.4.c,and as particles move in Brownian motion, their positions Rj(t) change, asdo the phase angles q · Rj(t).

The average value of the scattered field is

〈ES(q, t)〉 =∑

j=1,N

〈exp [−iq · Rj(t)]〉 = 0 ,�

�

�

�4.29

since the phase angles are uniformly distributed between 0 and 2π. Theaverage of the scattered intensity is

⟨

|ES(q, t)|2⟩

=∑

j=1,N

∑

k=1,N

〈exp {−iq · [Rj(t) − Rk(t)]}〉 =

=∑

j

1+

j 6=k∑

j=1,N

∑

k=1,N

〈exp [−iq · Rj(t)]〉〈exp [−iq · Rk(t)]〉 =

= N ,�

�

�

�4.30

where, in averaging the cross-terms separately, we have expoited the assump-tion that the suspension is dilute so that the particles positions are uncor-related. Eqs. (4.29) and (4.30) are well-known results for the random walk:the mean is zero since the walk is, on average, symmetrical about the origin;the mean-square displacement is proportional to the number of steps.

Substituting Eq. (4.28) into the definition of g(1)(q, t), Eq. (6.1), andusing similar manipulations as those which led to (4.30), gies

g(1)(q, τ) = N−1∑

j

〈exp{−iq · [Rj(0) − Rj(τ)]}〉 = 〈exp{−iq · [R(0) − R(τ)]}〉 .

�

�

�

�4.31

The last step follows from the fact that the average motions of identicalparticles are themselves the same.

The next step is to compute the scattered intensity time correlation func-

68


tion

I2(q, τ) = 〈I(q, 0)I(q, τ)〉 =

⟨

∑

j=1,N

∑

k=1,N

exp {−iq · [Rj(0) − Rk(0)]} ·

·∑

l=1,N

∑

m=1,N

exp {−iq · [Rl(τ) − Rm(τ)]}⟩

=

=∑

j

∑

k

∑

l

∑

m

〈exp {−iq · [Rj(0) − Rk(0) + Rl(τ) − Rm(τ)]}〉 .�

�

�

�4.32

Then we exploit again the independence of particles positions in a dilutesuspension. Consider first the terms for which one subscript, let’s say k,is different from all the others. Thus we can factorize the average to give〈exp[iq · Rk(0)]〉〈...〉 which, as discussed above, is zero. Next, consider theterms for which j = k = l = m. There are N of these, each of value 1. Inthe large N limit, these can be neglected since the dominant contributionsto the quadruple sum are O(N2), as we will see shortly. The only remainingcontributions are those for which pairs of subscripts are equal. One class ofthese, j = l, k = m, is also zero since 〈exp{iq · [Rj(0) + Rj(τ)]}〉 = 0. Thusnon-zero contributions to Eq. (4.32) only arise when j = k, l = m or j = m,l = k, giving (for N → ∞)

I2(q, τ) =∑

j

1∑

l

1 +

+∑

j

〈exp{−iq · [Rj(0) − Rj(τ)]}〉∑

k

〈exp{−iq · [Rk(0) − Rk(τ)]}〉 =

= N2 + N2 |〈exp{−iq · [Rj(0) − Rj(τ)]}〉|2 .�

�

�

�4.33

Using the definition of the normalized time correlation function g(2)(q, t), Eq.(4.9), together with Eqs. (4.30), (6.1), (4.31), (4.33) then gives

g(2)(q, τ) = 1 +∣

∣g(1)(q, τ)∣

∣

2,

which is again the Siegert relation Eq. (4.23), already obtained in Section4.2.1 in a more general context.

For a dilute suspension of identical non-interacting spheres, the field cor-relation function g(1)(q, t) is given by Eq. (4.31). If we define

∆R(τ) ≡ R(τ) − R(0) ,�

�

�

�4.34

the displacement of the particle in time τ , we see therefore that

g(1)(q, t) = 〈exp[iq · ∆R(τ)]〉 ,�

�

�

�4.35

and DLS provides information on the average motion of a single particle.

69


The displacement of a particle in Brownian motion is a (real) three-dimensional random variable having a Gaussian probability distribution:

P [∆R(τ)] =

[

3

2π 〈∆R2(τ)〉

]3/2

exp

[

− 3∆R2(τ)

2 〈∆R2(τ)〉

]

,�

�

�

�4.36

where the particle’s mean-square displacement in time τ is

⟨

∆R2(τ)⟩

= 6Dτ .�

�

�

�4.37

The ‘free-particle’ diffusion constant is given by the celebrated Stokes-Einsteinequation,

D =KBT

6πηR,

�

�

�

�4.38

where KB is Boltzmann’s constant, T the temperature, η the shear viscosityof the suspension medium, and R particles’ radius. Evaluation of the averagein Eq. (4.35) over the distribution of (4.36) gives

g(1)(q, t) = exp

[

−q2

6

⟨

∆R2(τ)⟩

]

= exp(−q2Dτ) = e−τ/τC ,�

�

�

�4.39

being τC = (Dq2)−1. in this simple case the autocorrelation function decaysexponentially. As the diffusion coefficient is linearly related to the parti-cles size, this result is used in a widespread application of DLS: measuringthe particles size in dilute suspensions. The normalized intensity correlationfunction g(2)(q, t) is given by Eq. (4.24), and it takes the form

g(2)(q, τ) = 1 + βe−2Dq2τ .�

�

�

�4.40

For completeness, we now estimate the size of a speckle in the far fielddiffraction pattern. One should calculate how much two points in the farfield must be separated in order to have the scattered electric field at the twopoints uncorrelated. If the two points are defined by scattering vectors q1

and q2, with ∆q ≡ q2 −q1 (see Fig.4.6), with use of Eq. (4.28) the followingrelation holds for the cross correlation function of the two fields:

〈ES(q1, t)E∗S(q2, t)〉 =

∑

j=1,N

∑

k=1,N

〈exp [−iq1 · Rj(t)] exp [−iq2 · Rk(t)]〉 .

�

�

�

�4.41

For a dilute suspension, the same arguments as those used above lead to

〈ES(q1, t)E∗S(q2, t)〉 =

∑

j=1,N

〈exp [i(q2 − q1) · Rj(t)]〉 =∑

j=1,N

〈exp [i∆q · Rj(t)]〉 .

�

�

�

�4.42

If the two points coincide, q1 = q2, Eq. (4.42) simply reduces to the intensity⟨

|ES(q)|2⟩

= N . If the two points are well separated, so that ∆q is large,the arguments applied to Eq. (4.29) give 〈ES(q1, t)E

∗S(q2, t)〉 = 0, implying

70

4.3. THE HETERODYNE CORRELATION FUNCTION

uncorrelated fluctuations. Since, in the average, the position Rj(t) of theparticle can lie anywhere in the scattering volume, an estimate of the value∆q at which correlations between the two fields is lost is given by

∆qLV ≈ 2π ,�

�

�

�4.43

where ∆q is the magnitude of ∆q and LV is the dimension of the scatteringvolume parallel to ∆q and perpendicular to the direction of the scattering(Fig. 4.6). Thus the angle ∆θ subtended by one speckle at the scattering

Figure 4.6: Propagation and scattering vectors describing light scattered to twodifferent points. When ∆θ is small, fluctuations in the two fields are correlated;when ∆θ is large enough for the points to be in different speckles, the fluctuationsare uncorrelated.

volume is

∆θ =∆q

kS

≈ 2π/LV

2π/λ=

λ

LV

,�

�

�

�4.44

where kS = 2π/λ is the magnitude of the propagation vector of the light andλ is its wavelength.

We see, therefore, that the angular width of a speckle is the same as thewidth of the diffraction maximum that would be obtained if an aperture ofthe same size of the cross-section of the scattering volume were illuminatedby coherent light. This is analogous to the case of a regular diffraction grat-ing, where the overall size of the grating determines the angular width of themaxima in the diffraction pattern. Finally, we note that the scattered lightfield can be interpreted as a spatial Fourier component, of wavevector q, ofvariations in the ‘density of scattering material’ in the sample. It is knownfrom the theory of Fourier analysis that the independent spatial Fourier com-ponents needed to describe, for example, density variations in a box of lineardimension LV are separated in reciprocal space by ∆q ≈ 2π/LV , the resultof Eq. (4.44). Thus one speckle in the scattered light field can be identifieddirectly with one spatial Fourier component of the sample.

4.3 The heterodyne correlation function

Homodyne and Heterodyne techniques measure two different correlation func-tions of the dielectric constant, and, under proper conditions, they can bring

71


different physical information. As anticipated in Section 4.2, in the hetero-dyne method, a small portion of the unscattered laser light (local oscillator)is mixed with the scattered light on the detector active area.

Although heterodyne method has its own importance in a wide spectrumof interesting applications – such as, citing one for all, local velocimetry mea-surements of systems under shear – our interest is mainly focused on thehomodyne technique, as the main instrument for the present achievements.However, among the causes for the presence of spurious signals in a homo-dyne PCS experiment there is the unwanted laser light due to reflections orflare that has not been scattered, but acts as a local oscillator. Thus, in thepresent Section, we aim at an understanding of the heterodyne correlationfunction.

One of the main requests for a heterodyne experiment is that, by properchoice of the experimental conditions, the amplitude of the local oscillatormay be much greater than the amplitude of the scattered field. We will neglectsuch requirement in the following derivation, looking at the way homodyneand heterodyne correlation functions can contribute to the resulting digitalcorrelation spectrum.

4.3.1 Heterodyne method

Let ELO(t) be the local oscillator field, supposed to vary at the laser frequency.ES(q, t) is, as usual, the scattered field at the detector in the far field. In thepresent method the time autocorrelation of the detected intensity

Ihe(q, t) = |ELO(t) + ES(q, t)+|2�

�

�

�4.45

is computed. Thus the heterodyne correlation function is

〈Ihe(q, 0)Ihe(q, τ)〉 =⟨

|ELO(0) + ES(q, 0)+|2 |ELO(τ) + ES(q, τ)+|2⟩

.�

�

�

�4.46

We suppose the fluctuations of the local oscillator to be negligible, andwe consider the two fields as statistically independent. Thus, terms such⟨

|ELO(τ)|2 |ES(τ)|2⟩

decouple, to give ILOIS, i.e. the product of the averagedintensities (IS ≡ 〈IS(τ)〉 and ILO ≡ 〈ILO(τ)〉 are time independent, and theq dependence will be omitted for simplicity). Moreover, averages of fieldssuch as 〈ES(τ)〉 are zero, since 〈eiωτ 〉 = 0. With these assumptions, of thesixteen terms coming from Eq. (4.46), eight are null and three are time-independent. Of the remaining terms, on can be recognized as the (nonnormalized) homodyne correlation function I2(τ), and we get

〈Ihe(q, 0)Ihe(q, τ)〉 = I2LO + 2ILOIS + I2(τ) +

�

�

�

�4.47

+ 〈[ELO(0)E∗S(0) + c.c.] [ELO(τ)E∗

S(τ) + c.c.]〉 .

The last term of Eq. (4.48), under the hypothesis of statistical independenceof the two fields, and recalling the definition of I1(q, τ) (Eq. (4.17)), becomes

〈ELO(0)E∗LO(τ)〉〈E∗

S(0)ES(τ)〉 + c.c. = 2ILORe [I1(τ)] ,�

�

�

�4.48

72

4.3. THE HETERODYNE CORRELATION FUNCTION

and, using the definitions for the normalized time correlations of the fieldand of the intensity Eq. (6.1) and (4.9), and reintroducing the dependenceon momentum q, Eq. (4.48) can be put in the form

〈Ihe(q, 0)Ihe(q, τ)〉 = I2LO+2ILOIS+I2

Sg(2)(q, τ)+2ILORe[

g(1)(q, τ)]

.�

�

�

�4.49

As anticipated, in an heterodyne experiment, the amplitude of the local oscil-lator is chosen to be much greater than the amplitude of the scattered field,so that

ILO >> IS ,�

�

�

�4.50

and in the left hand side of Eq. (4.49) only the first and the last terms are notnegligible. Thus the heterodyne method provides experimental knowledge ofthe scattered field time correlation g(1)(q, t), also called, for this reason, the‘heterodyne correlation function’. Under the Gaussian approximation (seeSection 4.2.1)) the Siegert relation Eq. (4.23) holds, and this is the reasonwhy it is often asserted in the literature that precisely the same informa-tion is contained in both homodyne and heterodyne experiments. However,in certain circumstances, the two techniques provide different insight intothe system’s dynamics. An example, concerning the simple case of discretescatterers, is given in next Section.

4.3.2 Discrete scatterers under flow

We consider now a suspension of identical, discrete scatterers, under a uniformshear flow with velocity v. We start from Eq. (4.35):

g(1)(q, t) = 〈exp[iq · ∆R(τ)]〉 ,

where now the displacement of a particle is still a Brownian motion, wherethe constant drift imposed by the shear, imply that the first moment of theparticle displacement ∆R(τ) ≡ R(τ) −R(0) is different from zero:

〈∆R(τ)〉 = vτ .�

�

�

�4.51

Thus, ∆R(τ) is a three-dimensional Gaussian variable, with distribution:

P [∆R(τ)] =

[

3

2π 〈∆R2(τ)〉

]3/2

exp

[

−3 (∆R(τ) − vτ)2

2 〈∆R2(τ)〉

]

,�

�

�

�4.52

where the mean square displacement 〈∆R2(τ)〉 is again related to the dif-fusion coefficient (Eq. (4.37). Evaluation of the average Eq. (4.35) on thedistribution (4.52) gives, neglecting pre-factors,

g(1)(q, τ) = exp[

−q2Dτ + iq · vτ]

.�

�

�

�4.53

Assuming the validity of the Gaussian approximation, we find the sameEq. (5.4), meaning that no information can be obtained on the drift velocity

73


v from a homodyne experiment. With use of the heterodyne method, we findinstead

〈Ihe(q, 0)Ihe(q, τ)〉 = I2LO + 2ILO e−q2Dτ cos(q · vτ) .

�

�

�

�4.54

The heterodyne correlation function oscillates with a period T = 2π/q ·v, and such oscillations are modulated with a decreasing exponential withcharacteristic decay time τC = (Dq2)−1.

74

5InfraRed Photon Correlation

Spectroscopy

Performing DLS on liquid chalcogen elements (S, Se, Te) and chalcogenideglasses, is a non-trivial task due to the low bandgap energy and the increasedabsorption at visible wavelengths: a common feature shared by these mate-rials (see Section 3.2.5).

As an example, we again consider the transmission of amorphous andliquid Selenium [105], which was reported in Fig. 3.6. In Fig. 5.1 the blackdashed line indicates the wavelength of the radiation used for the experiment,while green and red dashed lines show the wavelengths of the green and redlaser lines most commonly used in conventional PCS setups. The inset reportsthe absorption edge (in eV) for Se as a function of the temperature. Indeed,as Figure 5.1 shows, the intrinsic adsorption edge of liquid Se is located atenergies higher than the energy of the 1064 nm (1.16 eV) laser, used in thiswork, even up to temperatures as high as 350O C.

liquid

200 400 600

1.2

1.4

1.6

1.8

Photon

Energy [eV

]

Temperature [K]

630 K

Tm = 494 K

energy of 1064 nm

Figure 5.1: Transmission of amorphous and liquid Selenium [105]. The blackdashed line indicates the wavelength of the radiation used for the experiment, whilegreen and red dashed lines show the wavelengths of the green and red laser linesmost commonly used in conventional PCS setups. The inset reports the absorptionedge (in eV) for Se as a function of the temperature. The solid horizontal line showsthe energy of the source used in the present work. The melting point for Se is alsoreported as a dashed line.

75

CHAPTER 5. INFRARED PHOTON CORRELATION SPECTROSCOPY

As a further example, in Fig. 5.2 the transmittance in the infrared spec-trum of glasses belonging to different families is reported [180]. For glassysilica the transmission edge is in the near infrared, namely at about λ =400nm. For ZBLAN (a mixture of zirconium, barium, lanthanum, aluminum, andsodium fluoride) it shifts to 800 nm. Sulfide, selenide and telluride chalco-genide glasses share a wide transparency window in the infrared portion ofthe spectrum.

