UNIVERSIT `
A DEGLI STUDI DI ROMA ‘SAPIENZA’
FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALIPh.D. THESIS IN MATERIALS SCIENCE
Relaxation Dynamics in Amorphous
Chalcogenides probed by InfraRed Photon
Correlation Spectroscopy
Tutors:
Candidate:
Prof. Giancarlo Ruocco
Stefano Cazzato
Dr. 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 pour
The sunlight through the dark chambers of a house, You will perceive many minute bodies mingling, In a multiplicity of ways within those rays of light Throughout the entire space, and as it were In a never ending conflict of battle
Combating and contending troop with troop Without pause, maintained in motion by perpetual Encounters and separations; so that this
Should assist you to imagine what it signifies When primordial particles of matter
Are always meandering in a great void. To this extent a small thing may suggest A picture of great things, and point the way To new concepts. There is another reason Why you should give attention to those bodies Which are seen wavering confusedly
In the rays of the sun: such waverings indicate That beneath appearance there must be Motions of matter secret and unseen.
Contents
1 Introduction 7
I
Glassy dynamics and amorphous chalcogenides
11
2 Dynamics of supercooled liquids 13
2.1 Phenomenology of the glass transition . . . 13
2.1.1 Interplay of different time scales . . . 14
2.1.2 Free energy landscape . . . 18
2.1.3 Correlation functions . . . 18
2.1.4 The calorimetric glass transition . . . 20
2.1.5 Viscosity, structural relaxation time and fragility . . . 23
3 Amorphous chalcogenides 27 3.1 The chalcogens . . . 27
3.1.1 Sulfur (S) . . . 27
3.1.2 Selenium (Se) . . . 32
3.1.3 Tellurium (Te) . . . 34
3.2 Binary As-Se and As-S chalcogenides . . . 35
3.2.1 Structural properties . . . 36
3.2.2 Thermal properties . . . 39
3.2.3 Viscosity and structural relaxation . . . 42
3.2.4 The onset of an intermediate phase in binary chalco-genide glasses . . . 44
3.2.5 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 . . . 56
4.1.2 Dynamic Light Scattering . . . 59
4.2 Homodyne Photon Correlation Spectroscopy . . . 61
4.2.1 The Gaussiam approximation . . . 62
4.2.2 Discrete scatterers . . . 66
4.3 The heterodyne correlation function . . . 71
4.3.1 Heterodyne method . . . 72
4.3.2 Discrete scatterers under flow . . . 73
5 InfraRed Photon Correlation Spectroscopy 75 5.1 Setup description . . . 77
5.1.2 The sample environment . . . 79
5.1.3 The detectors . . . 81
5.2 Setup alignment and calibration . . . 82
5.3 Signal-to-noise ratio . . . 86
5.3.1 Effects due to finite intensity . . . 87
5.3.2 Effects due to finite experiment duration . . . 87
5.3.3 Effects due to unwanted scattered light . . . 88
5.4 The samples . . . 89
III
Results
91
6 Liquid chalcogens 93 6.1 Liquid sulfur . . . 93 6.1.1 Experimental . . . 93 6.1.2 Data treatment . . . 956.1.3 Temperature dependence of chain relaxation time . . . 96
6.1.4 Momentum dependence of chain relaxation time . . . . 100
6.1.5 Discussion . . . 100
6.2 Liquid selenium . . . 102
6.2.1 Experimental . . . 103
6.2.2 Results . . . 104
6.2.3 Discussion . . . 105
7 Chalcogenide glass formers 107 7.1 As-Se chalcogenides . . . 107
7.1.1 Experimental . . . 107
7.1.2 Data treatment . . . 108
7.1.3 Structural relaxation in the As-Se series . . . 113
7.1.4 Discussion . . . 115
7.2 As-S chalcogenides . . . 117
7.2.1 Experimental . . . 117
7.2.2 Data treatment . . . 118
7.2.3 Structural relaxation in the As-S series . . . 119
7.2.4 Discussion . . . 122
1
Introduction
The main subject of this experimental work is the development of a novel spectroscopic 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 of chalcogenide glasses (ChGs), in the supercooled liquid regime above the glass transition temperature.
ChGs are a group of inorganic glassy materials which contain one or more of the chalcogen elements: sulphur, selenium and tellurium (but not oxygen), in conjunction with more electropositive elements, most commonly arsenic and germanium, but also phosphorus, antimony, silicon, tin and other. Their use in a wide range of optical, electronic and memory applications [1] has stimulated a wide interest for a more thorough understanding of such ma-terials properties and of variations of such properties with composition. In particular, 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 oxide materials. Moreover, ChGs have been shown to exhibit a dazzling variety of 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 (optical gratings, 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 the dynamics in the warm and supercooled liquid phases. Both Se and S are well known to be easily obtainable in a glassy state and among the reasons that have stimulated an intense interest in investigating such systems there is also the notion that they constitute one of the simplest polymers possible. In par-ticular, vitreous selenium became of interest for the scientific community at the beginning of the 20th century when Wood [11] and Meier [12] reported the first research on the subject. Later on, in 1979, amorphous selenium has been referred to as “one of the best studied of all substances” [13]. Liquid S and Se have been recognized to be assemblies of living polymers [14], i.e. polymers whose lengths fluctuate and attain an equilibrium length distribution at any given temperature T.
