# Glass physics: Wikis

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The physics of glass is the science of the glassy or amorphous state of matter as seen from an atomic or molecular point of view. This article provides an overview of research into glass: a solid in which no significant crystallization has occurred. Thus there is no long-range ordering or extended formation of any Bravais lattice.

Amorphous silica particles (average particle diameter 600 nm). The short-range order evidenced in most liquids becomes thermally arrested or "frozen in" as a result of rapid cooling during the liquid → glass transition.

## Introduction

Schematic representation of a random-network glassy form (top) and ordered crystalline lattice (bottom) of identical chemical composition.

Generally speaking, the atomic or molecular structure of glass exists in a metastable state with respect to its crystalline form. That is, glass would convert into a more stable crystalline form, but the rate of this conversion is slow. This essentially reflects the dynamic nature of the glass transition and the formation of the elastic solid from a non-equilibrium supercooled liquid state.[1][2][3]

Much work has been done to elucidate the primary microstructural features of glass forming substances at both small (microscopic) and large (macroscopic) scales. One emerging school of thought is that a glass is simply the "limiting case" of a polycrystalline solid at small crystal size. Within this framework, domains, exhibiting various degrees of short-range order, become the building blocks of both metals and alloys, as well as glasses and ceramics. The microstructural defects of both within and between these domains provide the natural sites for atomic diffusion, and the occurrence of viscous flow and plastic deformation in solids. [4]

Since the early theoretical and experimental investigations on polymorphism and the various states of aggregation which can be assumed by a given substance, the glassy (or "vitreous") state of matter has been recognized as having the mechanical response of both solid and liquid states, depending on the time and length scale under consideration. The primary focus of this article is to understand how these ideas can be implemented within the scope of previous work done regarding the dynamics of the glassy state of matter. After an initial review of the fundamental concepts of viscosity as they pertain to glass-forming liquids, a closer look at the phenomenon of structural or stress relaxation will be augmented by a thorough treatment of much of the experimental work which has been done in the past in order to understand the nature of viscoelastic behaviour and viscous flow, or plastic deformation in solids. [5] [6]

The direction will then turn towards the process of vitrification and the glass transition, where we will begin by noting that the lattice network of a solid can be represented by the superposition of elastic waves of atomic displacement. This description of the lattice structure lends itself naturally to a description of density fluctuations or thermal phonons, whose interactions are described also in the theory of heat transport in both crystalline and non-crystalline solids. The final considerations will be those arising from the differences in the electronic structure of different glass forming materials.

## Structure of glass

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### Domains and defects

Henry Eyring was among the first to explain the effects of liquid structure on various modes of molecular diffusion. Utilizing the "hole" or "free volume" theory of non-associated liquids, Eyring attempted to explain viscosity, plasticity and diffusion in liquids as the formation and filling of molecular-sized holes. In a manner analogous to that proposed by Einstein for a crystalline solid in his theory of specific heat, each atom is regarded as moving independently of all other atoms. [7]

Eyring treated atoms as hard spheres and ascribed a certain "free volume" to each sphere, corresponding to the volume traced out due to thermal displacements of a particle from its equilibrium position. He accounts for the low entropy of fusion (latent heat divided by fusion temperature) of the alkali metals by assuming that the frequency of the atomic vibrations increases on melting, owing to the volume increase. However, Frenkel pointed out that the average oscillation frequency will be a decreasing function of the volume. A more satisfactory approximation was put forth by Lennard-Jones, who considered used interatomic forces in order to calculate the electric field surrounding a single atom.[8]

Mott and Gurney pointed out the absence in these theories of the concept of a varying degree of atomic or molecular order, suggesting the possibility of a continuous transition from the solid to the liquid phase. Bernal, however, had shown on geometrical grounds with various modes of particle packing that no continuous transition from the crystalline to the amorphous or liquid state is possible without passing through the polycrystalline state, in which regular regions are separated by regions of misfit. [9]

SEM micrograph of surface of colloidal solid. Structure and morphology consists of ordered domains with both interdomain and intradomain lattice defects.(Amorphous colloidal silica particles of average particle diameter 600 nm).
Highlighted image of surface of colloidal solid. Emphasis on microstructural defects to illustrate the defect/domain morphology typical of a simple one-component glass.