Figure 5.2: Relative transmittance of different kinds of glasses as a function ofradiation wavelength in the infrared spectrum. Figure from Ref. [180].

Even pure sulfur, which is a pale yellow liquid just above its meltingtemperature (Tm ∼ 120O C), becomes a dark red, viscous liquid above theso-called lambda-transition point Tλ ∼ 160O C, preventing any conventionalPCS investigation in that region (see Section 3.1.1).

Among other interesting applications of the IRPCS technique, there isthe investigation of macromolecular solutions of biological interest, in casesin which visible light absorption is too high. Since the scattered intensityIS ∝ λ−4, macromolecules are much worst scatterers in the infrared than inthe visible spectrum. As a drawback, high detection efficiency and/or signal-to-noise ratio of the detector is required. Despite that, there is the advantageof decreasing multiple scattering effects in turbid suspensions, thus openingthe possibility of studying more concentrated solutions than with conventionalPCS techniques.

In the present Chapter, the apparatus used for the InfraRed Photon Corre-lation Spectroscopy (IRPCS) investigations of supercooled chalcogenide glassformers and chalcogen elements, will be introduced. Some of the experimen-tal problems faced during the setup development, alignment and calibrationwill be also discussed.

76

5.1. SETUP DESCRIPTION

5.1 Setup description

5.1.1 General sketch of the IRPCS setup

In general terms, all light scattering experiments contain the same compo-nents; it is only in specific details that experiments designed to measuredifferent aspects of the dynamic properties of a solution or a melt are distin-guished. The major components are the light source; the spectrometer, whichcontains an optical system to define the scattering angle and also to limit thenumber of coherence areas; the detector; and the signal analyzer, which in atime-domain experiment is a correlator software. The need for use of a nearinfrared probe, together with the requirement that many scattering anglescould be accessed at the same time, drove the setup development towardscustomized solutions concerning basically all of its components.

In Fig. 5.3 the IRPCS setup is sketched. A solid-state Nd:Yag monomodelaser, with nominal output power of 500 mW, operates at a wavelengthλ=1064 nm (top left). The beam, with polarization perpendicular to theplane of the paper, is focused in the sample, with a beam waist of tipically50 to 30 µm (corresponding to a focal length of lens l1 from 125 to 75 mm,respectively). The sample environment comprises a scattering cell and athermoregulation setup which will be described in more detail in Section 5.3.The sample cuvette, generally chosen with a cylindrical shape, is lodged inthe scattering cell with its vertical axes perpendicular to the plane of thepaper.

The scattered intensity is collected at five different values of the scatter-ing angle θ, i.e., according to relation (4.3), at five different values of theexchanged momentum

q =4πn

λsin

θ

2,

�

�

�

�5.1

being n the sample index of refraction.The values θi (i=1,...,5) of the scattering angle can be chosen by displacing

five platforms, running on circular rails centered around the scattering cell.As shown in Fig. 5.4, each platform holds the optics and mechanics necessaryfor the scattered beam collection as well as collimation on an optical fiber.The minimum angular offset between adjacent collection devices is 20◦. Thescattered beam is collected by a biconvex lens l2 (see Fig. 5.4) with focus onthe illuminated sample volume, so that the scattering volume V is defined bythe intersection of gaussian beam profiles relative to lenses l1 and l2. Priorto be collimated on a monomode optical fiber, the scattered light crosses abandpass interferential filter, centered around the laser line (Fig. 5.4), thusallowing to cut undesired contributions due to visible light background (seeSection 5.1.3).

The five optical fibers are connected to as many thermoelectrically cooledsilicon avalanche photodiode detectors, with circular active area of ∼ 180 µm(see Section 5.1.3 for a more thorough description).

77


Figure 5.3: Sketch of the experimental setup. The laser beam of wavelength 1064nm is focalized by lens l1 onto the sample. The scattered field is collected at fivedifferent values of the scattering angle θ and then collimated on five monomodeoptical fibers. The collection/collimation devices (a sketch is reported in Fig. 5.4)can move on circular rails, so that the scattering angles can be chosen before theexperiment. Five avalanche photodiodes (APDs, see Section 5.1.3) are used asdetectors, and collected photon-counts are analyzed by the correlator software,which computes time autocorrelation functions.

The number of photons ni(t) detected at time t = k∆t (with integerindex k) from detector i (i=1,...,5) in the time interval [t, t + ∆t], is collectedby input-output digital cards from National Instruments (r), and the timeautocorrelation functions

Ci(t) = 〈ni(0)ni(t)〉�

�

�

�5.2

are computed, using a software correlator developed in our laboratories by R.Di Leonardo [181]. If ni

0, ni1, ... are the number of pulses appearing at input

channel i at times t0, t1 = t0 + ∆t, tj = t0 + j∆t, the correlator will estimatethe discrete time autocorrelation function of channel i as a function of thedelay time tau = k∆t as

Gi(τ = k∆t) =N−1∑

j=0

nijn

ij+k ,

�

�

�

�5.3

78


Figure 5.4: A collection/collimation unit. It consists of a plate which is anchoredto the circular rails reported in Fig. 5.3, holding the collection (left side) andcollimation (right side) optics as well as mechanics. Optics: A collection lens l2 isfocalized on the illuminated sample volume (f is its focal length). The collectedscattered field impinges on a bandpass filter, centered at the laser wavelength, and isthen collimated on an optical fiber. Mechanics: The Y -axes lies along the directionof propagation of the (collected) scattered field. The Z-axes is perpendicular to thescattering plane. Translation stages are used for fine alignment of both collectionand collimation optics. These are labelled as X, Y , or Z according to their work-axes.

which is a good approximation to the true autocorrelaton function wheneverthe change in the value of the correlation function during the time ∆t is small[182].

For the sake of exposition, we will refer – from now on – to the ensemble ofcollection, detection, and signal analysis devices, relative to one fixed valueof the scattering angle, as to a ‘channel’. Thus, expressions such as ‘thechannel at 90◦ scattering’ will be used. Finally, Fig. 5.5 shows a picture ofthe setup. A more thorough description of some relevant components will begiven during next sections.

5.1.2 The sample environment

In studying the long-time dynamics of glass formers near and above the glasstransition, the sample temperature is a critical experimental parameter, aschanges by few degrees can lead to changes of the structural relaxation timeby many orders of magnitude. Even more delicate is the situation for whatconcerns system undergoing living polymerization, since one can find temper-ature regions where the underlying dynamical time-scale changes appreciablyin a few tenth of a degree. Thus, particular attention has been devoted to

79


Figure 5.5: A picture of the experimental setup (view from top). The laser sourcelies outside the picture (top-left corner). The beam follows the direction of thearrow. It is focalized into the sample cuvette (S), which is lodged in the scatteringcell, by lens l1. The scattered field (two rays are drawn) is collected by five inde-pendent units (see also Figures 5.3 and 5.4). Lenses l2 and fiber collimators (C)are also indicated.

the temperature control apparatus.

A custom design has been adopted for the scattering cell, of which apicture is reported in Fig. 5.6. It is basically a cylinder made with anticorodal(an aluminum based alloy), with a hole along the vertical axes, made to allowthe replacement of cuvette-adaptors, to be selected as function of the samplecuvette shape and dimension. These are generally cylindrical, glass or silicacuvette, but different geometries can be accommodated. Along the scatteringplane, small apertures allow for the incoming beam to impinge on the sampleand propagate in the forward direction. A slit window opens at almost 180◦,thus permitting the collection of scattered radiation from a vast angular range.

A helical embed runs all around the external wall of the cell, in order to

80


Figure 5.6: A picture of the scattering cell (S in Fig. 5.5) The incoming beamfocalization lens l1 is visible on the left side. .

maximize thermal contact with a heater wire. This latter is a metal conductorwire, covered with ceramic, and again with a metal layer. The wire struc-ture allows for high temperature performance, high thermal conductivity, butcomplete electrical isolation from the external environment. It is connectedto a power-supply, which is controlled by a thermoregulation software.

The temperature-control software, has been developed with Labview (r),and is based on a Proportional-Integral-Derivative (PID) algorithm. Temper-ature can be monitored with the use of different probes, such as J-, K-typethermocouples or thermoresistors. These are brought almost in contact withthe sample cuvette, trough a proper aperture of the cell. Temperature read-ings are obtained by means of a commercial, high-precision input/output card(National Instruments r), equipped with built-in cold junction compensa-tion.

As a result, temperature control with a precision within 0.1◦ is obtainedfrom room temperature to about 350◦.

5.1.3 The detectors

Performing DLS experiments with a near-infrared probe is made possible bythe use of avalanche photodiodes (APD). These are photodetectors that canbe regarded as the semiconductor analog to photomultipliers. By applyinga high reverse bias voltage (typically 100-200 V in silicon), APDs show aninternal current gain effect (gain from 1 to 100 or more) due to impact ioniza-

81


tion (avalanche effect). In general, the higher the reverse voltage, the higherthe gain. APDs offer a combination of high speed and high sensitivity un-matched by PIN detectors, and quantum efficiencies at λ >400 nm unmatchedby Photomultiplier tubes (PMT).

Figure 5.7: Photon detection efficiency as a function of wavelength for the siliconAPDs used in the present work. Figure from Perkin Elmer.

In Fig. 5.7 we report the wavelength dependence of quantum efficiencyfor the silicon APD detector used for the IRPCS setup. Even if the APDquantum efficiency is only about 2% at 1064 nm, nevertheless, it is a farbetter performance than a PMT. The APD detectors are accompanied, in thepresent setup, by the use of bandpass interferometric filters (Fig. 5.4) withcentral wavelength of 1064±2 nm and full width at half maximum (FWHM)10±2 nm.

Two commercial photon counting modules (Perkin Elmer r) are employedas detectors in the IRPCS setup, one with a single APD (sigle detector), andone with four APDs (four detectors). Each module comprises a temperaturecontrol device in order to ensure performance stability. It also contains ahigh-voltage DC-to-DC converter and is powered from a single 5 V source.

Dark counts are 60 c/s for the single photon counting module, and about400 c/s for the 4 detectors array.1 The circular active area is 175±5 µmdiameter at the minimum detection power.

5.2 Setup alignment and calibration

In order to check the goodness of the setup, we conducted light scatteringexperiments with a standard sample, namely, a diluted water suspension oflatex microspheres of well defined diameter, ø = 110 nm. Once assumed the

1Since the single counting module, due to a lower dark count-rate, provides a slightlybetter signal-to-noise ratio, it was always coupled with the 90◦ scattering channel, becausethe related geometry is the less affected by spurious effects due to unwanted scatteredlight (see Section 5.3.3). The intent was to maximize de signal-to-noise-ratio for thisadvantageous configuration.

82

5.2. SETUP ALIGNMENT AND CALIBRATION

validity of the independent particles approximation, these colloidal particlesare supposed to perform Brownian motion as a consequence of constantlyoccurring collisions with the surrounding water molecules. As we have seenin Section 4.2.2, the expected normalized intensity correlation function isgiven, in the Gaussian approximation, by

g(2)(q, t) = 1 + βe−2Dq2t ,�

�

�

�5.4

where the diffusion coefficient of the microspheres in water is given by

D =KBT

6πη(T )R.

�

�

�

�5.5

where KB is the Boltzmann’s constant, T the absolute temperature, η(T ) the(temperature dependent) water viscosity, while R is the particle’s radius. Fora fix scattering angle θ, the value of q is fixed by Eq. (4.3), and the expected‘measured intermediate scattering function’

fM(q, t) = β−1/2√

g(2)(q, t) − 1�

�

�

�5.6

can be fitted with an exponential function of time, thus obtaining an estimateof the correlation time

τ =1

Dq2,

�

�

�

�5.7

and of the coherence factor β. The value of β is maximized by fine alignmentof the collection and collimation optics.

To ensure the validity of Eqs. (5.4) to (5.7), one must prevent any localheating of the solution, due to absorption at the probe wavelength, eventu-ally leading to convective flow and thermal lens effects, as well as unreliableestimate of the sample temperature. In principle, no variation of the homo-dyne correlation function Eq. (5.4), induced by a convective flow, would beobserved in Gaussian approximation (see Section 4.3.2). Anyway the relax-ation time τ varies with temperature according to Eqs. (5.5) and (5.7), andits dependence on the incident light power, P , is a signature of local heating.No power dependence of τ was ever encountered during the scattering exper-iments, provided that P < 10 mW. The solution concentration was chosenhigh enough to allow to work with source powers much less than 1 mW, stillremaining far within a single scattering regime.

From the knowledge of the scattering angle it would be possible to give anestimate of the particles radius, which should be equal to the value supplied bythe producer. This would be a test for the reliability of PCS data. Actually,this procedure is meaningful only if a squared cuvette, with planar walls, isused at a scattering angle of 90◦, or at different angles if correction is madefor refraction at the cell surface. This procedure led to an estimate of theparticles diameter ø = 112 ± 6 nm.

83


Figure 5.8: Scattering geometry for a cylindrical cuvette. The sketch representsa view along the scattering plane. The laser beam impinges on the cuvette in thedirection reported by the arrow. The collection lens is supposed to be able tomove along the x-axes. The effect of refraction trough the sample-air interface isconsidered. θ0 is the ‘nominal’ value of the scattering angle (here it is 90◦), whileθeff is its actual value once refraction effects are taken into account. See the textfor major details.

When it is desired to make measurements at a number of different angles,a cylindrical cell would seem to offer many advantages. It is however, diffi-cult to achieve the alignment necessary to realize the potential for accuracyinherent in the light scattering method. As a matter of fact, in this case thescattered beam refraction through the walls prevents a reliable knowledge ofthe scattering angle. The situation is illustrated in Fig. (5.8). The arrowpoints towards the incident beam propagation direction, and a section alongthe scattering plane of the cuvette is reported. At a first instance we ne-glect the effect of the cuvette walls an consider the effect of refraction at thesample-air interface. The sample refractive index is n, while in air it is 1. Thecollection lens l2 is able to move along the x axes, by means of a translationstage (stage X on left side of Fig. 5.4).

In case the lens is correctly aligned it is centered at zero x position, itsoptical axes lies along the dashed-dotted line, an the collected scattered beamimpinges normally on the walls (i.e. the sample-air interface). In this con-ditions the scattering angle is θ0 (90◦ in the figure), the same as the anglebetween the incoming beam direction and the lens l2 optical axes. When,on the contrary, the lens is not in the x = 0 position, the scattered beamfollows the red path, which is affected by refraction at the walls. Here, thescattering angle is θeff , different from θ0. Unfortunately, θ0 is the detectorangular position one can read on a graduated scale, while the real scatteringangle θeff is not known.

84

5.2. SETUP ALIGNMENT AND CALIBRATION

Nevertheless, once the position x (unknown) of the collection lens withrespect to the cuvette center is fixed, an estimate of θeff can be given, startingfrom the knowledge of the relaxation time τ of spheres with known radius, in asolvent with known viscosity and index of refraction, at a known temperature.This estimate follows from Eqs. (5.5) and (5.7), substituted into the invertedrelation (5.1):

θeff = 2 arcsin

λ

2n

√

3

2π

η(T )R

KBT τ

,�

�

�

�5.8

where λ is the beam wavelength. By measuring τ as a function of x oneobtains θeff (x). An example is given in Fig. 5.9 (open dots). A modeldescribing the expected behavior of θeff (x), once only refraction effects aretaken into account, can be derived, yielding:

θeff (x) = θ0 + arcsinx

ξ− arcsin

x

nξ.

�

�

�

�5.9

Here θ0 is the ‘nominal’ scattering angle, already defined, and ξ is the cuvetteradius (see Fig. 5.8) . A generalization of Eq. (5.9), also taking into accountthe refraction at the glassy cuvette wall can be done, once the wall thicknessdw and the glass index of refraction nw are known:

θeff (x) = θ0+arcsinx − x0

ξ + dw

−arcsinx − x0

nw(ξ + dw)+arcsin

x − x0

nwξ−arcsin

x − x0

nξ,

�

�

�

�5.10

where the parameter x0 accounts for the fact that the x axes origin corre-sponds to an arbitrary position of the translation stage. A direct comparisonof the model Eq. (5.10) with experimental results is reported in Fig. 5.9 (fullline). A best fit procedure allows an estimate of the two parameter θ0 ± δθ0

and x0 ± δx0. Finally, the lens l2 is aligned to the position x0, correspondingto the normal incidence configuration, with an effective scattering angle θ0.The procedure is repeated for each channel.

