Model for diffusion coefficient estimation of calcium ions in CaO–Al 2 O 3

Abstract

As a common component in metallurgical slag, CaO plays an important role in desulphurisation, dephosphorisation and absorption of non-metallic inclusions. In order to better understand the mechanism of the slag/metal reactions, the diffusion dynamics of calcium ions in CaO–Al₂O₃–SiO₂ slags were studied. It was found that there was almost a linear relation between the logarithms of pre-exponential factor and diffusion activation energy. By combining the relation between the diffusion activation energy and the optical basicity corrected in the CaO–Al₂O₃–SiO₂ slags, a simple model used to estimate the diffusion coefficient of calcium ion is proposed. The estimated values of the CaO–SiO₂ and CaO–Al₂O₃–SiO₂ systems agree well with the experiment data, with a mean deviation of 35·3 and 22·5% respectively.

Introduction

In metallurgical processes, diffusion phenomena occur in slag/metal reactions such as the desulphurisation and dephosphorisation of blast furnace slag and the absorption of inclusions in mould flux. The various reactions are completed by the diffusion and migration of ions in the slag and metal, but the reaction rate is always controlled by the diffusion in the liquid slag.¹ Because of its excellent desulphurisation and dephosphorisation properties, CaO is an important oxide in slags that could also combine with Al₂O₃ in liquid steel to form low melting point inclusions; hence, research into the dynamics of calcium ion diffusion would be a significant achievement.

Self-diffusion of various species present in the melts, where there are no chemical potential gradients, can occur by random motions. If the diffusion ion was replaced by its isotope, the diffusion process becomes tracer diffusion. Moreover, tracer diffusivity is considered to be equivalent to the self-diffusivity in liquids.² Towers et al. ^3,4 investigated the diffusion of Ca⁴⁵ in CaO–Al₂O₃–SiO₂ molten slags by the radioactive tracer technique and the capillary method. The results suggested that the diffusion coefficient is in the range of 10⁻⁷–10⁻⁶ cm² s⁻¹ in the temperature range of 1623–1823 K. Niwa⁵ improved Towers’ experiment device and also measured the diffusion coefficient of Ca⁴⁵, the results of which had the same order of magnitude with Towers’ work. Saito and Maruya⁶ used the semi-infinite medium method to obtain the diffusivity of Ca⁴⁵ ions, in which the behaviours of Al₂O₃ and MgO were also discussed, and the range of temperature and composition of slags became wider. Keller et al. ⁷ used the capillary technique to measure the diffusivity of Ca⁴⁵ ions in CaO–SiO₂ molten slags, which was determined in the temperature range of 1773–1973 K, but the values were higher than that obtained by Saito and Maruya.

Goto et al. ⁸ calculated the tracer diffusivities of multicomponent oxide slags, which were based on the Nernst–Einstein relation. Recently, Zhang and Chou⁹ used the same method to estimate the diffusion coefficient of calcium ions in CaO–Al₂O₃–SiO₂ slags and also used the relation between the calcium ion concentration and diffusivities to calculate the diffusion coefficient. Nevertheless, both results they calculated were not satisfactory and were much larger than the true values. Because the volume and electrical conductivity of molten slags were both needed in the calculation process, it is a pity that we could not always acquire the necessary data for them.

Diffusion coefficient like viscosity and electrical conductivity is also an important physical property, the value of which could describe the rate of slag/metal reactions. Data on viscosity and electrical conductivity are plentiful, but are rather scarce for the diffusion coefficient of ions. In order to have good knowledge of the diffusion process of ions in slags, more reliable diffusion data are essential. The present study aims at developing a model to estimate the diffusion coefficient of calcium ions in slags related to the structure of slag, but not the relation between diffusion coefficient and viscosity or electrical conductivity.