It was therefore proposed that an amorphous solid or glass is nothing more than the limiting state of a polycrystalline solid in which the individual grains or domains are so small that one cannot draw any sharp distinction between the surfaces or misfit. This suggestion was made in the interest of improvement over the interpretation made by Eyring of the entropy of fusion for the liquid state. Thus, the energy of the aggregate is assumed to be proportional to the total area of the surfaces of misfit, and the entropy approximated accordingly.[10] Subsequent papers by Lennard-Jones et al. on cooperative phenomena emphasized the order-disorder transition of melting, utilizing the statistical approach of the Bragg–Williams model. [11] [12] [13]

Frenkel formalized the notion of temporary (or temporal) domains in his classical text which includes a description of heterophase density fluctuations. In this context, fluctuations in particle density with configurations resembling that of the crystalline state will exist in a liquid at temperatures above the melting point. An equilibrium distribution of such fluctuations will be in a constant state of growth and dissolution, and will not require the simultaneous collision of the total number of particles in each domain. [14] [15]

Ookawa emphasized the crystallite theory of liquids and its consequences on melting and viscous flow. He points out that the specific heat is one of the macroscopic properties that truly reflects the mode of thermal motion of the primary particles in different states of aggregation. He interprets specific heat measurements as indicating that the thermal agitation of the constituent elements of a liquid is nearly vibrational (solid-like) at low temperature, but almost non-vibrational as high temperature.[16]

Ookawa also showed that the domain size is a decreasing function of temperature. Smaller domains are seen to be strongly coupled with a resulting cooperative motion. In systems with a finite compressibility, the degree of correlations between the displacements of neighboring crystallites will decrease with the distance of separation. Such correlated displacements are shown to be necessary in the consideration of the dynamic properties of the liquid under a sufficiently large external force.

Furthermore, a constant of proportionality exists (the viscosity) between the rate of shear strain and the applied force in normal Newtonian liquids, and consideration of correlations are unnecessary. But under sufficient external force, the proportionality is lost with non-Newtonian effects becoming manifest. In either case, translational motion at the inter-crystalline boundaries is accompanied by the relative shear movement of the crystallites. The motion of the boundary, which is assumed to be non-directional in the natural state, becomes predominant in one direction in a shear stressed state, resulting in a macroscopic flow of the system.

This interpretation of boundary motion is similar to that of Mott, who suggests that the fluidity of a liquid is due only to the presence of a large number of such mobile crystal defects. Mott's interpretation of melting or loss of rigidity of a polycrystalline mass is therefore based on the point at which boundaries between individual grains can move freely through the aggregate. This is similar to Frenkel's suggestion that the viscous flow of a liquid and its crystallization both take place by the same mechanism of diffusion.

### Dislocations and melting

Consider an edge dislocation in a crystal lattice. The Burgers vector is the amount by which the path around the singularity fails to close. The pattern would be a closed circuit on a perfect lattice.

The notion of inter-crystalline or grain boundaries being composed of proper arrays of dislocations led to the next logical step in the sequence of events leading to an understanding of flow mechanisms in liquids as well as in solids. MacKenzie and Mott suggested the possibility that melting results from a sudden proliferation of dislocations. [17] This led Shockley to establish a relationship between the mobility of a dislocation and the fluidity of a liquid in quantitative manner, by assuming a value for the effective dislocation concentration in the liquid state (combined with certain assumptions regarding dislocation width). [18] Sears subsequently proposed that viscosity anomalies in supercooled liquids result from the growth of solid-like nuclei via a screw dislocation mechanism. [19] Cahn elaborated on the conept of crystal growth from the liquid state by proposing that—in sufficiently supercooled systems—a liquid-solid interface can advance normal to itself in the absence of any such heterogeneities. [20]