The first works on ChGs were attributed to Frerichs in the early 50’s on the 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 1969 Abrikosov et al. [21] reported in their monograph the phase diagram for the As-S and As-Se systems. As-S alloys can be formed with an As content up to 46% while in As-Se this maximum content can be raised to almost 60%. These glass formulations can contain a non-stoichiometric amount of chalcogen and excess S and Se atoms can form chains, probably responsible of changes in cooperativity of structural relaxation with composition. The stability against crystallization in bulk As-S and As-Se alloys and their optical properties has stimulated activity in these materials, as well as an increasing interest in the ability 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 liquid dynamics of arsenic selenide and sulfide ChGs on approaching the glass tran-sition temperature Tg from above. This view from the liquid side allows the
experimentalist to follow the dynamical processes triggering structural arrest at Tg. Indeed, still lowering the temperature, the characteristic time-scale
of structural relaxation continues to increase, eventually falling outside any experimentally accessible window. The resulting glass phase can be viewed, in this respect, as a liquid phase in which molecules are so tightly packed that the 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, the subject of numerous investigations, both theoretically and experimentally, and the formulation of a coherent microscopic picture of the glass transition still remains a challenge for the scientific community [26, 27, 28, 29]. However, on a phenomenological ground, the rapid - often non-Arrhenius - increase of the shear viscosity η on decreasing the temperature of a supercooled liquid, approaching the glass transition temperature, has long ago been recognized as one of the salient features of the dynamical glass transition effect. This temperature behavior of viscosity has stood as the basis for the strong and fragile classification of supercooled liquids [30].
As suggested by many theoretical approaches, a good variable capable of describing the dynamics of supercooled liquids is the density time correla-tion funccorrela-tion, which encodes informacorrela-tion about the microscopic relaxacorrela-tion timescales in the material. Typically, a two step decay is observed: a fast relaxation accounts for local rearrangements of the particles, while a slow decay is related to the highly cooperative structural relaxation process. The timescales 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 of supercooled glass-forming liquids, as the scattered intensity signal provides
direct information on the timescales characterizing the decay of the den-sity fluctuations. In particular, homodyne photon correlation techniques are applicable to the measurement of dynamics evolving on timescales ranging from the µs to 100 s, corresponding to what, on a molecular scale, are long-distance, long-time phenomena, including the typical timescales of structural relaxation 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 of reaching the best experimental conditions for the investigation of the density-density time correlation function in systems offering a transmission maximum in the near IR. Through this technique we are allowed, for the first time, to investigate the nature of homodyne correlation function in supercooled liquid ChGs 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 q dependence of relaxation function can be accessed with the IRPCS setup, by performing acquisitions at five different values of the scattering angle at one time. This was made possible by the implementation of five independent ac-quisition channels, collecting the scattered signal from the sample at different scattering angles.
The thesis is divided into three parts:
1. The first part, split into two chapters, introduces the reader to the scientific problem. In the first chapter, the complex phenomenology related to the dynamical glass transition is introduced. In the second chapter structural, thermal, rheological and linear optical properties of ChGs belonging to the As-Se and As-S series, as well as of amorphous chalcogens, will be examined.
2. The second part, also divided into two chapters, presents the technique and the materials used in the experiments. In the first chapter, the basic theories for dynamic light scattering are presented. The second chapter is dedicated to the description of the experimental IRPCS technique and of samples preparation.
3. In the last part, divided into two chapters, the experimental results are presented and discussed in the light of the background provided in the first part. In particular, results on liquid chalcogens dynamics are reported in the first chapter, while the second chapter is dedicated to results on the structural relaxation dynamics in arsenic selenide and sulfide ChGs.
Part I
Glassy dynamics and
amorphous chalcogenides
Even the definition of glass is arbitrary: basically a rate of flow so slow that it is too boring and time-consuming to watch.
Kenneth Chang
The 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 falling temperatures and a ‘forming interval’ is passed, where the glass makers can work. This interval can be characterized by a viscosity range or by a cor-responding temperature range. Further lowering the temperature, the glass melt becomes a solid, i.e., glass. This continuous transition from a liquid to a glass is called the glass transition and is the subject of the present chapter. The glass transition is different from solidification by crystallization. The latter is a (first order) phase transition with a well defined thermodynamic transition temperature, the crystallization temperature. From a dynamic point of view the matter is not so simple. Crystal formation needs time to crystal nuclei of the new phase to form, and time for crystals to grow.
Crystallization is also relevant for the glass industry. The aim is usually to avoid crystallization in order to get a clear optical glass. Other industries, dealing with ceramics or enamels, for instance, are interested in a manageable interplay between the glass and phase transitions.
The glass transition is of general interest not only for manufacturing and industrial applications. It is also of general interest in natural science, ma-terials science, and the life sciences. As an example, the continuous nature of the glass transition, as opposed to the discontinuous one characterizing thermodynamic phase transitions, lays at the basis of the cryopreservation of biological tissues in nature. Many animals in winter survive freezing weather by producing glucose, or other natural sugars (e.g., trehalose), within their bodies to act as an antifreeze [31, 32]. The most commonly cited example is the wood frog who freezes solid in winter but returns to life during the spring thaw [33]. At the onset of cold temperatures, glucose produced by the liver circulates throughout the body. In addition to stabilizing proteins [34], the glass-forming properties of glucose reduces the likelihood that ice will form inside cells or capillaries. Should this crystallization occur, the rapid expansion would rupture the cell membrane or otherwise destroy the delicate internal components [33, 35].
tran-sition phenomenon, is that no generally accepted microscopic theory is so far available. Hence all pictures remain vague to a certain degree, and we must use general principles for the description. Such description is based on a view from the liquid side, in order to embrace the dynamic glass transition far above Tg. Apart from some picosecond/teraherz processes, the material is in
a state that cannot be understood from the solid standpoint. Moreover, the phenomena that will be described are typical for all liquids and disordered materials. 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 diffusive motion 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 measurable processes relax to equilibrium.
Cooling down from the liquid phase the slow degrees of freedom of the glass former are no longer accessible and the viscosity of the under-cooled melt grows several orders of magnitude in a relatively small temperature interval. As a result, in the cooling process, from some point on, the time effectively spent at a certain temperature is not enough to attain equilibrium.
