Calculation for density of molten slags using optical basicity

Abstract

A model for density calculation of multicomponent slags was developed in the present work. This model is an extension of a previous model proposed by the present author. Density values of molten slags were calculated from molar volume values, which were calculated from molar volume of pure component and mixing molar volume. Mixing molar volume is correlated to differences between reciprocals of optical basicities of network forming oxides and those of network modifying oxides. Charge compensation for AIO₄ ⁵⁻ was taken into consideration explicitly. An increase in molar volume due to Al coordination shift from sixfold to fourfold was modelled by defining molar volume of a hypothetical state of alumina in fourfold coordination in the present work. Density values of M₂O–SiO₂ (M = Na, K), CaO–Na₂O–SiO₂, MgO–Na₂O–SiO₂, Al₂O₃–CaO–SiO₂, Al₂O₃–Na₂O–SiO₂ and Al₂O₃–CaO–MgO–SiO₂ system were calculated by employing the present model. Comparisons between calculated and experimental values in literature were also made. The calculated values are in good agreement with experimental data with a mean deviation within 2%. The present work also showed that the addition of network modifying oxide with high optical basicity values would lead to a remarkable negative deviation of molar volume from ideal mixing, which could be interpreted by considering larger Si–O shortening caused by metal cations with lower electronegativity. In addition, the positive deviation for metal oxide and alumina join was also well explained in terms of molar volume increase due to Al coordination change in the present work.

Introduction

As one of the important properties of metallurgical molten slag, the density of the slag has many effects on metallurgical processes, e.g. metal–slag separation, foaming, etc. Accurate density values of molten slag are required in momentum, heat and mass transfer simulations for metallurgical processes. Knowledge of slag density is also essential to the understanding of slag structure, due to its sensitivity to structural changes. Despite the importance from a theoretical and practical aspect, experimental data for slag density are still very scarce due to many difficulties at high temperature measurements.

Models for density calculation of slag could be very useful tools to provide density values for users. Many attempts have been made to model the density of molten slag through years.^1–7 Molar volume, the reciprocal of density, is a fundamental thermochemical property and often employed to construct models for density calculation. A most quoted additive method proposed by Bottinga and Weil¹ calculates density from molar volume by employing partial molar volume of different components. In their first paper, Bottinger and Weil considered an ideal mixing of different components and used constant partial molar volumes for one component. Later, they considered further partial molar volumes of alumina–silica liquids to be composition dependent in their modified model.² A model in which partial molar volumes of Al₂O₃ and SiO₂ are polynomial functions of composition was proposed by Mills and Keen³ for density calculations of multicomponent slags. Although certain success in providing calculated data within 2% had been achieved, these models are only based on mathematical regression analysis, and structural information of slags was not taken into consideration explicitly.

The deviation from ideal mixing for molar volume has been realised by many researchers and excess or mixing molar volume has been employed to model density of silicate melts. A non-ideal model (SC) for CaO–Al₂O₃–SiO₂, MgO–Al₂O₃–SiO₂ and CaO–MgO–Al₂O₃–SiO₂ melts involving an excess volume term between SiO₂ and CaO was proposed by Courtial and Dingwell^4,5 and led the calculation of the liquid’s molar volume within 0·5% of uncertainties, except for Ca and Al rich compositions. Persson et al. ⁶ developed a model to calculated density based on correlation between relative integral molar enthalpy of mixing and mixing molar volume. The comparison between calculated and measured values showed good agreement. A method to calculate density of slag from mixing molar volumes, which is calculated using optical basicity through combining Toop–Samis thermodynamic model⁷ with the method by Ottonello et al.,⁸ was proposed by the present author.⁹ The densities of some binary MO–SiO₂ (MO is divalent metal oxide) and some ternary silicate slags MO–M^’O–SiO₂ have been calculated using this approach to get a good agreement with experimental data. However, this model was limited in calculation of narrow kinds of silicate slags within CaO–FeO–MgO–MnO–SiO₂ system and cannot be applied to calculate density of aluminosilicate slags.

