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Journal of Non-Crystalline Solids 353 (2007) 1221–1225

Compositional trends in glass transition temperatures ininorganic network glasses

Richard Kerner *

LPTMC, Universite Pierre et Marie Curie – CNRS UMR 7600, Tour 22, 4-eme etage, Boite 142, 4 Place Jussieu, 75005 Paris, France

Available online 26 March 2007

Abstract

We present a simple mathematical model of glass transition based on the analysis of molecular agglomeration in overcooled liquids.With simple assumptions concerning local configurations and their and bonding energies, and with elementary combinatorics we are ableto derive the dependence of the glass transition temperature on chemical composition in non-organic covalent glasses.� 2007 Elsevier B.V. All rights reserved.

PACS: 61.43.Fs; 64.70.f

Keywords: Glass transition

The success of the theory of crystalline solid state isbased on the use of dual space whose symmetry, imposedby the symmetry of crystal itself, enables one to quantizethe wave vectors of phonons, to introduce the Brillouinzones, and to derive many thermal, optical and electromag-netic properties of crystals. In structural glasses which canbe described by Zachariasen’s concept of the Continuous

Random Network (CRN), the use of dual space is not help-ful anymore because it becomes infinitely chaotic and intri-cate, although locally at least glasses do not differ verymuch from corresponding crystalline counterparts. Forexample, regular Si O1

2

� �4

tetrahedra are found in bothcrystalline and amorphous SiO2; similar structures can beobserved in GexSe(1�x) glasses. Other regular local struc-tures, six-membered boroxol rings or edge-sharing tetrahe-dra, appear in borate or chalcogenide glasses, respectively.This local order is rapidly destroyed in the medium rangedue to the relative weakness of angular constraints.

The Stochastic Agglomeration Theory (SAT) is basedon the analysis of stable local configurations in real space,and modelizes a progressive buildup of random networks

0022-3093/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.jnoncrysol.2006.09.063

* Tel.: +33 1 44 27 72 98; fax: +33 1 44 27 51 00.E-mail address: rk@ccr.jussieu.fr

during the glass transition process. Taking into accountthe energy landcape and the entropic factors SAT providesa new definition of glass transition temperature Tg relatingit with minimal fluctuation condition.

The formulae for Tg deduced from the stochasticagglomeration models have found a spectacular confirma-tion in recent years due to the novel experimental tech-niques introduced by Boolchand and collaborators [1,2].The new approach to the measurements of Tg is calledModulated Differential Scanning Calorimetry (MDSC).The trends in Tg behavior are parameter free and can bemeasured very reliably from the reversing heat flow bygoing up and down with T. The advantage is that the Tg

so defined is devoid of kinetic and aging effects. The MDSCmethod has also made possible the discovery of the revers-

ibility window in many network glasses i.e. the region ofcomposition in which glassy networks remain isostatic itsinternal structure evolving keeping the number of con-straints per atom constant.

The model of glass formation [3–7] is based on a detaileddescription of agglomeration and growth of micro-clustersthat already exist in hot melts about to undergo glasstransition. In order to predict structural features of theresulting random network, we must have a reasonablemodel imitating in which way these clusters grew and

Fig. 1. A cluster agglomerating from two types of blocks, mA = 2, mB = 3.

1222 R. Kerner / Journal of Non-Crystalline Solids 353 (2007) 1221–1225

agglomerated, as well as their own structure and relativestatistical weight. To do so, we must find first a descriptionof the irreversible process of agglomeration in terms ofvariables directly related to the configuration space of thesystem.

A hot melt at constant temperature contains a lot ofclusters of different sizes and shapes. Let us denote by k

the number of atoms in a cluster. At a given temperaturenegligibly few ones are bigger than kmax. The statistical dis-tribution of cluster sizes and shapes depends on the temper-ature. At a given T some of the clusters split into smallerunits, while some of the smaller ones agglomerate creatinglarger entities; however, on the average the balance is main-tained and the mean distribution of clusters in the space ofall possible configurations remains constant.

