Contents
List of Figures
 1 A schematic representation of a choice of local configurations. a) Possible SRO starlike configurations in glasses: a ”regular” tetrahedron [a structure] and an ”altered” tetrahedron [a structure] b) glass: a ”regular” atom and an ”altered” atoms Ge
 2 Two possible connections of and configurations. With labelled bridges, there are ways to form the same doublet
 3 An example of the arrest of the temporal fluctuations of structure. The plot represents the equation (2.5) with , , [, glass] and (solid line), (dashed line) and (dotted line)
 4 The phase diagrams of the equ.(6): a) (phase separation) b) (glass formation)
 5 a) A selenium chain b) a selenium chain with a crosslinking silicon atom
 6 Glass transition temperature [in K] versus concentration for typical glass systems, [], [] and []. The plotted values are reported in [20], [47] and [52]. The straight lines correspond respectively to the equations (solid line), (dashed line) and (dotted line).
 7 Glass transition temperature [in K] of [] and [] systems versus the concentration . The line represents the slope equation The plotted data are taken from [42] and [56].
 8 a) IVVI binary glasses. The SRO structures are the and tetrahedra and eventually the tetrahedra with two nonbridging atoms (, ). b) SRO structures of a boron glass. A triangle () and a fourcoordinated boron (N4 unit) (, ). c) SRO structure of phosphor based glasses () (,). d) Possible MRO in based glasses. The boroxol ring () and the tetraborate group ().
 9 Glass transition temperature [in K] versus the concentration in glasses with [], [] and []. The line represents the equation . The data are reported in [67] and [68].
 10 Glass transition temperature [in K] of glasses versus modifier concentration with [], []. The lines represented correspond to (solid line, ) and to (dotted line, ). The plotted data are given in reference [69] and [71].
 11 Glass transition temperature [in K] versus the concentration in glasses, with [], [], [], [] and []. The lines represent the equations (dotted line) and (solid line). The plotted values are taken from [41].
 12 Glass transition temperature [in K] of , [] [] and glasses []. The represented lines correspond to (solid line, pure cornersharing and ), (dotted line, pure edgesharing and ) and (shaded line, pure edgesharing and ). The data are taken from [88][90].
 13 Glass transition temperature of glasses. The solid lines correspond to the pure edgeand cornersharing situations, using the slope equation (2.16) and (3.12). Data are taken from [95]. Intermediate situations are plotted and correspond to (, , dotted line) and (, , dashed line).
 14 Glass transition temperature [in ] of glasses with [] and []. The solid line corresponds to the slope equation (3.27) with , , and (for sodium systems). Data are taken from [99]. The dotted line represents a modified slope equation using Martin’s correcting factor [92] for based glasses.
 15 The dithioborate group proposed as possible MRO structure in glasses [92]. This structure is made of cornersharing doublets ( species).
 16 Glass transition temperature [in K] of systems. The line represents has a slope (3.27) with , , and . Data are reported in reference [102].
THE SLOPE EQUATIONS: A UNIVERSAL RELATIONSHIP BETWEEN
LOCAL STRUCTURE AND GLASS TRANSITION TEMPERATURE
Matthieu Micoulaut
Laboratoire GCR  CNRS URA 769
Université Pierre et Marie Curie, Tour 22, Boite 142
4, Place Jussieu, 75005 Paris, France
Abstract:
In this article, we present a universal relationship between the glass transition temperature and the local glass structure. The derivation of the simplest expression of this relationship and some comparisons with experimental values have already been reported in a recent letter [1]. We give here the analytical expression of the parameter of the GibbsDi Marzio equation and also new experimental probes for the validity of the relationship, especially in low modified binary glasses. The influence of medium range order is presented and the unusual behavior of in binary and systems explained by the presence of modifierrich clusters (denoted by doublets).
Pacs: 61.20N81.20P
1 Introduction
It is wellknown that the formation of glasses requires cooling to a sufficiently low temperature below the glass transition without the occurrence of detectable crystallization. In treating this phenomenon, it is has been suggested by different authors that specific structural features or physical properties will result in glasses being formed [2][7]. Various models have been proposed in order to describe this transition, which can appear superficially to be a secondorder thermodynamic phase transition. These models involve generally factors (thermodynamic, structural, kinetic) which are viewed as decisive in the formation of glasses.