Model descriptions

For a diffusion ion of i, the relationship between temperature and diffusion coefficient can be described by the Arrhenius type equation (1) where D _i is the diffusion coefficient of i (m² s⁻¹), T is the temperature in kelvin, D ₀ is the pre-exponential factor (m² s⁻¹), R is the gas constant (8·314 J mol⁻¹ K⁻¹) and E is the diffusion activation energy (J mol⁻¹). Expressing E/R as B, equation (1) can be rewritten as (2) Basing on much experimental data,^3–7 it is found that there is almost a linear relation between ln D ₀ and B, as shown in Fig. 1. According to the regression analysis, the relation between ln D ₀ and B can be described by the following equation (3) and the values of m and n are 5·54×10⁻⁴ and −22·159 respectively.

Figure 1.

Relation between ln D ₀ and B for CaO–SiO₂ and CaO–Al₂O₃–SiO₂ systems

Correlation between B and optical basicity

The diffusion coefficient is closely related with the structure of slag. NBO/T, optical basicity, free oxygen, bridging oxygen and non-bridging oxygen are commonly used to describe the structure of the slag.^10–15 In the current model, the structure of the slag was described by the optical basicity, which is also successfully used to estimate other physical properties of slag.

In the CaO–SiO₂ system, CaO acting as the modifier is used to break down the network structure formed by SiO₂. When Al₂O₃ is present in the melts, the condition is changed. It is widely accepted that Al³⁺ has a strong preference for the tetrahedral coordination structure in silicate melts. However, the valence of Al³⁺ ion is lower than that of Si⁴⁺ ion in the tetrahedra, so alkaline or alkaline earth cations such as Ca²⁺ and Na⁺ are needed to keep the charge balance by forming [1/2Ca(AlO₄)]⁴⁻ or [NaAlO₄]⁴⁻.¹⁶ Consequently, the optical basicity of the CaO–Al₂O₃–SiO₂ system should be corrected, which can be calculated by the following equation¹² (4) where x _i is the mole fraction of each component, and the coefficients 1·0, 0·6 and 0·48 are the optical basicities of CaO, Al₂O₃ and SiO₂ respectively.

For the CaO–SiO₂ and CaO–Al₂O₃–SiO₂ systems, the parameter B is plotted as a function of corrected optical basicity in ^{Figure 2} Figs. 2 and 3. Furthermore, the approximate relation between Λ ^corr and B can be expressed in the following form (5) The a, b, c and d values for both systems were optimised using experimental data, which are shown in Table 1.

Figure 2.

Relation between optical basicity and B for CaO–SiO₂ system

Figure 3.

Relation between corrected optical basicity and B for CaO–Al₂O₃–SiO₂ system

Table 1.

Model parameters for CaO–SiO₂ and CaO–Al₂O₃–SiO₂ systems

System	a	b	c	d
CaO–SiO₂	−6·8865×10⁷	1·3109×10⁸	−8·3101×10⁷	1·7558×10⁷
CaO–Al₂O₃–SiO₂	1·7565×10⁹	−3·2773×10⁹	2·037×10⁹	−4·2172×10⁸

Results

The relative deviation δ _n and the mean deviation Δ were used to evaluate the performance of the current model in estimating the diffusion coefficient of calcium ion, which were calculated as follows (6) (7) where D _mea and D _est are the measured and estimated diffusion coefficients respectively, and N is the number of diffusion coefficient data. The comparison between measured and estimated values for the CaO–SiO₂ system is shown in Table 2. The experimental data are taken from Saito and Maruya⁶ and Keller et al. ⁷ It can be seen that the estimated values agree well with the data obtained by Keller et al., but are a little higher than the results determined by Saito and Maruya, which may be caused by the difference of the experimental method. The mean deviation Δ is ∼35·3%.

Table 2.

Comparisons between measured and estimated diffusion coefficients in CaO–SiO₂ system