An extended model to calculate density of slags containing alkali oxide and/or alumina will be presented in this work. A relationship between deviation from ideal mixing for molar volume and differences between reciprocals of optical basicity of network forming oxides and those of network modifying oxides will be established. Charge compensation of and coordination change of aluminium is to be taken into account. Density values of multicomponent slag are to be calculated using the present model and compared with experimental density values in literature. The deviation from ideal mixing for molar volume of molten slag will be discussed on account of structure information in detail.

Model description

Density values of slags ρ were directly related to molar volumes according to following equation (1) where V _m, x _i and M _i represents molar volume, mole fraction and mole mass of component i, respectively.

Molar volume of slag can be calculated from molar volume of pure components and molar volume of mixing (2) where V _i are molar volume of component i. V _mix is the molar volume of mixing.

In a previous work,⁹ a correlation between mixing molar volume with integral Gibbs free energy of mixing ΔG _mix were adopted as follows (3) Various λ values for binary systems were optimised in previous work. However, it was found that λ values are only slight varied for different binary systems. Accordingly, a constant value of λ = 0·025 is adopted in the present work.

A thermodynamic model proposed by Toop and Samis⁷ was employed to calculate values of ΔG _mix. It was assumed in the Toop–Samis model that three types of oxygen exist in silicate melts,¹⁰ bridge oxygen O⁰, non-bridge oxygen O⁻ and free oxygen O⁻² and the three oxygens are in a chemical equilibrium as follows The equilibrium constant K ₁ is (4) , and are the number of moles of bridge oxygen, non-bridge oxygen and free oxygen per mole of silicate melts. K ₁ is constant at a given temperature and characteristic of the cations present in any binary or ternary silicate melts. There are relationships among , , as follows (5) (6) The equations (4)–(6) can be solved to obtain the molar number of three types of oxygen. Then we get value of as follows (7) where is the molar fraction of silica.

Ottonello et al. ⁸ correlate the equilibrium constant K ₁ to optical basicity Λ¹¹ and obtain the following equation (8) where subscripts and represent network modifying and network forming cations, respectively. and are molar fractions of in all network modifying cations and molar fractions of in all network forming cations. and are optical basicities of and respectively. Covalent oxides as SiO₂, P₂O₅ and GeO₂ are common network formers in tetrahedral coordination. Metal oxide such as CaO, MgO, FeO, MnO, Na₂O and K₂O are considered as network modifiers.

Integral Gibbs free energy of mixing for binary silicate is calculated by the following equation (9) Combining equations (3), (7) and (9), we obtain the following equation. (10) Since the equilibrium constant K ₁ directly correlates to optical basicity according to equation (8), the above equation provides a way to calculate molar volume of mixing from optical basicity.

The optical basicity values¹¹ adopted in the present work were shown in Table 1. Optical basicity values for various oxides are well accepted, except values for some transition metal oxides. Direct measurement of optical basicity using UV spectroscopy cannot be performed on transition metal oxide due to strong UV absorption of these oxides. Optical basicity values can be calculated using Pauling electronegativities.¹² Some researchers also derived optical basicities of transition metal oxide according to sulphide capacities, and obtained quite different optical basicity values from those calculated by electronegativities. In the present work, optical basicity values calculated from Pauling electronegativities were adopted. It was found that these optical basicity values would provide a good fitting to molar volume deviation from ideal mixing for binary silicate.⁹ Especially for slag systems containing transition metal oxide, theoretical optical basicity values from electronegativities could provide a much better measurement on deviation from ideal mixing than values from sulphide capacity. For example, theoretical values for FeO from electronegativities is 0·51, then according to equations (8)–(10), mixing molar volume for FeO–SiO₂ is much weaker than that for CaO–SiO₂ (because of large optical basicity values of CaO (1)), which is in line with experimental data. In contrast, optical basicity value for FeO from sulphide capacity equals to 1, which would provide apparent wrong prediction that the same departure from ideal mixing for FeO–SiO₂ as CaO–SiO₂ system. The possible reason for that optical basicity values from electronegativities could have very good performance on prediction of mixing molar volume could be due to that there is a good correlation between electronegativities of metal cations and length of Si–O bonds, which will be discussed in detail in the section of results and discussion. Besides, these theoretical values were successfully applied to correlating phosphorous capacities with optical basicity¹³ and also provided a good description on scales of mixing Gibbs free energy according to work by Ottonello et al. ⁸

Table 1.