But let the temperature decrease, and the distribution ofprobabilities of finding a given number of clusters of thesize k will also evolve: the higher the temperature, thegreater the ‘population’ of clusters corresponding to lowervalues of k, and vice versa. It is quite obvious that when thetemperature slowly decreases, the ‘center of gravity’ of theoverall distribution is displaced slowly towards greater val-ues of k; at the same time, kmax is growing, too. We believethat the best way to get more insight into the structuralproperties of a glassy network is to reconstruct as preciselyas possible the very process of its creation, which occursduring the glass transition and stops at the temperature T

slightly under Tg, and consists essentially in the saturationof all bonds between the micro-clusters that already existedin the hot melt. The whole phenomenon is an off-equilib-rium thermodynamical process, during which, startingfrom some moment, clusters agglomerate with the expo-nentially growing speed. This results in an exponentialgrowth of viscosity and in a sharp variation of heat capac-ity Cp.

We suppose that the energies stored in a given clusterduring its agglomeration are additive i.e. the energy islocally conserved. This means that we can evaluate theprobability of observing a given cluster by summing up

Fig. 2. Typical Tg dependence on modifier concentration: (a) in a B2O3 + Na2OSe(1�x) glass (white).

all the agglomeration pathways that lead to its creationfrom elementary ‘singlets’, by multiplying the probabilitiesof each transition by corresponding statistical weights andBoltzmann factors.

Let us take as an example the binary As–Se glass. Con-sider an average cluster as shown in Fig. 1. There are twopossible kinds of sites on its rim: a selenium atom with oneunsaturated bond, and an As atom presenting one or twofree bonds; these are indicated by x = and y = Thereis an intermediate site z = , but it is bound to transformimmediately into an x or into an y, so it can be not takenunder consideration.

To each site a Se or a As atom can attach itself, repro-ducing one of the initial configurations, in the specific com-binations shown in the second column of Fig. 2. Eachagglomeration step is synonymous with complete satura-tion of all the bonds on the rim, so that only two typesof sites (denoted by x and y) observed, assuming that thegrowth is of dendritic type (no small rings present).

All possible agglomeration processes are represented inDiagram 1.

The Boltzmann factors depend on the energy barrierscharacteristic for the three types of bonds, A–A, A–B andB–B:

� ¼ EAA

kT; g ¼ EAB

kT; a ¼ EBB

kT: ð1Þ

glass, (b) in a SiO2 + Na2O glass and (c) in a Gex Se(1�x) (black) and Six

Diagram 1. States, steps and matrix entries for mA = 2, mB = 3 (e.g. Asc Se(1�c) glass) case.

R. Kerner / Journal of Non-Crystalline Solids 353 (2007) 1221–1225 1223

Consider the dendritic growth with only two types ofsites appearing all the time. Given an arbitrary initial state(px,py), the new state results from taking into account allpossible ways of saturating the bonds of the previousstate’s sites by the available external atoms. The un-nor-malized probability factors are displayed in the figure.The non-normalized probability factors can be arrangedin a matrix

2ð1� cÞe�� 8ð1� cÞ2e�2g þ 12cð1� cÞe�g�a

3ce�g 12cð1� cÞe�g�a þ 18c2e�2a

!: ð2Þ

The normalized transition matrix is written as

M ¼Mxx Mxy

Myx Myy

� �¼

Mxx 1�Myy

1�Mxx Myy

� �; ð3Þ

where the entries are obtained by normalizing the columnsof the matrix (2).

Mxx ¼2ð1� cÞn

2ð1� cÞnþ 3cand Myy ¼

3cl2ð1� cÞ þ 3cl

; ð4Þ

where we have introduced the abbreviated notation n = eg–�

and l = eg–a. The action of matrix M on the probability dis-tribution (px,py) symbolizes the new layer creation, with analtered probability distribution of sites x and y, ðp0x; p0yÞ.

