1 Introduction

Many criteria (dealing with structural and thermo-kinetic orders) were proposed to explain the oxides glass-forming ability (GFA). They are founded on crystallochemical concepts or refer to the criteria linked to the nature, the chemical bond strength and other characteristics of metal oxides (Goldschmidt 1926; Zachariasen 1932; Dietzel 1941; Stanworth 1946, 1948, 1952; Sun 1947; Smekal 1951; Rawson  1956, 1967; Turnbull 1969; Hrubý and Štourač 1974; Minaev 1978, 1980, 1983, 1991; Štěpánek and Hrubý 1980; Červinka and Hrubý 1982; Inoue et al 1990; Lu et al 2000; Stoch 2001; Lu and Liu 2003; Rao et al 2004). The main purpose is to predict and/or describe how any single oxide (and under what conditions) is able to form glasses. The structural theory of GFA was first proposed by Goldschmidt (1926) and completed by Zachariasen (1932) who studied all the oxides GFA theories and published his crystallochemical theory. This theory placed the understanding of glass structure and its relationship to composition on a chemical basis. Zachariasen (1932) claimed four rules for structure that allow oxides that tend to form glasses: (i) an oxygen atom is linked to not more than two glass-forming atoms, (ii) the coordination number of the glass-forming atoms is small, (iii) the oxygen polyhedra share corners with each other, not edges or faces and (iv) the polyhedra are linked in a three-dimensional network. From these considerations, following oxides are known to be glass formers: B2O3, SiO2, GeO2, P2O5, As2O3, Sb2O5, Sb2O3, In2O3, Tl2O3, SnO2, PbO2, P2O3, As2O5, Nb2O5 and Ta2O5 while TeO2, SeO2, MoO3, WO3, Bi2O3, Al2O3 and V2O5 are listed to be ‘conditionally’ glass formers. All other oxides are classified as modifiers (alkali and earth-alkali oxides) or intermediate (transition metal oxides).

Inspired by the atomic field of Dietzel (1941), Stanworth (1946, 1948, 1952) and Smekal (1951) added details of electronegativity and bonding nature between E atoms and oxygen atoms in E x O y oxide. Thermodynamical theory of glass formation was proposed as the ‘energy criterion’ by Sun (1947) who attempted to establish a correlation between the bond strengths (between the cation Ey +  and the anion O2 − ) and this aptitude. He proposed his own criterion, the ratio \(E_{\rm B}^{\rm Sun} = E_{\rm d} /I\) (kcal·mole − 1), where E d is the oxide dissociation energy and I the oxide crystal coordination number. Sun’s basic idea is that, when a melt is quenched to form a glass, the stronger the M–O bonds, the more difficult are the structural rearrangements necessary for crystallization and hence, the easier is glass formation. Stoch (2001), correlated this flexibility of structure with GFA in his chemical approach, assessing that the flexibility of the internal structure of solids stems from the free motion of their structural units without discontinuity of the structure as a whole. Lu et al (2000) reviewed some simple GFA parameters based on characteristic temperatures and other physical properties of oxides, oxide melts and bulk metallic glasses (BMGs). Among them, the reduced glass transition temperature, T rg (ratio of the glass transition temperature, T g and the liquidus temperature, T l), proposed by Turnbull (1969) with an assumption that the nucleation frequency and crystal growth of melt scales are inversely proportional to viscosity of the liquid. T rg indicator was developed for only monoatomic systems. Later on, Lu and Liu (2003) confirmed that T rg computed by T l/T g shows a better correlation with GFA than that given by T g/T m for different multicomponent alloy systems (T m is the melting point). Inoue et al (1990) proposed a new GFA indicator, called the supercooled liquid region, ΔT xg (the temperature difference between the onset crystallization temperature, T x and the glass transition temperature, T g) based on the considerations of supercooled liquid stability against crystallization. Both ΔT xg and T g/T l ratios are commonly used as GFA indicators for BMGs. Hrubý and coworkers (Hrubý and Štourač 1974; Štěpánek and Hrubý 1980; Červinka and Hrubý 1982) used DTA data analysis technique to value GFA of materials (that was for the approaching chalcogenide glasses and BMGs) in Ge–Si–S system. Sun criterion was enriched by Rawson (1956, 1967) and finally described as the ‘kinetic theory of glass formation’ by Turnbull (1969). Studying the oxides GFA, Rawson introduced the oxide melting temperature (T m) as a necessary parameter to link with its bond strength. He proposed the ratio \(E_{\rm B}^{\rm Rawson} ={E_{\rm B}^{\rm Sun}}\mathord{\left/{\vphantom {{E_{\rm B}^{\rm Sun}}{T_{\rm m} }}} \right. \kern-\nulldelimiterspace} {T_{\rm m} }={E_{\rm d} } \mathord{\left/{\vphantom {{E_{\rm d} } {\left( {I\cdot T_{\rm m} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {I\cdot T_{\rm m} } \right)}\) (kcal·mole − 1·K − 1) that is considered to be the thermal bond strength. Its value is high for all oxide formers. The lower the value, the harder the amorphization will be. Minaev (1978, 1980, 1983, 1991) partially agreed with Rawson and suggested a theoretical model to compute the oxide bond strength.