Optical basicity Λ values for various components

Oxide	CaO	FeO	MgO	MnO	Na₂O	K₂O	Al₂O₃	SiO₂
Λ	1	0·51	0·78	0·59	1·15	1·4	0·6	0·48

The method described above had been applied to calculate density of binary and ternary silicate melts and a good performance has been achieved. However, there exist some inherent difficulties when the method is extended to slag systems containing alumina. Application of equation (8) requires the classification of different oxides into network formers and modifiers. Alumina is a typical amphoteric oxide, which may act either as network former or network modifier. The structure role of Al₂O₃ in the aluminosilicate slag system has received large attention during many years.¹⁴ It has been widely accepted that aluminium has a strong preference for tetrahedral coordination in aluminosilicate melts. , is not as stable as and has to be compensated by metal cations Mⁿ⁺ (n is valency of M), such as Ca²⁺ and Na⁺ to form [M_1/n AlO₄] complexes. One mole of needs 1/n mole of Mⁿ⁺ to take part in charge compensation. If there are enough metal cations to compensate Al, i.e. Al<1/nM, all aluminium exists in tetrahedral coordination. Otherwise, some aluminium exists in octahedral coordination. There are still some controversial for the role of octahedral coordinated aluminium to entire network. It is presumed that octahedral coordinated aluminium forms isolated AlO₆ outside network and behaves neither as network former nor network modifier.

In the present work, charge compensation of metal cation on was taken into consideration explicitly in modelling the density of slag. If the total amount of metal oxide in slag is larger than that of alumina, aluminium is assumed to exist in tetrahedral coordination and considered to be network former. Since some metal cations have been employed to compensate , this part should be deducted from network modifier. If amount of metal oxides is not sufficient for charge compensation, charge compensated alumina behaves as network former, whilst the remaining aluminium (in octahedral coordination) exists as isolated AlO₆. Because all metal oxides are employed for charge compensation, deviation from ideal mixing due to mixing of network former and network modifier is zero.

Molar volume of slag is a direct measure of atomic packing density. It is naturally considered that coordination shift of aluminium from six to four in aluminosilicate should bring corresponding change of molar volume. Very early experimental density data measured by Safford and Silverman¹⁵ for the series Na₂O–CaO–SiO₂ glasses with Al₂O₃ added had provided clear evidence for change of molar volume due to coordination shift. Their data indicated that most aluminium is in tetrahedral coordination because partial molar volume for Al₂O₃ in glasses is much greater than that of corundum (in octahedral coordination).

Pure alumina which is not charge compensated and assumed to be in octahedral coordination could be denoted by Al(VI)₂O₃. A hypothetical state of alumina that is in tetrahedral coordination, denoted by Al(IV)₂O₃, could be defined in the present work for modelling of molar volume change due to coordination shift. The molar volume of Al(VI)₂O₃ could be obtained by experimental data for pure molten alumina, whilst molar volume of Al(IV)₂O₃ could be seen as a parameter that can be optimised by experimental data for molar volume of aluminosilicate slag.

The density of ternary silicate was calculated by assuming ideal intermixing of binary silicates according to method proposed by Richardson¹⁶ in our previous work. However, this method can only be applied to silicate slag free of Al₂O₃. In the present work, molar volumes of multicomponent slags are calculated from molar volumes of pure components and molar volume of mixing, and molar volume of mixing values are directly calculated according to equation (10). It should be mentioned that in equation (10) should be replaced by molar fraction of all network forming cations, e.g. for aluminosilicate systems. Calculation of K ₁ requires taking contributions of all network modifying oxide and network forming oxide into consideration.

Base on above analysis, the following formulas for molar volume calculation were obtained for the A_2/v1O–B_2/v1O–Al₂O₃–SiO₂ system (where A_2/v1O and B_2/v2O are two network modifier. v1, v2 is valency of A, B respectively).

If X(A_2/v1O)+X(B_2/v1O)>x(Al₂O₃), we have (11) and If X(A_2/v1O)+X(B_2/v1O)<X(Al₂O₃), then we have (12) and In above equations, , , and are molar volume of A_2/v1O, B_2/v1O, SiO₂ and Al₂O₃ respectively. , , , and are molar volume of A_2/v1O, B_2/v1O, SiO₂, Al₂O₃(IV) and Al₂O₃(VI) respectively. , , and are optical basicity values of A_2/v1O, B_2/v1O, SiO₂ and Al₂O₃ respectively.