p0xp0y

!¼

Mxx Mxy

Myx Myy

� �px

py

!: ð5Þ

The difference~p0 �~p can be interpreted as the derivative ofthe probability distribution with respect to time or anotherparameter describing the advance in agglomeration, likethe average number of layers at a given time. This schemecan be easily generalized to the case of arbitrary valence,say mA and mB. In that case, the stochastic 2 · 2 matrixhas the same form as (3), but with the entries given by

Mxx ¼ 1�Myx ¼mAð1� cÞn

mAð1� cÞnþ mBc;

Mxy ¼ 1�Myy ¼mAð1� cÞ

mAð1� cÞ þ mBcl:

The eigenvalues of this matrix are k1 = 1 and k2 = Mxx �Mxy = Myy �Myx, with jk2j < 1, so that in the long run,after the matrix M was applied many times (creating manyconsecutive layers), only the first eigenvector will survive.This stationary eigenvector (corresponding to the eigen-value 1) is

p1xp1y

!¼ 1

Mxy þMyx

Mxy

Myx

� �: ð6Þ

On the surface of clusters, when averaged over a greatnumber, px is the A concentration and py is the B concen-tration. The high homogeneity exhibited by known glassstructures suggests that even in relatively small clusters,deviations from the average modifier concentration c mustbe negligible. Thus, in the bulk, the B concentration shouldbe equal to c. Therefore, the condition of minimal fluctua-tions in the bulk concentration can be interpreted as theglass transition condition. The asymptotic state is thusfixed by the external concentration. It follows that theeigenvector (6) must be equal to the average distributionvector (1 � c,c), therefore it is enough to satisfy the follow-ing equation:

py ¼ c¼ Myx

MxyþMyx¼ mAmBcð1� cÞþm2

Bc2l

m2Að1� cÞ2nþ2mAmBcð1� cÞþm2

Bc2l:

ð7Þ

In the low concentration limit c� 1 and we can neglect thequadratic terms c2 in the formula (7), which leads to amuch simpler equation

py ¼ c ’ mAmBcð1� cÞm2

Að1� cÞ2nþ 2mAmBcð1� cÞ: ð8Þ

In the low concentration limit condition (7) leads to the fol-lowing equation for c,

cð1� cÞ ðm2An� mAmBÞ � cðm2

An� 2mAmB� �

¼ 0: ð9Þ

Eq. (9) has three solutions: the two obvious ones, c = 0 andc = 1, and the third one, which can be identified as the def-inition of the glass transition temperature:

1224 R. Kerner / Journal of Non-Crystalline Solids 353 (2007) 1221–1225

c ¼1� mA

mBn

2� mAmB

n: ð10Þ

In the case when the A-component (e.g. selenium) is a glassformer, the initial slope of Tg at c = 0, when the modifieratoms are still scarce, has a particularly simple form. Whenc! 0 and Tg! Tg0, (the glass transition temperature forpure selenium) in formula (10), we must have

njT¼T g0¼ e

EAB�EAAkT g0 ¼ mB

mA: ð11Þ

Taking the logarithm on both sides, we get

EAB�EAA

kT g0

¼ lnmB

mA

� �or EAB�EAA� kðlnmB� lnmAÞ¼ 0:

ð12Þ

Formula (12) can be interpreted as the well known relation-ship for the free energy minimum, dF = dU � TdS. As amatter of fact, when a single modifier atom (B) arrives inthe melt at c = 0, the energy variation dU equalsEAB � EAA. The entropy variation can be evaluated follow-ing Boltzmann’s definition of configurational entropy,S = klnW, where W is the number of configurations avail-able to a system. Here two atoms AA could create m2

A

bonds, whereas the couple of atoms B can create mAmB

bonds. The corresponding entropy variation will be

dS ¼ k lnðmAmBÞ � lnðm2AÞ

� �¼ k½ln mB þ ln mA � 2 ln mA� ¼ k½ln mB � ln mA�: ð13Þ

However, the interpretation of this formula is quite differ-ent than in classical statistical mechanics, instead of config-urations we consider different bond creation processes.