The best known structural model of glass formation and glass formation ability is that due to Zachariasen [8] and others [9, 10] who proposed a classification of oxide materials in terms of glassformers (e.g. ), modifiers (e.g. ) and intermediates (e.g. ). This led to the random network concept [11] which received support from xray diffraction studies of a variety of glasses, although these studies did not establish the model as unique representation of structure [12][14]. In these models, the relationship with the glass transition temperature is lacking.
Importance of thermodynamic factors in glass formation has been pointed out by Gibbs and Di Marzio and by Adam and Gibbs [15, 16]. These authors suggested that the glassy state is thus defined in terms of thermodynamic variables (temperature, volume,…) and related ones (bulk compressibility, heat capacity,…), but it is not necessarily implied that the glassy state is one of even metastable equilibrium (with reference to a possible crystalline phase). It can be stated that a glassforming material has equilibrium properties, even if it may be difficult to realize. The theory developed by Adam and Gibbs on this basis, was able to predict a secondorder phase transition and also a quantitative relationship between glass transition temperature and the crosslinking density in some linear molecular chains.
Nevertheless, when the glass transition temperature is measured under standard conditions (for example calorimetrically at a fixed heating rate), an important question concerns the relationship of with some other physical and measurable factors. Various proposals have been made in the past which suggested for example that scales with the melting temperature (the ’twothird rule‘ proposed by Kauzmann [17]), the boiling temperature, the Debye temperature of the phonon spectrum, etc. [18]. Besides the influence of these thermodynamic factors, attention has been devoted to structural factors, in particular to the valence of the involved atoms in the glassforming material. Tanaka [19] has given an empirical relationship between and , the average coordination number per atom: . This proposal agrees for various glassforming systems including chalcogenide materials and organic polymeric materials. However, the relationship between structural factors and becomes more complicated if the composition of the glassforming alloy is changing. For example, in network glasses, the glass transition temperature is not varying monotically with , and exhibit even some characteristic behavior (maximum at [20], which corresponds to the stoichiometric composition , anomaly at , where the average coordination number is equal to ). Obviously, is sensitive to the chemistry involved, and a maximum in at the stoichiometric composition may result from the formation of a chemically ordered network in which only the bonds are present. In other systems, a general accepted rule states that is increasing when the connectivity of the network is increasing, and viceversa. Besides these rather qualitative relationships between structure and glass transition temperature, there exists a firm rule for predicting in particular glassy materials, namely chalcogenide network glasses. The equation relating the glass transition temperature to some structural factor is that due to Gibbs and Di Marzio [15] and Varshneya and coworkers have shown recently that a close equation was particularly well adapted for predicting in multicomponent chalcogenide glass systems [21]. Indeed, one can consider the chalcogenide glass system as a network of chains (e.g. selenium atoms) in which crosslinking units (such as germanium atoms) are inserted. The increase of is produced by the growing presence of these crosslinking agents, which can be roughly explained by the grwoth in network connectivity. In the former version, Gibbs and Di Marzio applied successfully their equation in ordre to explain the data in polymers [22]. An adapted theory constructed by Di Marzio [23] has shown that for glass systems with some chain stiffness, the glass transition temperature versus crosslinking density could be expressed as:
(1) 
where is the glass transition temperature of the initial polymeric chain and a constant.
In this article, we shall present several relationships between the glass transition temperature and the local structure in glassforming materials, by using an agglomeration model, created by R.Kerner and D.M.Dos Santos [24] and applied with success to various glassforming systems [25]. The physical insight of the model can be found in references [26] and [27]. Different situations will be reviewed in this article and previous results, reported in [1], will be inserted for completeness. In section 2, we present the construction with starlike entities and obtain the first slope equation for cornersharing structures (single bonded network, i.e. absence of dimers). Comparison with experimental data is presented in section 3 for chalcogenide network glasses and for binary oxide glasses. The relationship with the GibbsDi Marzio equation is also given in section 3. Section 4 is devoted to other structural contributions, such as the edgesharing tendency between two local structures (twomembered rings, or dimers), which modifies slightly the slope equation, and to the influence of particular bonds, leading to a second set of slope equations. These equations can describe the unusual behavior of in and based glasses. Finally, section 5 summarizes the most important results of the paper.
2 Construction with starlike entities
2.1 A very simple structural consideration
For the reader’s convenience, we shall first present the simplest possible construction of the agglomeration model, with two starlike entities. The vocabulary introduced in this section will be illustrated, whenever possible, by two archetypal glass systems which are binary and glasses.