Molar volumes of pure components are assumed to be linear functions of temperature as V = V ₀+αT.Temperature dependence of molar volume for SiO₂ was obtained by linear fitting of density data for SiO₂ due to Bacon et al. ¹⁷ Experimental data for density of CaO cannot be found in literature due to its high melting point. A large discrepancy for experimental density of FeO was found in literature data, which mainly stems from existence of some Fe³⁺ or iron. Hara et al. ¹⁸ measured density values of CaO–Fe₂O₃–FeO system and found that linear equations could describe density values in this system. Molar volumes of CaO and FeO were extrapolated from their equations. Due to the lack of experimental data, partial molar volumes of MnO, K₂O and Na₂O given by Mill and Keen³ are adopted for the present model. Density data for Al₂O₃ due to Mitin and Nagibin¹⁹ were employed to generate a temperature dependence of molar volume for Al(VI)₂O₃. The partial molar volume of MgO proposed by Bottinger and Weil¹ was adopted. Temperature dependences of molar volumes for different components were summarised in Table 2.

Table 2.

Temperature dependences of density for pure components (V = V ₀+αT)

Component	V ₀/cm³ mol⁻¹)	α/cm³ mol⁻¹ K⁻¹
CaO	16·49	0·0027
FeO	13·54	0·00125
MgO	10·27	0·00122
MnO	12·83	0·00156
Al₂O₃(VI)	6·87	0·0118
Al₂O₃(IV)	15·8	0·00875
Na₂O	27·15	0·0033
K₂O	42·62	0·00518
Li₂O	15·82	0·0019
SiO₂	22·54	0·00275

Results and discussion

The present model has been applied to calculate molar volume of many slag systems from binary to quaternary systems. In order to have a quantitative comparison between calculated and experimental values, mean deviation Δ between calculated and experimental values is defined according to following equation (13) where (V _n)_cal and (V _n)_mea are the calculated and measured molar volume respectively. N is the number of samples.

In our previous work,⁹ density values of binary systems as CaO–SiO₂, FeO–SiO₂, MnO–SiO₂ and MgO–SiO₂ were calculated and good agreement was achieved. Different λ values for different binary systems were employed. In the present work, a constant λ = 0·025 value was used to calculate density of these binary systems. Figure 1 shows the comparison between calculated and experimental values^4,20–23 for CaO–SiO₂, FeO–SiO₂, MnO–SiO₂ and MgO–SiO₂ systems. It can be seen from the figure that the calculated values agree also well with measured values. This indicates that a constant λ value is sufficient to describe deviation of molar volume from ideal mixing. It can be also seen from the figure that CaO–SiO₂ system has most a significant deviation from ideal mixing for molar volume. In contrast, FeO–SiO₂ and MnO–SiO₂ system only have very slight departure.

Figure 1.

Comparisons between calculated and experimental values for CaO–SiO₂, MgO–SiO₂, MnO–SiO₂ and FeO–SiO₂ systems

Density values for some alkali silicate system, Na₂O–SiO₂ and K₂O–SiO₂ systems, were calculated using the present model. Calculated values were compared with measured values by Bockris et al. ²⁴ As shown in ^{Figure 2} Figs. 2 and 3, the values calculated by the present model are in good agreement with experiments. A significant departure from ideal mixing for molar volume of these systems could be also found from ^{Figure 2} Figs. 2 and 3.

Figure 2.

Comparisons between calculated and experimental values for Na₂O–SiO₂ system at 1673 K

Figure 3.

Comparisons between calculated and experimental values for K₂O–SiO₂ system at 1673 K

The present model was also applied to calculate the density of some alkaline earth oxide and alkaline oxide mixed silicate as CaO–Na₂O–SiO₂ and MgO–Na₂O–SiO₂ system. As shown in ^{Figure 4} Figs. 4 and 5, the values calculated by the present model are in good agreement with experimental values.²⁵ The mean deviation between the calculated and experimental values is 1·2%. The values calculated by assuming ideal mixing are also shown in ^{Figure 4} Figs. 4 and 5. It can be seen that the assumption of ideal mixing would bring larger errors for calculation.

