# Vitrification and structural differences between metal glass, quasicrystal, and Frank-Kasper phases

**ABSTRACT** The concept of icosahedral short-range order is extended to metallic glass, quasicrystal, and Frank-Kasper phases. The cluster

model, together with the theory of local structural fluctuations, explains the static elasticity of glass, which distinguishes

glass from liquid. An elastic peak of the dynamic structural factor indicates the possibility of transverse mode propagation

in glass. As opposed to dislocations and disclinations in crystals, those in glass are artificially introduced defects, which

serve as easily perceptible structural models. Thermodynamic relaxation theory may only be used for limited groups of vitrifying

compounds the same applies to representation of vitrification as the second-order phase transition. The structure of real

quasicrystals may not be adequately represented by Penrose tiling even after its decoration. This is associated with packing

defects, inclusions of other phases, and chemical inhomogeneities. Quasicrystals have specific defects in an icosahedrally

coordinated network of bonds, which distinguish them from Frank-Kasper phases. Criteria for isolating physically realizable

Penrose tiling from all possible mosaics of this type are suggested. Structural distortions that transfer the diffraction

rings of quasicrystalline samples into ellipses are explicable even in a linear approximation for the stress field created

by a phason. The term “long-range order” seems to be wrong even for ordinary crystals. For quasicrystals, the notion of “rotational”

order is more pertinent.

**0**Bookmarks

**·**

**50**Views

- Citations (26)
- Cited In (0)

- [Show abstract] [Hide abstract]

**ABSTRACT:**A metallic solid (Al-14-at. pct.-Mn) with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations, has been observed. Its diffraction spots are as sharp as those of crystals but cannot be indexed to any Bravais lattice. The solid is metastable and forms from the melt by a first-order transition.Physical Review Letters 11/1984; 53(20):1951-1953. · 7.73 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A defect description of liquids and metallic glasses is developed. In two dimensions, surfaces of constant negative curvature contain an irreducible density of point disclinations in a hexatic order parameter. Analogous defect lines in an icosahedral order parameter appear in three-dimensional flat space. Frustration in tetrahedral particle packings forces disclination lines into the medium in a way reminiscent of Abrikosov flux lines in a type-II superconductor and of uniformly frustrated spin-glasses. The defect density is determined by an isotropic curvature mismatch, and the resulting singular lines run in all directions. The Frank-Kasper phases of transition-metal alloys are ordered networks of these lines, which, when disordered, provide an appealing model for structure in metallic glasses.Physical Review B 11/1983; 28(10). · 3.66 Impact Factor - Acta Crystallographica 01/1959; 12(7):483-499.

Page 1

Journal of Structural Ctumdsuy, VoL 37, No. 1, 1996

VITRIFICATION AND STRUCTURAL

DIFFERENCES BETWEEN METAL GLASS,

QUASICRYSTAL, AND FRANK-KASPER PHASES

A. E. Galashev

UDC 541.66:541.161:541.251:539.213

The concept of icosahedral short-range order is extended to metallic glass, quasicrysta~ and Frank-Kasper

phases. The cluster mode~ together with the theory of local structural fluctuations, explains the static

elasticity of glass, which distinguishes glass from liquid. An elastic peak of the dynamic structural factor

indicates the possibility of transverse mode propagation in glass. As opposed to dislocations and

disclinations in crystals, those in glass are artificially introduced defects, which serve as easily perceptible

structural models. Thermodynamic relaxation theory may only be used for limited groups of vitrifying

compounds; the same applies to representation of vitrification as the second-order phase transition. The

structure of real quasicrystals may not be adequately represented by Penrose tilin~ even after its decoration.

This is associated with packing defects, inclusions of other phases, and chemical inhomogeneities.

Quasicrystals have specific defects in an icosahedral!y coordinated network of bonds, which distinguish

them from Frank-Kasper phases. Criteria for isolating physically realizable Penrose tiling from all possible

mosaics of this type are suggested. Structural distortions that transfer the diffraction rings of quasicrystalline

samples into ellipses are explicable even in a linear approximation for the stress field created by a phason.

The term "long-range order" seems to be wrong even for ordinary crystals. For quasicrystals, the notion

of "rotational" order is more pertinent.

INTRODUCTION

In everyday life and practical activities we often encounter materials called glasses. At the same time, many

solids with noncrystalline structure are called amorphous. Unfortunately, up to now there is no general agreement

regarding the pertinence of these terms. Sometimes glasses and amorphous solids are distinguished based on structural

differences. It is generally assumed [1] that neither system has long-range order but both have short-range order, which

differs between glasses and amorphous solids. In the latter, the short-range order is thought to be analogous to crystalline

order, whereas glasses a~-e characterized by the short-range order inherent in liquids. Thus amorphous solids are similar

to nanocrystals. According to another classification, amorphous substances are all disordered solids irrespective of

their background (method of preparation). Then glass is defined as [2] a noncrystalline solid obtained by quickly cooling

a liquid. This definition of glass is correlated to the method of its production.

It is interesting to cite the opinions of two authoritative bodies. The Americal Society on Material Research

adopted the following definition: "Glass is an inorganic product of melting that solidifies with~,at cryst~lliT~tion" [3].

The Commlr~ion on Terml-ology, USSR Academy of Sciences, suggested a more detailed definition: "Glass is any

X-ray amorphous solid obtained by overcooling a melt with any chemical composition and solidification temperature

range that gradually becomes viscous and acquires mechanical properties of a solid; the process of vitrification of the

liquid must be reversible" [4]. Some authors [3] believe that glass need not be obtained from a melt, since glassy

compounds may be prepared, e.g., by drying gels from solutions or by chemical gas-phase deposition.

Thus the notion of an "amorphous solid state" of a substance is wider than the notion of a "glassy state."

Glasses are always amorphous, but not all amorphous substances are glasses. A glassy substance is formed when

Institute of Thermal Physics, Ural Branch, Russian Academy of Sciences. Translated from Zhurnal Strulaurnoi

Khimff, Vol. 37, No. 1, pp. 138-158, January-February, 1996. Original article submitted February 17, 1995.