2 A relative approach to GFA

Focusing on single oxides GFA, literature analysis showed that all the aforementioned criteria appeared to be useful for the general understanding of GFA but (except Rao et al (2004)) have found no use for direct comparative estimation of this phenomenon. The non-dimensional method for experimental data analysis permitted Rao et al (2004) to study the glass formation tendency (GFT) of a wide range of materials. Author proposed the criterion G, linked with bonding (bond energy and ionicity), structure (constraints and size asymmetry) and viscosity (configurational entropy) factors and assessed ‘...G is a product function for simple materials and is intended to be like a figure of merit’ (Rao et al 2004). Rao ranked the most used oxides as following: B2O3, SiO2, P2O5, GeO2, MoO3, TeO2, V2O5, WO3, PbO2, SnO, PbO, Al2O3, AS2O3, Li2O, MgO, Na2O and CaO.

From a unique thermodynamical point of view, Sun criterion, \(E_{\rm B}^{\rm Sun} ={E_{\rm d} } \mathord{\left/{\vphantom {{E_{\rm d} } I}} \right. \kern-\nulldelimiterspace} I\), is unable to sort oxides as formers on the one hand and intermediate and modifiers on the other. Indeed, the obtained values are \(E_{\rm B}^{\rm Sun}>79\,\textrm{kcal}\cdot\textrm{mole}^{-1}\), (60–79) kcal·mole − 1 and \(E_{\rm B}^{\rm Sun} <63\textrm{~kcal}\cdot\textrm{mole}^{-1}\) for glass former, intermediate and modifier oxides, respectively. The obtained values for this criterion are ambiguous because they do not lead to the three oxide groups (sorted by their own distinctive interval values). Al2O3 is a real intermediate oxide that may be a former one, when considered with a coordination number I = 4 \(({79<E_{\rm B}^{\rm Sun} <101\,\textrm{~kcal}\cdot\textrm{mole}^{-1}})\), while being a modifier, if I = 5 \(({63<E_{\rm B}^{\rm Sun} <73\,\textrm{~kcal}\cdot\textrm{mole}^{-1}})\) or intermediate, if I = 6 \(( {53<E_{\rm B}^{\rm Sun} <67\,\textrm{~kcal}\cdot\textrm{mole}^{-1}})\). That is why it never forms glass by itself but contributes to any form of melting of other oxide formers. Rawson criterion (F b = F L/T m) is more suitable for the definitive sorting between the former and the other (modifiers and intermediate) oxides. A real oxide former as SiO2 (F L = 106 kcal·mole − 1·K − 1) is definitively separated from an intermediate one such as zirconium dioxide (F L = 81 kcal·mole − 1·K − 1). Because of its higher melting point, ZrO2 possesses a low Rawson criterion value and consequently to Rawson’s considerations cannot form glasses. In contrast, Rawson criterion value of B2O3 is very high and thus, this oxide never crystallizes.

Minaev confirmed this criterion and enriched the model to compute the bond strength of oxides and their melts. Moreover, researchers have often faced contradictions. Indeed, according to Stanworth (1946) electronegativity criterion, Si, Ge, Sn and Sb possess an electronegativity of 1·8. However, oxides of the first two elements are good glass formers, while Sb2O3 forms glasses only at very high cooling rates and SnO2 never form glasses (Červinka and Hrubý 1982).