The preparation, indeed, plays a fundamental role to get a glass out of a liquid, thus avoiding the crystallization of the substance. Depending on the material, the ways of obtaining a glass are very diverse, and consist not only in the cooling of a liquid but also include compression, irradiation of crystals with heavy particles, chemical reactions, polymerization, evaporation of sol-vents, deposition of chemical vapors, etc. Many kinds of materials present a glass phase at given external conditions if prepared in the proper way. For an exhaustive literature, the reader can refer to [26, 36, 37, 38].
The crystal state is always at lower energy than the glassy state, but the probability of germinating a crystal instead of a glass during the vitrification process is negligible when cooling fast enough: the nucleation of the crystal phase is practically inhibited. In a nucleation event, a small but critical number of unit cells of the stable crystal combine on a given characteristic time-scale, the nucleation time τnuc.
In order to clarify this point, a simplified picture of a nucleation event can be given. Consider a system which, at a certain time, is found in a metastable state M (see Fig. 2.1.a), and in which thermal fluctuations has brought to the formation of a bubble of the stable state S. Let l represent the linear size of the bubble, and fM, fS be the free energies per unit volume pertaining to
the metastable and stable states respectively; thus fM > fS. The energetic
cost to be payed for the nucleation of the stable state has a negative volume contribution −(fM − fS)l3 = −δfl3 playing in favor of nucleation, and a
2.1. PHENOMENOLOGY OF THE GLASS TRANSITION
Figure 2.1: (a) Schematic representation of the nucleation of a stable state bubble S within a metastable phase M. (b) The energetic cost for the formation of a nucleus of size l of the stable S phase: Cost(l) = −δfl3+σl2. This function has a maximum for l∗= 2σ/3δf .
surface contribution accounting for the two phases interaction, which can be described in terms of a surface tension σ. The latter contribution will be positive, thus playing against the stable state nucleation. The net energetic cost 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 the
opposite case, this will tend to implode. Thus there is a critical size for crystal nuclei, generated by spontaneous thermal fluctuations, to give rise to stable configurations capable of accreting. The nucleation time τnuc is the
time necessary for thermal fluctuations to give rise to a stable crystal nucleus. In a good glass former, the number of molecules involved in the nucleation must be much larger than the number of molecules cooperating in the struc-tural relaxation of the glass phase, composing what is called a ‘cooperative rearranging region’, yielding, in this way, a nucleation time much longer than the 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 and the behavior of the glass former.
Imagine to follow a liquid glass former during a cooling procedure, starting from a high temperature (look at Fig. 2.2 as a guide, starting from the left side). Already in the warm liquid, different processes occur at different time scales. At a given temperature, that we will simply denote as Tcage [29], the
thermal movement of particles is slow enough for the diffusion to be hindered by the formation of cages. A cage is, in this sense, a dynamic concept relative to each particle in the liquid, whose motion is constrained to occur next to
other particles around it, with which it collides. This is at difference with the purely collisional motion taking place in the warm liquid. Cooling further, a first bifurcation of time scales takes place between the relative fast rattling time 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, that becomes longer and longer as the temperature decreases. This process is called in many different ways in the literature. Staying close to the notation of Ref. [28, 29], we chose the name αβ. The reason will become clear in a short 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 temperature Tcage. Then, at Td, a second bifurcation occurs, more important for the onset of the glass formation, in the dynamic crossover region where the dynamic glass transition 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 particle while it is rattling with a high frequency among its neighbor particles forming a 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 a global rearrangement of the system, tanks to a large cooperativeness of the particles (we will call them α processes), and processes that involve only a limited number of molecules in a local, microscopically small, rearrangement, thus not contributing to the structural relaxation of the glass former. The latter are usually called β processes. We reserve the label α for the slowest processes, needing a huge cooperativeness to occur below the crossover region, stressing that, in general, different molecular mechanisms may be responsible for the lower temperature α process and the higher temperature αβ processes.
2.1. PHENOMENOLOGY OF THE GLASS TRANSITION
In the crossover region, thus, αβ processes bifurcate in α processes – with time scale the structural relaxation time – and β processes with a much shorter characteristic time: this way we are in the presence of a separation of time scales, that becomes more and more enhanced as temperature is lowered.
In the literature the temperature at which the crossover takes place is named either as Tc (crossover or critical) [28], as Td (dynamic) [39] or as
Tmc(mode coupling) [40]. Traces from dielectric susceptibility measurements
on two distinct polymer systems are reported on an Arrhenius plot – i.e. a plot of the typical relaxation frequency (the inverse of relaxation time) vs the inverse temperature – in Fig. 2.3.
Figure 2.3: Dielectric traces for dynamic transition in the Arrhenius diagram for two glass forming liquids. (a) Temperature dependence of the dielectric peak frequency for bis-methoxy-phenyl-cyclohexane (BMPC) [41]. The open symbols refer to the dielectric results for the α- (◦) and β-process (△). The dashed lines are a Vogel-Fulcher-Tammann (VFT, this model function will be introduced later in Section 2.1.5) fit to the α-relaxation for T < Tβ, and an Arrhenius fit to the β-relaxation data. Tβ ≈ 270K marks the bifurcation temperature regarding the α-and β-relaxation. (b) Temperature dependence of the dielectric peak frequency for ortho-terphenyl (OTP). The open symbols refer to the dielectric results for the (◦, [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 line represents scaled viscosity data of OTP. The diamonds refer to photon correlation spectroscopy (PCS) data and the triangles represent Brillouin light-scattering data (see Ref. [41] and references cited therein). Tβ ≈ 290K marks the bifurcation temperature regarding the α- and β-processes. Figure from Ref. [41].
The descriptive stile used so far in the present section, should not induce the reader to believe that all the points raised here have been brought to a stage of full understanding [29, 28]. They should serve instead as a necessary starting point on approaching a convenient description of such a problematic topic as the relaxation dynamics of glass forming systems.
