The ability to infer the mechanical properties of glass from its structural composition is important in the glass industry, where accurate prediction of properties such as viscosity, hardness, and chemical durability have obvious benefits . So far, the main method for deducing the optimal composition to maximize certain mechanical properties has been empirical [1]. For example, in 1915, scientists used empirical analysis to discover a very durable, heat and shock resistant glass known commonly today as Pyrex®[18]. While this method does arrive at the answer, a theoretical understanding of the connection between a glass’s composition and its mechanical properties is of more interest in modern glass science.

One direct approach for explaining a glass behavior based on its structure is Molecular Dynamics. In short, this process examines the Coulomb interactions between networks of atoms. Since there are no other types of interactions between atoms, examining the Coulomb forces is certain to work. However, the computation forces due to atomic bonding requires a tedious numerical integration with a time step of 10-15 s [2]. These computational limits preclude the analysis of sufficiently large networks.

In order to get more useful methods of analysis, some simplification needs to be made. Topological Constraint Theory [4,6,14] does not calculate or examine any atomic forces, rather, it examines the mere existence or absence of a bond. This eliminates the need for a tedious strength

calculation. Constraint Theory makes a further simplification by discarding any extraneous forces and only calculating the main influences on atomic motion, called constraints. This idea of considering constraints was first suggested by Maxwell [5] and was later developed into a breakthrough theory by Phillips in 1979 [4]. These atomic constraints fall into the categories: α Constrains and β Constraints (Fig 1.) α constraints describe forces associated with bond stretching, while β constraints describe forces associated with bond bending.

Depending on the number of constraints each atom experiences, the nature of its motion can be deduced. If an atom has two few constraints, it is said to be floppy, or under-constrained and if has atom has too many, it is said to be rigid or over-constrained. Extra constraints in a rigid system are called redundant bonds. There is also a critical number of constraints where the atom is neither over-constrained nor under-constrained. Such an atom is said to be isostatically constrained [6] (Fig 2.). The number of constraint determines the nature of the topological connection. Since we want information about the network, not just one atom, we need to figure out the number constraints of some average particle in the network. We define the mean coordination number with this in mind.

m=x N cn ( A )+ y N cn (B )+(1− y−x) N cn(C )

This is a simple weighted average, defined here for a 3 component system with N cn denoting the coordination number (number of neighboring atoms) and x and y denoting the molar percentage of A and B, respectively.

β

α

Fig 1. Types of Intermolecular Constraints

Floppy

Next we must discover how many constraints are affecting this average particle. The number of atomic constraints in the system can be expressed as

N co=m2

+2m−3

First term: Since each bond connects two atoms, each bond adds two α constraints to the system, here we consider one half constrain per α constraint because we are only focused on the constraints of one average particle, not two. Second term: This term counts angular constraints. When an atom has a coordination number of two, there in one β constraint, and each addition atom bonded adds two β constraints [4].

Since rigid bonds add the network rigidity, and floppy modes do not, it is useful to express to fraction of floppy modes as a function of the p, the probability that there will exist a bond between two nodes. The initial approximation made by Maxwell [5] is

f =dN−¿¿

where d is the number of dimensions, N is the total number of atoms in the network, z is the average coordination number, p*=2d/z. This equation defines the useful value nr, the number of redundant bonds per degree of freedom. dN basicalls gives you to total number of bonds, and the ¿ indicated how many of them form constraints, hence the difference between these two values gives you the number of non-constraining bonds. The left-hand simplification shows that f approaches 0 linearly as p is increased, further accuracy can be achieved by using the series expansion

f =3(1−p)5−2 (1−p )6+O((1−p )8)

which also approaches zero as p increases, but this time the approach is asymptotic. This approximation closely fits computer simulations of percolation thresholds [8].

The nature of the floppy and rigid constraints has a profound impact on the behavior of the network. For now we will relate this behavior to its glass forming ability. Although glasses are not the most favorable thermodynamic state, they do represent local minima in free energy (Fig 3.). That is to

say, if a glass is to become a crystal, it must reconfigure its topology into an ordered network, and that takes some amount of activation energy. If the network is mostly floppy, it takes almost no energy to reconfigure the network, and the network can easily overcome the low energy barrier to form glass. When a network is over-constrained, the redundant bonds serve only to add internal stresses to the system [8]. All these stresses provide enough energy to overcome the activation barrier and, again, the network becomes a crystal. This processes is called rigidity percolation [14]. The two extreme ends of Fig. 4 represent regions where glass is not easily formed. So as N co is varied from one extreme to another, if passes a critical point. This point