120 0022-4766/96/3701-0120515.00 y Plenum Publishing Corporation

Page 2

crystalliTztion is avoidable and the substance undergoes vitrification as a necessary step. Glass is characterized by a

certain temperature range of softening. Currently, the available glass-forming substances are: one-component systems

(S, Se, As, P), oxides (SiO2, GeO2, B203, P205, As203, Sb203, FeO2, V205, etc.), chalcides (As2S3, As2Se3, As2Te3,

GeS 2, GeSe 2, CdAs 2, etc.), aqueous solutions (H2Oz, H2SO 4, HCI, etc.), and also some halides and carbonates. Recently

(1959), metallic glasses were discovered.

In 1984, a metallic alloy was obtained [5] whose structure had orientational long-range order with an icosahedral

symmetry group. This is a strange structure from the viewpoint of classical crystallography. For geometrical reasons,

a crystal, which is a system with a long-range order, cannot have fivefold rotational symmetry. The new substances

were termed quasicrystals. More recently, a consistent theoretical model of quasicrystals appeared. The model is based

on the tessellation theory, which solves the problem of space filling with the three-dimeusional Penrose tiling. The

best equiprobable packing of species around a sphere is [6] an icosahedral arrangement of the nearest neighbors. In

the 1950S, it was assumed that small regions of icosahedral symmetry should be widespread in dense overcooled liquids.

Later the icosalaedral short-range order was adopted as an important structural element in metallic glasses. The

structure of metallic glasses and crystalline melts may be represented as a packing of tetrahedra joined by faces. An

edge represents a chemical bond between two atoms and belongs to several tetrahedra. When a melt is cooled, a

regular network of bonds between the atoms of the same component, distributed among six tetrahedra, is formed. The

atoms of other components occupy icosahedral nodes. The solidified melt with this structure represents a crystal phase.

For example, the slowly cooled melt of Mg32(AI, Zn)49 becomes a structure with global cubic symmetry but with slightly

locally distorted icosahedral order. Crystal phases with short-range order formed of tetrahedra are called Frank-Kasper

phases. Crystal structures with icosahedral short- (but not long-) range order also belong to Frank-Kasper phases.

Thus the icosahedral short-range order of atomic arrangement is found in overcooled melts, glasses, quasicrystals,

and crystal phases. The presence of the icosahedral order may not serve as a criterion to identify these phases. Formation

of the icosahedral short-range order merely reflects the tendency of the system toward equilibrium. A local icosahedral

order is characterized by the minimal energy among all newly formed regions of the same size. Clearly, additional

characteristics are needed for an unambiguous determination of the above-mentioned states.

The absence of static elasticity with respect to shear strains is the main distinguishing feature of a liquid.

According to Bernal [7], a simple liquid has statistical close packing of spherical species. The icosahedral short-range

order of melts seems to be a particular case of this type of packing. Metallic glasses A4B also have [8] regions of

short-range order, in which the smaller B atoms, located in the voids, are surrounded by 8 or 9 larger A atoms. Glass

is a frozen overcooled melt (metastable state) that has a homogeneous or microheterogeneous structure. The short-range

order in glass may be icosahedral, and there is no long-range order.

At present, many glassy alloys of transition metals with nonmetals and semimetals as well as other types of

metallic glasses are available. The search for new types of glasses is in progress. The currently applied rates of cooling

are -10 l~ K/s. The microscopic processes in glassy systems are understood better when we use structural models.

Glass possesses [9] elastic properties that are inherent in solids. The vitrification theory, which explains the elastic

properties of glass by the existence of locally ordered regions, is preferable to traditional thermodynamic theories, e.g.,

to the theory of [10].

Vitrification reflects the effect of gradual freezing of a structure. The lower bound of the vitrification range

Tg is determined by the intersection points of the temperature dependences of specific volume, enthalpy, refractive

index, viscosity, diffusion coefficient, etc., for liquid and glassy states. Tg is generally called the vitrification point.

Kau~mann formulated [11] an empirical rule that relates Tg to the melting point:

Tg/Tmelt = 2/3.

(I)

Relation (I) is satisfied for certain classes of compounds at a cooling rate of 10-I-10 2 K/s. Sometimes the state of a

glassy substance is described using [12] the notion of the "imaginary" or structural temperature T h i.e., the temperature

of a sample that undergoes an isostructural transition from a nonequilibrium (frozen) state to the equilibrium one. As

a rule, a set of structural temperatures defines the structural state of a substance more accurately.

Quasicrystals have a quasiperiod of structure recurrence and thus possess long-range order. Moreover, a

quasicrystal as an entity is characterized by a symmetry axis that is not coincident with translational symmetry. Despite

121

Page 3

strong ordering, the quasicrystal radically differs from the ordinary crystal. From the viewpoint of crystallographic

geometry, a crystal structure may only be built from a finite number of atoms of different varieties, i.e., atoms that are

distingttished in one way or another. The construction of a crystal lattice is reduced to "depersonification" of atoms.

Lattice atoms are geometrically indistinguishable. On the contrary, in a quasicrystal the number of atoms of different

varieties is always infinite. The atoms have no identical environments, and in this sense all atoms are geometrically

distinguishable. While the ordinary crystal may be called a system with translational order, the quasicrystal is characterized

by rotational order, which is not typical of regular structures. The quasicrystal has a five-, seven-, or multiplefold

symmetry axis. The Penrose tiling is a planar analog of a quasicrystal. Both in two- and three-dimensional space, there

is an infinite number of different Penrose quasilattices composed of rhombs (rhombohedra) of several types. Various

quasicrystalliae structures, which have slightly different diffractograms, may be obtained by decorating (filling) each

rhombohedron in an appropriate stoichiometric ratio.