Figure 4.

Comparisons between calculated and experimental values for CaO–Na₂O–SiO₂ system (9·6CaO–40Na₂O–50·4SiO₂ in mol.-%)

Figure 5.

Comparisons between calculated and experimental values for MgO–Na₂O–SiO₂ system (9·6MgO–40Na₂O–50·4SiO₂ in mol.-%)

The molar volume values of some ternary aluminosilicate systems (Al₂O₃–CaO–SiO₂ and Al₂O₃–Na₂O–SiO₂) were calculated using the present model. Experimental data for Al₂O₃–CaO–SiO₂ system measured by Courtial and Dingwell⁴ using double-bob Archimedean method with Pt or Ir bobs and crucibles, were employed for comparison. For the Al₂O₃–Na₂O–SiO₂ system, experimental data due to Riebling²⁶ using high temperature counterbalanced sphere viscometer–densitometer were adopted to check the performance of the present model. The calculated results are shown in ^{Figure 6} Figs. 6 and 7. It can be seen from the figures that the calculated values are in good agreement with experimental data. The mean deviation is about 1·42% for Al₂O₃–CaO–SiO₂ system and 0·7% for Al₂O₃–Na₂O–SiO₂ system.

Figure 6.

Comparisons between calculated and experimental molar volume values for Al₂O₃–CaO–SiO₂ system (No. 1, 61·77SiO₂–14·22Al₂O₃–23·47CaO; No. 2, 40·61SiO₂–19·61Al₂O₃–38·35CaO; No. 3, 33·69SiO₂–24·36Al₂O₃–41·8CaO; No. 4, 42·8SiO₂–36Al₂O₃–20·72CaO; in mass-%)

Figure 7.

Comparisons between calculated and experimental molar volume values for Al₂O₃–Na₂O–SiO₂ system (No. 1, 60SiO₂–13Al₂O₃–27Na₂O; No. 2, 60SiO₂–7Al₂O₃–33Na₂O; No. 3, 67·2SiO₂–4·9Al₂O₃–27·9Na₂O; No. 4, 66·8SiO₂–10·3Al₂O₃–22·9Na₂O; in mass-%)

As a very important system in process metallurgy, quaternary Al₂O₃–CaO–MgO–SiO₂ system is a base system for various types of slags, such as blast furnace slags, ladle slags, etc. The present model was also applied to calculated molar volume of this system. The experimental data by Courtial and Dingwell⁵ were employed to compare calculated values with measured values. As shown in Fig. 8, the present model can predict accurate molar volume values that are consistent with experimental data. The mean deviation between calculated and experimental data for this system is 1·57%.

Figure 8.

Comparisons between calculated and experimental molar volume values for Al₂O₃–CaO–MgO–SiO₂ system (No. 1, 34SiO₂–40Al₂O₃–16CaO–10MgO; No. 2, 43SiO₂–21Al₂O₃–21CaO–15MgO; in mass-%)

It has been illustrated clearly in this study that there is a negative deviation from ideal mixing for molar volume of binary metal oxide–silica system. This behaviour is in line with many experimental investigations. Henderson et al. ²¹ measured the density of FeO–SiO₂ system and found a positive deviation of density from ‘ideal density’ calculated for mechanical mixtures of FeO and SiO₂. Tomlinson et al. ²³ investigated the density of MgO–SiO₂ and CaO–SiO₂ system and found a more obvious deviation for CaO–SiO₂ than MgO–SiO₂.

The deviation of molar volume from ideal mixing behaviour may be related to distribution of anions in silicate melts. Molar volume of slag would be dependent on the ionic size and packing density. Variation of Si–O bond length would have some attributes on the negation deviation effect. It has been found by many crystallographic researches on silicates, that the length of Si–O bond connected to non-bridging oxygen is shorter than the bond to bridging oxygen.^27,28 This could be interpreted by the Cruickshank’s d–p π bonding model.²⁹ The Si–O bond connected to non-bridging oxygen has a high order of π bonding and shorter length compared to Si–O bond connected to bridging oxygen. Thus, in ametal oxide–silica system, the non-bridging oxygen brought by metal cations would decrease the Si–O bond length and thereby lead to smaller molar volume compared to values calculated by ideal mixing. Besides, the angles of Si–O–Si, Si–O–M and M–O–M would be variable with environmental of Si, which could also have some attributes on the non-ideal mixing.