Kurosawa's study suggested that the contribution of lattice defects is more important than that of lattice vibrations in the melting of ionic crystals. [21] Kuhlmannn-Wilsdorf, emphasizing that the state of a dislocation core is more liquid than solid-like, suggested the pairing of dislocations whose Burgers vectors were of equal magnitude but opposite sign. In this "dislocation dipole", the long-range strain fields of the two members of the dipole cancel each other, resulting in a loss of all resistance to shear forces. [22]

Consider an elementary dislocation on a triangular lattice. The Burgers vector is the amount by which the path around the singularity fails to close. (The pattern would be a closed circuit on a perfect lattice). The 6 coordinated dislocation can be broken down into individual 5 and 7 coordinated disclinations which are all separated by one lattice spacing.

The heterogeneous free volume description (domains vs. defects) was considered also by Turnbull and Cohen in order to describe molecular transport in liquids and glasses. Utilizing the free volume model developed by Doolittle to describe viscous flow in Newtonian liquids, a relationship between the diffusion coefficient and the free volume was proposed based upon the concept that statistical redistribution of the free volume occasionally opens up voids large enough for diffusive displacement of a molecule which is typically confined within a molecular cage.

Glass formation occurs when the rate of cooling exceeds the time necessary for such diffusion to occur, thus limiting the molecular motion. This approach has been generalized by Cohen and Grest in the development of a formal treatment of relaxation processes in liquids and glasses. Other defect-diffusion models have also been developed which are based on the proposed existence of one mechanism for relaxation. Thus, at any molecular site, relaxation can only occur when a defect diffuses to the site. This approach has been used in order to describe volume-controlled relaxations in viscoelastic media. [23] [24][25] [26]

## Dynamics of glass

### Viscoelasticity

Even most simple liquids will exhibit some elastic response at frequencies or shear rates exceeding 5 x 106 Hz. Alternatively, if the vibrational period of the force is large (low frequency) compared with the relaxation time, then the vibrational motion of the body will partially degenerate into translational motion, and the resulting displacement will be evidenced by viscous flow. Materials which respond to mechanical disturbances by both viscous (or plastic/irreversible) and elastic (or reversible) behavior under distinct ranges of deformation (and rate deformation, or frequency) are referred to as viscoelastic.

Thus, when a mechanical force is applied suddenly to a fluid, the fluid responds elastically at first, just as if it were a solid body. Whether the rigidity or the fluidity predominates in a material under given conditions is therefore determined by the time scale of the experiment relative to the characteristic time of structural relaxation of the material. Zwanzig and Mountain calculated the high-frequency elastic moduli of simple fluids by considering the pressure and internal energy of the fluid. They concluded that the initial response to a sudden disturbance can be characterized by two quantities:

1. The high-frequency limit of the shear modulus G (or modulus of rigidity)
1. The high-frequency limit of the bulk modulus K (or modulus of compression).

The connection between a viscous and an elastic response is made by considering the stress for a disturbance varying periodically in time with a frequency, υ. For consideration of shear flow, it is supposed that the shear viscosity q(υ) is a function of the frequency, and is related to the relaxation time t, which is characteristic of the transition from elastic to viscous response. High-frequency disturbances are identified as those relating to elastic behavior, while low-frequency behavior is identified as ordinary viscous flow. Thus the frequency-dependent viscosity coefficient q(υ) is capable of describing both viscous and elastic phenomena, and can be related to the frequency-dependent elastic moduli, K(υ) and G(υ).