Gibbs Free Energy

Amount of Atomic Order

Fig 3. Energy Barrier to Crystallization

Energy Barrier

Crystal

Glass

represents the topological prediction of a network with ideal glass-forming ability, as introduced by Phillips [4]. Phillips reasoned that ideally constrained system has enough constraints to prevent motion but not too many as to stress the network rigid. Therefore the optimal glass forms whenN co=d , the number of dimensions. Setting d = 3, we obtain ideal glass behavior at m =2.45. As the Topological Theory was developed further [1, 9], ideal glass-forming ability was discovered to occur in a critical range, not at a single point. Glasses in this region are said to be in a Boolchand Intermediate Phase, named after its discoverer [10]. Glasses in the Boolchand Phase have interesting properties, such as extremely low minimum cooling rates (easily forms glass) and no internal stresses (no extra stress from redundant bonds), which are in accordance with the predictions made by Topological Theory.

The average number of constraints per atom, N co, does not only apply to glass-forming ability. Up until now, we have only considered network at absolute zero, since thermal energy has not yet entered the discussion. Mauro and Gupta [2,3] worked to expand topological theory to account for the current temperature of the network. They reasoned that, for a system of N atoms and d dimensions, there exists many local minima, where local energy considerations are minimized, each representing a mechanically stable configuration. In theory, atoms arrive at these local minima, guided by the gradient of the Nd-dimensional hyperspace. This descent guided by the slope of the surface is termed basin, and there exists only one basin for every stable mechanical configuration. The hyperspace is independent of temperature, but the way in which the network navigates this space does depend on temperature. At high temperatures, the system can flow freely among basins, but as the system is cooled down, inter-basin travel is reduced, and the point at which there is no inter-basin travel is dubbed the glass transition temperature. This reasoning was the basis for the temperature considerations of topological theory. We introduce q(T) the network rigidity, and a function of temperature.

q (T )=[1−e−∆F

kT ]vt obs

∆ F is the activation energy needed for breaking a constraint, k is Boltzmann’s constant, v is the vibrational attempt frequency, and t obs is the timescale of the observation. This function is reminiscent

of the classic Fermi energy distribution function, where q is rigid at T =0 (limT → 0

q (T )=1) and the network

becomes floppy as it gets hotter ( limT → ∞

q (T )=0). Since vtobs is generally large, q(T) also has the discreet

form

q (T )=θ (T q−T )={1 ,T <T q

0 ,T >T q

where Tq is the temperature of rigidity percolation. Both forms of q(T) are used, depending on the needs of the situation.

When the thermal energy in the system is accounted for, topological theory can be used to predict the viscosity and glass transition temperature of glass networks [1-3]. Also, knowing N co also tells you how many bonds are present in the system, which correlates directly to Shear and Elastic Moduli [2].

As Applied to Specific Networks

Now we shall apply the principles of Topological Constrain Theory to several specific glass networks. A topological analysis begins with the identification of the main configurational units in the network, called network-forming species. One the units are identified, knowledge of the network behavior much be applied. This knowledge either comes from direct reasoning in simple cases, such alkali/alkali earth silicates[16], or NMR data in more complex systems. Borate classes are complex and their behavior is still under debate[12], but some models such as the Yun-Bray model for binary and ternany borosilicate glasses have been developed with some success [17]. After the proper fractions of network-forming species can be mathematically deduces as a function of x (the change in the molar composition), constraint counting is applied and can be fit to experimental parameters with good agreement [11-13].

Alkali Silicates

We begin with an analysis of binary alkali silicate glasses of the composition

x R2O ∙ (100−x ) SiO2

For x = 0, pure vitreous silica, the structure consists exclusively of bridging-oxygen-containing SiO4

tetrahedra which lack long range order [11-13]. Each oxygen is then considered a ‘bridging-oxygen’ (BO) linking two Si atoms. Note: Silica tetrahedral occur in pairs, disorder is introduces into the network by a random bond angle between the tetrahedra. As network modifiers are introduced into the system, four coordinated Si (Si4) become three coordinate (Si3). To molar amount of oxygen is used to deduce to ratio of Si4/Si3 in the following way[16]. First we assume that only Si4 or Si3 units occur in the network. For relatively low values of x, that is a safe assumption [17]. Next, count the number of oxygen atoms in the network, one for each unit of R2O and two for each unit of SiO2.

N (O )=x (1 )+(100−x)(2)

The number of Si3 and Si4 groups sum to the total amount of Si,

Si4+Si3=100−x

We can also deduce the number of oxygen atoms from the connectivity of each Si unit

N (O )=2Si4+2.5Si3

With NBO counting as a full oxygen and a BO counting as half an oxygen atom because it will ultimately be counted twice. Rearranging the above equation leads to

Si3=2x

Si4=100−3 x

This implies that each addition of an R2O group creates two Si3 groups and takes away three Si4 groups.