In this work, a relationship was established between negative deviation from ideal mixing and difference of reciprocal of optical basicities for network forming and modifying oxide. Optical basicity is found to correlate strongly to the deviation from ideal mixing behaviour of silicate melts. The order of the degree of deviation from ideal mixing for MO–SiO₂ are CaO–SiO₂>MgO–SiO₂>MnO–SiO₂>FeO–SiO₂ (see Fig. 1), which is the same as order of corresponding optical basicity values: CaO>MgO>MnO >FeO (see Table 1).The correlation between optical basicity and negative deviation could be interpreted by bond ionicity and electronegativity of metal cations. Optical basicity values adopted here were calculated using Pauling electronegativities according to the following equation¹² (14) whereas, bond ionicity could be measured using fractional ionic character of bonds f, which is calculated by electronegativities according to Pauling¹² (15) Therefore, a direct connection between optical basicity and bond ionicity could be established in terms of electronegativities. Higher optical basicity value of oxides corresponds to high ionicity of bond. Brown and his coworkers²⁷ proposed a correlation between the Si–O(nbr) bond length and the average electronegativity of the non-tetrahedral cations bonded to O(nbr); longer Si–O(nbr) bonds are found to be associated with the more electronegative cations. When connected to more electronegative cations, Si–O tends to lose its π bonding potential by forming more covalent bonds and lengthen due to π bonding order decrease. Although developed for crystal structure of silicates, their correlation is consistent to one between optical basicity value and molar volume of molten slag in the present work. Higher optical basicity reflects lower electronegativity of metal cation and higher ionicity of M–O bonds, thereby leads to a decrease in Si–O length and thereby a negative deviation from ideal mixing.

Apart from negative deviation from ideal mixing for metal oxide and silicates, positive deviation was found in many slag systems with metal oxide and alumina. Experimental data for CaO–Al₂O₃ at 2473 K and MgO–Al₂O₃ at 2523 K by Elyutin et al. ³⁰ showed a large positive deviation for molar volume. This positive deviation could be well interpreted in terms of coordination changes with composition. The introduction of metal oxide into alumina melts would lead to charge compensation of and induce Al transition from sixfold coordination to fourfold coordination. The decrease in coordination looses the pack density of melts, therefore increases molar volume of melts. It should be noted that, in the present model, molar volume of hypothetical alumina in fourfold coordination is larger than molar volume of alumina in sixfold coordination. This model parameter guarantees the ability to reproduce this positive deviation behaviour.

The mean deviation for the present calculations is within 2% totally, which is roughly identical to the experimental uncertainty. Although many models based on multilinear fitting can also provide calculation within 2%, the present model has a stronger structural basis and could give a clearer description for interrelation between molar volume and structure. Although deviations from ideal mixing for silicate melts have been found in many experimental works, there is no thorough modelling work and clear interpretation on these deviations before the present work.

Summary

A density calculation model for molten silicate and aluminosilicate slags, which is an extension of our previous model for silicate slags, was developed in this work. Molar volumes of molten slags were calculated using molar volume of mixing, which is correlated to differences between reciprocals of optical basicities of network forming oxides and those of network modifying oxides. Charge compensated by metal cations is considered to have ability of network forming. Shift of coordination from octahedral to tetrahedral for aluminium in molten slag would lead to change of partial molar volume. Molar volume for a hypothetical state of alumina in tetrahedral coordination was defined in the present model to describe this behaviour. The molar volume of M₂O–SiO₂ (M = Na, K), CaO–Na₂O–SiO₂, MgO–Na₂O–SiO₂, Al₂O₃–CaO–SiO₂, Al₂O₃–Na₂O–SiO₂ and Al₂O₃–CaO–MgO–SiO₂ system. A comparison between calculated values and experimental values has shown that the present model provides accurate values for molten slags with a deviation within 2%. A significant negative deviation from ideal mixing for molar volume of some metal oxide and silica join was found. The negative deviation from ideal mixing for metal oxide and silica join was interpreted by considering effects of optical basicity, bond ionicity and electronegativity of cations on Si–O bonds length. The positive deviation for metal oxide and alumina join was also well explained in terms of coordination change of aluminium in structure.