In this paper, using IUPAC and NIST data (Pauling 1970; Burdett 1995; Lide 2009) and obeying to the main rules of the mathematical non-dimensional analysis, we suggest to take into account of an omitted thermodynamical characteristic: the isobaric heat capacity (C p) valued at the melting temperature (it is defined to be an extended value that allows measuring the faculty of all chemical components to absorb/restore the energy during all transformations with a temperature variation). This was the objective of Rawson (1956, 1967) who confounded between temperature and heat. At the melting temperature, the available heat depends on C p according to the following: \(C_{\rm p}^{{\rm E}_{x} {\rm O}_{y} } =a+b\cdot 10^{-3}\cdot T_{\rm m} \) (kcal·mole − 1·K − 1), where a and b are constants (Pauling 1970). We propose to add C p computed at T m temperature to Rawson’s formula denominator, which makes it non-dimensional. The mathematical non-dimensional analysis permits to describe any physical phenomenon with a single non-dimensional equation (with no measurement units, as atoms of electronegativity). It gives the number of non-dimensional equations E as the difference P − U, where P is the number of the studied properties and U, the number of measurement units (S.I.). In our considerations, GFA is the studied phenomenon, while P = 4 (E d; I; T m; C p) and U = 3 (cal; mole; K). Thus E = P − U = 4 – 3 = 1; that means that oxides GFA can be described using one and only one equation (that is consequently a non-dimensional one). Considering any oxide E x O y , its thermodynamical relative GFA will be:

$$ \textrm{ThRGFA}^{\textrm{E}_x \textrm{O}_y }=E_\textrm{B}^{\textrm{Rawson}} /C_\textrm{p} =E_\textrm{B}^{\textrm{Sun}} /\left( {T_\textrm{m}\cdot C_\textrm{p} } \right). $$

3 Results and discussion

Table 1 reports the needed physicochemical data of selected oxides. Sun and Rawson GFA criteria values are given and oxides are sorted according to the decrease of their corresponding computed ThRGFA (see also in figure 1). The new model allows the ranking of oxides in three groups (formers, intermediates and modifiers) according to their specific values: ThRGFA ≥ 1·709 × 103, 1·569 × 103 ≥ ThRGFA ≥ 0·812 × 103 and ThRGFA ≤ 0·743 × 103, respectively. In the first (oxide former) group, B2O3 reappears while being absent in this group according to Sun and/or Rawson criteria. According to the aforementioned classification, ZrO2 and Al2O3 were inadequately supposed to be former oxides regarding their high T m. The introduction of C p (as unavoidable characteristic) in the new model permits to reclass it in the adequate (intermediate) group of oxides according to their respective (1·15 and 1·14) ThRGFA values. Thus, ambiguities of Sun and Rawson classification are definitively raised. The confusion regarding B2O3 and (Al2O3, ZrO2) classification is resolved: the first one is a real former oxide while alumina and zircon are intermediates. Moreover, the intermediate value of oxides ThRGFA [6·51–1·70] × 103 is in accordance with Angell’s idea that sorted glass former oxides as strong and fragile ones (Bohmer et al 1993; Angell 2008, 2010). The obtained results are in agreement with the mixed approach of Rao et al (2004) observation that brillantly ranked the most used oxides as following: B2O3, SiO2, P2O5, GeO2, MoO3, TeO2, V2O5, WO3, PbO2, SnO, PbO, Al2O3, AS2O3, Li2O, MgO, Na2O and CaO (few differences are because of considering only thermodynamic parameters).

Table 1 Parameters and GFA criteria values of some oxides.
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Figure 1
figure 1

Computed ThRGFA values of some oxides.

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4 Conclusions

According to Sun and Rawson, the thermodynamical aspects of oxides GFA led to dimensional values and were linked with the bond strength and associated to their melting points. None of these criteria completely sorted the oxides in the consensually approved three groups; indeed, alumina and zircon were incorrectly listed as former oxides. All the computed non-dimensional ThRGFA values reflect a real ability; such a word is devoted to non-dimensional characterization (as elements electronegativity does). Using the latest thermodynamical data and methods to compute them, ThRGFA model definitively agreed between theoretical and technological considerations. All prior GFA criteria were appropriate, but the misunderstandings that appeared were related to the chosen data (all omitted the isobaric heat capacity), their values and mathematical analysis.