Alkali Borates and Alkali Borosilicates

For borates there is a similar compositional formula.

x R2O ∙ (100−x ) B2O3

As mentioned earlier, these systems of glasses are still under study and debate [12,17]. The Yun-Bray model [17] and random pair model [15] seem to be conventional choices for a mathematical production of network-forming-species ratios. The random pair model is used here to demonstrate a theoretical binary borate glass calculation, and relies of the knowledge of the possible network forming species of borate systems, which are given in Fig. 6. Table 1 shows the Yun-Bray Prediction [17], which can be found experimentally using NMR analysis and deviates minimally with experimentation for extreme values of K [12] (refer to Fig.7).

Table 1. Yun-Bray Prediction of Four Coordinated Boron (N4)

K≤8, Rmax = K/16 + .5 K≥8, Rmax = 1

R ≤ Rmax B4 = R (or .95R) B4 = R (or .95R)

R ≥ Rmax B4 = Rmax – 0.25(R – Rmax)/(1+K) B4 = 1

The random pair model uses similar reasoning as before to derive to following concentrations of two, three and four coordinated Boron units. There are fewer explanations because the process is similar for SiO2.

N (O )=2B4+ 32

B3=x+3(100−x)

B3+B4=2 (100−x )

Fig 7. Ternary Diagram for Na2O-B2O3-SiO2 system. R ≡ Na/B and K ≡ Si/B.

K

SiO2

Na2O

B2O3

R

Fig 6. The Configurational Units of a Borate Glass. Clockwise from top, two B4 tetrahedra (this is the only instance of B4s

[15]), Boroxol Ring, B3 Unit.

Therefore

N ( B4 )=2x

N ( B3 )=200−2 x

And it follows that the number of bridging oxygens must be

N (OB )=300−2x

And so the total is given by

N (OB+B4+B3)=5−4 x

Calculating the fractions of each network-forming species follows. The boron anomaly must be part of our theoretical considerations [3,16]. As more network modifiers are added to a borate network, the coordination initially increases, due to B3 to B4 modifications, but at a certain point (x=.33), the network runs out of available space to host B4 units, this occurs when there are the same number of B3 units as there are B4 units. This is due to random pair model [15] , which states that the only instance of a B4 units is attached at the corner to only one other B4 unit. This implies that for every double B4 unit there are 6 other bonds made with non B4 units, or for every B4 unit, there is one B3. Note that it is possible to have concentrations of B4 higher than .5, as shown by the Yun-Bray model. After the x=.33 point, the further addition of network modifying cations transform B3 species into B2 which is a three-coordinated boron atom with two BOs and one NBO. This is again due to random pair model theory, which states that NBO can only occur on the three coordinated Boron atoms. The existence on NBOs must be considered for x ≥.33. This leads to the follows piecewise equations crossing the x = .33 critical value (the derivations are left as exercises for the reader).

f ( B4 )={ 2 x5−4 x

,∧x≤13

6−8 x31−38 x

,13

≤∧x≤12

f ( B3 )={ 4−2 x5−4 x

,∧x ≤13

10−20x31−38x

,13

≤∧x≤12

f ( B2 )={ 0 ,∧x ≤13

18 x−631−38 x

,13

≤∧x ≤12

f (OB )={ 3−2 x5−4 x

,∧x≤13

21−28 x31−38 x

,13

≤∧x ≤12

For concentrations above x = 0.5, it would be appropriate to use silica network topological calculations. Fractional network-forming species of other glass systems (ternary or quaternary) follow similar methods, and would perhaps be the subject of future reports.

Topological Prediction of Mechanical Properties

Now that the fractions of each network forming species are known as a function of x, or x and y as in the Yun-Bray model. We first employ constrain counting to find the average number of constraints for a given network. We will start by counting the number of α constraints. To do this we recognize that each oxygen is necessarily bonded to two non-oxygen atoms, so counting the α constraints on the oxygens alone will account for every linear constraint. The number of α constraints is not affected by the BO/NBO properties of the oxygen. Angular constraints are counted using B2, B3 and B 4 units, having one, three and five constraints, respectively. Hence the average number of constraints is

n ( x )=2 (3−2 x )+f (B2)+3 f ( B3 )+5 f (B4)

Now linearization of some arbitrary property A can be used to predict the compositional dependence property A.

Linearization of A:

A ( x )=dAdn

(n ( x )−ncrit )

Where dAdn

is determined empirically and ncrit is chosen to be in the Boolchand intermediate phase,

usually around n = 2.4. These methods are verified experimentally by Smedskjaer [11-13].

References

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