The experimental indetermination on the scattering angle can be esti-mated according to the following considerations. After the alignment, thereference sample is replaced with the sample to be investigated. The align-ment procedure should ensure that the optical axes of lenses l2 relative tothe five channels, are crossing at the center of the new cuvette. This is actu-ally not exactly the case, and an offset ∆x is introduced between the cuvettecenter position during and after the alignment.

We define the quantities ρ = ∆x/ξ, σ = dw/ξ. If both ρ and σ are small,then

∆θ ∼ ρ

[

n − 1

n− σ

nw − 1

nw

]

.�

�

�

�5.11

The other contribution to ∆θ comes from the fact that, even if the offset∆x is zero, the estimate δθ0 found during the alignment with the referencesample is no more a good estimate, once a sample with index of refraction

85


Figure 5.9: Estimate from experimental data (◦) of the effective scattering angleθeff computed according to Eq. 5.8 as a function of the collection lens displacementx − x0, defined as from Fig. 5.8. A comparison with the expected trend Eq. 5.10is made (—) from which the parameters x0 and θ0 are obtained. The value of x0

(reported with respect to an arbitrary chosen reference) is then used to align thel2 lens.

n′ > n has been replaced. Again, if we define the (supposed) small quantityρ0 = δx0/ξ, then a relation analog to Eq. (5.11) holds, with ρ0 replacing ρ.Then, by differentiation of Eq. (5.1) one obtains

∆q

q∼ 1

2(ρ0 + ρ)

[

n − 1

n− σ

nw − 1

nw

]

cotθ

2.

.1

2(ρ0 + ρ)

[

n − 1

n

]

cotθ

2,

�

�

�

�5.12

where the last estimate corresponds to neglecting the cuvette walls contribu-tion.

5.3 Signal-to-noise ratio

Consideration of the signal-to-noise ratio is complicated by the large numberof factors that enter into the final answer. We recognize at the outset threesources of noise that limit our ability to measure the properties of the scat-tered radiation with arbitrarily high precision. They are effects due to thefinite intensity of the scattered light; effects due to a finite duration of theexperiment; and effects due to light scattered by unwanted effects (dust, forexample).

86

5.3. SIGNAL-TO-NOISE RATIO

5.3.1 Effects due to finite intensity

A contribution to fluctuations in the scattered light intensity is caused by thefact that the number of photons detected during each sample time is finite.If the instantaneous intensity of the scattered light corresponds to N photonsper sample time, we expect most of the time to detect from N−

√N to N+

√N

photons during a sample time. The number detected will obey a Poissondistribution law. The expected uncertainty in the correlation function due tothis effect is given by

√G, where G is the number given by Eq. 5.3. This

contribution to the noise may be reduced by increasing the laser intensity,or by focusing the laser beam to a smaller diameter. In case of a solution ofdiscrete scatterers, the solute concentration can be increased, or the scatteringangle can be reduced.

5.3.2 Effects due to finite experiment duration

A second limitation in accuracy is due to the fact that data are collected fora finite number of decay times of the correlation function. If the correlationfunction decays as

G(t) = 1 + e−Γt�

�

�

�5.13

and the total duration of an experiment is T , then the number of decay timesduring the experiment is ΓT . The corresponding signal-to-noise ratio, evenif the detected light level is infinite, is

S/N =√

ΓT .�

�

�

�5.14

The only way to improve this contribution to the signal/noise ratio is toincrease the duration of the experiment. In case of a diffusive dynamics, an-other way is to increase Γ which requires that light be scattered at a largerangle. This is not possible when investigating the structural relaxation ofsupercooled melts, since no momentum-dependence of the decay time is ex-pected.

Very long acquisition times are often required when a supercooled meltapproaches the glass transition since in this case, by definition, the relaxationtime τα approaches 100 s. If, for instance, at a certain temperature, τα ∼ 5s, then for a reasonable value of S/N ∼ 102, the experiment will take abouthalf a day. Generally a lower S/N is a reasonable compromise if one aims atreaching such slow dynamical regimes, and many hours of acquisition can beenough, also because during such long times, the probability of rare, unwantedscattering events increases. 2

In such cases performing several acquisitions at a time, can be of somehelp. Since decays related to the structural relaxation dynamics are expected

2Think for instance to a small bubble in a viscous melt. Small bubbles can be formedduring melt quenching, due to a sudden volume decrease. For long acquisition times,nothing prevents the bubble, which was initially far from the scattering volume, fromentering it, thus becoming largely the most efficient scatterer.

87


to be q-independent, different channel will measure in principle the samequantity. The difference from one channel to another lays in the coherencefactors, and in statistical fluctuations of the quantity G (Eq. (5.3).

5.3.3 Effects due to unwanted scattered light

The presence of unwanted signals in the scattered light provides the thirdmajor limitation to the quality of the light scattering results, and yet itprovides the ultimate limit in signal-to-noise ratio for the vast majority ofreal experiments. Included in this category are such effects as

• Fluctuations in laser intensity;

• Unwanted laser light due to reflections or flare that has not been scat-tered but acts as a local oscillator;

• Convection currents in the scattering cell;

• Dust, air bubbles, or other foreign matter in the solution or melt;

• Light scattered at the wrong angle present because of reflections in thecell;

• Crystallites, impurity molecules or other artifacts resulting from im-proper or inadequate sample preparation;

The principal difficulty with many of these contributions to the noisesignal is that systematic effects take place so that the measurements areconsistent from experiment to experiment but unfortunately give the wronganswer. Some of the previous points will be discussed in the following.

The nominal power stability supplied by the laser producer is 5% rootmean square after warm-up. Tests on the source stability have been per-formed by use of a metal plate, allocated inside the scattering cell, in orderto provide reflection of the incident beam to the detector. Homodyne corre-lation functions have been recorded, and no decay has been detected in theaccessed time window.

As to the second effect, i.e. the presence of a local oscillator componentdue to a small amount of unscattered light (small compared to the real scat-tered light), a careful consideration is needed. In this case the correlationfunction will take the form already considered in Section 4.3, Eq. (4.49). Forsimplicity we consider first the case of discrete monodisperse scatterers. Thecorrelation function will contain two exponentials, one with a decay rate of2Dq2 and another at Dq2, proportional in amplitude to the intensity of thelocal oscillator. If it is fitted to a single exponential, the calculated decay ratewill differ from the correct result according to [183]

∆Γ

Γ= −16

9

ILO

IS

,�

�

�

�5.15

88

5.4. THE SAMPLES

being ILO and IS the local oscillator and scattered intensities, respectively.Anyway, as we considered in Section 4.3.2, oscillating contributions from un-wanted heterodyne scattering may also be expected under particular circum-stances. One of these circumstances is when a flow in the liquid is induced,for instance, by convection phenomena. These phenomena are particularlyprevalent when a thermal control bath is poorly designed, and a proper designof the temperature control system virtually eliminates the problem, but theycan be triggered by local heating due to absorption at the laser wavelength.

Another source of oscillations in the heterodyne term are vibrations. Ho-modyne experiments are almost immune to this tipe of problem. The reasonis that all the sources of light wich will ultimately interfere at the detec-tor are located within the relatively small scattering volume, and because ofthe relatively incompressible nature of the solvent, typical laboratory sourcesof vibration are incapable of causing relative motions of the various macro-molecules by sufficient amount to be important. Experiments that are moreprone to interference from vibration are those employing a local oscillator. Inthese cases, relative motion of the two parts of the experiment by a quarterwavelength of light or less, can cause large, entirely erroneous signals.

For melt samples, propagating density fluctuations, eventually due to vi-brations, can also contribute as oscillating heterodyne terms to the correlationfunction. This topic is extensively treated in references [184, 185, 186].

We also point out that, since microspheres suspensions are relatively goodscatterers, no heterodyne contribution may be detected during the alignmentprocedure with the reference sample, even in presence of a (relatively) weakparasite local oscillator field, which could instead affect correlation functionsfrom the sample if the latter is a particularly weak scatterer.

Another source of unwanted scattering, strongly influencing the abilityto investigate the structural relaxation in viscous liquids, is the presenceof unwanted particulate in the sample, due to impurities, but also to thepresence of clusters of atoms, or crystallites in the melt. Sample preparationis a matter of chief concern, which will be described in the next section.

5.4 The samples

The samples investigated in the present work are pure selenium and sulfur,as well as several compositions from the arsenic selenide (AsxSe100−x) andsulfide (AsxS100−x) series. In all these cases, living apart pure elements, theglass transition temperature is above room temperature, thus, after meltquenching, these materials are all amorphous solids in normal conditions.

Samples were prepared at the Institute of Chemical Engineering and HighTemperature Chemical Processes, Foundation for Research and Technology-Hellas, Patras (Gr); by the group of Prof. Spyros N. Yannopoulos. Compo-nent elements (S, Se, As: Cerac, 5N purity) have been subjected to furtherpurification by distillation. According to the desired compositions, proper

89


Figure 5.10: Samples from the As-Se and As-S series; the arsenic content x (at.%) is also reported. Note that in all the composition range the AsxSe100−x samplesare completely opaque.

AsxSe100−x series AsxSe100−x series

Se S

As10Se90 As10S90

As20Se80 As20S80

As30Se70 As30S70

As40Se60 As40S60

Figure 5.11: The samples investigated.

amounts were loaded in carefully cleaned with hydrofluoric acid, quartz tubeswith dimensions 6 mm o.d. - 4 mm i.d. After loading the cells, each samplewas subjected to heating under vacuum in order to remove possible water oroxide traces and then was sealed, still under vacuum. A picture of represen-tative As-S and As-Se samples is shown in Fig. 5.10, where the As content xis also reported for each sample. As can be noted, Selenides are completelyopaque whatever the amount of As. On the contrary, sulfides absorb visiblelight in the shorter wavelength region, thus looking deep orange, tending todark red as the As content is increased.

90

Part III

Results

91

6Liquid chalcogens

6.1 Liquid sulfur

Perhaps the most characteristic example of inorganic polymerization is thatof liquid sulfur. Above its melting point, Tm ∼ 119◦C, sulfur is a low viscos-ity molecular liquid, where molecules are predominantly S8 units. Graduallyraising the temperature the liquid undergoes a sudden increase of its vis-cosity near the λ-transition at Tλ=159◦C . This temperature is believed tomark the transition between the molecular liquid and a polymer solution ofsulfur chains (Sµ) coexisting with S8 molecules. Evidences for the existenceof this transition have been examined in Section 3.1.1. We also stressed inthat Section, that the validity of the Maxwell relation, Eq. 2.8, relating vis-cosity to the relevant time-scale of dynamical processes triggering the liquidresponse to a shear stress, have been tentatively tested for this liquid withdynamic light scattering (DLS) techniques. However, no evidence for theunderlying dynamics was found since recently, with the first IRPCS inves-tigation on liquid sulfur [79], where the use of infrared radiation allowed toovercome absorption effects in this system. These results were obtained froma temperature study of homodyne correlation functions at a fixed scatteringangle θ = 90◦. The improved IRPCS setup described in Chapter 5 has beenemployed for a temperature and exchanged momentum study of the chainrelaxation dynamics in liquid sulfur around the λ-transition. The aim of thepresent section is to report on the results so far obtained.

6.1.1 Experimental

A sketch of the experimental IRPCS setup is reported in Fig. 5.3. Thefive acquisition channels were aligned with the use of a reference sample – awater solution of latex beads of known diameter – following the proceduresummarized in Section 5.2. The values of the scattering angles were thus fixedto θ=29◦, 45◦, 66◦, 91◦and 100◦. However, during the alignment procedure,spurious effect probably due to reflection of the incoming laser beam ontothe scattering cell, prevented a reliable alignment at θ=100◦. Thus, in thepresent report, data obtained from this configuration will be omitted.

The sample was loaded in a sealed quartz tube with dimensions 8 mm

93

CHAPTER 6. LIQUID CHALCOGENS

outer diameter and 6 mm inner diameter. Sample preparation, described inSection 5.4, is a critical experimental variable in the case of sulfur. In fact, thepresence of contaminating impurities strongly affects the extent of variationof viscosity, or of the related relaxation time, around Tλ. Moreover, ourexperience revealed that the scattering efficiency from density fluctuations inthe PCS momentum- and time-window is quite lower for sulfur than for otherchalcogenide systems investigated. This means that: (i) the presence of dustparticles, due to preparation procedure or large clusters of atoms diffusing inthe liquid, can strongly reduce the contrast of the ms-range chain relaxationdecay of interest; and (ii) spurious reflections can more efficiently affect thesignal collected by the detectors, resulting in heterodyne contributions to PCScorrelation functions. The first inconvenience can be eliminated by filteringthe sample, while the second can be reduced – but not completely eliminated– by careful alignment, and will be discussed later.

The sample temperature was monitored by an iron-constantane, j-typethermocouple (see Section 5.1.2). Before each measurement the liquid waskept at least 20 minutes at each temperature for equilibration. Several acqui-sitions runs were performed at increasing and decreasing temperature aroundTλ.

In Fig. 6.1 we report selected normalized IRPCS correlation functionsfrom liquid sulfur at two different values of the scattering angle θ=91◦and29◦from the first experimental run. Temperatures ranging from 153◦C to250◦C are considered. Full circles denote data obtained for T < Tλ, whileopen circles are used for T > Tλ. These two data sets are quite similar as

1E-5 1E-40,0

0,2

0,4

0,6

0,8

1,0

1E-5 1E-4 1E-30,0

0,2

0,4

0,6

0,8

1,0

C(g

(2)(t)-1)

159.5 OC

T[OC]

153 158 159.5 162 166 175 185 200 220 250

C(g

(2) (t

)-1

)

time [s]

250 OC

S

θ = θ = θ = θ = 91OS

θ = θ = θ = θ = 29O

159.5 OC

time [s]

250 OC

Figure 6.1: Reduced homodyne intensity correlation functions obtained for liquidsulfur are reported at different temperatures around the λ-transition (◦ for T > Tλ

and • for T < Tλ). Two different values of the scattering angle are considered.θ ∼ 90◦in panel (a), and θ ∼ 30◦in panel (b). The temperature dependence ofthe chain relaxation function is indicated by the arrows. Full lines report KWWmodel equations 6.2 as obtained from a best fit procedure.

far as the temperature behavior is considered. The previously observed [79]

94

6.1. LIQUID SULFUR

trend is confirmed: upon heating above the melting point the relaxation timefirstly increases reaching its longer time-scale at Tλ, then it decreases up tothe higher measured temperature T=250◦C . Moreover, the relaxation timeis found to increase of about one order of magnitude moving from the higherto the lower q investigated.

Sample q-dependence of scattered intensity time correlation functions arereported for two temperatures, around (◦) and above (•) the λ transitionin Fig. 6.2. For each temperature, correlation functions are shown at four

1E-5 1E-4 1E-3 0,01

0,0

0,2

0,4

0,6

0,8

1,0

0,0

0,2

0,4

0,6

0,8

1,0

29o

91o

29o

C

[g(2

) (t)-

1]

time [s]

T = 175.0ΟC

91o

T = 159.5ΟC

Figure 6.2: Momentum dependence of homodyne time correlation functions isreported for two temperatures, around (◦) and above (•) the λ-transition . Valuesof the scattering angle θ are, from lower to higher relaxation times, θ ∼ 90◦, 65◦,45◦and 30◦.

values of the scattering angle. Moving from lower to higher relaxation times,θ decreases from ∼ 90◦to ∼ 30◦. Such q behavior will be quantitativelyanalyzed later.

6.1.2 Data treatment

We recall here – as it was already discussed in Section 4.2.1 – that the quantitywhich brings physical information is the scattered field correlation functiong(1)(t), directly related to the density-density time correlation function for anisotropic medium in polarized scattering geometry:

g(1)(q, t) =

(

∂ε

∂ρ

)2

〈ρ(q, t)ρ∗(q, 0)〉 /⟨

|ρ(q, t)|2⟩

,�

�

�

�6.1

being ε the dielectric constant of the scattering medium. Instead, the mea-sured quantity in a homodyne experiment is the homodyne intensity correla-tion function g(2)(t). These two quantities are related by Eq. 4.24 in Gaussian

95


approximation. To extract the relevant parameters, the reduced homodyneintensity autocorrelaton functions g(2)(t) have been modelled with a stretchedexponential function, according to the equation:

C[

g(2)(q, t) − 1]

= Φ(q, t) =[

e−(t/τ)β]2

,�

�

�

�6.2

where Φ(q, t) is the normalized density autocorrelation function, and C is anormalization factor.