In 1985, a decagonal phase in the AI-Mn system was discovered [13]. EXAFS studies showed [14] that the

central Mn atom in the decagonal phase is surrounded by eight AI atoms (on the average). In the icosahedral phase,

there are ten such atoms. The icosahedral long-range order may be represented [15] via projections of the six-dimensional

cubic lattice to the three-dimensional space. To solve the structure of the A1-Mn decagonal quasicrystal, it was sufficient

[16] to analyze the five-dlmensional space via the Patterson function defined as a mapping of all possible weighted

mean vectors between atoms of the crystal structure. Using this analysis, it was shown that the quasicrystalline structure

of the AI-Mn decagonal phase is formed by six nonequidistant layers. Two layers are asymmetric with respect to each

other, and the other layers result from rotation of the former around the tenfold screw axis and reflection in the plane

perpendicular to this axis. Consequently, it is difficult to predict the structure of the quasicrystal. The model may be

improved using decorations in the three- and six-dimensional space. However, there are objective reasons for which

the model may not be made adequate. These involve the presence of a chemical inhomogeneity [17] in the icosahedral

phase and the instability of the quasicrystal against annealing. Thus the annealed NiTi 2 quasicrystal acquires [18] local

translational order, signaling that an intermediate state for the transition from a quasicrystal to an equlibrium crystal

has been achieved. The quasicrystal concept is undoubtfitlly useful, but the correspondence of model quasiperiodic

structures to real structures of alloys is still an open question.

o

CLUSTER MODELS OF GLASS AND LOCAL STRUCTURAL FLUCTUATIONS

The structural models of glasses are constructed using X-ray and neutron diffraction data, EXAFS, NMR,

and M6ssbauer data, and also the information on the density, elasticity, magnetic susceptibility, etc. Let us consider

the main principles of constructing a duster model based on some structural element, which is generally an icosahedron.

This model reflects the topological disorder and is often used to describe the structure of metallic glasses.

The principle of translational invariance may not be extended to irregular packing. Thus in the case of

two-dimensional space, it is impossible to cover a plane with close-packed regular pentagons. Neither can def'mitely

arranged regular tetrahedra f'dl the three-dimensional Euclidean space completely. The principal coordination

polyhedron in the duster model of glass is an icosahedron. The icosahedron may be built from 20 distorted tetrahedra

surrounding a common vertex. The tetrahedra are distorted due to reduction of all radial distances by a factor of 1.048

compared to the lengths of edges forming the triangular faces of the icosahedron. Since the space cannot be close-packed

with icosahedra, the model should contain dislocations. At low temperatures, the topological conditions are such that

the icosahedral dislocation lines may not intersect. We can assume that when a metallic melt is quickly cooled, these

lines are entangled as are molecules in polymer melts, as a result of which metallic glass is formed. In the two-dimensional

space, a geometrical analog of the icosahedron is a hexagon. In the tessellation theory, the icosahedron differs from

the hexagon in that the latter can be used to completely cover the plane.

Regular tetrahedral close packing may only be obtained [19] in a curved space. The two-dimensional analog

of this packing is a dodecahedron, which represents a sphere fiUed with pentagons. With the curved space close-packed

with icosahedra, the packing density for the similar species is 0.774. This is close to the value (0.7796) observed [20]

in the static honeycomb model. Evidently, the simple model of curved space reflects a real structure up to sizes smaller

than 2R, where R is the radius of space curvature. To increase the characteristic size of the model, it is necessary to

reduce the space curvature. A change in th6 curvature demands that a defect be introduced into the packing of

122

Page 4

structural units. One can include a disclination. A disclination is created by cutting the packing and adding (or

withdrawing) the contents between the two edges of the cut (a wedge of the given substance). A symmetry operation

here is rotation, whereas for an ordinary dislocation this is translation. As dislocations, disclinations introduce a stress

field into the material. The deformation introduced by a disclination may be expressed via a change in the space

curvature. Addition of a wedge to the packing section decreases the curvature but creates structural defects along the

edges and faces of the wedge. If two faces of the wedge are equivalent with respect to rotation for the given symmetry

group, the defects concentrate near the edge. In this way a perfect disclination line is created. Formation of a disclination

on a coordination icosahedron is illustrated by Fig. 1, in which it is shown that an introduction oF this defect results

in a change of the coordination number of the central atom from 12 to 14.

In the theory of local structural fluctuations [21], the physical state of a small region of a substance is

characterized by stresses created by atoms in local volumes. This theory is best applicable to metallic glasses, in which

local internal stresses are relatively high. According to [21], the inner energy of a system may be represented in an

elastic mean-field approximation as

where E 0 and V 0 are the energy and volume of a system that is free of internal stresses; B is the bulk modulus; G is

the shear modulus; Ks a and Ks r are the constants determined by the theory [22] of elastic inclusions, with both constants

being close to 2; p is hydrostatic stress; 31 and T z are shear stresses acting in the directions of principal stresses.

In a local interaction approximation, the second moments (,p2) and (z 2) are proportional to the temperature

[2]. In this case, the vitrification transition may be defined as freezing of these moments at nonzero values. Both in

glass and crystals, structural defects are divided [21] into thermal and athermal defects. Thermal defects in a crystal

are point defects, and athermal defects are dislocations, disclinations, and failure surfaces. Thermal defects in glass

correspond to deviations of local stresses from their equilibrium distributions. Athermal defects in glass are extended

and have long-range elastic fields. The structure of amorphous solids is often represented as dislocations and disclinations.

In these cases we have very high concentrations of defects, so that an energy treatment becomes impossible and we

have to resort to the topological concept. In the icosahedral model of glass, dislocations and disclinations may not be

considered complete analogs of crystal defects. Rather, they should be regarded as some geometrical elements using

which we can construct a structural model of glass.

The applications of cluster models are not confined to metallic glasses. The models are widely employed for

structural representations of glasses whose atoms form virtual groups in view of the weakness of molecular forces such

as van der Waals forces. In particular, this applies to oxide and chalcide glasses. The cooling rate during the preparation

of such glasses is low (-1-10 -2 K/s). During the cooling process, atomic positions and rates in the overcooled liquid

become mutually adapted. The structure of the resulting glass is irregular but with minimal topological disorder. In

other words, the free energy of the overcooled liquid increases to a minimal exterit with respect to the energy of the

corresponding crystal. The enormous cooling rates (up to 101~ K/s) used for the preparation of metallic glasses hinder

the structural relaxation of the melt. Metallic glass experiences local internal stresses, which may not be completely

removed even after prolonged znnealing. Internal stresses decrease the stability of glass and, among other reasons,

hinder the glass-melt transition. Another reason for cryst~lliTation of heated metallic glasses, which is associated with

their high heat conductivity, is treated at the end of the next section.