In the study of the high-frequency dynamics of simple liquids and solids near their melting points, the particular condition of zero vibrational frequency has been referred to as the "thermodynamic limit" (υ → 0). The conclusions of inelastic light scattering studies near the melting point is that there is no discernible difference between the liquid and solid vibrational spectra at sufficiently high frequencies. Thus, on the short time and length scales probed by these experiments, melting causes no discontinuous change in the microscopic dynamics of the substance. The lower the frequency, the larger the discontinuity between liquid and solid behavior—so that in the thermodynamic limit (zero frequency) the transition is first order. [27]

## Vitrification

### Early work

Since the early theoretical and experimental investigations on polymorphism and the various states of aggregation which can be assumed by a given substance, the vitreous state of a substance has been recognized as having the mechanical response of both solid and liquid, depending on the time and spatial scale under consideration. The atomic arrangement of network-forming oxides in the vitreous or glassy state was initially described as exhibiting the disorder (or short-range order) of its liquid precursor and therefore defying description through any crystalline hypotheses.[28] [29][30][31][32]

A multitude of substances including polymers, network forming oxides, and metallic alloys exhibit glass-forming characteristics, and it is of interest to discuss the criterion which separates the glass forming ability from the typically observed nucleation and growth of a crystalline phase.

It was initially suggested that the ultimate condition for the formation of a glass is that the substance can form extended three-dimensional networks lacking periodicity with an energy content comparable with that of the corresponding crystal network. Viscous flow characteristics of liquids near the glass transition temperature Tg were then interpreted in terms of configurational changes at the molecular level. [33]

Other authors pointed out that the metastable nature of the glassy state is associated with the freezing-in of certain internal degrees of freedom in the liquid, and proceeded to formalize a treatment of the thermodynamics and kinetics of the irreversible approach to equilibrium in glasses in terms of such macroscopic thermodynamic variables as the volume, enthalpy and entropy. Such treatments led to controversy over the use of irreversible thermodynamics in the treatment of glass formation, and the order (first or second) of the macroscopic phase transition. Such questions remain highly controversial in scientific circles today. [34] [35]

The difficulties involved in developing a fundamental understanding of the nature of the glass transition itself are similar to those involved in the understanding of any rate-controlled phenomena, such as chemical reactions. A distinction should be drawn between an empirical description of the kinetics of the reaction and a description in terms of its macroscopic variables (enthalpy, entropy, etc.), and a physical understanding of the reaction itself. The latter involves a picture of its mechanism, i.e., the path of the reaction.[36][37]

This point is emphasized in a dynamic interpretation of vitrification. Thus, the kinetic treatment of glass formation by Uhlmann which is based upon the kinetic equations for phase changes developed by Avrami should be contrasted, for example, with a mechanical description of vitrification. The latter emphasizes the localized nature of the distribution of microscopic stress and strain upon vitrification. [38][39][40]

### Elastic waves

Thus, it may be wise to note that the dynamics of any rate-controlled phenomena can only be understood if one considers the dynamic nature of its primary constituents. In this particular case, thermal motion in liquids can be decomposed into elementary longitudinal waves (or acoustic phonons), while transverse waves (or shear waves) were originally described (and observed) only in the crystalline state. This is the fundamental reason why simple liquids cannot support a shearing stress, but rather yield via plastic deformation and macroscopic flow. The inadequacies of this conclusion, however, were pointed out by Frenkel in his revision of the theory of elasticity of liquids. This revision follows directly from the continuous character of the transition from the liquid state into the solid one when this transition is not accompanied by crystallization and long-range atomic and/or molecular ordering (or self-assembly).

Longitudinal (acoustic) compression wave in a 2-dimensional lattice.
Transverse (optical) plane wave

It can be concluded from these observations on continuity that transverse vibrations can be propagated not only in crystalline bodies, but also in liquids. The fact that this conclusion is not verified experimentally in the case of ordinary liquids is due to the short time scale of relaxation compared to the period of vibrations which can be obtained with the help of modern opto-acoustic methods: lasers and ultrasonics. Under such conditions, the transverse vibrations must be strongly damped.