The most simplest way of describing short range order (SRO) in glasses can be made on the basis of starlike entities (we shall also call singlets or local configurations [27]). These local configurations share a central atom and they have clear, unambigous experimental evidence and a welldefined coordination number (fig. 1). The nature of the coordination can eventually be revealed by Xray or inelastic neutron diffraction techniques [28][29] which exhibit sharp and characteristic peaks at the corresponding bond length and give information about the number of nearest possible neighbors of a central atom. Typical examples are: the tetrahedron which is the lowest possible SRO structure in glasses such as or [29] and their binary compounds (fig. 1), or a fourvalenced germanium atom in network glasses [30]. Starting from a structure with a single type of singlet, called “ regular ” local configuration, one can modify the structure by adding a second kind of starlike entity, denoted by “ altered ” local configuration.
The probability of finding a “ regular ” local configuration with coordination number can be denoted by (e.g. tetrahedra in systems and of course ), whereas the one related to an “ altered ” atom with coordination number can be denoted by . The coordination number can correspond either to the valence of the atom B, or to the number of covalent bonds which connect the local configuration to the rest of the glass network. In IVVI based glasses, the number of coordination can be obtained by considering spectroscopy patterns [31, 32]. For example, when adding the modifier in , the number of covalent bonds decreases, because of the creation of socalled ”nonbridging atoms” (NBO) due to the high ionicity of lithium (ionic bonds). The local configuration B should be the tetrahedron [31], and obviously (figure 1). The creation of such a new local structure can be proposed on the basis of experiments. The peaks attributed to the tetrahedra in (assigned to structures in notation where ”4” stands for the number of bridges) are slightly shifted with addition of , and a typical chemical shift occurs due to the presence of species (the tetrahedron [31]) This chemical shift is compared to the one obtained from typical crystalline compounds (here the disilicate ) and identified. Finally, the local structure in binary systems can be described in terms of functions over the whole concentration range [33] (where is the number of NBO’s on a tetrahedron). With these examples, it is easy to see that the local configurations and can describe very well the structural change induced by the addition of in silica based glasses, at least for small concentrations of . In the network glasses ; the number of coordination can be given by the rule (where is the number of outer shell electrons)[34].
We shall show how the glass transition temperature changes as a function of and .
The aim of the construction is to evaluate the time dependence of the fluctuations of the local configuration probability, and derive an equation which imposes the minimization of local fluctuations [24]. One can reasonabely assume that these fluctuations remain important as long as the system is in the liquid or the supercooled state, where the low viscosity allows still the (A,B) configuration interchange by movement, bond destruction and creation or cation switching (in case of the presence of alkali modifier). Therefore, this should produce a variation of the local probability with respect to time and temperature. In a high viscosity state, one should expect a vanishing of the fluctuations, i.e. when the local probabilities reach a stationary value (no more fluctuations of and ). This can be identified with a stable (crystal) or a metastable solid (glass). We do not consider here relaxation processes taking place just before or at , as we are looking at a stationnary régime for , and . When is very small (corresponding to a system with high proportion of configurations), there are only two possible elementary processes of single bond formation (fig. 2), i.e. and , the second doublet being identical to .
The probabilities of these doublets may be proportional to the products of the probabilities of singlets, a Boltzmann factor which takes into account the energy of creation of the respective bond formation (i.e. for and for ) and a statistical factor which may be regarded as the degeneracy of the corresponding stored energy, because there are several equivalent ways to join together two coordinated local configurations [27]. In the single bond formation, the statistical factor is simply the product of the corresponding coordination numbers (see fig. 2). The probability of finding the constructed doublets is then:
(2) 
(3) 
where stands for with the Boltzmann constant and is the normalizing factor given by:
(4) 
We have not considered here the possibility of creation of a bond with two ”altered “ configurations . We shall indeed first focus our interest either on glass systems with a very low amount of local configurations or on systems which do not possess the ability in creating these bonds. These situations are observable in glasses with a small concentration of modifier, e.g. systems with [existence of and species only [31]] or glasses with [existence of and bonds only [20]]. The influence of bonds will be presented in a forthcoming section. We have also excluded the possibility of a simultaneous creation of two bonds or which leads to the formation of twomembered rings (dimers); such possibility does exist in binary chalcogenide glasses (e.g. based glasses), and will be also taken into account below. Anyhow, the analysis remains exactly the same. The new doublets may create a local fluctuation in the statistics, which can be evaluated as:
(5) 
If one denotes the average time needed to form a new bond by , we can evaluate the time derivative of due to the above fluctuation as
(6) 
We have neglected the dependence of the cooling rate in the equation (6) because network chalcogenide glasses or binary oxides such as or based glasses form very easily, and have critical cooling rates of the order of [35][37]. For these systems, it seems reasonable to neglect an additional cooling term.