The viscosity r/ of the cooled glass-forming liquid increases much faster than that of an easily cryst~lliTable

liquid. This is qualitatively understandable if we assume that at low temperatures atoms unite into large dusters. Since

the average number N O of atoms in clusters increases, the motion of the clusters is retarded. This and a reduction of

the number of free atoms (not involved in clusters) leads to an increase in 7/. If the clusters have a crystal structure,

they can serve as crystal nuclei. When kinetic effects hinder the formation of a crystal structure, there arises a problem:

How to form an overcooled liquid from large clusters with noncrystalline topology? This problem may be resolved,

e.g., if we assume that an overcooled liquid is a mixture of rigid and flexible clusters.

123

Page 5

I

i

I

i

I

b

a.

Fig. 1. Disclination line through the coordination icosahedron;

a) 14-atomic coordination polyhedron obtained from the icosahedron,

b) polyhedron with the coordination number 14.

Mechanical stability of glass is represented [23] by the criterion

Nf (,0 = = a, (3)

where d is the dimensionality of the space; Nf (d) and N r are the average numbers of the frozen and retained degrees

of freedom per atom, respectively.

The structure of glass is unstable if Nf < N r, because an excess of degrees of freedom facilitates crystalliT:~fion.

If Nf >> N~, the system has a large excess energy owing to noncompemated internal stresses and many broken bonds.

This destabilizes the structure and promotes crystalliTation. Inside a separate cluster, atoms are tightly bonded by

intermolec, lar forces, but the atoms of different clusters interact weakly with each other. For atoms belon~ng to rigid

clusters, the relation N t (d) > N r is satisfied; for atoms of flexible dusters, vice versa: Nf (d) < N r The homogeneous

overcooled liquid corresponds to an infinite flexible cluster; glass or solid is characterized as an infinite rigid cluster.

This concept of the vitrification and crystalli~tion mechanism was borrowed from percolation theory, Here ~,ariants

of mechanical rigidity may be considered as a vector property. Therefore in this case we can speak about vector

percolation. According to (3), the capacity for vitrification is essentially reduced to mechanical stability of a material.

Criterion (3) is not a sufficient condition for the existence of glass, because no restrictions on the chemical order in

~,la~ are imposed.

124

Page 6

The mechanical stability of a bond network in a muiticomponent system is provided by a certain short-range

order in the atomic arrangement of different components. For simplicity let us consider a binary amorphous alloy

AxBl_x . The alloy may have interatomic bonds such as AA, BB, and AB. Let us denote the concentrations of the

bonds (yet without discriminating between rigid and flexible bonds) by CAA, CBB, and CAB, respectively, so that CAA +

CBB + CAB = 1. The chemical stability of the alloy at the given composition x is provided by the maximal possible value

of the chemical order cAB. Let {P} -=- (PAA, PBB, PAB) represent the probabilities of formation of the corresponding

rigid bonds. Then in a mean-field approximation we have [23]

(x; {e}) = x + 0 - x) fB(ie}),

= 1 - (4)

e z + (d - 1) (Ze e),

where ~ is the average coordination number, determined by the bonds in the melt; P is the percolation threshold.

Analogous effective coordination numbers ~A and ~B for atoms of the A and B components are found using

the probability of formation of rigid bonds {P}, for example,

~A ----" ZA(PAACAA + PABCAB) / (CAA + CAB), (5)

where z A is the coordination number found from the composition (x) of the melt.

The set of relations (2)-(5) leads to a generalized mechanical criterion of stability of an amorphous alloy [23]:

x~ A + (1 -x)2 B = ~(d),

= 2) = 2,

(6)

= 3) = 2.4.

Relation (6) defines the line in thex-{P} plane that corresponds to the most stable structure of the alloy at an optimal

value ofx = xf ({P}) or with the vector percolation threshold P = Pf (x) for the given values of {P} and x, respectively.

In particular, if all AB bonds in the alloy are rigid, the system will have a perfect chemical order with

Nf (xf, {P} = 1) = d. As a rule, the structures of real glu~ses have no perfect chemical order.

The three-dimensional bond network in glass is stable for 2 -< ~ -< 2.4.

THERMODYNAMIC AND KINETIC ASPECTS IN THE VITRIFICATION THEORY

The literature on glass is so extensive and various that it is difficult to give a consistent classification of the

existing theories. The widespread ideas about the transition from liquid to glass are based on the concepts of free

volume, freezing of translational, rotational, and vibrational degrees of freedom, second-order phase transition,

cooperative structural relaxation processes, diffusion model, etc. A critical analysis of these theories is found in [24].

In recent years, thermodynamic relaxation theory has found wide acceptance [25]. In addition to external

parameters, for example, pressure p and temperature T, the theory uses an equivalent internal parameter of state

(there may be several such parameters). As a rule, the ~ parameter is a dimensional quantity characterizing the structure

of the given state. The nonequilibrium state (glass) has a value of ~ that corresponds to a certain equilibrium state,

for example, to the initial state. Due to the incomplete relaxation of the system during vitrification, ~ acquires a "frozen"

value. Theoretically as ~ we can choose the structural temperature T[ of a nonequilibrium system. The derivative

dTf/dT can serve as an order parameter (dimensionless quantity).

The structural temperature is usually determined [26] by projecting some structure-sensitive property Q on an

extrapolated dependence of this property for an equilibrium liquid, as shown in Fig. 2. The scheme reflects the hysteresis

effect on the behavior of Q during cooling and heating in the region of the vitrification transitions. The values of T/

differ for cooling and heating processes. The hysteresis for Q(T) is associated with peculiarities of structural relaxation

[23] during the direct and reverse vitrification transition. The cooling and heating curves also diverge for the dependence

of Tf on the rate of the process. Since heat capacity is cp -- dTf/dT [23], one can predict the hysteresis behavior of

cp(T). Figure 3 schematically presents the behavior of heat capacity cp in the region of the vitrification transition. In

general, the structural temperature can take different values depending on the property Q (entropy s, thermal expansion

125

Page 7

a , r ) r

Fig. 2. Scheme for determination [26] of structural temperature Tf

from the quality Q; 1) cooling, 2) heating, 3) extrapolation of the

property for an equilibrium liquid in the range of low temperatures.