Experimental verification of these conclusions have been obtained using molecular dynamics computer simulation studies in monatomic liquids and glasses where it has been shown that, at short wavelengths, monatomic liquids can support a propagating shear wave, i.e. a collective excitation which is analogous to the transverse phonons found in solids. The onset of this viscoelastic behavior is linked to the fact that as the wave number increases the rigidity of the liquid becomes an important factor. [41] [42] [43] [44] [45] [46][47]

Mechanisms of attenuation of high-frequency shear modes and longitudinal waves were considered with viscous liquids, polymers and glasses. [48] The subsequent work led to a fresh interpretation of the glass transition in viscous liquids in terms of a spectrum of structural relaxation phenomena occurring over a wide range of spatial and temporal scales. Experimentally, the use of dynamic light scattering experiments (or photon correlation spectroscopy) allows to study molecular processes down to time intervals of 10−11 sec. This is equivalent to extending the available frequency range to 109 Hz or greater. [49] [50]

Thus we see the intimate correlation between transverse acoustic phonons (or shear waves) and the onset of rigidity or vitrification. The frequency dependence of this phenomenon becomes apparent when one considers the increasing wavelength over which such rigidity can be observed. The relationship between these transverse waves and the mechanism of vitrification has been described by Chen et al. who proposed that the onset of correlations between such phonons results in an orientational ordering or "freezing" of local shear stresses in liquids, yielding the glass transition. [51]

The representation of thermal motion in liquids as a superposition of elastic sound waves was first introduced by Brillouin. Atomic motion in condensed matter can therefore be represented by a Fourier series of standing waves whose physical interpretation consists of a superposition of supersonic longitudinal and transverse atomic displacement waves (or density fluctuations) with varying directions and wavelengths. With respect to sound wave propagation, the speed of longitudinal or compression waves will be limited by the bulk modulus of compressibility. The square root of the ratio of the bulk modulus K to the density p will be equal to the velocity of propagation of longitudinal phonons. In the case of transverse vibrations or shear waves, for which the density remains constant, the speed of such waves will be limited by the shear modulus or rigidity. [52]

The square root of the ratio of the shear modulus G to the density will be equal to the velocity of transverse phonons. Thus, the wave velocities will be given by:

$V_{long}=\rho \sqrt{K}$
$V_{trans}=\rho \sqrt{G}$

where the constant of proportionality ρ in both cases is the particle density or reciprocal specific volume.

### Heat transport

The velocities of longitudinal acoustic phonons in condensed matter are directly responsible for the thermal conductivity which levels out temperature differentials between compressed and expanded volume elements. The thermal properties of glass are interpreted in terms of an approximately constant mean free path for lattice phonons. Furthermore, the value of the mean free path is of the order of magnitude of the scale of structural (dis)order at the atomic or molecular level. [53][54][55]

Thus, heat transport in both glassy and crystalline dielectric solids occurs through elastic vibrations of the lattice. This transport is limited by elastic scattering of acoustic phonons by lattice defects. These predictions were confirmed by the experiments of Chang and Jones on commercial glasses and glass ceramics, where mean free paths were limited by "internal boundary scattering" to length scales of 10−2 cm to 10−3 cm. [56][57]

The phonon mean free path has been associated directly with the effective relaxation length for processes without directional correlation. Thus, if Vg  is the group velocity of a phonon wave packet, then the relaxation length $l\;$ is defined as:

$l\;=V_g t$

where t is the characteristic relaxation time. Now, since longitudinal waves have a much greater group or "phase velocity" than that of transverse waves, Vlong is much greater than Vtrans, the relaxation length or mean free path of longitudinal phonons will be much greater. Thus, thermal conductivity will be largely determined by the speed of longitudinal phonons. [56][58]

Regarding the dependence of wave velocity on wavelength or frequency (aka "dispersion"), low-frequency phonons of long wavelength will be limited in relaxation length by elastic Rayleigh scattering. This type of light scattering form small partilces is proportional to the fourth power of the frequency. For higher frequencies, the power of the frequency will decrease until at highest frequencies scattering is almost frequency independent. Such higher frequency phonons may be scattered by "defect clusters".