2.1.2 Free energy landscape
In terms of a free energy landscape description of the phase space, where metastable states are represented by local minima and stable states by global minima, a two level structure appears: some minima of the free energy are separated by very small barriers and between them β processes take place; group of those minima are contained in larger basins separated by barriers requiring a much bigger free energy variation to be crossed. To make the system go from a configuration in one of these basins to another configuration in another basin, i.e., to have an α process, a longer time is needed. Indeed, the typical crossing time τeqis related to the free energy barrier ∆F separating
the 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 formation
of the solid glass) still very short in comparison with the observation time. Below Td the system is, thus, still at thermodynamic equilibrium. The phase
is disordered but the number of minima of the free energy increases, and some local minima become deeper. The dynamics of the slowest process (αβ for T > Td, or α for T < Td) displays a huge slowing down, but the temperature is
nevertheless high enough for the system to attain equilibrium on experimental time scales [29, 28].
2.1.3 Correlation functions
We consider in the present section the time behavior of relaxation functions in supercooled liquids. Our system comprises N elementary units, i.e. atoms, ions, molecules, with positions ri(t) and momenta pi(t), varying with time
according to the classical mechanics equations related to the Hamiltonian
H = 1 2 X i p2 i 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 by
time correlation functions of dynamical variables, and among all such possible variables there are certain which play a major role in the theoretical as well as experimental description of relaxation phenomena in liquid systems. We consider for the present purpose the number density variable ρ(r, t), defined as ρ(r, t)=. √1 N X i δ(r − ri(t)) , 2.1
We can also define its spatial Fourier transform
ρ(q, t) = √1 N X i eiq·ri(t), 2.2
2.1. PHENOMENOLOGY OF THE GLASS TRANSITION
which will turn to be useful since, generally, scattering experiments provide information on the reciprocal q space to the real space (see Chapter 4). The number density time autocorrelation function is defined as (we now specialize on samples which are isotropic on average, so that the average quantities depend only on the modulus q of the scattering vector)
F (q, t) = hρ(q, t)ρ∗(q, 0)i , 2.3 also called intermediate scattering function. It is experimentally accessible with inelastic (or dynamic) scattering techniques and provides dynamical information on the system under study. In Eq. (2.3), the symbol h· · · i denotes the statistical ensemble average
h· · · i = Z
Γ
dpNdrN(· · · )P(pNrN)
being P(pNrN) the system probability distribution over the entire phase space
Γ. 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 of the density. It is experimentally accessible with static scattering techniques and provides structural information on the system under study.
The normalized time autocorrelation of the density
Φ(q, t)=. hρ(q, t)ρ ∗(q, 0)i |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 a non increasing function of time.
The simplest relaxation model assumes stochastic dynamics. It was used by Maxwell in its viscoelastic theory for the shear relaxation. In the present context it is usually named after Debye, since he used this model for the discussion of dipole relaxation in liquids:
Φ(q, t) = exp
−τ (q)t
, 2.6
being τ (q) the characteristic decay time of density fluctuations. Structural relaxation (α) processes in supercooled liquids, are often associated with a fairly broad spectrum of relaxation times. A simple expression that describes the spectrum reasonably well over a wide range of time is the ‘stretched exponential’, or Kolraush-Williams-Watts (KWW) expression:
Φ(q, t) = exp " − t τ (q) β# . 2.7
Here τ is again a characteristic relaxation time, and β ≤ 1 is an exponent that can vary with temperature. β = 1 implies monoexponential relaxation (i.e. a spectrum with a single relaxation time), while a smaller β corresponds to a wider spectrum.
The stretching phenomenon was observed already in the 18th century by Kohlraush [43]. For the description of his relaxation data for the electrical polarization of glassy materials he modified the stochastic law Eq. (2.6) by a stretching exponent. The stretched exponential function entered the modern literature on glassy dynamics as an empirical fit formula for α relaxation of mechanical [44], electrical [45], or other variables (compare Ref. [46] for a compilation of examples).
Such a stretched behavior of relaxations has been sometimes interpreted as arising from the distribution of relaxation times associated with different exponentially relaxing regions or domains of different size or structure (the so-called cooperative rearranging regions). Anyway there is lack of a general consensus about the origin of nonexponential relaxations and several models have been proposed [27].
The KWW expression is convenient, but not sacrosanct; other expressions can be used to approximate the spectrum. However stretched exponential relaxations are predicted by Mode Coupling theory [40].
It is usually observed experimentally, mainly with dielectric relaxation or light scattering techniques [27, 47], that the relaxation evolves from a Debye exponential at T > Td to a two step process at lower temperature, that is
more and more enhanced as T → Tg. This corresponds to the bifurcation at
Td between structural, α, and fast, β, processes underlined in Section 2.1.1.
Also the β process follows a stretched exponential behavior. In terms of correlation functions, this means that they first decay rather quickly to a plateau 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 sample phenomenology 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. Consider a solid which has been heated to well above the melting point. When such a melt is gradually cooled its volume decreases continuously down to its freezing point Tm. At Tm the volume generally decreases abruptly
due to crystallization. Upon further cooling the volume again decreases continuously, but with a reduced slope, which is characteristic of the
2.1. PHENOMENOLOGY OF THE GLASS TRANSITION
Figure 2.4: Typical correlation function for undercooled liquids. A first decay occurs on the time scale of β-relaxation, and a further decay on the time scale of α-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 as to bypass crystallization, the volume below Tm continues to decrease
at the same rate as above Tm. At low enough temperature, a change
occurs in slope of variation of the volume, and the now-rather-viscous melt solidifies. The expansivity of this amorphous solid is similar to that 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 unique
temperature 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 volume
of the glass is slightly higher that that of the parent crystal and this is almost always the case. The regime of temperatures between Tm and
Tg is referred to as ‘supercooled region’.