The dashed lines show the projection method.

0-.

/

i 2

7"

Fig. 3. Schematic representation of the behavior [26] of heat capacity

cp of a vitrifying liquid Ca(NO3)2•

arrow shows the vitrification temperature Tg.

1) heating, 2) cooling; the

coefficient ap, compressibility/fiT)- The phenomenological relaxation theory using Ts is simple and is still used [26] in

studies on the vitrification transition.

The sharp variations in Cp, ap, fT, and the static shear modulus F0 near Tg give all impression that the

vitrification transition is similar to the thermodynamle second-order phase transition. However, it is inappropriate to

identify vitrification with an Ehrenfest type transition for a crystal [27], because the role of nonequilibrium factors is

neglected. The nonequilibrium character of the vitrification transition shows itself in the dependence of Tg on the

cooling rate and also in hysteresis effects for cp, ap, fiT, and i, to.

In the two-level model of structural and chemical transformations, the reaction route is determined [25] by

the value of the potential barrier between the changing free energy levels of the two states. In this model, the KanTrnann

paradox [11] is nonexistent in view of the extremely small values of configurational entropy at low temperatures. The

126

Page 8

temperature dependence of the entropy of a liquid in the region of overcooled states has an inflexion point after which

the curve slowly approaches the crystalline branch above, without ever intersecting it.

The law of thermodynamic similarity is inapplicable to vitrification. Some properties such as density, heat

capacity, and shear viscosity vary differently during vitrification. This depends on the specific features of the material.

Angell et al. [28, 29] proposed a scale for classification of vitrifying liquids, which extends from "strong" to "fragile"

liquids. "Resistant" liquids typically have a narrow region of distribution of relaxation times, and the shear viscosity

obeys the Arrhenius law:

r/= r/0 exp(A / 73, ~/0,A > 0, (7)

whereas the heat capacity changes slightly during vitrification. SiO 2 is a typical example of a "strong" liquid. The

structure of "fragile" liquids quickly changes with temperature. The relaxation times have more complex temperature

dependences. The behavior of the viscosity of "fragile" liquids disobeys the Arrhenius law, and the heat capacity drops

in passing from an overcooled liquid to glass. An example of a "fragile" liquid is ortho-terphenyl. The temperature

dependence of the viscosity of a "fragile" liquid is given by the Tammann-Vogel-Fulcher equation,

r/(T) - r/0 exp[A / (T - To)], t/0,A, T O > 0. (8)

The difference in the behavior of "strong" and "fragile" liquids may be explained by using the concept of

many-dimensional configurafional space. It is assumed [30] that for "strong" liquids the topography of the potential

energy hypersurface is homogeneous throughout the configurational space. The depth of the principal minimum relative

to the other minima of the energy relief is small The homogeneous roughness of the topography of a "strong" liquid

seems to be determined by the comparatively independent arrangement of defects in the bond network of the liquid

and by the nearly constant activation barriers for formation and exchange of defects. On the contrary, for "fragile"

liquids, the topography of the potential energy hypersurface is inhomogeneous. At relatively high temperatures, the

main minimum is among the minima with relatively low barriers. As the temperature decreases, the roughness of the

topography increases. The minima become more distinct from the saddle points, and very far minima appear. This

change in topography may be due to the higher content of regions with flaw-free packing of molecules, which correspond

to the lowest energy minima. This inevitably enhances the disconnectedness of defect-free regions. In the limiting case,

one high-ordered noncrystalline region, i.e., a quasicrystal, is formed.

It is impossible to understand the nature of vitrification without considering kinetic effects and relaxation

processes in glass. Adum and Gibbs [31] developed the theory of mean relaxation time, which describes the time

dependences of the vitrification transition. The main result of the theory is represented as

r = v'exp [C / (krsa)],

(9)

where s' a is the configurational entropy of an amorphous solid; 3' and C are positive constants.

The relaxation processes are controlled by the corresponding alternative states at a given temperature. The

deceleration of the kinetics during cooling is associated with a decrease in configurational entropy. The Adam--Gibbs

relation qualitatively describes the behavior of many real vitrifying systems.

From an acoustic viewpoint, the medium passes into a glassy state if it has a propagating transverse elastic

mode with a wavelength of 2rrk -1, exceeding the radius Rin t of interaction between species. The criterion for vitrification

is formulated [32] as

/9 Rint09 0 / r/<< 1, (10)

where p is the density of the medium, and co o is the characteristic frequency.

The vitrification transition may be studied by investigating the behavior of collective modes in a liquid. The

information about the whole set of modes in the medium is contained in the dynamic structural factor S(k, 09). In a

limited frequency region, the function S(k, co) may be recorded [33] as a sum of three Lorentz distributions,

Aozo 2(ARz, + (09 + + (AR z, - Ai 09,) 09s

[(09 _ ,o )2 +

zS(k, co) - ~ 2~z-------- + ,

(11)

[(09 + 09 )z +

127

Page 9

03

2-

3

7

cl

1

20-

10.

co, 10 r3 1/S

Fig. 4. Dynamic structural factor of metallic glass NgToZn3o; a) k = 2.17 A -1,

b) k = 72 A-l; 1) experiment [34], 2) calculation by Eq. (11).

where A 0 and z 0 are the amplitude and the width of the central peak of S(k, co); A R • iA i are the complex amplitudes

of the side peaks that correspond to the frequencies cos •

Figure 4 shows the functions S(k, co) calculated according to (11) in [33] and obtained in [34] by inelastic

scattering of thermal neutrons for metallic glass MgToZn30. An elastic peak for the dependence S(k, co) is evidently

present at k 1 = 2.17 A -1 and is missing at k 2 = 7.2 A -1. At extremely small wavelengths 2 < 2z~ / k 2 = 0.87 ~ the

collective excitations do not propagate in the MgToZn30 glass, i.e., glass approaches liquid in its acoustic properties.

The presence of an elastic peak indicates that glass contains groups of particles that remain almost invariable with

time. In the molecular dynamic model of a one-component system, the function S(k, co) for glass may not be calculated

[35] in view of the strong increase in the relaxation time.