When the local fluctuations are vanishing at and , the above expression should be , which amounts to finding the stationary or singular solutions of the differential equation (6) This leads to:
(7) 
There are always two singular solutions at the points and ; but there can exist also a third solution, given by the following expression:
(8) 
The solution represents a probability, it is therefore physically acceptable only if . Moreover, we want it to be an attractive point. Close to , every fluctuation should indeed vanish for , hence the righthand side of the linearized equation (6) in the vicinity of should be negative. This is possible only if:
(9) 
When condition (9) is not satisfied, the stable (attractive) solution is found at , which means that at the microscopic level the agglomeration tends to separate the two kinds of configurations, and .
Note that due to the homogeneity of the expression (8), only one energy difference is essential here: and that the absence of a doublet yields automatically a repulsive solution for .
The phase portraits of the differential equation (6) are shown in fig. 3.
2.2 The first slope equation
In order to give a more realistic meaning to the solution (8), we have to relate the probability of finding an ” altered “ local configuration , with the modifier concentration . In case of network glasses (e.g. ), the identification is obvious: . For binary glass systems (or following the notation), we have to recall a charge conservation law [27, 38]. The concentration of cations must be equal to the anionic contribution located on each particular local configuration, expressed in terms of :
(10) 
where and are the anionic contributions of the and configurations. For example, in low modified systems (), equation (10) becomes:
(11) 
where is the probability of finding species and the reduced concentration [38]. Since and are now related, there should be various glass formers displaying the tendency towards the solution in a wide range of modifier concentration, also when it tends to zero (, i.e. ). In this limit, we can obtain from (8) a condition concerning the energy difference
(12) 
where is the glass transition temperature at , i.e. in the limit when the modifier concentration goes to zero, corresponding to a pure A glass ( and ) or a pure network former ( and ).
This equation exhibits a relation between the statistical and energetic factors that are crucial for the glass forming tendency to appear. It tells us that in good binary glass formers whenever (and ), one should expect , and vice versa, in order to satisfy (9). This condition is what should be intuitively expected. Indeed, when a system displays the tendency towards amorphisation, it behaves in a ”frustrated” way in the sense that the two main contributions to the probabilities of doublets act in the opposite directions. Whenever the modifier raises the coordination number (), thus creating more degeneracy of the given energies (i.e. much more possiblities of linking the two entities and ). and increasing the probability of agglomeration, the corresponding Boltzmann factor is smaller than for the nonmodified atoms, , reducing the probability of agglomeration, and vice versa.
Recalling that the stable solution corresponding to the glassforming tendency defines an implicit function, , via the relation
(13) 
we can evaluate the derivative of with respect to the reduced concentration :
(14) 
In the limit () the result has a particularly simple form:
(15) 
Inserting condition (12), we obtain the first slope equation which is a general relation for binary glasses and which can be regarded as a universal law:
(16) 
For network glasses , the formula has been obtained in [1]:
(17) 
Equations (16) and (17) give the mathematical transcription of the wellknown rule mentioned at the beginning of this article [39]. The glass transition temperature increases with the addition of a modifier that increases the local coordination number () [e.g. based glasses [40] or [20]], and decreases with the addition of a modifier that decreases the local coordination number () [e.g. based glasses [41]].
3 Application and discussion
The equations (16) and (17) can be quite easily compared with experimental data. We compare the slope at the () origin to a set of experimental data among which the value and a measurement for the lowest possible concentration (or ), in order to produce approximate linearity, to be compared with the constant slope of (16) and (17). Therefore, we have tried to find, whenever possible, reported glass transition temperatures of glassforming systems which were composed of a very high fraction of ”regular” local configurations (e.g. species in or atoms in ). The lowest possible concentration is close to in most of the situations, so that we can roughly approach the limit value of formula (16) and (17). From the and values, we can compute an “ experimental value ” of and compare it to the theoretical one, deduced from purely structural considerations. In Reference [42], R. Kerner has demonstrated a good agreement of a close relationship with the behavior of at for the alkaliborate glass and the silicate glass . He predicted in the first case, which leads to a positive derivative, and giving a negative one in the second one. Nevertheless, the formula was incomplete because the factor was missing, but the obtained values were correct since in these investigated systems.