• Let us examine the specific heat plot (CP vs T , Fig. 2.5.b) for the same
typical case as in Fig. 2.5.a. On cooling the melt, its CP decreases very
little till when at Tm it drops abruptly to the CP value of the crystal.
But when so cooled as to bypass crystallization, the supercooled melt continues to follow the same heat capacity behavior of the melt above Tm. The supercooled melt, therefore, always has a higher heat capacity
than the crystal. On cooling further, however, the supercooled melt exhibits an almost abrupt decrease in CP at Tg where it solidifies into
a glass. The glassy state heat capacity is only slightly higher than that of 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 specific
volume does (Fig. 2.5.a), to the value characteristic of the crystal. When crystallization is bypassed, entropy decreases down to Tg where
it is closed to, but slightly higher than, the entropy of the crystal. If the cooling rate is slow, the slope changes at temperatures still closer to the entropy curve of the crystalline solid. But it never crosses the entropy curve of the crystal itself because that would be a thermodynamic ab-surdity, whereby a supercooled melt would possess lower entropy than the crystalline solid itself. This is often referred to in the literature as the ‘Kautzmann paradox’ [48].
Figure 2.5: Thermodynamic properties of glucose, as an example of the thermal glass 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 with discontinuity at Tm) or the liquid can be supercooled (solid line continuous at Tm). The thermodynamic glass transition (region) corresponds to the change of slope in V or to the jump in CP. Note that lower cooling rates bring to lower glass temperatures (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 too long observation times and/or well above Tg, parameters practically do not
show any time dependence and the amorphous solid seems, therefore, to be in a properly defined thermodynamic state. However, even in this case, the glass and the liquid phase cannot be connected by any path in the time independent parameter space, nor adiabatically slow state change can ever bring from the liquid phase to the glass phase below Tg [29]. Time will always
play a fundamental role in the formation and description of the glass, the fundamental reason simply being that it is not an equilibrium state. Glassy substances 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 freedom
leading the structural relaxation, like the diffusion processes that make the material flow, are frozen. This implies, for example, that in a given time tw
2.1. PHENOMENOLOGY OF THE GLASS TRANSITION
(the time waited after the quench to the glass state), only a limited region of the configuration space, connected to the initial configuration, can be visited in the evolution of the system. The system cannot explore all energetically accessible parts of the configuration space in this time. As a consequence, ensemble average and time average are no longer equivalent, at least on the time 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 operative determination. Indeed, a convention has to be established to fix the proper cooling and heating rates since the glass transition is kinetic in origin. Finally a further convention identifies which point of the smeared vitrification step, in specific heat, one has to adopt, because no true cusp, strict discontinuity or 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 certain molecular disorder the velocity of thermal fluctuations dramatically slows down with falling temperatures. Assuming that no phase transition intervents that would kill or change the disorder, we then have, for the present, no typical times. Instead the fluctuation time τeq increases continuously from 10−12 s(picoseconds) to 102 s (minutes), i.e., about fifteen orders of magnitude (see
the relaxation time vs. temperature path reported in Fig. 2.2). This variation is out of all proportion with the concomitant changes in density or structure. If the fluctuation time τeqarrives at a typical experimental time, the substance
vitrifies when cooling is continued.
A glass is then an amorphous solid formed from the melt by cooling to rigidity without crystallization. Rigidity means zero or weak steady response to a permanent shear stress. In the liquid we observe such a response: the viscosity η. To a certain approximation, it is connected to the fluctuation time 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 in
fluctuation 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 glass makers (105− 108Pa s), and other intervals. From a viscosimetry experiment,
the glass temperature is conventionally defined as the temperature at which the viscosity attains 1013 poise, which is an immense value if compared to
viscosities of normal, warm, liquids. In normal liquids the diffusive dynamics takes place on time scales of some percent of a picosecond, thus, considering
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], log η 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 formally η → η∞, and for η → ∞ we have formally T → T0 > 0, the Vogel temperature.
Both asymptotes are far away from the range of application: the viscosity in the working interval is many orders obove η∞, and Tg is several tens on
kelvins 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 corresponding to the working interval of viscosity. For a given cooling rate ˙T = dT /dt, they have a lot of time for glasses with large ∆T . These are the ‘long’ glasses, and the others the ‘short glasses’. This property of a glass-forming melt, can be characterized by one parameter [52] that may be determined from properties near Tg. This parameter was called fragility by Angell [53, 54]. There is
much misunderstanding outside the glass transition community because this concept has, in general, nothing to do with fragility of glasses in the common sense, i.e., breaking easily, or being brittle. Instead, Angell’s pristine inter-pretation was related to molecular pictures for dynamics or thermodynamics in the liquid.
Fragility is usually characterized [36] by a dimensionless ‘steepness index’ related to T = Tg by m = lim T →Tg d log10η dTg/T . 2.10
An Arrhenius plot of viscosity versus inverse temperature for a variety of glass forming liquids is reported in Fig. 2.6. For most of the liquids the viscosity seems to extrapolate to 10−4 poise for T → ∞. In this plot, the
typical Arrhenius behavior of an activated process would be represented by a straight line. For the majority of glass formers this is actually not the case, in fact they can be conveniently represented with a VFT behavior. The fragility Eq. (2.10) can be estimated for a VFT, Eq. (2.9), thus leading:
mV F T = 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 F T ≈ 16
Tg
Tg − T0
2.1. PHENOMENOLOGY OF THE GLASS TRANSITION
Figure 2.6: Viscosity vs. inverse temperature for glass-forming liquids, showing behavior classified as ‘strong’, typified by open tetrahedral networks, to ‘fragile’, typical of ionic and molecular liquids. Here Tg is defined by the criterion that η(Tg) = 1013 poise. For most of the liquids, the viscosities seem to extrapolate to a common value of around 10−4 poise at high temperatures, corresponding to a fundamental molecular vibrational frequency of around 1013 s−1 . Figure from Ref. [30].