Vitrification is a kinetic transiti,.~r, in which the final state depends not only on the character of chemical bonds

but also on the cooling rate. According to the kinetic concept [36], for each substance there is a certain critical cooling

rate, qcr This is a minimal cooling rate when glass is still formed. Using the data on the nucleation and growth of the

crystal phase, we can construct [37] a diagram "transformation time - transformation temperature" (Fig. 5). The diagram

relates to the start of mass crystalliTafion, when the content of the crystalliTed substance is not more than 0.01. The

of collective modes.

7" NS

1

bs "

. ,bl

rg

"bn f:

Fig. 5. "Transformation time- transformation temperature" diagram for the

start of isothermal crystalliT~tion, when the fraction of the crystallized

substance is 0.01 (for the explanation, see the text).

128

Page 10

m;nirnal time t N during which a certain portion of the melt crystallizes corresponds to the position of the extremum

of the function t(T), denoted by T N. The critical cooling rate is calculated according to the equation

qcr -- (Zmelt- TN) / tic"

(i2)

The vicinity of the minimum of t(T) and the part of this curve that approaches Tmelt correspond to the stationary

process of nucleation (section 1, Fig. 5). The a--a line is a continuation of this branch into the range of low temperatures;

it describes the nucleation of a hypothetical residual and a nonvitrifying liquid. Section 2 of the curve corresponds to

nonstationary nucleation, which leads to sharp acceleration of crystallization in a relatively narrow temperature range.

Below Tg the kinetics of the process depends on the number of "frozen-in" crystal nuclei. Let the number of such

ready-made nuclei be N i in one limiting case and N 2 in another, with N 2 > N 1. Then at T < Tg, the nucleation is faster

in the second case, and the corresponding dependence T(t), denoted as b--b for N2, is farther to the left than b'--b'

for N 1 (Fig. 5). Region 3 (hatched in Fig. 5) defines the density of possible "frozen-in" centers.

This estimation of the tendency toward vitrification does not always give satisfactory results. This is especially

true for pure metals, where stable chemical bonds are absent and the high heat conductivity leads [38] to thermal

shielding of densely located crystal nuclei. As a result of shielding, the crystal nucleus grows faster. The heat of

crystzlliTation of the local crystalllne region does not dissipate in the material but is spent to maintain further

cryslzlliTation. Fast heating of the frontier amorphous region, together with unification of the growing crystzlline regions,

lead to an explosive process, in which the rate of mass crystalliration reaches --1 m/s.

This brief discussion of vitrification models, certainly, cannot give answers to many questions, but makes one

realize the importance of different approaches to investigation of this complex phenomenon.

QUASICRYSTALS

Quasicrystals are often called shechtmanites, after the name of one of the researchers (D. Shechtman) who

discovered them. Shechtmanites have crystallographic planes, but their diffraction patterns indicate that they have

fivefold symmetry. The quasicrystal has a noncrystalline and nonamorphous order. The structure of a quasicrystal differs

significantly from that of glass. The structure of shechtmanite is characterized by a common fivefold symmetry axis

around which the quasicrystal is rotated. The structure of metallic glass may be ascribed the whole set of fivefold

symmetry axes, but in this case each axis is determined only by a small set of atoms. One can construct quasicrystals,

e.g., with a sevenfold symmetry axis but not with icosahedral symmetry. Examples of two-dimensional five- and sevenfold

Penrose tiling are given in Fig. 6. The first of these tessellatious (Fig. 6a) is constructed from rhombs of two types

with acute ~, angles: Jr/5 and 2.,r/5. To construct the second mosaic, one needs rhombs of three types with ~, equal to

3r/7, 2~/'7, and 3.,r/7.

The available experimental facilities do not always permit us to establish whether the five- or sevenfold symmetry

axis extends throughout the whole crystal or is met in it only locally, with the main part of the space occupied by the

crystal structure. The same diffraction pattern (as in quasicrystals) is observed, e.g., in cubic crystals with a large

translation period, i.e., in crystals built by the twin law. The icosahedral long-range order in shechtmanite is explained

by tessellation theory.

As a rule, the Pertrose tiling is taken as a starting structure for constructing the model of atomic structure of

quasicrystals. An infinite three-dlmensional Penrose tiling is formed from rhombohedra of two types (Fig. 7). The ratio

of the number of prolate rhombohedra to that of oblate rhombohedra is equal to the golden number:

30 = (1 + V~)/2 = 2 cos(~r / 5). (13)

The structure of shechtmanite is represented using Penrose tiling only when the rhombohedra forming the

quasicrystal contain atoms of each element in a proper ratio. Modeling a particular alloy generally requires decoration

of each rhombohedron with atoms of definite variety. By analogy with ordinary crystals, in shechtmanites we can

sometimes isolate sublattices.

As Kasper-Frank phases, quasicrystals may be described using defects in an icosahedrally coordinated network.

The difference between these states is illustrated by a two-dimensional system. Here a pentagon is a symmetric analog

129

Page 11

t

r

Fig. 6. Five- (a) and sevenfold (b) Perurose tilings.

of an icosahedron. F'~mres 8 and 9 show arrangement of pentagons on a Pcnrose tiling composed of rhombs of two

types with acute angles z~/5 and 2x/5. The packing density of the two-dimcusional analog of the quasicrystal (F'~ 8b),

n = r 0 / 2 = 0.809, is smaller than n of the corresponding prototypes of Kasper-Frank phases (Fig. 9). Random close

packing of hard disks is nrcp = 0.82 [39]. F~are 9a, e shows the packings in which the centers of the pentagons form

a triangular lattice with the ratios of the sides of the wedging triangles -1.539 (Fig. 9a) and 1..577 (Fig. 9c).

In these cases, the packing densities are v'-5~ --" 0.854 and (2V"5 / 3)r0 ~ 0.921. The packing shown in F'~ 9c

is pentagonal dose packing in two-dimensional space. The pentagonal packing presented in F'~ 9a, c differs from the

quasicrystalline packing (Fig. 8b) in having a smaller fraction of the length of the edge boundary. If we continue to

diminish the length of this boundary, we obtain an optimal packing shown in Fig. 9b. This arrangement of pentagons

has the same packing density as the packing of pentagons shown in Fg. 9a. However, its short-range order corresponds

better to the order in a qn:~icrystal (Fig. 8b). The packing presented in Fig. 9d demonstrates an arrangement of

pentagons in the form of bipentagonal dusters. The packing density n = 0.828 approaches the value of nzt. p.