Below Tλ and for scattering angles different from ∼ 90◦, a heterodyne con-tribution to the intensity autocorrelation function was sometimes revealed,probably due to spurious reflections of the incoming laser beam. In suchcases a different model was used taking into account the presence of oscilla-tions from the heterodyne component of measured relaxation functions. Theemergence of such oscillating contributions was discussed in Section 5.3, whilethe correct model for describing such contributions is given in Section 4.3, Eq.(4.54). An example is reported in Fig. 6.3.

1E-5 1E-4 1E-3 0,01 0,1

0,0

0,5

1,0

C

[g(2

) (t)-

1]

time [s]

T=158.5OC

θ = 45O

Figure 6.3: An example of the occurrence of heterodyne scattering conditionsduring the IRPCS experiment on liquid sulfur, for T < Tλ (◦). This was probablydue to the decreased scattering efficiency in this temperature region, which canbe seen as a consequence of the reduced polymer content. In these conditions themodel function Eq. 6.2 has been accordingly modified, also taking into accountoscillatory contributions ( , see the text for details).

6.1.3 Temperature dependence of chain relaxation time

In Fig. 6.4 the average relaxation time τc = 〈τ〉 = β−1Γ(β−1)τ (Γ is the Eulergamma function) and the stretching parameter β are reported vs. temper-ature. The behavior of τc indicates its tight relation with the λ-transition:it rapidly increases below Tλ, reaching its maximum at the transition point,where the extent of polymerization Φ(T ) is found to start its rapid increase(Fig. 3.3.b). Then, at still increasing temperatures, τc more slowly decreases.

96

6.1. LIQUID SULFUR

160 180 200 220 240

0,05

0,10

0,15

0,20

0,25

160 180 200 220 240

0,65

0,70

0,75

0,80

0,85

0,90

0,95

Temperature [0C]

Ch

ain

re

laxa

tion

tim

e τ c [

ms] θ = 91ο

(a)

(b)

βK

WW

Temperature [

0C]

Figure 6.4: (a) Temperature behavior of the chain relaxation as measured byIRPCS at a scattering angle θ ∼ 90◦. Two acquisition runs from the same sulfursample are reported. Full symbols are from the first run, while open symbols havebeen obtained after several thermal cycles at increasing and decreasing temper-atures around the λ-transition . (b) Temperature dependence of the stretchingparameter β in the same scattering conditions, and for the same acquisition runsas in (a). Vertical line marks the λ-transition (Tλ= 159◦).

If by one hand, the underlined behavior of the chain relaxation time qualita-tively agrees with that reported in Ref. [79], on the other hand, quantitativeagreement is less satisfactory, as can be inferred by looking at Fig. 6.5. Thepresent findings (•) show a broader peak of the τc(T ) path with respect tothe previous investigation (full dots plus line), and the time variation spansa narrower time interval. As a possible explanation we hypothesized a de-pendence of this quantity on the sample thermal history. The sample usedfor the present measurements had been, in fact, subjected to thermal cyclesaround Tλ, prior to be used for the present investigation. In order to ratio-nalize this picture, we performed several acquisition runs at increasing anddecreasing temperatures across Tλ. Open symbols in Fig. 6.5 show the av-erage relaxation time and stretching parameter as obtained during the lastrun, after four thermal cycles. Thus, increasing the number of temperaturecycles across the λ-transition experienced by the sample, makes the transitionbroader, up to the point at which no clear peak can be identified (◦). Anyway,the polymer solution region above Tλ shows a perfect reproducibility even fordistant runs. This could be seen as the signature of the presence of residuallong polymer chains or ring-like structures below Tλ after several cycles. Alsothe stretching parameter β shows a similar behavior, evolving to a broader Tdependence as the number of cycles increases (see Fig. 6.4).

Finally, Fig. 6.6 reports the same quantities, τc and β, for the remainingscattering angles. The same runs as in Fig. 6.4 are considered, confirmingthe underlined trends.

97


140 160 180 200 220 240

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

Temperature [0C]

Ch

ain

re

laxa

tion

tim

e τ c [

ms] θ = 90ο

Figure 6.5: Temperature behavior of the chain relaxation time reported fromRef. [79] (times a constant factor 0.6). Two acquisition runs from the same sulfursample are reported. Full symbols are from the first run (same as Fig. 6.4.a), whileopen symbols have been obtained after several thermal cycles at increasing anddecreasing temperatures around the λ-transition .

98

6.1. LIQUID SULFUR

0

1

2

3

4

140 160 180 200 220 240 160 180 200 220 240 260

0,5

0,6

0,7

0,8

0,5

0,6

0,7

0,0

0,4

0,8

1,2

160 180 200 220 240 260

0,5

0,6

0,7

140 160 180 200 220 2400,0

0,2

0,4

0,6

θ = 29ο

Temperature [0C]

Cha

in r

elax

atio

n tim

e τ c [m

s] θ = 29ο

βK

WW

Temperature [0C]

βK

WW

θ = 45ο

Cha

in r

elax

atio

n tim

e τ c [m

s] θ = 45ο

βK

WW

θ = 66ο

Temperature [0C]

Temperature [0C]

Cha

in r

elax

atio

n tim

e τ c [m

s] θ = 66ο

Figure 6.6: Same as Fig. 6.4 for scattering angles θ = 29◦, 45◦and 66◦. Left

panels: Temperature behavior of the chain relaxation for two acquisition runs.Full symbols are from the first run, while open symbols have been obtained afterseveral thermal cycles at increasing and decreasing temperatures around the λ-transition . Right panels: Temperature dependence of the stretching parameterβ in the same scattering conditions, and for the same acquisition runs as in leftpanels. Vertical lines mark the λ-transition (Tλ= 159◦).

99


6.1.4 Momentum dependence of chain relaxation time

In Fig. 6.7 we report τc as a function of the inverse squared momentum1/q2 for selected temperatures above and below Tλ. Values of the exchangedmomentum q are obtained from Eq. (5.1), the index of refraction, n, is takenfrom Ref. [65]. Within experimental uncertainty the points follow a lineartrend once a fixed temperature is considered. With a linear regression (fulllines in Fig. 6.7) we estimated the angular coefficient G as a function oftemperature, according to the relation

τc(q) =G(T )

q2.

�

�

�

�6.3

The parameter G(T ) is reported in Fig. 6.8. As expected from the underlined

Figure 6.7: τc is reported vs. inverse squared momentum q for several temperaturesabove and below the λ-transition . Linear regression curves are reported. Inset :The high q region is reported on an enlarged scale.

diffusive-like behavior (i.e. 1/q2 proportionality of τc) of the chain relaxationtime, G(T ) follows the trend imposed by τc(q, T ), with a maximum at thetransition temperature.

6.1.5 Discussion

According to the Maxwell relation, Eq. (2.8), the abrupt increase in viscosity(see Fig. 3.3.a) observed at Tλshould trigger an equivalent increase of thestructural relaxation time τα. G∞ in this equation is the “unrelaxed” shearmodulus of the liquid, i.e. the value of G measured in the short wavelengthlimit. In an ideal system, characterized by harmonic interactions, G∞ wouldbe temperature independent. Practically it mildly decreases with tempera-ture and it is found to be a rather sharply defined quantity, ranging for manymolecular liquids between 109 and 1010 Pa. The validity of this picture, as

100

6.1. LIQUID SULFUR

Figure 6.8: The parameter G(T ) of Eq. 6.3, obtained from linear regressions ofτc(T, 1/q2) data shown in Fig. 6.7, is reported as a function of temperature.

already stressed in Section 3.1.1, would imply the emergence of a ∼ 10 nsrelaxation time at the λ-transition . This would reflect in an increase in themeasured sound velocity and acoustic absorption in Brillouin light scattering(BLS) spectra. None of the two effects has ever been detected by BLS ex-periments on sulfur. On the contrary, these data testify for a decrease of thestructural relaxation time at increasing temperature, i.e. the orthodox be-havior in ordinary liquids [78, 77]. The situation becomes even more puzzlingconsidering ultrasonic experiments [187, 188, 72], performed in the 1-10 MHzdomain, when the timescale of the probe is much longer than the 10 ns ex-pected for the relaxation time. In this relaxed, ωτ << 1, regime one shouldmeasure the viscosity directly from the attenuation of an ultrasonic pulse.Surprisingly, no critical temperature dependence of the acoustic attenuationhas ever been observed, as recently emphasized [189].

As already stated in Ref. [79], the discovery of this new feature allows rec-onciling the unusual behavior of sulfur with the usual viscoelastic framework.A two relaxation process scenario emerges, where the relevant time-scales areτc and the one related to ordinary structural relaxation, τα, which in the nor-mal liquid regime above the melting point is expected to be in the picosecondrange. These two relaxation time-scales are characterized by very differenttemperature and momentum behaviors.

Thus, the Maxwell relation around and above Tλ can be reformulated as[79]:

η = (G∞ − Gc)τα + Gcτc ,�

�

�

�6.4

where Gc is the shear modulus related to the chain relaxation process. Theincrease of viscosity with temperature, in this picture, would be triggered bythe appearance of the low frequency relaxation, largely compensating for thedecrease of τα [79]. Since τc >> τα ∼ 10−12s, the first term of Eq. (6.4)

101


becomes progressively less important as temperature is raised above Tλ. Onecan estimate the magnitude of Gc as [79]:

Gc =

[

η − G∞τα

τc − τα

]

T>Tλ

≈[

η

τc

]

T>Tλ

,�

�

�

�6.5

where the approximation follows also from η >> G∞τα, being G∞ ≈ 108 ÷109Pa [75]. As can be seen from Fig. 6.9, Gc extracted through Eq. (6.5) dis-plays an abrupt temperature dependence growing by five orders of magnitudein ten degrees above Tλ and asimptotically reaches values in the narrow range4 ÷ 8 Pa. The presence of this additional relaxation, leading to the reformu-

Figure 6.9: Shear modulus Gc as determined by the ratio η/τc. The viscosity η istaken from Ref. [64], while τc is from Ref. [79]. Figure is taken from Ref. [79]

lation of the Maxwell relation given in Eq. (6.4), manifests itself through theexistence of this “intermediate” plateau of the real part of the shear modulus.More importantly, at variance with the α-relaxation plateau, the plateau ofGc turns out to have significant temperature dependence. Hence, the largeincrease in viscosity observed at Tλ is brought about by both the temperaturedependencies of the relaxation time τc and of the plateau Gc, with the latterlargely dominating [79]. Moreover, the observed momentum dependence ofthe slow relaxation process underlines its different nature with respect to thestructural relaxation dynamics.

6.2 Liquid selenium

We report in the present section on the study of the slow dynamics of super-cooled liquid and molten selenium by means of IRPCS.

102

6.2. LIQUID SELENIUM

Liquid Selenium is one of the most interesting elemental liquids owing toits twofold coordination property that is the basis for the formation of one-dimensional (chain-like) polymeric molecules. Entanglements between longchains leads to an unusually high viscosity of the liquid above the meltingpoint. Chains (trans configuration) are not the only molecular species in themelt and the glass. Se8 rings (cis configuration) are also present in an extentconsiderably lower than that of the Sen chains [84, 190]. Further, ring-likefragments may also exist as parts of a long chain and they contribute tothe structural features accounting for the Se8 isolated rings. It was foundexperimentally [84] that the population of ringlike fragments decreases withtemperature as was predicted by simulations. The polymer content was foundto be of about 85% around the glass transition temperature with a light in-crease with temperature rise. At high temperatures, at ∼ 400◦C, experiments[191] have shown that the Se8 rings are almost absent from the liquid. Coor-dination defects that inevitably will change the dynamics of liquid seleniumappear with increasing temperature. In particular, one-fold and three-foldcoordinated selenium atoms are the main defects where the fraction of theformer is expected to increase faster with temperature and the latter can beconsidered responsible for the formation of clusters in the liquid state.

By measuring the scattered intensity autocorrelation function, the averagesize of the selenium clusters in the liquid state can be extracted, and thetemperature dependence of the related diffusion coefficient, which was foundto exhibit Arrhenius behavior, can be determined.

6.2.1 Experimental

Performing DLS on glassy selenium is a non-trivial task due to the lowbandgap energy and the increased absorption at visible wavelengths. Ab-sorption problems become more important at elevated temperatures abovethe melting point of selenium. Indeed, as Fig. 5.1 shows, the intrinsic ad-sorption edge of liquid selenium is located at energies higher than the energyof the 1064 nm (1.16 eV) laser, used in this work, even up to temperaturesas high as 350 ◦C. As already pointed out at the beginning of Section 3.1.2,the structural relaxation in liquid selenium has been investigated by means ofdifferent techniques such as viscosimetry [192, 93], and dielectric relaxation[102]. A comparison of these data with light scattering dynamics of seleniumhas been prevented, up to now, by the optical properties of this system, whichin its amorphous and liquid states – for temperatures higher than, let’s say,200 ◦K – is completely opaque (see Fig. 5.1, [105]).

Samples were hold in sealed quartz tubes with 6 mm o.d. and 4 mmi.d. Light scattering data were recorded by lowering the temperature in stepsof 10◦C over the temperature range 270◦C to 160◦C, which included boththe normal molten state as well as the supercooled liquid state. Before eachmeasurement the liquid was kept at least 20 min at each temperature forequilibration. Several acquisitions were performed a glass transition temper-

103


ature Tg∼ 35 − 40◦C (depending on the quenching rate) while crystallizationsets in at (depending on the quenching rate) while crystallization sets in atabout 90 − 100◦C; the crystal melts at 221◦C.

6.2.2 Results

A representative set of the measured correlation functions is shown in Fig.6.10(open symbols). As is clear at a first sight, the relaxation time increases astemperature decreases, coherently with the expected behavior for a coop-erative structural relaxation, but the observed relaxation process reaches atimescale of about 100 s already at a temperature of 160◦C. This result clearlyreveals that the relaxation process we monitor with PCS at this particulartemperature range does not correspond to the structural relaxation of su-percooled selenium which is associated to glass transition dynamics. Indeed,at these temperatures the structural relaxation is much faster than the timewindow of PCS and hence density fluctuations associated with the primaryalpha-process are completely relaxed. Most of the correlation functions, asis shown in Fig. 6.10, are satisfactorily fitted with a single exponential re-laxation function (full lines). However, it has been reported [193, 103] that

Figure 6.10: Representative normalized intensity autocorrelation functions at se-lected temperatures (circles). Continuum lines represent the model function: asingle exponential decay. The whole measurement was performed on cooling theliquid, at T=160 0C the system probably started to crystallize.

the relaxation process of supercooled selenium just above the glass transitiontemperature is described by a stretched exponential function with β ∼0.5.

The relaxation time as obtained from the best fit procedure is reportedas a function of the inverse temperature in Fig. (6.11.a). The same figurealso reports arbitrarily scaled viscosity data from [93], showing that at hightemperatures there is no matching between the trends of viscosity and relax-ation time, meaning that these two quantities are not related as should be

104

6.2. LIQUID SELENIUM

Figure 6.11: (a) Relaxation time vs. temperature as measured by IR-PCS (opencircles). Also reported arbitrarily scaled viscosity data (continuous line and fullcircles Ref. [93]). (b) Effective radius of the Selenium clusters vs. temperature.The melting point of crystalline selenium is also reported (−−−).

if we were dealing with a structural relaxation process. This fact, togetherwith the exponential behavior of time correlations functions, and the incon-sistency between the calorimetric Tg and the relaxation times observed in thepresent experiment, allows us to speculate on a different origin of the processwe observed. Specifically, we may hypothesize the formation of clusters ofselenium atoms, whose self-diffusion gives origin to the observed signal. It iswell known that a self-diffusive dynamics of clusters of a fixed size in a viscousmedium, reflects in the homodyne PCS technique as a purely exponential de-cay of g(2)(q, t), with a characteristic time given by τ = 1/2Dq2, being D theself-diffusion coefficient (see Section 4.2.2). The temperature dependence ofthe diffusion coefficient is given by the Einstein relation

D =kBT

6πη(T )R,

�

�

�

�6.6

being η(T ) the temperature dependent viscosity of the selenium medium, andR the mean radius of diffusing clusters. This allowed – within the hypoth-esis made – an estimate of the effective radius of clusters as a function oftemperature, as reported in Fig. 6.11.b.