Glass formers can thus be classified according to their fragility parameter, as follows [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 show
a high resistance to structural changes, usually small jumps of specific heat (with the exception of cases where hydrogen bond plays a major role), their vibrational spectra and radial distribution functions show little reorganization in a wide range of temperature and the free energy hyper-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 seen from 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 presents a large deviation from the Arrhenius law and the viscosity pattern is phenomenologically 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,
with little thermal excitation, it is able to reorganize itself in structures with different particle orientations and coordination states. In terms of the free energy landscape the fragile glass presents much more degen-erate minima, separated by sensibly smaller barriers than the strong glass, see Fig. 2.7. Their fragility varies in a wide spectrum, and can be as high as 200 for very fragile glasses. Some examples of fragile glasses are K+Ca2+NO3−, K+Bi3+Cl−, orto-terphenyl (OTP), toluene,
chlorobenzene, but also selenium and sulfur. In general these are liquids characterized by simple, non-directional Coulomb interactions or, else, van der Waals interactions [55]. The most fragile substances known are polymeric [36].
Figure 2.7: One dimensional pictures of the free energy landscape F in fragile and strong glass formers. The horizontal axis represents the one dimensional projection of the configurational coordinates of the degrees of freedom {r}. The crystal global minimum 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 is the 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 = 1 was originally given by Adam and Gibbs [51]. An alternative explanation for this choice is provided in the framework of the so-called Random First Order Transition (RFOT) theory [56]. However these studies do not exclude exponents γ > 1, always compatible with data, merely affecting the width of the fitting interval. On the contrary, analitic approaches [57] yield γ = 2 in three dimensions. The same broadening can be implemented for strong glass formers, 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 beyond the scope of the present exposition. Further insight can be found in references [28] and [29].
Giallo `e un color che si chiama orpi-mento (...) ed `e propio tosco. Ed `e di color pi`u vago giallo resimigliante all’oro, che color che sia.
Cennino Cennini
Il 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 on arsenic selenide and sulfide ChGs (see Chapters 6 and 7), as well as on amor-phous chalcogen elements S and Se. These compounds properties will be summarized in the present Chapter, without any intent of completeness. For each 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 be left apart, mainly because this topic is not of direct interest for our present achievements. 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 of sulfur, selenium and tellurium with electropositive elements or with organic radicals. 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. The valence electronic shell has 6 electrons with the disposal 3s23p4. Sulfur can
be found in the following oxidation states: -2, 0, +2, +3, +4, +5, +6. This element exhibits several crystalline and non-crystalline forms. In order to explain the atomic scale structure in the solid and liquid phase we must take into account some peculiarities of the chemical bonding.
Sulfur forms di-covalent bonds. It has two unpaired p electrons and can form σ-type bonds. The p orbitals are oriented reciprocally at 90◦ angles.
char-acteristic angle for the sp3 hybridization. Starting from these bonds it is
possible 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 chalcogen bonds. 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 the atoms of a given configuration are called dihedral angles. These angles are situated 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 special situation of two types of configurations appears due to the contribution of the π-bonds between the p-electron pairs on neighboring atoms. The ring molecules 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 be
packed in the solid state of sulfur. The study of elemental sulfur offers an attractive challenge because of the unique diversity of stable molecules that it can form in the gaseous, liquid and solid state and because the chemical conversion of molecular species occurs at moderate temperature conditions [58, 59, 60, 61].
The stable crystalline modification at room temperature is orthorhombic sulfur (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 sulfur
melts at about 119◦C, forming a light yellow liquid of relatively low viscosity.
According to an interpretation of IR and Raman experiments [62, 63], this liquid is made up mainly of S8 ring molecules, too.
Liquid sulfur exhibits an anomalous dependency on temperature of the viscosity [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 oil
in n.c.), following the behavior of most other liquids. Then, still continuing to raise the temperature the viscosity increases abruptly at Tλ = 159◦C and the
liquid color becomes dark-brown (see Fig. 3.4). At about 187◦C the sulfur
viscosity reaches its maximum (about 930 poise, comparable to the viscosity of peanut butter) and, thereafter, gradually decreases so that at 300◦C it
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 this temperature range, has led this transition to be called a ‘λ-transition’ (Fig. 3.3.a). At the transition, other physical properties, such as the refractive index, 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 openinvolv-ing of sulfur rinvolv-ings to form diradical S∗
8 chains (initiation step)
and (ii) the concatenation of S∗
8 to long chains Sµ (propagation step). The
transition is thermoreversible and hence S8 rings form again upon cooling
be-low Tλ. These macromolecule chains do not appear at low temperatures due
to the sulfur tendency to the bond in cis (eclipsed) configuration. Neverthe-less, when the temperature is raised, the probability for the bonding in trans configuration increases. This leads to an equilibrium for the concentration of S8 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 transipolymeriza-tion is still a matter of debate due to the inherent experimental dif-ficulties. The first available reliable set of experimental data is that provided by Koh and Klement [71] where Φ(T ) was estimated by rapidly quenching liquid 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 CS
2 gave the weight
frac-tion of Sµ as the insoluble portions.