Reconstruction of the atomic structure of a quasicrystal based on Penrose tili,g demands deter~i-ation of

symmetry, packing pattern, and unit cell decoration. For the quasicrystal, there is an infinite number of variants of unit

cell formation. Unfortunately, there is no method for direct determi-ation of the class of local isomorphism, i.e, packing

type and unit cell decoration method, from diffa'action data. In [40], three criteria for identification of physically

realiTable dasses of isomorphism are suggested. The first criterion demands that the packing be reproducible. This is

130

Page 12

a 6

Fig. 7. Prolate (a) and oblate (b) rhombohedra. The spherical angles

at the vertices of the rhonbohedra are multiple to the angle 3r/5.

b

Fig. 8. Packing of pentagons corresponding to a representation of

a quasicrystal as a two-dimensional Penrose tiling; a) decoration

of Pearose rhombohedra, b) arrangement of pentagons on the

Penrose tiling,

achieved by setting certain rules of selection that fix allowed unit cells within definite boundaries. The second criterion

reflects the ability of the packing to grow;, using local a~,~egation rules, a quasicrystal may be grown from the seed

given in the unit cell The third criterion suggests the poSsl'bility of quick growth of quasicrystalline packing that is

close to fixed packing at a rate comparable to the growth rate of the ordinary crystal

Reproducibility is a necessary condition if the packings of the given class of local isomorphism are energetically

stable. The capability for growth and the possibility of rapid growth are necessary to obtain physically attainable states.

These criteria may be satisfied, which is shown [40] by directly projecting a set of points from a five-dimensional cubic

lattice to the two-dimensional space orthogonal to the direction 6 = (1,1,1,1,1). The projections of the neighboring

points of the hyperlatfice give rhomb edges. The tessellation was constructed by joining oblate and prolate rhombs.

According to the given rules, the quasicrystal nucleus can grow only up to a certain size. Surpassing the dead lim;t

requires certain changes in the packing rules. The dead surface cannot appear during quick growth of the cluster

formed by oblate and prolate rhombs if the cluster has a packing defect with tenfold symmetry. The Pem'ose tiling

resulting from such growth containg a point defect.

The structure of a real quasicrystal is generally distorted with respect to the Penrose tiling. These distortions

are explained by the presence of an elementary excitation - a quasiparticle - in mnlticomponent systems. The

quasicrystal prepared by quickly cooling a melt may contain inclusions of other phases. In this case, an elementary

excitation in it is called a phason. The existence of phasons is atm'buted to fluctuation of concentrations of one of the

system components around any charged particle, for example, around an electron. The phason may be stable and may

move together with the charged particle. The question arises of the effect of phason-induced stresses on the Penrose

tiling projected from many-dimensional space.

131

Page 13

I:':'~

i ,~

c d

Fig. 9. Crystalline packings of pentagons. Packing densities:

a), b) 0.854; c) 0.921; d) 0.828.

The six-dimensional space is divided [41] into two three-dimensional spaces. One of them is an invariant

subspace of the icosahedral point group, i.e., the Peurose tillnE is obtained by projecting the points of the hyperlattice

of the six-dimensional space to the subspace. For the phason-induced stress field, linear approximation is sufficient to

explain [42] the distortions in the diffraction pattern associated with the transformation of the rings into ellipses. The

phason stress field shifts the diffraction peaks and decreases their intensity. In the same approximation, it has been

shown that the global fivefold symmetry may be distorted to form many clusters that give local fivefold symmetry around

their centers. Consequently, states resembling Frank-Kasper phases have been obtained.

Currently, specification of quasicrystnnine structures has not yet achieved the level of traditional crystallography

[43]. However, the characteristic features of these structures obtained in diffraction experiments have shown that 90%

atomic positions correspond to perfect icosahedral quaasicrystals for such systems as A1FeCu [44] and AIPdMn [45].

The phase diagram for the first system in the range of pressures from 0 to 0.5 MPa was obtained in [4ol, and structural

transformations in the decagonal quasicrystalline phase of A1CuCoSi and the icosahedral quasicrystalline phase of

AICuCoFe are investigated in [47].

Recently, a model of quasicrystals based on hierarchical packing of clusters was proposed [48]. The Mackey

pseudoicosahedron (MPI) was used as an elementary structural unit of the model. On the external layer of the cluster

there are 42 atoms (12 icosahedron vertices plus 30 icosidodecahedron vertices), and inside the cluster there may be

8 or 9 atoms. The MP1 differs from the ordinary Mackey icosahedron in that inside it contains a small dodecahedron

whose vertices and center are partially fdled with atoms. Another structural element of the model is the "binding unit."

These units may be parts of the MPI. The "binding units" are arranged in layers (shells) having the same d::;.fity as

the MPI. Formation of a quasicrystal from clusters is based on an extension concept: in some extending step -.';:~ MPI

centers are surrounded in the same way as the MPI centers of the preceding generation. In the adjacent e::encling

steps, a change in scaling is r 3 where r 0 is a golden number. In addition, the extending clusters should reflect the

same magic electron numbers as the starting free clusters. When a quasicrystal is built on a duster basis, an icosahedral

132

Page 14

W

Fig. 10. Global star of the point A of the Delaunay system.

structure of the cluster is used, which is confu-med both in mass-spectrometric investigations of cluster beam~ [49] and

in computer simulations [50].

Representation of quasicrystals as a hierarchy of clusters is an extension of so-called separation analysis [51].

This representation is also consistent with the analysis [52] of electron transition through the internal boundary. However,

neither the exact chemical composition nor the local homogeneity may be fitted by this procedure to real alloys, which

are generally imperfect crystals.

Finally, let us consider the problem of correctly using the term "long-range order." In the introduction it was

noted that in the case of quasicrystals it is more appropriate to speak about the type of rotational order, which is not

observed in crystals. Let us now show that the term "long-range order" is inadequate even for an ordinary crystal.