3.1 Singlebonded systems
3.1.1 Chalcogenide network glasses
Various experimental probes for equation (17) can be obtained in very simple simple glass forming systems, namely chalcogenide network glasses, for which numerous experimental data are available in the literature, and . For completeness, we report some values which have already been presented elsewhere [1]. We have extended the list of investigated systems in order to prove beyond any doubt that the impressive agreement of (17) with experimental data is not a matter of coincidence. Within the ranges (close to , whenever possible) found in various references, the formula agrees very well with the experimental data for the sulphur, tellurium and selenium based glasses (Table I and II). Indeed, the computation of the slope of at obtained from the measured glass transition temperatures, leads to experimental values of , which in turn can be compared to the predicted ones, e.g. in glasses or in systems. The structural change induced by the growing proportion of a modifier atom is here obvious. At , the twovalenced atoms () of the group form a network of chains with various length; this is particularly verified in vitreous selenium. For very low concentration, the atoms of modifier (Ge, As, Si,…) produce crosslinking between the chains as suggested by Boolchand and Varshneya [52][53], thus creating a new stable structural unit with coordination number for germanium or silicon atoms (fig. 4), or for arsenic.
For the computation of the rates , we have used standard values, found in the literature, or averaged over a set of reported glass transition temperatures, so for selenium [39], [39] for sulfur and for tellurium [44]. As mentioned above, the sign of the derivative of with respect to depends on the sign of . In all reported chalcogenide glasses, the rate is greater than one, the systems display therefore an increase of the glass transition temperature with increasing , starting from . Different systems are represented in figure 5 and show that the linear approximation of (17) can give a correct estimation of up to in certain glass formers, e.g. , whereas the substantial increase of in systems yields a satisfying description only for .
A certain type of problem arises in the systems for which the glassforming region can not be extended towards (existence of a minimal value of for the glassforming region); it is obvious that our formula can not be applied when glass can not be formed in the limit . Nevertheless, the formula can sometimes be extrapolated down to , when the variation of versus is linear for greater values of . For example, the value shown in Table I for glasses has been obtained by this method, because exhibit linearity over a wide range of and can be extrapolated linearly down to [45]. The constant slope allows then a comparison with (17).
3.1.2 Binary glasses
As pointed out in ref. [42], the slope equation (16) seems to be also verified in binary glassformers. The most common singlebonded systems are the oxide binary glasses which use the typical network formers such as , , or . The structural change which occurs when adding a modifier is well understood in terms of the modified random network concept [54]. It suggest that when alkali oxide is introduced into the glass former, the network is depolymerized through the formation of sites bearing nonbridging oxygens (NBO). Indeed, each molecule of the modifier [] creates in the network two ionic sites, thus converting the covalent or partly covalent network into a sizedecreasing structure. For large amounts of modifier, the structure reduces generally to isolated ionic species [e.g. units, the amorphous analogue of orthosilicates [33]].

based glasses
In silica based glasses, the creation of species with three covalent bridges () is proposed when adding a modifier. Therefore in these systems. Table III show a very good agreement of (16) with the experimental data for various types of modifier.
Since this glass can form continously down to (), it is possible to find measurements for very low concentrations, yielding a better agreement with the predicted value of reported in the table. On can also remark that the dramatic decrease of the glass transition temperature when adding a very few modifier can be mathematically explained by the presence of the factor in the slope equation (fig. 6).
The higher the initial glass transition temperature of the network former, the more pronounced will be the decrease of . Of course, silica based glasses, which has one of the highest value among glass materials, show very well this caracteristic feature. The fabrication of glass by ancient Egyptians is due to this fact. With the heating techniques of that time, it was impossible to form glass from the desert sand, made almost of . Nevertheless, with the addition of to of , obtained from the ashes of burned algae, they could produce it quite easily, because of the sharp decrease of the glass transition temperature.