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) Temperature dependence of the extent of polymerization Φ(T ) from different experimental as well as theoretical works. Full circles: data from [70]; crossed squares: data taken from 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 Wheeler et 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 demonstrate the increasing trend of this quantity but failed in providing a quantitative and accurate description of this parameter. More recently [70], a new Raman investigation 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 increase of the structural relaxation time of liquid sulfur. Considering the ideal be-havior of a temperature-independent shear modulus G∞, at the λ-transition
one should expect the relaxation time to rise from ∼ 1 ps to ∼ 10 ns. Since the frequency position of the resonance is ruled by the condition ωτ ∼ 1, this should imply the detection of relaxation effects in the GHz frequency domain by 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 in liquid sulfur [77, 78]. However, a recent investigation with InfraRed Photon
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 relaxaabsorp-tion process with characteristic time in the ms range. The temperature trend of the average chain relaxation time is found to abruptly increase just below Tλ, reaching its maximum value
at the transition. The present findings will be discussed more thoroughly in Section 6.1, where a further study of the chain relaxation process, as a function of both temperature and exchanged momentum, will be reported.
Theoretical efforts made to relate the dynamics of liquid sulfur to the underlying polymerization phenomenon are mainly related to thermochemi-cal models [80, 81, 82], and provide hypothesis [82] on physithermochemi-cal mechanisms affecting this fluid behavior. A simple equilibrium polymerization theory of liquid sulfur has been proposed by Tobolsky and Eisemberg [73], who defined two equilibrium constants associated to chemical reactions of polymerization initiation 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 interaction
between 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 trantran-sitions described by non-classical exponents, and that the Tobolsky-Eisemberg model represents the mean-field limit of their theory. Despite these theoretical efforts, an accurate quantitative description of the observed viscosity and degree of polymeriza-tion is still lacking. Predicpolymeriza-tions for Φ(T ) from the Tobolsky-Eisemberg and Wheeler models are reported in Fig. 3.3.b.
3.1.2 Selenium (Se)
Selenium is the element with Z = 34 in the Periodic Table, and its atomic mass is 78.96. The configuration of the valence electrons is 4s24p4. The
oxidation states of selenium are -2, 0, +2, +4, +6. The sp3 hybridization is
less stable than in sulfur.
The particular features of the crystalline states of selenium are based on the tendency of the selenium atoms to have the trans configuration more expressed than sulfur atom. The most stable crystalline form of selenium is the hexagonal (also known as grey or metallic Se). The lattice is made from parallel helical chains (see Fig. 3.5). Every atom has two neighbors in its own chain and four neighbors situated in neighboring chains. The radius of the helix is 0.98 ˚A. Within the selenium chains the atoms are bonded by covalent bonds and between chains act molecular forces of the Van der Walls type [84]. The hexagonal unit cell contains three atoms. The bond angle is about 103◦ and the torsion (dihedral) angle is about 100◦. However, a
weak covalentlike attraction between chains arising from the partial overlap of 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 the intermolecular distances are appreciably smaller than the distances dictated by the Van der Walls interactions . Monoclinic and rhombohedral Se are the corresponding molecular crystals where the structural building blocks are the eight-membered Se8 and the six-membered Se6 rings, respectively [85, 1].
Figure 3.5: Structure of hexagonal selenium. (a) Chain configuration in the unit cell. (b) The atom chain (view along the c-axis). Figure from Ref. [1].
Amorphous selenium (a-Se) is a dark-grey solid, presumably the only monatomic glass known, at ambient temperature and pressure conditions. It was suggested that the amorphous phase would be built from disordered chains and rings of di-covalent atoms. The covalent distance, the valence an-gle and the dihedral anan-gle in non-crystalline selenium seem to be very similar
3.1. THE CHALCOGENS
to those from the hexagonal selenium crystal. The same Van der Walls forces act between the chains. The density deficit of ∼ 10% in the amorphous phase suggests that the packing of the structural units is far from close packing [1]. Diffraction data on this system could not answer the question concerning the relative population of ring-like and polymer-like conformations, an issue that 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 able to model equally well the obtained radial distribution functions [86]. The same argument has also been noticed by Ref. [87] where it was shown that the static structure factor reflects mainly the properties of the short-range structural order that is dominated by two- and three-body correlations; it was quite insensitive, however, to a medium-range structural order that is related to more than four-body correlations.
Thus, in order to firmly establish the constitution of amorphous selenium chemical methods for analysis are needed. Brieglieb [88] dissolved vitreous selenium in CS2 and demonstrated on the basis of selective dissolution that
this type of selenium is made of a mixture of chains and rings. The proportion found between chains and rings are closely dependent on the preparation conditions. Different experimental techniques have been used to estimate the chain 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 atoms to 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 of means including calorimetric [97], dilatometric [98], ultrasonic [99, 100, 101], and dielectric [102] techniques. B¨ohmer and Angell [103] reported a study of the stress relaxation modulus in a-Se. The mechanical correlation functions of stabilized supercooled liquid Se have been found to exhibit two distinct decay channels, attributed by the authors to the existence of a cooperative transition, near 300 K, between ringlike and chain elements. The authors do not exclude a different interpretation, representing the step in the correlation function as the entanglement plateau, which is a well-known feature in the mechanical spectra of high molecular weight polymers [104].
More recently, Raman spectroscopy has been employed to study the effect of temperature on the medium-range structural order of a-Se trough its glass transition [84]. In this study the authors assert that a-Se is found to undergo a 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 developed tight-binding [106, 107, 108] and ab initio [109, 110, 111, 112] molecular dynamics simulations, while a theoretical analysis taking into account the medium-range structure of this system is presented by Nakamura and Ikawa
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 photodarkening phenomena [113, 114, 112] have received much attention in semiconductor technologies.
3.1.3 Tellurium (Te)
Tellurium has the atomic number Z = 52, the atomic mass 127.60 and the configuration of the valence shell is 5s25p4. It is a hard solid with metallic
aspect. The oxidation states in compounds are +2, +4 and +6. Because the cis configuration is not favored in tellurium, it exists only one crystalline state at normal pressure. This form is called α-tellurium, it exhibits hexagonal symmetry 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 the volume electrons, so much reduced that the transition in the glassy state can be reached only trough cooling rates as high as 1010 K/s [115]. The
amor-phous state is obtained by evaporation and deposition on solid substrates maintained at very low temperatures. Stuke [116] suggested that amorphous tellurium should have a distorted chain structure where the interchain bond-ing is weaker than in the hexagonal tellurium but the bonds within the chains are longer and nearer to the covalent bond.