Footnotes

Acknowledgements

The financial supports from NSFC (no. 50704002 and no. 51174018) are gratefully acknowledged.

References

Bottinga

and Weill

D. F.

: Am. J. Sci., 1972, 272, 438–475.

Bottinga

Weill

D. E.

and Richet

: Geochim. Cosmochim. Acta, 1982, 46, 909–919.

Mills

K.C.

and Keen

B.J.

. Int. Mater. Rev., 1987, 32, 1–120.

Courtial

and Dingwell

D. B.

: Geochim. Cosmochim. Acta, 1995, 59, 3685–3695.

Courtial

and Dingwell

D. B.

: Am. Mineral., 1999, 84, 465–476.

Persson

Matsushita

Zhang

and Seetharaman

: Steel Res. Int., 2007, 78, 102–108.

Toop

G. W.

and Samis

C. S.

: Trans. Metall. Soc. AIME, 1962, 224, 878–887.

Ottonello

Moretti

Marini

and Zuccolini

M. V.

: Chem. Geol., 2001, 174, 157–179.

Shu

: High Temp. Mater. Process, 2007, 26, (5–6), 341–348.

10.

Fincham

C. J. B.

and Richardson

F. D.

: Proc. R. Soc. (Lond.), 1954, 233, 40–62.

11.

Duffy

J. A.

and Ingram

M. D.

: J. Non-Cryst. Solids, 1976, 21, 373.

12.

Pauling

: ‘The nature of chemical bond’, 98; 1960, Ithaca, NY, Cornell University Press.

13.

Gaskell

D. R.

: Trans. Iron Steel Inst. Jpn, 1982, 22, 997–1000.

14.

Mysen

B. O.

: ‘Structure and properties of silicate melts’; 1988, Amsterdam, Elsevier Science.

15.

Safford

H. W.

and Silverman

: J. Am. Ceram. Soc., 1947, 30, 203–211.

16.

Richardson

F. D.

: ‘Physical chemistry of melts in metallurgy’, Vol. I; 1974, London, Academic Press.

17.

Bacon

J. F.

Hasapis

A. A.

and Wolley

J. W.

: Phys. Chem. Glasses, 1960, 1, 90–98.

18.

Hara

Irie

Gaskell

D. R.

and Ogino

: Trans. Jpn Inst. Met., 1988, 29, 977–989.

19.

Mitin

B. S.

and Nagibin

Y. A.

: Russ. J. Chem. (Engl Transl.), 1970, 44, (5), 741–742.

20.

Lee

Y. E.

and Gaskell

D. R.

: Metall. Trans., 1974, 5, 853–860.

21.

Henderson

Hudson

R. G.

Ward

R. G.

and Derge

: Trans. Metall. Soc. AIME, 1961, 221, 807–811.

22.

Segers

Fontana

and Winard

: Eletrochim. Acta, 1978, 23, 1275–1280.

23.

Tomlinson

J. W.

Heynes

M. S. R.

and Bockris

JO’M

: Trans. Faraday Soc., 1958, 54, 1822–1823.

24.

O’M Bockris

Tomlinson

J. W.

and White

J. L.

: Trans. Faraday Soc., 1956, 52, 299.

25.

Adachi

Ogino

and Toritani

: Technol. Rept. Osaka Univ., 1960, 10, 149.

26.

Riebling

E. F.

: J. Chem. Phys., 1966, 14, (8), 2857.

27.

Brown

G. E.

Gibbs

G. V.

and Ribbe

P. H.

: Am. Mineral., 1969, 51, 1044–1061.

28.

Brown

G. E.

and Gibbs

G. V.

: Am. Mineral., 1969, 51, 1528–1539.

29.

Cruickshank

D. W. J.

: J. Chem. Soc., 1961, 1077, 5486–5504.

30.

Elyutin

V. P.

Mitin

B. S.

and Anisimov

Y. u. S.

: Izv. VUZ. Tsvet. Metall., 1974, 4, 42.