6.2.3 Discussion

Coordination defects in liquid selenium are found to depend strongly on thetemperature. In particular at 570 K defects are associated with one-fold andthree-fold atoms (C1 and C3 defects) in bound pairs, even though their con-centration is low. At high temperatures defects are isolated and are mainly

105


of C1 type [194]. The size of clusters is found in the present work, to increaseas temperature decreases, ranging between 50 nm and 350 nm (Fig. 6.11.b).The behavior of the clusters size, strongly increasing with decreasing tem-perature, would thus be in agreement with the interpretation of clusters asoriginating mainly by C3 type defects.

In conclusion, we exploited IRPCS to study the slow dynamics of thenon-transparent glass-forming liquid, selenium. We were able to measure therelaxation time associated to the single relaxation process observed in thisliquid at different temperatures covering the normal as well as the supercooledregimes. The evolution of the correlation time with temperature has beenascribed to the self-diffusion dynamics of selenium clusters whose effectiveradius has been estimated as a function of temperature. In perspective webelieve that a study of the size of clusters at varying exchanged momentum,as well as in function of the aging, would result in a better understanding ofthe dynamics of such material in the liquid state.

106

7Chalcogenide glass formers

7.1 As-Se chalcogenides

In the present section we report on an IRPCS study of binary chalcogenideglasses (GhGs) of composition AsxSe100−x with arsenic content x=10, 20, 30,40 at.%. A comparison of existing data on structural relaxation in thesesupercooled melts obtained by means of dilatometry and recoverable shearcreep compliance as well as viscosimetry data (Section 3.2) with dynamiclight scattering (DLS) data has been prevented, up to now, by the opticalproperties of these compounds, which, in their amorphous and liquid statesare completely opaque, as far as visible radiation is considered, in all the glasscomposition range.

7.1.1 Experimental

The experimental technique used to perform DLS was described in detail inChapter 5. During the present investigation the setup operated at a fixedscattering angle of 90 degrees. This should not be considered as a limitingworking condition, since the structural relaxation dynamics is expected tobe angle-independent on the exchanged momenta, q, probed by light scatter-ing. Furthermore, the q dependence of homodyne correlation functions waschecked for a limited range of compositions and temperatures, confirming theexpected q-independent behavior of relaxation functions. The samples wereloaded in sealed quartz tubes with dimensions 6 mm outer diameter and 4mm inner diameter. Sample preparation procedures were described in Section5.4. The sample temperature was monitored by an iron-constantane, j-typethermocouple (see Section 5.1.2). Light scattering data were recorded in thesupercooled liquid regime over a temperature range from just above Tg up tothe first signs of crystallization, occurring at different temperatures for theprobed compositions. Before each measurement the liquid was kept at least20 minutes at each temperature for equilibration. Several acquisitions wereperformed to ensure reproducibility of the results.

A representative set of the measured correlation functions is given inFig.7.1 for all the investigated As concentrations (open circles).

107

CHAPTER 7. CHALCOGENIDE GLASS FORMERS

10-4 10-3 10-2 10-1 1000.0

0.2

0.4

0.6

0.8

0.0

0.2

0.4

0.6

0.8

1.010-4 10-3 10-2 10-1 100

10-4 10-3 10-2 10-1 1000.0

0.2

0.4

0.6

0.8

10-4 10-3 10-2 10-1 100

0.0

0.2

0.4

0.6

0.8

1.0

210OC

270OC

x = 40

T

140OC

80OC

x = 10

AsxSe

1-x

t [s]

norm

aliz

ed

g(2) (t

)

x = 20

130OC

190OC

norm

aliz

ed

g(2) (t

)

t [s]

x = 30

165OC

220OC

Figure 7.1: Normalized intensity autocorrelation functions from supercooledAsxSe100−x are reported for all the compositions investigated (◦). Continuum linesrepresent the model function Eq. 7.2. Relaxation times increase as temperaturedecreases, as schematized by the arrow on the top-left panel. For each compositionthe higher and lower experimentally accessed temperatures are reported. Temper-ature steps are each 10 degrees for x=10 and 40. For x=20 sample temperaturesare, starting from long relaxation times (i.e. from right to left in the panel) T=130,135, 140, 145, 150, 160, 170, 180 and 190◦C. For x=30, following the same criterion,T=165, 170, 175, 180, 190, 200, 210 and 220◦C.


As is clear at a first sight, the relaxation time strongly increases as tem-perature decreases, coherently with the expected behavior for a cooperativestructural relaxation process. While relaxation functions for the stoichiomet-ric composition (As2Se3, x=40) and for x=30 display a one decay behavior,for the samples with lower As content a broader distribution of relaxationtimes can be recognized.

108

7.1. AS-SE CHALCOGENIDES

Kolraush-William-Watts analysis

A model with several stretched exponential (or KWW functions) decay chan-nels, describing the scattered field correlation function as

g(1)(t) =∑

i=1,N

Ai exp[

−(t/τi)βi]

,�

�

�

�7.1

with N = 1 for x=30, 40, and N > 1 for x=10, 20, was tentatively used toreproduce the data, with poor results for x<30%. In particular, for x=10 apeculiar, almost logarithmic decay is found for the intensity autocorrelationfunctions, characterized by an extremely broad relaxation times distribution,hardly conceivable with a superposition of KWW contributions (Fig. 7.1).

0.81 0.84 0.87 0.90 0.93 0.96 0.99-6

-4

-2

0

2

-6

-4

-2

0

2

As20Se80

CONTIN ANALYSIS

τα

τ2 , τ3

KWW ANALYSIS

τslow , τfast

Lo

g[<

τ>/s

]

Tg/T

Figure 7.2: Full symbols: Average relaxation times τslow (•) and τfast (N) obtainedfor the As20Se80 sample from a double KWW model (Eq. (7.1) with N=2). Opensymbols: average relaxation times obtained for the same sample, from an ILTanalysis of homodyne correlation functions, according to Eq. (7.2). The dashedline shows that the extrapolated behavior of the slower relaxation time reaches 100s at the calorimetric glass transition Tg taken from Ref. [138].

For x=20 a single KWW model proved to be ineffective in reproducingthe shape of relaxation functions, while a double stretched exponential modelprovided better results. However, even in this case, no clearly distinct decaysare definitely recognizable, and a least square algorithm comparing experi-mental data with Eq. 7.1 (with N=2) does not provide convergence unlessone of the model parameters βi, (or one of the model parameters τi) is fixeda priori. Fig. 7.2 shows the temperature dependence of relaxation timesτi (i = 1, 2) found with the aforementioned procedure by fixing one of thestretching coefficients to a reasonable but arbitrary value (• and N).

For x=30 and 40 the density correlation function conforms to a KWWfunction. The stretching parameter β at varying temperature are reportedfor these two concentrations in Fig. 7.3. For the stoichiometric sample wefind β ∼ 0.65, while for x=30 β ∼ 0.56.

109


160 180 200 220 240 260 280

0.5

0.6

0.7

0.8

0.5

0.6

0.7

0.8

As30

Se70

β KW

W

Temperature [°C]

As40

Se60

Figure 7.3: Stretcing coefficient from KWW analysis for AsxSe100−x with x=30(◦) and 40 (•). Average values are also reported by the arrows.

Regularized inverse Laplace analysis

Alternatively, the scattered field time correlation function g(1)(t) was analyzedas a weighted sum of independent purely exponential contributions, i.e.

g(1)(t) =

∫

S(τ)exp(−t/τ)dτ =

∫

S(lnτ)exp(−t/τ)dln(τ) ,�

�

�

�7.2

where the second equality is the logarithmic representation of the relaxationtimes. The distribution of relaxation times S(lnτ) was obtained by the inverseLaplace transform (ILT) of g(1)(t) using the CONTIN algorithm [195, 196].

The application of the ILT method Eq. (7.2) to experimental data, issubjected to a regularization procedure, where a regularization parameter,α, must be chosen. In fact, solving equation (7.2) is generally an ill posedproblem. This means that, even for arbitrarily small, but non-zero, noise levelin the g(1)(t), there still exist a large (typically infinite) set of solutions S(τ)that all fit the g(1)(t) to within the noise level [195, 196]. The imposition of aconstraint to the ordinary least-square solution to the problem, provides thepossibility of selecting one particular member S(τ) from the set.

More specifically, the integral on the right hand side of Eq. 7.2 will berepresented as a weighted sum

g(1)k ≈

Nx∑

j=1

Akjxj, k = 1, ..., Ny

�

�

�

�7.3

where the data g(1)k contain experimental noise, and hence the ≈ sign, the

Akj are known, and the xj are to be estimated. The ordinary least-squaresolution is the set of xj satisfying

V AR ≡Ny∑

k=1

wk

(

g(1)k −

Nx∑

j=1

Akjxj

)2

= minimum�

�

�

�7.4

110


being wk a set of weighting parameters allowing to take into account theexperimental uncertainties relative to each g

(1)k . CONTIN can handle cases

where the equations (7.3) are ill-conditioned or linearly dependent ore evenwhere there are more unknowns than equations (i.e., Nx > Ny). In suchcases the least-square problem Eq. (7.4) admits several solution sets xj,and additional information is to be introduced by imposing constraints. Theconstrained regularized solution is the set xj that satisfies

V AR + α2

Nreg∑

k=1

(

ri −Nx∑

j=1

Rkjxj

)2

= minimum.�

�

�

�7.5

The term added to VAR is called the ‘regularizor’. Its form is determined byspecifying the arrays r and R; its strength is determined by specifying α, the‘regularization parameter ’. The regularizor penalizes a solution for deviationfrom behavior expected on the basis of a statistical a priori knowledge, oron the bases of the ‘principle of parsimony’. Absolute a priori knowledge(as, e.g., non-negativity of g

(1)k ) can be incorporated into the regularizor by

imposing explicit constraints to the problem, thus contributing to specify thearrays r and R. It can eliminate a large subset of the ensemble of solutionsS(τ). The principle of parsimony says, of all solutions S that have not beeneliminated by constraints, chose the solution that reveals the least amount ofdetail or information that was not already known or expected. The CONTINalgorithm automatically sets R and r to penalize deviations from smoothness.In addition to smoothness it also includes the request of minimum numberof peaks in the S(τ) [195, 196]. The choice of the regularization parameteris left to the user and is mainly subjected to a compatibility criterion of theresulting solution with experimental data within the noise level. The greaterthe choice of α, the smoother the model function Eq. 7.2, the broader thefeatures of the relaxation times distribution S(τ) and the smaller the numberof peaks in this function. Many test CONTIN analysis were performed onexperimental data at varying the regularization parameter, and the optimalmodel was chosen as the smoothest one still correctly reproducing the decay ofrelaxation functions. We found the best compromise between smaller numberof peaks in S(τ) and meaningful model Eq. 7.2 for α = 2 · 10−2.

The aforementioned choice lead to an estimate of the relaxation timesdistribution (RTD) reported for the sample composition As20Se80 in Fig.7.4. This is indeed an interesting example, since, as previously reported,the KWW model Eq. 7.1 provided two average relaxation times at eachtemperature, and a comparison of these two models can be attempted.

First look at Fig. 7.4. The RTDs all display the same features, simplyshifting to lower times as temperature increases. The strongest peak is ob-served at long times; it reaches almost 1 s at 130◦C. Looking at table 3.13the calorimetric Tg for the present composition is expected at ∼ 90◦C, thus,this strong feature in S(τ) is expected to track the structural relaxation timeτα(T ) for this sample. Two more, less intense and comparable features are

111


0.0

0.2

0.4

1E-5 1E-4 1E-3 0.01 0.1 1

0.090.180.27

0.0

0.1

0.2 0.00

0.15

0.30

0.00

0.12

0.24

0.00

0.15

0.30

0.00

0.15

0.30

0.00

0.12

0.24

0.00

0.15

0.30

τ3

As20Se80

α = 2*10-2Temperature [

OC]

τ [s]

130τ

ατ

2

190

150

145

170

135s(

τ)140

180

160

Figure 7.4: Relaxation time distribution S(τ) is reported at different temperaturesfrom regularized ILT analysis of As20Se80 homodyne correlation functions. Threemain peak features are found, shifting to lower times as temperature increases.Corresponding relaxation times are indicated as τα, τ2 and τ3. Theire averagevalues are reported in Fig. 7.2 (open symbols).

found in RTDs at decreasing times, tagged as τ2 and τ3 in the figure. Theaverage relaxation times 〈τα〉, 〈τ2〉 and 〈τ3〉 are extracted as the first mo-ments of the single peak distributions. These are compared with the KWWaverage relaxation times 〈τslow〉 and 〈τfast〉 in Fig. 7.2. While 〈τα〉 is rec-ognized to fairly extrapolate to 100 s at the calorimetric Tg, an its estimateprovides agreement with the KWW parameter 〈τslow〉, 〈τ3〉 accounts for thefast stretched exponential decay provided by model 7.1. The emergence ofa further relaxation time-scale, namely 〈τ2〉, points out the problem of thevalidity of the ILT approach. On one hand, agreement between ILT modelEq. 7.2 and experimental data is quite good, as shown in Fig. 7.1 (full lines,x=20). On the other hand a KWW analysis with two decay channel is notfully reliable, as a consequence of the smooth nature of correlation functions,showing no distinct steps. In any case a structural relaxation time-scale τα

can be evidenced, as well as a broader relaxation spectrum than a “classical”single KWW. Whatever the way estimates are obtained, additional relax-ation times are found to follow the same temperature path of τα, shifted tolower times (higher temperatures) thus underlying a common origin relatedto highly cooperative relaxation processes. The same picture holds for x=10,

112


complicated by the fact that a multiple-KWW analysis is even much harderthan the case x=20; however, the RTD most prominent feature again tracksthe relaxation time-scale attaining 100 s around the calorimetric glass tran-sition. Thus, in the chalcogen rich region the spectrum of relaxation timesbroadens with respect to the simple KWW behavior holding for x ≥ 30. Forx=30 and 40 good agreement is obtained between KWW parameters and ILTextracted relaxation times. In such cases the stretching parameter β couldbe also estimated.

In the next section the behavior of structural relaxation time τα withtemperature will be considered. All the relaxation times data are derivedfrom ILT analysis.

7.1.3 Structural relaxation in the As-Se series

The structural relaxation time τα as a function of the inverse temperatureis reported for all the investigated As-Se compositions in Fig. 7.5.a (opensymbols). Viscosity data for these systems, reported in Fig. 3.12.a, have

0.85 0.90 0.95 1.00

-3

-2

-1

0

1

2

1.8 2.0 2.2 2.4 2.6 2.8

-4

-2

0

2

m=16 x = 10 x = 20 x = 30 x = 40

T

g(x) / T

AsxSe

100-x

Lo

g [<

τα >

/s]Lo

g [<

τ α >/s

]

1000/T [K-1]

Figure 7.5: (a) Logarithm of structural relaxation time τα vs. inverse temperaturefor supercooled liquid AsxSe100−x samples, with x=10 (◦), 20 (△), 30 (◦) and 40(⋄) As at.%. Relaxation times are derived from a regularized ILT analysis ofIRPC correlation functions shown in Fig. 7.1 (see text for details). Vogel-Fulcher-Tammann (VFT) model curves are reported as full lines ( and for x=10, 20respectively). Arrhenius model functions are also displayed ( and for x=30,40 respectively). Model functions for viscosity data (dotted curves) obtained forthe same compositions x=10, 20, 30 and 40 (see Fig. 3.12.a), are compared withthe structural relaxation time by normalization. The obtained shear modulus G∞

is shown in Fig. 7.6. (b) Fragility plot for AsxSe100−x. Experimental data arereported together with the same model curves as in (a), extrapolating to the glasstemperature limit of 100 s at Tg/T = 1. The extreme of strong behavior with alimiting fragility m = 16 is referenced ( ).

been fitted to Vogel-Fulcher-Tammann (VFT, Eq. 2.9) or Arrhenius models,

113


depending on the composition (curves in Fig. 3.12). These model curves havebeen normalized to fit the structural relaxation data (dotted curves in Fig.7.5.a). In fact, the mechanical Maxwell relation, Eq. 2.8, provides

τα =η

G∞

.�

�

�

�7.6

The behavior of τα is found to fairly reproduce the viscosity vs. temperaturepath, with the only exception of the selenium richer composition, which seemsto display a more pronounced non-Arrhenius character than expected fromviscosimetry. Estimates for G∞, ranging around 108 Pa, are reported infigure 7.6. Experimental data have been subjected to a best fit procedurewith a VFT model function for x < 30, while an Arrhenius model is moreappropriate for x ≥ 30, as also noticed by Bernatz et al. [93] and Malek etal. [142]. The model curves are reported as full lines in Fig. 7.5.a.