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-S and As-Se binary series. The properties relative to the binary and ternary systems formed by these latest are extensively described by Popescu [1].
The best studied are amorphous As2Ch3 systems, where Ch denotes a
chalcogen 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 known
as 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 state are known besides As2S3 and As2Se3. The compounds As4Ch4 and As4Ch
were 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 it
crystallizes in the monoclinic system. The compound As4S is known as a
mineral under the name of duranusite, and it exhibits a layered crystalline structure related to that of As. Its elementary cell is orthorhombic.
Figure 3.7: Arsenic sulfide minerals. (a) An orpiment sample (As2S3). (b) A realgar sample (As4S4).
For binary series AsxSe100−xand AsxS100−xthe glass-forming ability strongly
varies with composition. Alloys can form glasses for 0 < x < 65 in the case of AsxSe100−x, and for 0 < x < 45 in the case of AsxS100−x. It is remarkable
that, although As4S4 and As4Se4 are isostructural, As4Se4 is very stable in
the glassy state, while As4S4 cannot be obtained in the glassy state even by
very rapid quenching; only rapid quenching accompanied by high pressure can determine the formation of glass.
For increasing concentration of arsenic in the system As-S, the stability against crystallization increases. The glass with the content of 6 at.% As crystallizes at room temperature in a day with the formation of rhombic
sulfur, while the glass of composition As2S5 (x=28.6) cannot be crystallized
by thermal annealing. The glasses situated in the range 5÷16 wt.% As usually crystallizes by long time annealing at 60◦C with the formation of rhombic
sulfur. 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 K
for ten days [117]. In the system As-Se the glasses crystallize under pressure and high temperatures in all the composition range.
3.2.1 Structural properties
The local structure of binary ChGs, considering both the bulk and thin film forms, was extensively studied using neutrons and x-rays in the 1960s and 1970s. 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 until recently.
Goriunova and Kolomiets pointed out the importance of covalent bonding in ChGs as the most important property to guarantee stability of these glasses [118]. Vaipolin and Porai-Koshits reported x-ray studies in the beginning of the 1960’s [119, 120] of vitreous As2S3 and As2Se3 and a number of binary
glass compositions based on these two compounds. These glasses were shown to contain corrugated layers, which were deformed with increasing size of the chalcogens. The character of the bonds was also shown to become more ionic when at equimolar compositions. Nevertheless, resolution in both reciprocal and real space was very often not sufficient to obtain details of the glass network organization.
The emergence, during the last decade, of third generation synchrotron as well as spallation neutron sources allowed new possibilities to be explored. On one hand, x-ray absorption near edge structure (XANES) and extended x-ray absorption fine structure (EXAFS) have exposed researchers to a more thorough understanding of the electronic and structural properties of these amorphous materials, and especially in As2S3 [121, 122, 123]. These two
pre-vious techniques are element-specific and allow one to investigate the short to medium range structure around the absorbing atom [124, 125]. On the other hand high resolution neutron diffraction measurements over a wide range of exchanged momenta, Q, with or without isotopic substitution, anomalous x-ray scattering, and high energy x-ray diffraction experiments were carried out, 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 glass formation region. These glasses are confirmed to remain homogeneous on mesoscopic scale over the entire vitreous domain. Moreover, the lowest-Q diffraction peak of the static structure factor S(lowest-Q) exhibits remarkable changes with composition which reflect changes in the glass network on both the short- and intermediate-range scales.
3.2. BINARY AS-SE AND AS-S CHALCOGENIDES
More involved is the investigation of short- and medium-range structure in AsxS100−xsystems, mainly because the S-rich binaries are believed to phase
separate. Kawamoto and Tsuchihashi [128] were the first who observed two types 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 soluble species, interpreted as molecular sulfur rings, appearing at x < 20. The solu-ble species was actually recognized as being constituted of S8 rings after later
Raman spectroscopy investigations [129, 62]. A simple model for sulfur-rich structure 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-rich AsxS100−xglasses: (a) A fragment of the stoichiometric As2S3 glass with two CS-AsS3/2 pyramidal units. (b) Transformation of bridging sulfur into sulfur dimer in the domain 25 < x < 40. (c) A fragment of the glass structure in the domain x< 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); see
Fig. 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 supposed that a ‘saturated’ composition exists, below which all the AsS3/2 pyramids
become isolated, i.e., they do not share any corners and are separated by three S dimers. This saturated composition will be x = 25 (i.e. independent AsS3 pyramids linked by S-S bonds). For x < 25 S8 rings start forming in the
glass network (Fig. 3.8.c).
Phase separation in As-S glasses is the subject of a recent experimental study 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 are found not to support the simple structural model previously introduced. In particular, for what concerns the ‘saturated’ composition, the characteristic vibrations of sulfur rings are found to appear at x . 28; Raman spectra are reported in Fig. 3.9.a. Nevertheless, the existence of the two suggested
![Figure 3.6: Transmission of amorphous and liquid Selenium. Figure from Ref. [105].](https://thumb-eu.123doks.com/thumbv2/123dokorg/2906814.14139/34.892.256.644.117.393/figure-transmission-amorphous-liquid-selenium-figure-ref.webp)
![Figure 3.11: Summary of MDSC results for As x Se 100−x (a, b, c from Ref. [138]) and As x S 100−x (d, e from Ref](https://thumb-eu.123doks.com/thumbv2/123dokorg/2906814.14139/41.892.186.771.252.754/figure-summary-mdsc-results-se-ref-s-ref.webp)