based glasses
Another system behaves very similarly to the silica based glass, namely systems. This system has been extensively studied because of its unique physical properties, among which the socalled “density anomaly”. Ivanov and Estropiev first reported [58] that the density of these glasses increase with addition of alkali oxide and further studies showed that density, as well as refractive index reach a maximum around added and then decrease [59][61]. Numerous structural studies have been carried out in order to elucidate the reason of this anomaly, among which investigations who infered the presence of octahedra within the network [62] in order to explain the density maximum, although recent EXAFS studies [63] have clearly shown that pressureinduced coordination changes in are reversable and that should not observed at room pressure. Other authors believe that the anomaly should result from an alternative structural reorganization [64]. MicroRaman experiments have been performed and results have been obtained in this sense: the increase of the density with a low amount of modifier can be related to the existence of rings of particular sizes, namely 4 and 3membered rings, the growing proportion of the latter one being responsible of the density anomaly [65].
From the data which are available, we can obtain the rate which is very close to in most of the systems (Table IV). This suggests that for a very small concentration of modifier , the coordination number of the basic tetrahedra changes into , as in the silicate glass (fig. 7a). But in contrast with this latter glass system, the proportion of structures is not increasing any more when is growing, as shown indirectly in fig. 8. Indeed, the glass transition temperature shows a minimum around and then increases. The positive derivative of versus for , means that the local coordination number of the “ altered ” configuration is now greater than the one of the “ regular “ configuration (supposed to be composed of a mixture of and structures), thus confirming the growing presence of octahedra (with ).

based glasses
Other glass systems display the same agreement between the predicted value of and the one obtained from experimental data [68][70]. This is realized in based glasses. The available measurements of systems are represented in figure 9 and compared to the equation should be equal to in these glass systems (fig. 7c). The basic network former is made of phosphor tetrahedra with four bonds, among which one is double bonded, so that it is not connected to the rest of the network, therefore [ structure]. At the beginning, the addition of a modifier produces the usual creation of one , as in silicate and germanate glasses [ structure, ]. The rate of this structure is increasing with the modifier concentration and reaches unity for , yielding the metaphosphate chain structure, made of cornersharing polymeric tetrahedra [71]. On this basis, we predict for the glass. . The rate of

based glasses
All the presented data up to now, exhibit a decrease of with growing modifier concentration , in agreement with the slope equation (16) and the currently accepted rule which states that the increase of the coordination number (i.e. the conectivity of the network) produces an increase of the glass transition temperature. The wellknown symmetrical example of this rule is given by the based glass, which presents a positive derivative of at the origin [40]. The addition of [] transforms the triangles (), which represent the basic SRO structural unit of , into tetrahedra (, N4 species [72], fig. 7b). The linear increase of versus the modifier concentration in these glasses is explained by the conversion of a threevalenced network into a fourvalenced one, thus increasing the connectivity [73]. The nature of the cation seems not to have some influence for low concentration, when [40]. Systems with the same concentration but with a different modifier cation, display still very close data (figure 10). The rate of species is growing linearly with up to whatever the involved cation [38]. For , the glass tranition temperature decreases due to the growing presence of triangles sharing one NBO () [40]. When comparing the rate obtained from experiments with the theoretical one derived from the local structure consideration, one obtains instead of .
Nevertheless, an alternative structural proposal supports the experimental value of . The structure of is indeed a typical example of wellcharacterized mediumrange order. There is a strong experimental evidence for the existence of larger structural groups than the local SRO triangles, namely the boroxol ring , made of three connected triangles [74][77]. The spectroscopic patterns of Raman investigation, and neutron diffraction exhibit for this compound sharp and wellcharacterized peaks, which can be attributed either to the breathing modes of the oxygens inside the boroxol ring (in case of Raman studies, at [78, 79]) or to the bond length inside a boroxol ring (in the case of diffraction [28, 76]). Whatever the proportion of this structural unit in the network former ( is the currently accepted value [74][77]), the coordination number remains . The addition of the modifier leads to the creation of the socalled tetraborate group, even at the very beginning [72]. This structural group is made of several threemembered rings (as the boroxol group) sharing N4 species, with coordination number . This yields a rate of (fig. 7d).
3.2 Relationship with the GibbsDi Marzio equation
We have mentioned at the beginning of this article that Gibbs and Di Marzio have given a formula relating to some structural factor, on the basis of thermodynamical considerations. Varshneya and coworkers have modified this equation in order to test its validity on chalcogenide network glasses [21, 80]. These systems satisfy all the required conditions of Gibbs and Di Marzio’s model, namely the presence of polymeric atomic chains (as chains in ) which can be crosslinked by other atomic species, such as germanium. They have expressed in terms of the network average coordination number , rather than the concentration . is widely used for the description of network glasses since Phillips has introduced this concept in his constraint theory [81]. These authors have redefined for multicomponent chalcogenide glasses the crosslinking density of Gibbs and Di Marzio equation (1) as being equal to the average coordination number of the crosslinked chain less the coordination number of the initial chain, i.e.: , and the GibbsDi Marzio equation can in this situation be rewritten as:
(18) 
where is a system depending constant, whereas it was suggested that the constant of the initial equation (1) is universal [23]. Sreeram et al. fitted the constant to their measurements by leastsquares fit [80] and obtained a value which depends on the considered system and the involved atoms.