10 15 20 25 30 35 40107

108

109

AsxSe

100-x

G [P

a]

As at%

Figure 7.6: Values for the glass shear modulus G∞ for AsxSe100−x systems, derivedas η/τα.

An estimate for the glass transition temperature of our samples can beobtained by extrapolating to an average relaxation time of 100 s the VFTor Arrhenius laws describing each data set. Armed with this estimate ofTg we can now construct a fragility plot, analogous to that reported in Fig.2.6; that is, a plot of Log10 〈τα〉 against reciprocal temperature scaled tosystem dependent Tg. Fig. 7.5.b shows our data for AsxSe100−x plotted inthis manner, together with the model curves (the same that appears as fulllines in Fig. 7.5.a) extrapolating up to 100 s. From the slope of the data nearTg, we find the fragility index (Eq. 2.10) and its dependence on concentration.

The limit of strong-Arrhenius behavior, characterized by a steepness orfragility index m ∼ 16, is also reported for clarity in Fig. 7.5.b as a dashedline. From this figure the AsxSe100−x series appears to be characterized bya fragile to intermediate behavior, with a parameter m decreasing from thechalcogen rich region on increasing the arsenic content up to a minimumaround the 30 at.%, and then again slightly increasing at 40 at.%. A summary

114


on compositional trends of fragility and glass transition temperature is shownin Fig. 7.7. The present IRPCS results are represented by open circles, whileprevious Tg measurements from MDSC are reported as full black dots (Fig.7.7.b, [138]) . Fragility estimates from viscosimetry, taken from references[93] and [142], are reported as full blue symbols (Fig. 7.7.a). See also Tab.3.13 for a summary on previous MDSC and viscosimetry results on this series.

0 10 20 30 40 5040

80

120

160

200

30

40

50

60

70

80

AsxSe

100-x

(b)

fra

gilit

y

Tg

[OC

]

As at%

from IRPCS from MDSC& from viscosimetry

(a)

Figure 7.7: (a) Fragility parameter for the series AsxSe100−x is reported versusthe As concentration x, from the present (◦) as well as from previous viscosimetryinvestigations (N from [93] and • from [142]). (b) Tg is reported from MDSCmeasurements (•, [138]) and from the present IRPCS investigation (◦).

IRPCS results provide good agreement with both chalorimetric and vis-cosimetric data. Systems with compositions x≥ 30 are confirmed to be strongglass-forming liquids, while the fragile character is enhanced as the chalcogencontent increases, with a fragility index even exceeding the value reported forpure selenium at x=10.

7.1.4 Discussion

Amorphous AsxSe100−x systems are dynamic networks of threefold coordi-nated As atoms and twofold coordinated Se atoms. Following the simplemodel described in section 3.2.1 for amorphous As sulfides, the stoichiomet-

115


ric composition can be though of as an assembly of pyramidal AsSe3/2 unitssharing their corner selenium atoms. Upon further addition of the chalcogenelement bridging Se atoms are transformed into Se dimers, and then intochains, resulting in a structure in which all the AsSe3/2 pyramids becomeisolated. At variance with the AsxS100−x systems, in AsxSe100−x networksformation of Se8 ring molecules as a separated phase is not expected. Theseglasses, in fact, are found to remain homogeneous on mesoscopic length-scales[127].

The distribution of relaxation times in the supercooled liquid regime forthese chalcogenide systems is found in the present IRPCS investigation, tobecome broader and broader on approaching the Se-rich region of composi-tions, while homodyne correlation functions, reflecting the behavior of thedensity correlator, start to decay in a complicated fashion in the same range,with the emergence of several decay processes, up to the almost logarith-mic decay found for the 10 at.% As sample near the glass temperature Tg

(Fig. 7.1, x=10). Structural relaxation in these systems, at least in thechalcogen-rich region, may be thought to arise from dynamical rearrange-ment of cross-linking networks, whose structure (Se chains length betweenAsSe3/2 pyramids) is mainly determined by the average coordination numberof the mixtures, and continuously vary with composition.

A connection between glassy network rigidity and liquid dynamics wasenvisaged in Section 3.2.4. In that context, a parallelism among the com-positional trends of glass MDSC non-reversing heat flow, ∆Hnr, and super-cooled liquid fragility was established. The former quantity, ∆Hnr, is re-ported for AsxSe100−x in Fig. 3.11.b. Three different regions, correspondingto floppy (x < xc(1)), isostatically rigid (xc(1) < x < xc(2)) and stressed rigid(x > xc(2)) glassy networks are evidenced. In particular the isostatically rigidregion corresponds to a vanishing ∆Hnr, and values for the transition con-centrations xc(1) = 29 and xc(2) = 37 are reported [138]. A comparisonbetween fragility and non-reversing heat-flow in AsxSe100−x can be found inFig. 3.16.e. Those glasses belonging to the floppy and stressed rigid compo-sition ranges are fragile, while those belonging to the isostatically rigid (orintermediate) region are strong.

The samples investigated in the present work, would thus present onemember (x=40) from the stressed rigid region, one from the intermediate(x=30) region, and two (x=10, 20) from the floppy region. Once acceptedthis picture, it turns out that floppy AsxSe100−x networks are characterized byrelaxation function not conforming to the simple KWW model, with broaderrelaxation time distributions. On the contrary, both isostatically and stressedrigid networks are found to conform to a one-decay KWW model. The stretch-ing parameter β is found to decrease in passing trough xc(2) = 37 on increas-ing Se content, from β ∼ 0.65 at x=40 to β ∼ 0.56 at x=30.

The connection between glassy network rigidity and supercooled liquid dy-namics, seems to emerge – in AsxSe100−x systems – with an increased strength

116

7.2. AS-S CHALCOGENIDES

in view of the presented IRPCS results. We may conjecture on a deeper for-mulation of this connection than that exposed in Section 3.2.4, based onthe comparison between fragility and non-reversing heat-flow. Glass As-Secompositions in the intermediate phase give rise to strong liquids, charac-terized by stretched exponential relaxation functions. Floppy glasses giverise to fragile liquids, characterized by relaxation functions which behave ina non-conventional KWW fashion. Rigid glasses give rise to fragile liquidscharacterized by single KWW decays.

In perspective, an IRPCS study of more AsxSe100−x compositions in thefloppy, intermediate and rigid region would eventually provide more evidencefor the relation between dynamics in the supercooled liquid phase and networkrigidity in the glass phase. Investigation of different chalcogenide systems,such as Ge-Se or Si-Se, would serve to check for system-dependence of theunderlined phenomenology.

7.2 As-S chalcogenides

In the present section we will report on an investigation of representativesystems belonging to the series AsxS100−x in the supercooled liquid state,analogous to the one presented in Section 7.1 for As-Se chalcogenides. Onone hand, the experimental procedure as well as data treatment developedon the same line followed during the aforementioned investigation, and manydetails will be taken for granted. On the other hand, results obtained forthe present systems will need a separate discussion, since peculiar long-timerelaxation mechanisms are found.

Samples with arsenic content x=10, 20, 30, 40 at.% have been studied. Asusual, a comparison with calorimetry and viscosimetry results will be done.

7.2.1 Experimental

During the present investigation the setup operated at two fixed values of thescattering angle, θ ∼ 90◦ and ∼ 65◦. The relaxational dynamics was foundto be angle independent, and results from the 90◦ scattering angle will bepresented here, unless otherwise specified. The samples were loaded in sealedquartz tubes with dimensions 6 mm outer diameter and 4 mm inner diameter.Light scattering data were recorded in the supercooled liquid regime and theliquid was kept at least 20 minutes at each temperature for equilibration,before each acquisition. Several acquisition runs were performed for eachsample in order to check reproducibility.

A representative set of the measured correlation functions is given inFig.7.1 for all the investigated As concentrations (open circles).

117


10-4 10-3 10-2 10-1 100 101

0.0

0.2

0.4

0.6

0.8

1.0

10-4 10-3 10-2 10-1 100 101 1020.0

0.2

0.4

0.6

0.8

0.0

0.2

0.4

0.6

0.8

1.010-4 10-3 10-2 10-1 100 101 102

10-4 10-3 10-2 10-1 100 1010.0

0.2

0.4

0.6

0.8

t [s]t [s]

norm

aliz

ed

g(2) (t

) x = 40

x = 30

AsxS

1-x

170OC

270OC

τ(2)

α

τ(1)

α

x = 20

155OC

280OC

τ(1)

α

τ(1)

α

τ(2)

α

τ(1)

α

norm

aliz

ed

g(2) (t

)

x = 10

155OC

310OC

220OC

280OC

Figure 7.8: Normalized intensity autocorrelation functions from supercooledAsxS100−x are reported for all the compositions investigated (◦). Continuum linesrepresent the model function Eq. 7.2. For each composition the higher and lowerexperimentally accessed temperatures are reported. Temperature steps for x=10and 20 are each 10 degrees up to 160◦C; for x=30 T -steps are each 5◦C from 170◦Cto 210◦C, and then each 10◦C. For x=40 T -steps are each 10◦C.


Reduced homodyne correlation functions (Fig. 7.8), as for the As-Se series,are found to display a one decay behavior at the stoichiometric composition(As2Se3, x=40) and for x=30. For x=10 and 20 a more complex scenariois found. Nevertheless, at variance with the results for As-Se, relaxationfunctions display here two well separated decay channels in the chalcogenrich region.

A model with several stretched exponential decay channels, given by Eq.7.1, with N = 1 for x=30, 40, and N > 1 for x=10, 20, was used to reproducethe data. However, poor results for the average relaxation times are obtainedfor x=10 and 20. In this case, a regularized inverse Laplace transform (ILT)of g(1)(t) using the CONTIN algorithm provided better results (see Section7.1.2). The RTD presents two peak features for x< 30, corresponding to the

118


relaxation rates of the decay steps in g(2)(t). Average relaxation times will

be indicated as < τ(1)α > and < τ

(2)α >, corresponding to shorter and longer

time-scales respectively (see Fig. 7.8 for x=10, 20).For x=30 and 40 the density correlation function conforms to a KWW

function. Alternatively the regularized ILT analysis provides good estimatesof the average relaxation time at these concentrations, in agreement with esti-mates from KWW parameters. Model functions for g(2)(t), Eq. 7.2, obtainedfrom ILT analysis are reported as full lines in Fig. 7.8. The unique averagerelaxation time will be indicated, in this case, as < τ

(1)α > (Fig. 7.8 for x=30,

40). The reason for this will become clear in the following section.

7.2.3 Structural relaxation in the As-S series

The average relaxation time < τ(1)α > is reported for all the investigated

compositions in Fig. 7.9 against the inverse temperature (open symbols).Conveniently normalized viscosity data from Ref. [143] are also reported(line plus full symbols) for the compositions x=12.5, 20, 30, 40.

The behavior of < τ(1)α > is found to fairly reproduce the viscosity vs.

temperature path. Estimates for the modulus G1 (G∞ in Eq. 7.6), rangingaround 108 Pa, are reported in figure 7.11 (◦). A best fit procedure witha VFT model function for x < 40, or with an Arrhenius model for x=40,provided appropriate description for this relaxation time dependence on tem-perature. These model curves are reported as full lines in Fig. 7.9. A specialcase is that of As10S90, for which a reliable estimate of the VFT parameterswas prevented by the experimental uncertainty, as well as by the rather scat-tered distribution of 〈τ 1

α(T )〉. In this case the Vogel temperature T0 was fixedto the value reported in Tab. 3.13 for x=12.5, derived from viscosity data.This provided good agreement with experimental data on the investigatedtemperature range. However we should bare in mind, in the following, thatany extrapolation made for x=10 brings some degree of approximation.

At this point, we hypothesize that the relaxation rate < τ(1)α > is the one

driving structural arrest at the glass transition. Evidences for this identifi-cation will be given in the following. The glass transition temperature forthe present systems, obtained by extrapolating < τ

(1)α > to 100 s according

to the underlying VFT or Arrhenius laws, is reported in Fig. 7.12.b (◦) to-gether with the calorimetric MDSC results (•) and estimates of this quantityfrom viscosimetry (N) (see Section 3.2.2). Our extrapolated glass temper-

ature, that will be addressed to as T(1)g , follows the compositional trend of

the calorimetric glass transition also indicated as T(1)g (it is the lower MDSC

temperature (•) below x∼ 25). This circumstance, if on one hand is somehowexpected for x≥30, where a single relaxation step is found, on the other handposes the question of an interpretation of the slower relaxation rate < τ

(2)α >

for x< 30.Consider now the temperature dependence of < τ

(2)α >, reported in Fig.

119


0.6 0.7 0.8 0.9 1.0

-5

-4

-3

-2

-1

0

1

2

1.8 2.1 2.4 2.7 3.0-6

-4

-2

0

2

4

m=16 x = 10

x = 20 x = 30 x = 40

T (1)

g (x) / T

AsxS

100-x

τ(1)

α

Lo

g [<

τ(1)

α >

/s]Lo

g [<

τ(1)

α >

/s]

x=40

x=30

x=12

.5

1000/T [K-1]

x=20

Figure 7.9: (a) Logarithm of average relaxation time < τ(1)α > vs. inverse temper-

ature for supercooled liquid AsxSe100−x samples, with x=10 (◦), 20 (△), 30 (◦)and 40 (⋄) As at.%. Relaxation times are derived from a regularized ILT analysisof IRPC correlation functions shown in Fig. 7.8. For x≥ 30 this is the only re-laxation time-scale found in the PCS window. Instead, for x< 30 τ1

α characterizesthe faster decay step of IRPCS relaxation functions (see Fig. 7.8). Vogel-Fulcher-Tammann (VFT) model curves are reported as full lines ( , and forx=10, 20 and 30 respectively). An Arrhenius model was used to reproduce exper-imental data for x=40 ( ). Viscosity data (full symbols plus full lines) for thecompositions x=12.5, 20, 30 and 40 (see Fig. 3.12.b) are compared with the struc-tural relaxation time by normalization. The obtained shear modulus G1 is shownin Fig. 7.11 (black symbols). (b) Fragility plot for AsxS100−x. Experimental dataare reported together with the same model curves as in (a), extrapolating to theglass temperature limit of 100 s at Tg/T = 1. The extreme of strong behavior witha limiting fragility m = 16 is referenced ( ).