Composition/mol.-%	Temperature/K	D _mea/×10⁶ cm² s⁻¹				D _est/×10⁶ cm² s⁻¹	Relative deviation/%									Reference
CaO–SiO₂	Temperature/K	D _mea/×10⁶ cm² s⁻¹				D _est/×10⁶ cm² s⁻¹	Relative deviation/%									Reference
0·366–0·634	1773	1·7				2·01	18·2									7
	1823	2·2		2·1		2·61	18·6				24·3
	1873	3·0		2·8		3·34	11·3				19·3
	1883	2·7		3·0		3·5	29·6				16·7
	1923	3·2	3·3		3·8	4·22	31·9		27·9					11·1
0·413–0·587	1833	3·0		2·8		2·70	−10·0				−3·6
	1873	3·6				3·22	−10·6
	1923	3·8	4·5		4·2	3·98	4·7		−11·6					−5·2
0·470–0·530	1823	3·1	3·4		3·6	2·60	−16·1		−23·5					−27·8
	1873	5·0	4·0		5·1	3·33	−33·4	−16·8					−34·7
	1973	6·4		7·4		5·23	−18·3					−29·3
0·512–0·488	1823	4·2				2·61	−37·9
	1833	4·7		4·2		2·75	−41·5					−34·5
	1873	5·1	5·7		5·3	3·37	−33·9			−40·9					−36·4
	1923	6·9				4·29	−37·8
0·552–0·448	1773	4·3	4·7		4·5	2·06	−52·1			−56·2					54·2
	1873	5·7	6·8		6·2	3·16	−44·6			−53·5					−49·0
	1973	10·5		9·9		4·65	−55·7					−53·0
0·569–0·431	1758	0·71				2·08	192·4									6
	1783	0·87				2·23	156·7
	1803	1·1				2·36	115·0

For the CaO–Al₂O₃–SiO₂ system, the measured values are from Towers et al.,^3,4 Niwa⁵ and Saito and Maruya.⁶ The comparison between measured and estimated values is shown in Table 3, with the mean deviation Δ of ∼22·5%. The relative deviation of each estimated value is also very low. Thus, it was concluded that the estimated diffusivity of calcium ions fit well with the measured values in the CaO–Al₂O₃–SiO₂ system. In a slag of CaO–20 wt-%Al₂O₃–40 wt-%SiO₂, the comparison between the measured diffusion coefficients of calcium ions with the calculated values by different researchers is shown in Fig. 4. It can be seen that the calculated values of Goto et al. ⁸ and Zhang and Chou⁹ are a little larger than the measured ones, and that the present model provides a more accurate estimation for diffusivity values of calcium ion.

Figure 4.

Comparisons of measured diffusion coefficients with estimated values

Table 3.

Comparisons between measured and estimated diffusion coefficients in CaO–Al₂O₃–SiO₂ system

Composition/mol.-%	Temperature/K	D _mea/×10⁷ cm² s⁻¹		D _est/×10⁷ cm² s⁻¹	Relative deviation/%		References
CaO–Al₂O₃–SiO₂	Temperature/K	D _mea/×10⁷ cm² s⁻¹		D _est/×10⁷ cm² s⁻¹	Relative deviation/%		References
0·496–0·283–0·221	1623	3·9		3·74	−4·1		6
	1668	6·9		6·13	−11·2
	1713	10·0		9·8	−2·0
0·522–0·249–0·229	1713	8·0		8·80	10·0
0·522–0·249–0·229	1783	19·0		18·90	−5·3
0·453–0·125–0·423	1623	3·9	3·3	3·14	−19·5	−4·8	3,6
	1673	7·0	6·8	5·72	−18·3	−15·9
	1723	10·5	13·0	10·6	1·0	−18·5
0·481–0·111–0·407	1623	5·2		8·39	61·3		5
	1673	6·4		11·43	78·6
	1723	9·5		15·28	60·8
0·484–0·273–0·243	1623	3·5		2·64	−24·6		4
	1773	21·0		16·67	−20·6
	1813	34·0		25·88	−23·9