The slope equation (17) can be related to the GibbsDi Marzio equation in the pure chalcogen limit and gives, after identification, an analytical expression for .
According to Phillips [81], one can express the average coordination number in terms of the coordination number of the covalently bonded atoms, i.e. the coordination numbers and of the (chalcogen atom) and configuration (modifier atom).
(19) 
The slope at the origin, where (and ) is then:
(20) 
In the vicinity of the pure chalcogen region (), a first order development of the GibbsDi Marzio equation has the following form:
(21) 
which leads by identifying (20) and (21) to an analytical expression of the constant , involving only the coordination number of the modifier atom.
(22) 
The value of can now be computed for different glass systems for which the coordination number of the modifier atoms are wellknown, e.g. for chalcogenide based glasses. The possible values for are (for ), (for ) and (for ). The latter situation corresponds to the glass and the agreement of with the value obtained by a leastsquares fit of the glass transition temperatures data versus , is very good and close to the measurements of Varshneya and coworkers. Other IVVI systems behave very similarly, as seen in Table V. However, expression (22) is valid for binary network glasses only, but we believe that it can be generalized for multicomponent chalcogenide network glasses, involving at least three different types of atoms.
3.3 Other structural contributions
3.3.1 Edgesharing character
One of the possible corrections of the equation (16) can be produced by the influence due to the edgesharing character of the local configurations. Binary chalcogenide glasses are the most representative systems of such a tendency. They form indeed very easily twomembered rings (dimers) and the fraction of dimers can be either very low (such as in [82]) or very high as in based glasses [32]. Indeed, the proposed long range structure in these latter systems is a chain of polymeric edgesharing tetrahedra which are crosslinked by cornersharing tetrahedra [83, 84]. Thus, one can consider that the local glass structure of and is made of pure edgesharing tetrahedra. This result has been given by Tenhover on the basis of spectroscopy [83], but also obtained by Sugai (Raman investigation and modelization) [85], Vashishta and coworkers (molecular dynamics) [86] and Gladden and Elliott (radial distribution function calculation) [29]. We have seen that the statistical factors which appeared at the very beginning of the construction in the expression of the probabilities of doublets (2)(4) are responsible for the presence of the term in (16) and (17). If there is an edgesharing tendency, the number of ways in joining together two singlets will be different and will modify the logarithmic expression.
The number of ways of joining by edges a singlet with coordination number with a singlet with coordination number is in three dimensions and the probabilities can be rewritten in the pure edgesharing situation as:
(23) 
(24) 
where is an energetical correcting factor which takes into account the fact that the energy stored in an edgesharing doublet is not equal to a single bond energy .
(25) 
The construction is performed along the same scheme as equations (5)(LABEL:27). The amorphous singular solution is given by:
(26) 
which exists only if:
(27) 
and the general feature of the phase diagram (fig. 5) remains the same in this situation. The solution can be, as before, examined in the limit condition ( or ), where the pure Anetwork exists and . This yields the energy difference:
(28) 
where is still the glass transition temperature of a pure A configuration glass, but with pure edgesharing local configurations. One should note that the energetical correcting factor does not appear in (26) and in the forthcoming equations. The slope at the origin is then consequently modified, but the derivation of the slope equation remains similar to the one presented above:
(29) 
Unfortunately, there are very few experimental data at our disposal in systems displaying a strong edgesharing tendency, because they seem very difficult to form for very small modifier concentrations [87][89]. This is due to the strong edgesharing tendency which is also responsible for crystallization ease [32].