Translational order does not require that an arbitrarily remote atom occupies its strictly defined position. The most

general laws of arrangement of atomic centers in an infinite space are given using the (R, r)-system introduced by

Delaunay [53]. In crystallographic geometry, a Delaunay system is defined [54] as a set of points of Euclidean space

that satisfies two requirements: 1) discreteness axiom: the distance between any two points of the set should exceed a

segment of a t-mite length r;, 2) covering axiom: the distance from any point of the set to the nearest point should be

less than some fixed segment of the length R.

The first requirement prevents an extremely dense arrangement of the points, and the second requirement

does not allow the points to be too sparse. When both requirements are satisfied, the points are nearly uniformly

distributed in space. In a physical system, atoms cannot approach to arbitrarily small distances due to repulsion, and,

on the other hand, mutual arrangement of atoms is regulated by the law of entropy growth. Let us connect an arbitrary

point A of a Delaunay system with all of its other points. The resulting system of segments is called [54] a global star

of the point A with respect to the given system (Fig. 10). If all points of the Delaunay system possess equal global

stars, the system is called regular.

It was shown [54] that even in the most unfavorable case a regular two-dimensional system is completely

determined by the positions of points in a circle of the radius 4R. For three-dimensional space, the domain of a regular

point system does not fall beyond the sphere of the radius 10R. Thus the order in an ordinary crystal may be represented

as a translation in some limited region of space but not in arbitrarily large regions. A reduction of the domain of the

regular point system in the two-dimensional case with respect to the three-dimensional one means that in low-dimensionai

space the liquid geometrically approaches the crystal.

CONCLUSION

In recent years, our understanding of possible structures of substances has changed significantly. Before the

discovery of quasicrystalline symmetry, there were only two radically different definitions of the structure of condensed

phases, dividing all solids and liquids into ordered and disordered media; now this classification needs to be developed

in more detail. It was traditionally accepted that the long-range order is the main characteristic of the crystalline state,

133

Page 15

and the long-range order in substances was associated with geometrical memory. In other words, it was assumed that

the position and properties of atoms in a crystal are determined not only by its nearest neighbors (short-range order)

but also by the increasing quantity of remote atoms. However, studies by Delannay [53] have altered this understanding

of the long-range order in crystals. An atomic arrangement in space is ascribed the long-range order if a vector system

of all global stars [54] corresponding to a Delaunay system is discrete. Such point systems give discrete diffractograms

and are physically represented by perfect and flawed crystals and quasicrystals. It was shown [54] that if a vector system

of global stars is a Delauuay system, we obtain a discrete diffractogram. In this case it is not suggested that the system

is perfect. Therefore it is not paradoxical that the quasicrystal possessing a long-range order is characterized bythe

fivefold symmetry axis that is not coincident with translational symmetry, i.e., with strict periodicity. The absence of

periodicity is not caused [55] by chaotic disorder; it is caused by an overlapping of at least two incommensurate

periodicities whose period ratio is an irrational number. Irrational numbers associated with quasicrystaUine structures

are always algebraic, i.e., they are roots of polynomials with integer coefficients. Thus the golden number 30, which

characterizes the fivefold symmetry, is a solution of the quadratic equation

z 2-r 0-1=0, (14)

and any power of T O may be represented as a linear combination of the numbcz itself and unity with integer coefficients.

Suppose a linear chain of atoms (one-dimensional crystal) is modulated by superposing a wave with an irrational period

with respect to the period of the starting chain; then the new chain will be nearly periodic because it is nearly exactly

mapped on itself. The higher the degree of approximation of an irrational number by a rational fraction describing

the approximation of the atomic arrangement in the new chain, the better the coincidence of atoms when the new

"original" is superposed on its rational "copy." The Penrose parquet evidently refers to the case with two

incommensurate basis periods [56]. Thus quasicrystalline structures may be regarded [57] as some kind of interpolation

of crystalline structures like an irrational number is interpolated by two rational numbers. In other words, a quasicrystal

is a generalized crystal in which the long-range translational order is replaced by rotational order.

An ordinary crystal is defined as a system with translational symmetry, which is expressed as an infinitely

recurrent motif. Periodicity in three directions is described using three-dimensional space groups. The reason for the

broken translational order in the crystal is not only thermal motion. In solids, there may be static disorder caused by

chaotically distributed own defects or support defects. This disorder may be present [58] in a sample at T = 0 K, when

the orientational order in molecular arrangement is retained. In amorphous solids, the orientational order may be kept

at large distances [20]. This is most likely associated [59] with the geometrical properties of a system.

The picture of structural changes occurring during vitrification is still unclear. Along with the cluster model

involving an icosahedral short-range order and the disclination model of glass, there are network [3], displanation [60],

and other models. In [60], it is suggested that glass be considered as an original state that is independent of the structure

of a liquid and the relaxation rearrangements during cooling. There were numerous attempts to reduce vitrification to

a change in a certain parameter that defines chemical bonding. Subsequently it always appeared that such approaches

are only useful for classification of certain groups of vitrifying compounds. We'can state with confidence that the

tendency toward vitrification is inherent in compounds in which the short-range order is determined by the character

of chemical bonding.

Vitrification may be considered as freezing of an equilibrium state. Then glass is a nonequilibrium system that

very slowly returns to the equilibrium state. The frozen motions (degrees of freedom) may be any type of motion, but

fgation of structural changes is most important. The structure-sensitive properties in vitrifying systems involve: molar

volume, viscosity, vitrification temperature Tg, thermal expansion coefficient, and elasticity constants. Thus vitrification,

i.e., freezing of certain degrees of freedom, makes it possible to differentiate glass from another amorphous substance.

The vitrification phenomenon is universal. For example, the concentration of thermal defects in crystals is vitrified.

The spin momenta are frozen. Different equilibria of structural type can exist [61] in one-component vitrifying systems.

This is explained by the geometrical effect in a packing. Thus in the elastic sphere model, three energy minima exist

only in the case of irregular packing [62].

Vitrification is not comparable to the second-order phase transition. It occurs in a definite temperature range,

but not at some taxed temperature. A description of vitrification in the form of the Ehrenfest transition leads to

irresoluble contradictions, due to which the ki0etic factors of transformation may not be taken into consideration. Ideal

134