Similar arguments were subsequently generalized to many glass forming substances using Brillouin scattering (thermal generation of phonons with detection by light scattering). [59][60]

Some experiments have been contrasted with these theories interpreting thermal resistance as being due to scattering of sound waves by lattice defects in crystals. Diffuse scattering of phonons has been attributed rather to the general "roughness" of the amorphous structure which is said not to contain such well defined lattice defects.[61] [62]

The thermal phonon mean free paths or relaxation lengths of a number of glass formers have been plotted versus the glass transition temperature, indicating a linear relationship between the two. This has suggested a new criterion for glass formation based on the value of the phonon mean free path. [63]

### Electronic structure

The influence of thermal phonons and their interaction with electronic structure is a topic which was appropriately introduced in a disussion of the resistance of liquid metals. Lindemann's theory of melting is referenced, and it is suggested that the drop in conductivity in going from the crystalline to the liquid state is due to the increased scattering of conduction electrons as a result of the increased amplitude of atomic vibration. Such theories of localization have been applied to transport in metallic glasses, where the mean free path of the electrons is very small (on the order of the interatomic spacing). [64] [65]

The formation of a noncrystalline form of a gold-silicon alloy by the method of splat quenching from the melt led to further considerations of the influence of electronic structure on glass forming ability based on the properties of the metallic bond. [66] [67] [68] [69]

Other work indicates that the mobility of localized electrons is enhanced by the presence of dynamic phonon modes. One claim against such a model is that if chemical bonds are important, the nearly free electron models should not be applicable. However, if the model includes the buildup of a charge distribution between all pairs of atoms just like a chemical bond (e.g., silicon, when a band is just filled with electrons) then it should apply to solids. [70] [71]

Thus, if the electrical conductivity is low, the mean free path of the electrons is very short. The electrons will only be sensitive to the short-range order in the glass since they do not get a chance to scatter from atoms spaced at large distances. Since the short-range order is similar in glasses and crystals, the electronic energies should be similar in these two states. For alloys with lower resistivity and longer electronic mean free paths, the electrons could begin to sense that there is disorder in the glass, and this would raise their energies and destabilize the glass with respect to crystallization. Thus, the glass formation tendencies of certain alloys may therefore be due in part to the fact that the electron mean free paths are very short, so that only the short-range order is ever important for the energy of the electrons.

It has also been argued that glass formation in metallic systems is related to the "softness" of the interaction potential between unlike atoms. Some authors, emphasizing the strong similarities between the local structure of the glass and the corresponding crystal, suggest that chemical bonding helps to stabilize the amorphous structure. [72] [73]

Other authors have suggested that the electronic structure yields its influence on glass formation through the directional properties of bonds. Non-crystallinity is thus favored in elements with a large number of polymorphic forms and a high degree of bonding anisotropy. Crystallization becomes more unlikely as bonding anisotropy is increased from isotropic metallic to anisotropic metallic to covalent bonding, thus suggesting a relationship between the group number in the periodic table and the glass forming ability in elemental solids. [74]

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## Further reading

• R. Zallen (1998). The Physics of Amorphous Solids. Wiley Interscience.
• S.R. Elliot (1990). The Physics of Amorphous Materials (2nd ed.). Longman.
• N. Cusack (1987). The Physics of Structurally Disordered Matter: An Introduction. IOP Publishing.
• N.H. March, R.A. Street, M.P. Tosi, Eds., (1985). Amorphous Solids and the Liquid State. Springer.
• D.A. Adler, B.B. Schwartz, M.C. Steele, Eds. (1985). Physical Properties of Amorphous Materials. Springer.
• A. Inoue, K. Hasimoto, Eds. (1985). Amorphous and Nanocrystalline Materials. Springer.

## External links

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