The derivative of c with respect to the temperature T iseasily found, and its inverse gives the ‘magic formula’ fordT g

dc at c = 0: This defines the slope of the function Tg(c),which is an important measurable quantity:

Fig. 3. Other examples of the initial slope

dT g

dc

� c¼0

¼ T g0

ln mBmA

� � ; ð14Þ

where we used the fact that dndT ¼ � 1

T n ln n, and njc=0 = (mB/mA).

The formula valid for any concentration is morecomplicated:

dT g

dc¼ T g

1� mAmB

n� �

þ ð1� mBmA

lÞh i2

mAmB� l

� �n ln n� mB

mA� n

� �l ln l

: ð15Þ

Discussion

Formula (14) has been checked against the experimentvery successfully, in more than 30 different compounds.In some cases the formula does not seem to work well; usu-ally it comes from the change of valence of certain atomsprovoked by the influence of the surrounding substrate.

Other interesting cases are shown in Fig. 3.The singularity in this formula when mA = mB (i.e. a

mixture of two different glass formers with the same coor-dination number) can be easily dealt with as was shown byM. Micoulaut. If the glass transition temperature of thepure glass-former A is Tg0, and that of the pure glass-for-mer B is Tg1, then the ‘pure states’ c = 0 or c = 1 representstationary solutions of (7) and can be factorized out. Thenon-trivial condition for the glass forming is given by (7)Now, using the limit conditions at c! 0, Tg = Tg0 andc! 1, Tg = Tg1, the expressions n and l at the arbitrarytemperature T can be written as

nðT Þ¼ eEAB�EAA

T g0�T g0

T ¼ mB

mA

� �T g0T

; lðT Þ¼ eEAB�EBB

T g1�T g1

T ¼ mA

mB

� �T g1T

:

ð16Þ

formula agreement with experiments.

Fig. 4. Tg(c) curves for S–Se and Te–Se binary glasses.

R. Kerner / Journal of Non-Crystalline Solids 353 (2007) 1221–1225 1225

Substituting these expressions into (14) and taking the limitc! 0, we get

dT g

dc

c¼0

¼T g0 1� mB

mA

� �T g0�T g1T g0

" #

lnðmBmAÞ : ð17Þ

Even when mA = mB, this formula has a well definedlimit. Indeed, set mB

mA¼ 1þ �, and then develop the numera-

tor and the denominator of the above equation in powersof �. In the limit when �! 0 we arrive at a simple lineardependence which is in agreement with common senseand with experiment as well, namely (see Fig. 4)

dT g

dc

c¼0

¼ T g1 � T g0: ð18Þ

This formula is also confirmed by many experiments, e.g.performed on selenium–sulfur mixtures (wheremA = mB = 2). The deviations from the linear law (18) ob-served in the Se � Te binary glass are explained by the factof the chemical properties of tellurium, which changes itsvalence from 2 to 3 in presence of selenium.

References

[1] P. Boolchand, D.G. Georgiev, M. Micoulaut, Nature of glasstransition in chalcogenides, J. Optoelectron. Adv. Mater. 4 (4) (2002)823.

[2] P. Boolchand (Ed.), Insulating and Semiconducting Glasses, WorldScientific Publishing Co., Singapore–London–New York, 2001.

[3] R. Kerner, Stochastic matrix and self-organization in glasses, in: M.F.Thorpe, L. Tichy (Eds.), Proceedings of the NATO Advanced StudyInstitute on Properties and Applications of Amorphous Materials,Kluwer, 2001, p. 177.

[4] R. Kerner, D.M. dos Santos, Nucleation and amorphous andcrystalline growth: a dynamical model in two dimensions, Phys. Rev.B 37 (8) (1988) 3881.

[5] R. Kerner, G.G. Naumis, Stochastic matrix description of glasstransition in ternary chalcogenide systems, J. Non-Cryst. Solids 231(1998) 111.

[6] R. Kerner, M. Micoulaut, On the glass transition temperature incovalent glasses, J. Non-Cryst. Solids 210 (1997) 298.

[7] R. Kerner, A theory of glass formation, in: R. Elliott, M. Balkanski(Eds.), Atomic Diffusion in Disordered Media, World Scientific, 1998,p. 25.