7.10 as open symbols. Experimental trends for this quantity have been mod-elled with a VFT function for x=20 ( ) and with an Arrhenius law for x=10( ). Scaled viscosity data are also reported as full symbols, and the corre-sponding glass shear modulus G2 is reported in Fig. 7.11 (◦). The agreement

between temperature paths of < τ(2)α > and normalized viscosity points out,

even in this case, the validity of an hypothetical Maxwell relation. However,comparing shear moduli G1 and G2 relative to relaxation times τ

(1)α and τ

(2)α

for x< 30 with the unique glass modulus G1 for x≥30 (Fig. 7.11), we obtaina second indication for the consistency of our hypothesis: the glass modulusrelated to the fast relaxation time for x<30 is of the same order of magnitudeof the unique glass modulus for x≥30. Moreover, < τ

(1)α > and < τ

(2)α > share

the same temperature behavior, with < τ(2)α >∼ 103 < τ

(1)α >. Extrapolation

of the slow relaxation time to 100 s provides an estimate of the glass tem-perature T

(2)g , whose meaning becomes less obscure if reference is again made

to Fig. 7.12.b. Here T(2)g values are reported as grey open circles, and are

120


1.8 2.1 2.4 2.7 3.0

-2

0

2

4

6

Log

[< τ(2

)

α >

/s]

AsxS

100-x

x=20

x=12

.5

τ(2)

α

1000/T [K-1]

Figure 7.10: Logarithm of average relaxation time < τ(2)α > vs. inverse temper-

ature for supercooled liquid AsxSe100−x samples, with x=10 (◦) and x=20 (△)As at.%. Relaxation times are derived from a regularized ILT analysis of IRPCcorrelation functions shown in Fig. 7.8. τ2

α characterizes the slower decay step ofIRPCS relaxation functions for x< 30(see Fig. 7.8). A VFT model has been usedto reproduce the data of x=20 composition ( ), while an Arrhenius model ( )provided better results for x=10. Viscosity data (full symbols) for the compositionsx=20 and 12.5 (see Fig. 3.12.b) are compared with the structural relaxation timeby normalization. The obtained shear modulus G2 is shown in Fig. 7.11 (◦).

comparable to the high calorimetric transition temperature from Ref. [140](•).

Thus, two glass transition temperatures T(1)g < T

(2)g are found in the

chalcogen rich region below x=x, with 20 < x < 30, in agreement with MDSCresults (where x ∼ 25, see Fig. 7.12). Although the IRPCS overestimates

T(1)g with respect to calorimetry of about 30 degrees, we point out that these

estimates may be affected by the statistical quality of data, mostly for x=10,in which case the underlying VFT law, used to extrapolate the structuralrelaxation time to 100 s, brings a priori information as the Vogel tempera-ture T0 was derived from viscosimety data. However the trend of a stronglydecreasing glass temperature T

(1)g upon adding chalcogen to the system is

confirmed. The two glass transitions are related to structural arrest of somedegrees of freedom occurring when the temperature of the supercooled meltis lowered. The dynamical processes characterizing these degrees of freedomoccur on two well distinct time-scales, τ

(1)α < τ

(2)α , separated by three orders

of magnitude, which seem to share a common cohoperative nature, as theirtemperature behaviors share the same fate. These dynamical processes mir-ror in PCS homodyne correlation functions as two stretched exponential stepsA discussion on the possible origins of relaxation processes in the sulfur-richregion will be made in the following section.

121


10 15 20 25 30 35 40104

105

106

107

108

109

G1 from τ(1)

α

G2 from τ(2)

α

AsxS

100-x

As at%

G1, G

2 [Pa]

Figure 7.11: Values for the glass shear moduli G1 and G2 for AsxS100−x systems,

derived as Gk = η/ < τ(k)α >.

With our estimate of the glass temperature T(1)g , we can construct a

fragility plot for the relaxation rate τ(1)α , which is reported in Fig. 7.9.b.

From the slope of the data near Tg, we find the fragility index (Eq. 2.10),shown in Fig.7.12.a (◦). The fragility parameter m estimated from viscosime-try (see Tab. 3.13) is also reported (blue triangles). The IRPCS fragility islower than its viscosity-derived counterpart, with an offset of order 10 ormore, along the entire composition range. At a first sight, such inconsistencywould seem to match against the accordance between average relaxation timean scaled viscosity depicted in Fig. 7.9. Let us turn again to Fig. 7.9.a. Av-erage relaxation times are extrapolated to 100 s, i.e. the dashed horizontalline, in order to extract T

(1)g . Viscosity data, instead, are extrapolated to

1013 poise (see Fig. 3.12.b), which corresponds to the higher viscosity limitsreported in Fig. 7.9 (lines plus full symbols) in scaled units (i.e. in s, onceη/G1 is considered). By extrapolation of these scaled viscosity paths to 100 s,

we obtain T(1)g and m1 values reported as diamonds in Fig. 7.12. As expected,

the agreement with IRPCS estimates would be highly improved in this case.In Fig. 7.12.b the fragility parameter m2 estimated from extrapolation of

slow average relaxation time < τ(2)α > to 100 s is reported (◦). These values

are not much higher than the strong limit m = 16.

7.2.4 Discussion

The two-step relaxation functions (Fig. 7.8) obtained in this work on AsxS100−x

for x < 30, provide clear evidence of two independent relaxation processes.In principle, there exist several possibilities for the assignment of these pro-cesses. Since liquid sulfur is well known to form chains, one might be temptedto compare the long-time step, characterized by a slower dynamics than thestructural relaxation one, to the same feature appearing in relaxation spectraof high molecular weight polymers [197, 198]. However, phase separation of

122


10 20 30 40

40

80

120

160

200

20

30

40

50

T(2)

g

T(1)

g

AsxS

100-x

(b)

fra

gilit

y

T

g [O

C]

As at%

from IRPCS τ(1)

α

from IRPCS τ(2)

α

from MDSC from viscosimetry from rescaled viscosity

T(1)

g

m1

(a)

m2

Figure 7.12: (a) The fragility m1 parameter for the series AsxS100−x is reportedversus the As concentration x, from the present work (◦) as well as from viscositydata [143] (N are obtained from η extrapolation to 1013 poise, blue diamonds areobtained by extrapolation of η/G1 to 100 s). The fragility parameter m2 is also

reported (◦). (b) T(1)g and T

(2)g from MDSC ([140], •) and present work (◦ and

◦ respectively). Estimates of T(1)g from viscosimetry are also reported. Triangles

are obtained from extrapolation to 1013 poise, blue diamonds are obtained byextrapolation of η/G1 to 100 s.

the sulfur-rich binaries has been revealed by different experimental investi-gations [130, 140] for x . 28.5. Below this composition, excess sulfur atomsare believed to form rings. This issue has been addressed in Sections 3.2.1and 3.2.2. The coexistence of S8 rings and As-S networks of compositionnear x=28.5, could be at the origin of the double-step decays observed inIRPCS correlation functions. Moreover, phase separation could give a rea-sonable interpretation for the emergence of two glass temperatures in MDSCfor compositions below x ∼ 25 (black dots in Fig. 7.12.b). PCS average re-laxation times extrapolate to glass temperatures which reasonably agree withthe MDSC results.

In this picture, different molecular mechanisms would be involved in thetwo parts of the complete process. The slow part, characterized by an averagerelaxation time < τ

(2)α >, gives rise to arrest of some related degrees of free-

dom at a temperature, T(2)g , which is roughly composition independent (Fig.

7.12.b) below some 20 < x < 30, and near to the glass temperature for the28.5 As at.%. Thus, the corresponding molecular mechanism may be thought

123


as involving rearrangement of the As-S network phase. The fast part, char-acterized by an average relaxation time < τ

(1)α >, should reasonably involve

cooperative processes in which the sulfur-rich phase molecular mechanismsplay some role.

124

8Conclusions

Infrared Photon Correlation Spectroscopy (IRPCS), a novel experimentaltechnique developed in our laboratories, allowed us for the first time, toperform Dynamic Light Scattering experiments on supercooled chalcogenideglass formers and liquid chalcogens in the time domain (1µs to 100 s) typicallyprobed by PCS.

Among chalcogen elements, liquid sulfur and selenium have been objectof investigation in this work. Quite peculiar relaxation processes were probedin these systems, related to different dynamical origins. In the case of liquidsulfur, homodyne correlation functions have been recorded above the meltingpoint at varying exchanged momentum, in a temperature region across the so-called λ-transition point, Tλ∼159◦C. This temperature marks the transitionbetween a molecular liquid phase (for T < Tλ) and a polymeric phase (for T >Tλ), in which a certain extent of S atoms form diradical chains. A peculiarrelaxation process is characterized by a typical decay-time in the ms range,which is found to abruptly increase just below Tλ, attaining its extremal valueat the transition, and then to decrease in a wider temperature range. Thisresult confirmed a previous IRPCS investigation of this system [79], in whichthe existence of this ‘chain relaxation time’ τc was firstly revealed. However,a certain dependence on the sample thermal history was observed for the τc

vs. temperature path. This has been hypothetically ascribed to the presenceof residual polymer chains below the transition temperature. Moreover, an1/q2 dependence of the chain relaxation time has emerged from the presentstudy.

In the case of selenium, we were able to measure the relaxation timeassociated to the single relaxation process observed in this liquid at differenttemperatures covering the normal as well as the supercooled regimes. Theevolution of the correlation time with temperature has been ascribed to theself-diffusion dynamics of selenium clusters whose effective radius has beenestimated as a function of temperature. The size of clusters is found toincrease as temperature decreases, ranging between 50 nm and 350 nm.

Among ChGs, four concentrations (x=10, 20, 30 and 40) from each oneof the AsxSe100−x and AsxS100−x series have been investigated in the super-cooled liquid phase above the glass temperature Tg. In AsxSe100−x systems,homodyne intensity correlation functions are characterized by a distribution

125

CHAPTER 8. CONCLUSIONS

of relaxation times which becomes broader and broader as the chalcogen-richregion is accessed. Relaxation functions are characterized by a stretched ex-ponential behavior for x≥30, in the composition region where the glass phaseis characterized by rigid (isostatically or stressed) networks. For composi-tions x < 30, where the glass phase is characterized by underconstrained,floppy networks, relaxation functions display more complex behaviors, in-volving multiple relaxation rates. Glass temperature and fragility parametershave been estimated, providing good agreement with previous viscosimetryinvestigations.

In AsxS100−x systems homodyne intensity correlation functions display aone-step stretched exponential behavior for x≥30, while a two-step stretchedexponential behavior is found for x<30. The two relaxation rates in thesulfur-rich region both follow, at varying temperature, the same behavior ofviscosity. Extrapolation of these average decay times to 100 s yields twoglass temperatures, T

(1)g and T

(2)g , providing reasonable agreement with the

two glass temperatures revealed by calorimetry experiments for x. 25. Thecomplex scenario holding in the chalcogen rich region has been ascribed tophase separation on the mesoscopic length-scale. The two phases are thoughtto be constituted by excess molecular sulfur coexisting with As-S networks oflower chalcogen content than the nominal concentration.

126

Bibliography

[1] M. A. Popescu, Non-crystalline chalcogenides (Kluwer Academic Publisher, Dor-drecht/Boston/London, 2001).

[2] A. V. Kolobov, editor, Photo-induced metastability in amorphous semiconductors

(WileyVCH, Berlin, 2003).

[3] M. Popescu et al., Chalc. Lett. 1 (2004).

[4] D. Lezal, J. Pedlikova, and J. Zavadil, Chalc. Lett. 1 (2004).

[5] G. Curatu, Design and fabrication of low-cost thermal imaging optics using precisionchalcogenide glass molding Vol. 7060, p. 706008, SPIE, 2008.

[6] J. Kanka, Design of microstructured optical fibres made from highly nonlinearglasses for fwm-based telecom applications Vol. 6990, p. 69900E, SPIE, 2008.

[7] D. Ielmini, Phys. Rev. B 78, 035308 (2008).

[8] H. L. Rowe et al., The application of the mid-infrared spectral region in medicalsurgery: chalcogenide glass optical fibre for 10.6 µm laser transmission Vol. 6852, p.685208, SPIE, 2008.

[9] S. A. R. Hilton, J. McCord, R. Timm, and R. A. L. Blanc, Amorphous materialsmolded ir lens progress report Vol. 6940, p. 69400Q, SPIE, 2008.

[10] D. Grobnic, S. J. Mihailov, C. W. Smelser, and R. Walker, Bragg gratings madewith ultrafast radiation in non-silica glasses; fluoride, phosphate, borosilicate andchalcogenide bragg gratings Vol. 6796, p. 67961K, SPIE, 2007.

[11] R. W. Wood, Phil. Mag. 3, 607 (1902).

[12] W. Meier, Phil. Mag. 31, 1017 (1910).

[13] G. Czack, G. Kirschstein, and H. Kugler, editorsSelen, Gmelin Handbuch der Anor-

ganischen Chemie Vol. Ergnzungsband A3, 8th edition ed. (Springer, Berlin Heidel-berg, 1981).

[14] J. C. Wheeler, S. J. Kennedy, and P. Pfeuty, Phys. Rev. Lett. 45, 1748 (1980).

[15] R. Frerichs, Phys. Rev. 78, 637 (1950).

[16] R. Frerichs, J. Opt. Soc. Am. 43, 1153 (1953).

[17] G. Dewulf, Rev. Opt. 33, 513 (1954).

[18] A. W. Klein, Verres et Rfractaires 9, 147 (1955).

[19] B. T. K. N. A. Goriunova, Zhurnal Tekhnicheskoi Fiziki (Russian) 25, 2069 (1955).

[20] S. R. Ovshinsky, Phys. Rev. Lett. 21, 1450 (1968).

[21] N. K. A. et al., Semiconducting II-VI, IV-VI and V-VI Compounds (Plenum Press,New York, 1969).

[22] J. K. Zakis and H. Fritzche, Phys. Stat. Sol. B 64, 123 (1947).

[23] S. R. Elliott, Physics of amorphous materials (Longman, London/New York, 1984).

[24] M. F. Churbanov, J. Non-Cryst. Solids 184, 25 (1995).

[25] J. S. Sanghera and I. D. Aggarwal, J. Non-Cryst. Solids 256-257, 6 (1999).

[26] C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, and S. W. Martin, J.Appl. Phys. 88, 3113 (2000).

127

BIBLIOGRAPHY

[27] W. Gotze and L. Sjogren, Rep. Prog. Phys. 55, 241 (1992).

[28] E. Donth, The glass transition, relaxation dynamics in liquids and disordered mate-

rials (Springer, Berlin, 2001).

[29] L. Leuzzi and T. M. Nieuwenhuizen, Thermodynamics of the glassy state Series incondensed matter physics (Taylor and Francis, New York/Oxon, UK, 2007).

[30] C. A. Angell, J. Phys. Chem. Sol. 49, 863 (1988).

[31] J. H. Crowe, J. F. Carpenter, and L. M. Crowe, Annu. Rev. Physiol. 60, 73 (1998).

[32] J. H. Crowe and L. M. Crowe, Science 223, 701 (1984).

[33] K. B. Storey and J. M. Storey, Annu. Rev. Ecol. Syst. 27, 365 (1996).

[34] S. N. Timasheff, Annu. Rev. Biophys. Biomol. Struct. 22, 67 (1993).

[35] D. L. Sidebottom, Phys. Rev. E 76, 011505 (2007).

[36] R. Bohmer, K. L. Ngai, C. A. Angell, and D. J. Plazek, J. Chem. Phys. 99, 4201(1993).

[37] B. Mysen and P. Richet, Silicate glasses and melts, properties and structure, Devel-opments in Geochemistry Vol. 10 (Elsevier, Amsterdam, NL, 2005).

[38] K. Rao, Structural chemistry of glasses (Elsevier, Amsterdam, NL, 2002).

[39] M. Mezard, G. Parisi, and M. Virasoro, Spin glass theory and beyond (WorldsScientific, Singapore, MAL, 1987).

[40] W. Gotze, Liquids, freezing and the glass transition, Les Houches Summer School ofTheoretical Physics Vol. session LI (1989) (North-Holland Amsterdam, 1991).

[41] C. Hansen, F. Stickel, T. Berger, R. Richert, and E. W. Fischer, J. Chem. Phys.107, 1086 (1997).

[42] G. P. Johari and M. Goldstein, J. Chem. Phys. 53, 2372 (1970).

[43] F. Kolraush, Pogg. Ann. Phys. 12, 393 (1847).

[44] J. D. Bast and P. Gilard, Phys. Che./ Glasses 4, 117 (1963).

[45] G. Williams and D. C. Watts, Trans. Faraday Soc. 66, 80 (1970).

[46] K. L. Ngai, Comm. Sol. State Phys. 9, 127 (1979).

[47] S. N. Yannopoulos, G. N. Papatheodorou, and G. Fytas, Phys. Rev. B 60, 15131(1999).

[48] W. Kauzmann, Chem. Rev. 43, 219 (1948).

[49] H. Vogel, Phys Z. 22, 645 (1921).

[50] G. Tammann and G. Hesse, Z. Anorg. Allg. Chem. 156, 245 (1926).

[51] G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965).

[52] C. A. Angell and W. Sichina, Ann. New York Acad. Sci. 279, 53 (1976).