Discussion

For the CaO–SiO₂ system, it is found that the diffusion coefficient decreases with increasing SiO₂ (Table 2). This may be explained by the degree of melt polymerisation. Zhang and Jahanshahi¹⁷ calculated three types of oxygen by cell model for the MO–SiO₂ (M = Ca, Mg, Mn) system, and the results are shown in Fig. 5. It can be seen that (fraction of bridging oxygen) increases quickly at , and (fraction of free oxygen) is almost zero. In others words, the degree of melt polymerisation increases as the concentration of SiO₂ rises. Hence, the higher degree of melt polymerisation will make a bigger viscosity at specific temperatures. According to the Eyring relationship^18,19 (8) where k is Boltzmann’s constant, D is the diffusion coefficient, λ is the mean translation distance for each diffusion jump of diffusing species and η is the viscosity. If the jumping distance λ of calcium ions is assumed to be constant, the rise of the SiO₂ concentration makes the diffusion coefficient of calcium ions decline. Amini et al. ²⁰ suggested that the diffusion of species in the melt is claimed to be controlled by large anions. Consequently, these tetrahedra are joined together in chains or rings by bridging oxygen at a high SiO₂ concentration, of which the bigger volume and weaker mobility also could enhance the diffusion resistance and the jumping distance.

Figure 5.

Composition dependence of fractions of three types of oxygen calculated by cell model for MO–SiO₂ system

In general, the more network breaking oxide in the molten slag, the larger the optical basicity, and the lower the viscosity. Consequently, the diffusion activation energy should decline as the optical basicity increases; however, the data do not agree with that. As shown in ^{Figure 2} Figs. 2 and 3, with the optical basicity increasing, the diffusion activation energy initially decreases and then increases, but for the CaO–SiO₂ system, the diffusion activation energy decreases again at Λ>0·65. In the present study, the content of calcium oxide varied between 0·366 and 0·569 (mole fraction). Combining the phase diagram of CaO–SiO₂ (Fig. 6), it is found that the liquidus temperatures of the CaO–SiO₂ slags have the same changing trend with the value of the diffusion activation energy with the content of calcium oxide increasing, and the values of optical basicity 0·605 and 0·653 are the turning points respectively. Moreover, Eyring et al. ²¹ suggested that the driving force for diffusion came from the concentration gradient of diffusion ions, and the standard free energy changed with the jumping distance, the relation of which can be described by the curve in Fig. 7. Consequently, the increasing content of CaO may make the structure and physical properties of molten slags change, which results in the changing of the jumping distance of calcium ion and also the value of the diffusion activation energy.

Figure 6.

Phase diagram of CaO–SiO₂ system

Figure 7.

Relation between standard free energy and jumping distance

From a diffusion activation energy value point of view, the value of the diffusion activation energy of the CaO–Al₂O₃–SiO₂ system is about 140–320 kJ mol⁻¹, and 140 kJ mol⁻¹ is almost the maximum value of the CaO–SiO₂ system (115–145 kJ mol⁻¹). In the CaO–Al₂O₃–SiO₂ slag, the units need Ca²⁺ to keep charge balance, so this may make the concentration of free Ca²⁺ reduced, which results in the rise of diffusion activation energy and a lower diffusivity. Zhang and Chou⁹ also found that there is a linear relationship between the concentration of Ca²⁺ and the logarithm of diffusion coefficient, which increased with the concentration of Ca²⁺ increasing. Furthermore, due to keep charge balance, the mobility of these calcium ions in [1/2Ca(AlO₄)]⁴⁻ would become lower, and the units may increase the jump distance λ of calcium ion.

Conclusions

A simple method for the diffusion coefficient estimation of calcium ions in the CaO–Al₂O₃–SiO₂ slags is proposed. In the model, there is an increasing function relationship between the logarithm of pre-exponential factor and the diffusion activation energy calculated by optical basicity. The comparisons between estimated and experimental values show good agreement. With increasing polymerisation degree of molten slag, the diffusion coefficient of calcium ions declines. Because of the charge balance in the CaO–Al₂O₃–SiO₂ slag, the decreasing of free Ca²⁺ concentration and the weaker mobility of these calcium ions in [1/2Ca(AlO₄)]⁴⁻ result in a lower diffusion coefficient.

Footnotes

Acknowledgements

The financial support from the National Science Foundation of China (grant no. 51090384) is gratefully acknowledged.

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Model for diffusion coefficient estimation of calcium ions in CaO–Al 2 O 3 –SiO 2 slags

Abstract

Introduction

Model descriptions

Correlation between B and optical basicity

Results

Discussion

Conclusions

Footnotes

Acknowledgements

References