Nevertheless, some data are represented in figure 11 and they concern and based glasses. As explained above, these systems possess a high amount of dimers in the basic network former and some of the previously cited theoretical and experimental studies propose an approximate fraction of dimers of , in terms of functions [83][85]( is identified with a tetrahedron sharing common edges with its neighbors, hence runs from to ). The currently accepted repartition of the functions for () is: , , . Figure 11 displays the available experimental data about the binary chalcogenide systems. The different straight lines using the slope equation (29) and corresponding to possibilities of structural modification are also plotted. The solid line corresponds to a pure cornersharing situation (with and ), whereas the dotted and dashed lines represent a pure edgesharing situation slope equation with respectively and , and and . Although precise information about the glass transition temperature is lacking in the very low modification regime (low concentration) [87][89], one can observe that a conversion (solid line in fig. 11) using pure cornersharing tetrahedra [slope equation (16) with and ] seems not adapted for the description of at the origin . The two other possibilities seem more accurate and also support what is proposed on the basis of spectroscopy by Eckert [90] and Martin [91]. In glasses, the addition of lithium sulphide produces the conversion of into species (a structure with two nonbridging sulfur atoms, i.e. a tetrahedron ), thus producing a growing rate of the phases, identified with edgesharing dimers (the HT form of [90]) and cornersharing polymers (the LT form of [90]). At , the edgesharing tendency remains important, the ratio of the HT and LT phases is [90]. No characteristic signature of a lithium dithiosilicate phase is observed on the spectroscopy patterns (the crystalline phase), even at a concentration where this compound should be expected (at , if one refers to the oxide analogous glass [31]). The same happens for the selenide glass. The structural modification imposed by the presence of is similar to the sulphide system. The basic tetrahedra are converted into tetrahedra and a phase occurs, made of almost cornersharing tetrahedra [90].
For these two systems, the slope equation (29) with and seems best adapted and figure 11 confirms moreless this structural scenario.
In contrast with the lithium glass, it is possible to observe a spectroscopic signature of a phase in the glasses, confirming the presence of units (a tetrahedron), as suggested by Pradel [92]. On this basis, a reasonable structural conversion is for very low concentration (dashed line in fig. 11). The rate of edgesharing structures is not decreasing when is growing and it is still equal to at [89]. Therefore, one should propose for these binary systems the pure edgesharing slope equation with and (shaded line in fig. 11), which seems to agree with the experimental data of systems. However, a glass transition measurement for the concentration , or lower, is missing, but it should certainly be useful in order to give information about the local structural modification. On the basis of what has been described above, we propose for , in the sodium sulfide glass and in the lithium sulfide one.
In other chalcogenide glasses, the rate of edgesharing structures is much lower. Typical glasses displaying a nonnegligible edgesharing tendency are the [93], [94] or () [95] binary glasses, for which numerous experimental measured glass transition temperatures are also available. In the case of a mixture of corner and edgesharing structures, one must use the doublet probability:
(30) 
where , , and are the probabilities of doublets which have been defined above. The slope equation is obtained as previously:
(31) 
where uses the energetical correcting factor , if one assumes that the rate of corner and edgesharing structures remains roughly constant in the lowmodified régime. can be related to the rate of edgesharing doublets:
(32) 
Figure 12 shows a typical example of such intermediate systems and gives information about the rate of edgesharing structures (dimers) in the chalcogenide binary glasses . In the based glass, the line using the slope equation (31) with has best agreement with experimental data. The rate of dimers should therefore be about in this glass, according to figure 12. As before, we believe that a measurement for should give more precise information and improve the estimation of .
3.3.2 Influence of BB bonds. The second slope equations
In the first consideration of section 2, the model has been constructed only with and doublet agglomeration. It corresponds to the most common situations where only a very few “ altered ” configurations should be expected when the starting structure made of almost “ regular ” configurations is slightly modified (low concentration). Therefore, no bonds were considered. The presence of these bonds at the very beginning of the modification (i.e. the tendency of a system to create such bonds, even when there are very few configurations) can of course change substantially the thermal behavior of the glass (and the final ) and modify the slope equations. The construction presented in the previous section corresponded to a situation where a defect (the configuration created by a modifier) was diluted insided the whole structure, thus leading to and doublets only. This seems adapted for the description of binary oxide glasses or network glasses. If the oxide ion is replaced by a larger and more polarizable ion, such as the sulfide or the selenide ion, the local environment of a configuration, composed of electronrich ions (,) and modifier cations () may favour the occurence of local bonds.
With the notations introduced in section 2, the probability of finding a pure singlebonded doublet is:
(33) 
where is the bond energy and the the new normalizing factor:
(34) 
In the pure edgesharing situation, the probability has the following expression:
(35) 
The stationary solution of equation (6) is changed and depends on the energy . Two energetical differences are now involved: and .
(36) 