[53] C. A. Angell, Relaxations in complex systems (National Technical Information Ser-vice, Springfield, 1985), p. 3.

[54] I. M. Hodge, J. Non-Cryst. Solids 202, 164 (1996).

[55] I. Hodge and J. O’Reilly, J. Phys. Chem. B 103, 4171 (1999).

[56] T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes, Phys. Rev. A 40, 1045 (1989).

[57] T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. A 35, 3072 (1987).

128

BIBLIOGRAPHY

[58] R. Steudel, Top. Curr. Chem. 102, 149 (1982).

[59] B. Meyer, Chem. Rev. 76, 367 (1976).

[60] M. Schmidt, Angew. Chem. 85, 474 (1973).

[61] B. Meyer, Elemental Sulfur (New York: Interscience, 1965).

[62] A. T. Ward, J. Phys. Chem. 72, 4133 (1968).

[63] R. Steudel and H.-J. Mausle, Z. Anorg. Allg. Chem. 478, 139 (1981).

[64] R. F. Bacon and R. Fanelli, J. Am. Chem. Soc. 65, 639 (1943).

[65] R. Sasson and E. T. Arakawa, Appl. Opt. 25, 2675 (1986).

[66] A. Donaldson and A. D. Caplin, Phyl. Mag. B 52, 185 (1985).

[67] S. Hosokawa, T. Matsuoka, and K. Tamura, J. Phys. Cond. Matt. 6, 5273 (1994).

[68] E. D. West, J. Am. Chem. Soc. 81, 29 (1959).

[69] A. M. Kellas, J. Chem. Soc. 113, 903 (1918).

[70] A. G. Kalampounias, K. S. Andrikopoulos, and S. N.Yannopoulos, J. Chem. Phys.118, 8460 (2003).

[71] J. C. Koh and W. Klement, J. Phys. Chem. 74, 4280 (1970).

[72] V. K. Kozhevnikov, W. B. Payne, and J. K. Olson, J. Chem. Phys. 121, 7379 (2004).

[73] A. V. Tobolsky and A. Eisenberg, J. Am. Chem. Soc. 81, 780 (1959).

[74] L. Crapanzano, PhD thesis, 2006.

[75] S. C. Greer, J.Phys. Chem. B 102, 5413 (1998).

[76] A. T. Ward and M. B.Myers, J. Phys. Chem. 73, 1374 (1969).

[77] F. Scarponi, D. Fioretto, L. Crapanzano, and G. Monaco, Submitted to Phil. Mag. .

[78] A. D. Alvarenga, M. Grimsditch, S. Susman, and S. C. Rowland, J. Phys. Chem.100, 11456 (1996).

[79] T. Scopigno et al., Phys. Rev. Lett. 99, 025701 (2007).

[80] R. Powell and H. Eyring, J. Am. Chem. Soc. 65, 648 (1943).

[81] F. J. Touro and T. K. Wiewiorowski, J. Phys. Chem. 70, 239 (1966).

[82] A. Eisenberg, Macromolecules 2, 44 (1969).

[83] S. J. Kennedy and J. C. Wheeler, J. Chem. Phys. 78, 1523 (1983).

[84] S. N. Yannopoulos and K. S. Andrikopoulos, Phys. Rev. B 69, 144206 (2004).

[85] J. Donohue, The structure of elements (Wiley, New York, 1974).

[86] P. Jovari and R. G. D. amd L. Pusztai, Phys. Rev. B 67, 172201 (2003).

[87] K. Nakamura and A. Ikawa, Phys. Rev. B 67, 104203 (2003).

[88] S. Brieglieb, Z. Phys. Chem. A 144, 321 (1929).

[89] A. Eisemberg and A. V. Tobolsky, J. Polym. Sci. 46, 19 (1960).

[90] R. C. Keezer and M. W. Bailey, Mater. Res. Bull. 2, 185 (1967).

[91] J. Deprez, J. Rialland, and J. Perron, J. Phys. Paris Lett. 39, L142 (1978).

[92] H. Krebs and W. Morsch, Z. Anorg. Allgem. Chem. 263, 305 (1950).

129

BIBLIOGRAPHY

[93] K. M. Bernatz, I. Echeverrıa, S. L. Simon, and D. J. Plazek, J. Non-Cryst. Solids307-310, 790 (2002).

[94] M. Misawa and K. Suzuki, Trans. Japan. Inst. Metals 18, 427 (1977).

[95] C. Massen, A. Weijts, and J. Poulis, Trans. Faraday Soc. 60, 427 (1964).

[96] J. W. W. Warren and R. Dupree, Phys. Rev. B 22, 2257 (1980).

[97] R. B. Stephens, J. Non-Cryst. Solids 20, 75 (1976).

[98] E. Kittinger, Phys. Status Solidi A 44, K35 (1977).

[99] S. Etienne, G. Guenin, and J. Perez, J. Phys. D 12, 2189 (1979).

[100] E. Kittinger, J. Non-Cryst. Solids 27, 421 (1978).

[101] E. Kittinger, Z. Naturforsch 32a, 946 (1977).

[102] M. Abkovitz, D. F. Pichan, and J. M. Pochan, J. Appl. Phys. 53, 4173 (1982).

[103] R. Bohmer and C. A. Angell, Phys. Rev. B 48, 5857 (1993).

[104] J. D. Ferry, Viscoelastic properties of polymers, 3rd ed. (Wiley, New York, 1980).

[105] R. S. Caldwell and H. Y. Fan, Phys. Rev. 114, 664 (1959).

[106] C. Bichara, A. Pellegatti, and J. P. Gaspard, Phys. Rev. B 49, 6581 (1994).

[107] D. Molina, E. Lomba, and G. Kahl, Phys. Rev. B 60, 6372 (1999).

[108] E. Lomba, D. Molina, and M. Alvarez, Phys. Rev. B 61, 9314 (2000).

[109] D. Hohl and R. O. Jones, Phys. Rev. B 43, 3856 (1991).

[110] F. Kirchhoff, G. Kresse, and M. J. Gillan, Phys. Rev. B 57, 10 482 (1999).

[111] F. Shimojo, K. Hoshino, M. Watabe, and Y. Zempo, J. Phys.: Cond.Matt. 10, 1199(1998).

[112] X. Zhang and D. A. Drabold, Phys. Rev. Lett. 83, 5042 (1999).

[113] A. V. Kolobov, H. Oyanagi, K. Tanaka, and K. Tanaka, Phys. Rev. B 55, 726 (1997).

[114] A. V. Kolobov, M. Kondo, H. Oyanagi, A. Matsuda, and K. Tanaka, Phys. Rev. B58, 12 004 (1998).

[115] H. A. Davies and J. B. Hull, J. Mat. Sci. 9, 707 (1974).

[116] J. Stuck, J. Non-Cryst. Solids 4, 1 (1070).

[117] G. C. Das, N. S. Platakis, and M. B. Bever, J. Non-Cryst. Solids 15, 30 (1974).

[118] N. A. Goriunova, B. T. Kolomiets, and V. P. Shilo, Soviet Phys. Tech. Phys. 3, 912(1958).

[119] A. A. Vaipolin and E. A. Porai-Koshits, Soviet Phys. Solid St. 2, 1500 (1961).

[120] A. A. Vaipolin and E. A. Porai-Koshits, Soviet Phys. Solid St. 5, 497 (1963).

[121] C. Y. Yang, M. A. Paesler, and D. E. Sayers, Phys. Rev. B 39, 10 342 (1989).

[122] G. Pfeiffer, J. J. Rehr, and D. E. Sayers, Phys. Rev. B 51, 804 (1995).

[123] V. Kolobov, K. Tanaka, and H. Oyanagi, Phys. Solid State 39, 64 (1997).

[124] A. M. Flank et al., J. Non-Cryst. Solids 91, 306 (1987).

[125] A. J. Leadbetter and A. J. Apling, J. Non-Cryst. Solids 15, 250 (1974).

[126] Q. Ma, W. Zhou, D. E. Sayers, and M. A. Paesler, Phys. Rev. B 52, 10025 (1995).

130

BIBLIOGRAPHY

[127] E. Bychkov, C. J. Benmore, and D. L. P. and, Phys. Rev. B 72, 172107 (2005).

[128] Y. Kawamoto and S. Tsuchihashi, J. Am. Ceram. Soc. 54, 131 (1971).

[129] G. Lucovsky, F. L. Galeener, R. C. Keezer, R. H. Geils, and H. A. Six, Phys. Rev.B 10, 5134 (1974).

[130] E. Bychkov, M. Molishova, D. L. Price, C. J. Benmore, and A. Lorriaux, J. Non-Cryst. Solids 352, 63 (2006).

[131] M. Kastner, Phys. Rev. B 7, 5237 (1973).

[132] M. Tanaka, M. T, and M. Hattori, Jpn. J. Appl. Phys. 5, 185 (1966).

[133] R. blachnik and A. Hoppe, J. Non-Cryst. Solids 34, 191 (1979).

[134] M.B.Meyers and E. J. Felty, MRS Bull. 2, 535 (1967).

[135] P. Boolchand, G. Lucovsky, J. C. Phillips, and M. F. Thorpe, Phil. Mag. 85, 3823(2005).

[136] T. Wagner, S. O. Kasap, and K. Maeda, J. Mat. Research 12, 1892 (1997).

[137] E. C. Weeks, J. R. Croker, A. C. Levitt, and A. S. amd A. Weitz, Science 287, 627(2000).

[138] D. G. Georgiev, P. Boolchand, and M. Micoulaut, Phys. Rev. B 62, R9228 (2000).

[139] T. Wagner and S. O. Kasap, Philos. Mag. B 74, 667 (1996).

[140] T. Wagner, S. O. Kasap, M. Vlcek, A. Sklenar, and A. Stronski, J. Non-Cryst. Solids227-230, 752 (1998).

[141] S. Dzhalilov and S. Orudzhevo, Russ. J. Phys. Chem. 40, 1148 (1966).

[142] J. Malek and J. Shanelova, J. Non-Cryst. Solids 351, 3458 (2005).

[143] S. V. Nemilov, Soviet Phys. Sol. State 6, 1075 (1964).

[144] J. Malek, Thermochim. Acta 311, 183 (1998).

[145] A. V. Tobolsky, W. MacKnight, R. B. Beevers, and V. D. Gupta, Polymer 4, 423(1963).

[146] B. Ruta, Studio della dinamica ad alta frequenza in un vetro di zolfo attraversola diffusione anelastica di raggi x, Master’s thesis, Universita degli studi di Roma“Sapienza”, A.A. 2005-2006.

[147] B. G. Aitken and C. W. Ponader, J. Non-Cryst. Solids 256-257, 143 (1999).

[148] B. G. Aitken and C. W. Ponader, J. Non-Cryst. Solids 274, 124 (2000).

[149] K. Tanaka, Phys. Rev. B 39, 1270 (1989).

[150] P. Boolchand, D. G. Georgiev, and B. Goodman, J. Opt. Adv. Mat 3, 703 (2001).

[151] J. C. Phillips, J. Non-Cryst. Solids 34, 153 (1979).

[152] M. F. Thorpe, J. Non-Cryst. Solids 57, 355 (1983).

[153] P. Boolchand, D. Selvanathan, Y. Wang, D. G. Georgiev, and W. J. Bresser, Prop-

erties and applications of amorphous materials (Kluwer Academic Publisher, Dor-drecht/Boston/London, 2001), p. 97.

[154] X. W. Feng, W. J. Bresser, and P. Boolchand, Phys. Rev. Lett. 78, 4422 (1997).

[155] M. Zhang and P. Boolchand, Science 266 (1994).

131

BIBLIOGRAPHY

[156] K. L. Chopra, K. S. Harshavardhan, S. Rajgopalan, and L. K. Malhotra, Sol. StateComm. 40, 387 (1981).

[157] D. Selvanathan, W. J. Bresser, and P. Boolchand, Phys. Rev. B 61, 15061 (2000).

[158] J. A. Savage, Infrared optical materials and their antireflection coatings (AdamHilger, Bristol, 1985).

[159] J. Tauc, Amorphous and liquid semiconductors (Plenum Press, London and NewYork, 1974), p. 209.

[160] J. K. Zakis and H. Fritzche, Phys. Sta. Solid. B 64, 123 (1974).

[161] A. B. Seddon, J. Non-Cryst. Solids 184, 44 (1995).

[162] J. S. Sanghera and I. D. Aggarwal, Infrared fiber optics (CRC, Boca Raton, FL,1998), chap. 9.

[163] K. Tanaka, J. Non-Cryst. Solids 35-36, 1023 (1980).

[164] E. Skordeva, J. Optoel. Adv. Mat. 1, 43 (1999).

[165] K. Tanaka and S. Nakayama, J. Optoel. Adv. Mat. 2, 5 (2000).

[166] M. Kitao, Jpn. J. Appl. Phys. 11, 1472 (1972).

[167] R. Zallen, R. E. Drews, R. L. Emerald, and M. L. Slade, Phys. Rev. Lett. 26, 1564(1971).

[168] G. Lucovsky and R. M. Martin, J. Non-Cryst. Solids , 185 (1972).

[169] J. Tauc, Amorphous and liquid semiconductors (Plenum Press, London and NewYork, 1974).

[170] M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, Appli. Phys. Lett. , 1153(1992).

[171] M. Asobe, Opt. Fiber Tech. 3, 142 (1997).

[172] F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, J. Non-Cryst. Solids274, 232 (2000).

[173] R. Pecora, J. Chem. Phys. 40, 1604 (1964).

[174] R. Bergman, L. L. Borjesson, L. M. Torell, and A. Fontana, Phys. Rev. B 56, 11619(1996).

[175] L. Comez et al., Phys. Rev. E 60, 3086 (1999).

[176] S. Yannopoulos, G. Papatheodorou, and G. Fytas., Phys. Rev. E 53, R1328 (1995).

[177] P. N. Pusey, Neutrons, x-rays and light: scattering methods applied to soft condensed

matter (Elsevier, Amsterdam, NE, 2002), chap. 1 and 9.

[178] B. Berne and R. Pecora, Dynamic light scattering (John Wiley and Sons, New York,1976).

[179] M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).

[180] J. S. Sanghera et al., J. Optoel. Adv. Mater. 3, 627 (2001).

[181] F. Ianni, R. D. Leonardo, S. Gentilini, and G. Ruocco, Physical Review E (Statistical,Nonlinear, and Soft Matter Physics) 75, 011408 (2007).

[182] V. A. Bloomfield et al., Dynamic light scattering (Plenum, New York, 1985).

[183] N. C. Ford, J. Chem. Scr. 2, 193 (1972).

132

BIBLIOGRAPHY

[184] S. L. Brenner, R. A. Gelman, and R. Nossal, Macromolecules 11, 202 (1978).

[185] R. Nossal and S. L. Brenner, Macromolecules 11, 207 (1978).

[186] R. Mao, Y. Liu, M. B. Huglin, and P. A. Holmes, Polym. Internat. 45, 321 (1998).

[187] D. L. Tomrot, M. A. Seredniskaja, and T. D. Chkhikvadze, Sov. Phys. Dokl. 29, 961(1984).

[188] V. F. Kozhevnikov, J. M. Viner, and P. C. Taylor, Phys. Rev. B 64, 214109 (2001).

[189] G. Monaco et al., Phys. Rev. Lett. 95, 255502 (2005).

[190] S. N. Yannopoulos and K. S. Andrikopoulos, J. Chem. Phys. 121, 4747 (2004).

[191] A. I. Popov, J. Phys. C 9, L675 (1976).

[192] R. B. Stephens, J. Appl. Phys. 49, 5855 (1978).

[193] D. Caprion and H. R. Schober, Phys. Rev. B 62, 3709 (2000).

[194] G. Kresse, F. Kirchhoff, and M. J. Gillan, Phys. Rev. B 59, 3501 (1999).

[195] S. W. Provencher, Comp. Phys. Commun. 27, 229 (1982).

[196] S. W. Provencher, Comp. Phys. Commun. 27, 213 (1982).

[197] H. Kriegs et al., J. Chem. Phys. 124, 104901 (2006).

[198] K. Adachi and T. Kotaka, Macromolecules 18, 466 (1985).

133

relaxation dynamics in amorphous chalcogenides probed by...

Documents