VISCOSITY MODEL FOR ALUMINOSILICATE MELT

The structurally based viscosity model proposed in our previous study is extended to include more components, e.g. SiO 2 , Al 2 O 3 , FeO, MnO, MgO, CaO, Na 2 O and K 2 O. A simple method is proposed to calculate the numbers of different types of oxygen ions classified by the different cations they bonded with, which is used to characterize the influence of composition on viscosity. When dealing with the aluminosilicate melts containing several basic oxides, the priority order is established for different cations for charge compensating Al 3+ ions, according to the coulombic force between cation and oxygen anion. It is indicated that basic oxides have two paradox influences on viscosity: basic oxide with a higher basicity decreases viscosity more greatly by forming weaker non-bridging oxygen bond; while it increases viscosity more greatly by forming stronger bridging oxygen bond in tetrahedron after charge compensating Al 3+ ion. The present model can extrapolate its application range to the system without SiO 2 . Furthermore, it could also give a satisfy interpretation to the abnormal phenomenon that viscosity increases when adding K 2 O to CaO-Al 2 O 3 -SiO 2 melt within a certain composition range.


Introduction
Aluminosilicate melt plays significant roles in the fields of glass-making, pottery-making, geological research and pyrometallurgy etc., with viscosity as one of the most important physical properties. Therefore, accurate viscosity values for aluminosilicate melts are necessary not only for optimizing and improving the productive processes, but for studying the structure of aluminosilicate melts. Estimating viscosity by model may be a useful method considering experimental measurements are both time consuming and sometimes inaccurate for the difficulty of high temperature operation.
Many viscosity models are proposed to estimate viscosity of aluminosilicate melts [1][2][3][4][5][6][7][8][9][10][11]. Though these models are successful to some extent, many shortcomings still exist: (i) The application ranges of these models are very narrow. They can only obtain good estimation results in or near the composition (or temperature) range, in which the model parameters are optimized, while the results will be inaccurate when away from the range. (ii) For aluminosilicate melts containing several basic oxides, these viscosity models can not well represent the viscosity variation tendency with composition. Sometimes, an inverse tendency with the experimental finding may be given. For instance, it is found that viscosity increases when adding K 2 O to CaO-Al 2 O 3 -SiO 2 melt [12], while all the viscosity models can only give a decreasing tendency. (iii) The calculation results of these models for oxide systems without SiO 2 are very bad (e.g. Al 2 O 3 , CaO-Al 2 O 3 and CaO-FeO-Al 2 O 3 systems, etc..), in other words, they can only be applied to the system containing SiO 2 . In view of these points, much work still should be done on the viscosity model.

Points must be considered in viscosity model
At constant pressure, viscosity is determined by temperature and chemical composition of molten slags. The main object of modeling viscosity of aluminosilicate melt is finding a proper relation to correlate viscosity with composition and temperature. Generally, the temperature dependence of viscosity expressed by Weymann equation [13] , or Arrhenius equation [14] which is same as Eyring equation [15]. The Weymann equaiton is used by Riboud et al. [1], Urbain [6], Kondratiev et al. [7], Zhang et al. [8], Ray et al. [5] and Shu [9] etc; while the Arrhenius equation (or Eyring equation) is used by Iida et al. [10], Nakamoto et al. [4], KTH [3], NPL [2], and Shankar et al. [16]. It was pointed out by Shankar et al. [16] that both the two types can well describe the variation of viscosity with temperature. Then, the problem of incorporating the influence of composition on viscosity is the central issue. Generally, viscosity model must fulfill the following four requirements: (i) Viscosity is sensitive to the structure of aluminosilicate melts, and different components (acidic oxide, e.g. SiO 2 ; basic oxide, e.g. CaO and MgO; amphoteric oxide, e.g. Al 2 O 3 ) have different influences on the structure. Thereby, viscosity model should be built based on a reasonable description to the structure of aluminosilicate melts. However, the empirical models by Riboud et al. [1], Urbain [6], Kondratiev et al. [7], NPL [2], Ray et al. [5], Shankar et al. [16], etc., didn't consider the influence of melt structure on the viscosity.
(ii) Generally, in the Arrhenius or Weymann equation, there is a linear relation between the logarithm of pre-exponent factor lnA and activation energy E which is called the temperature compensation effect and is a common rule in the fields of viscosity, kinetics, electrical conductivity and diffusion coefficient [17]. Therefore, the compensation effect should also be taken into consideration in the viscosity model, while it is not considered in viscosity models of Riboud et al. [1], NPL [2], KTH [3], Nakamoto et al. [4], Iida et al. [10], etc..
(iii) After considering the Arrhenius or Weymann equation as well as the temperature compensation effect between lnA and E, the remaining task of modelling viscosity is to find a proper expression to relate the activation energy E and composition. It has been pointed out that the activation energy of viscosity is a non-linear function of composition [18]. A reasonable viscosity model should consider this nonlinear behavior. Among the different viscosity models, the polynomial functions with the highest order of "n" were utilized by different authors to describe this behavior: n=1 in Riboud et al. model [1]; n=2 in Shu model [9]; n=3 in Urbain model [6], Kondratiev et al. model [7] and Zhang et al. model [8].
(iv) When Al 2 O 3 exists in the melts, Al 3+ can form tetrahedron, and replace the position of Si 4+ ion when there are enough metal cations (e.g. Ca 2+ , and Na + ) participating into the charge compensation of Al 3+ ions. It is found that different cations have different abilities of compensating Al 3+ ions [19], so viscosity model should distinguish the different priorities. However, no model has done this except NPL model [2] which achieves this goal according to the different optical basicity values of different basic oxides. But this way has defect for the reason that in NPL model many basic oxides have the same optical basicity values, for instance optical basicity of Li 2 O, CaO, FeO and MnO using the same value of 1.0, but actually the compensating abilities of these cations are different.
New viscosity model considering all the four requirements should be developed. This work has been done in our previous paper [17,20,21]. New model is structural based, and the model parameters for silicate melts (absence of Al 2 O 3 ) are closely related to the bond strength [17,20]. In the presence of Al 2 O 3 , viscosity of CaO-MgO-Al 2 O 3 -SiO 2 system is also well modeled [21]. However, the more complex or involving other basic oxides systems (containing Al 2 O 3 ) are not considered, meanwhile, the abnormal phenomenon of viscosity increasing as adding K 2 O to CaO-Al 2 O 3 -SiO 2 melt [12] is still unresolved. In this work, the model will be extended to resolve these problems.

Model
Details of the model have been given elsewhere [17,20,21]. Only a brief description of the model will be given here. The temperature dependence of viscosity is calculated by Arrhenius equation, where µ is the viscosity, Poise; A is the preexponent factor, Poise; E is the activation energy, J/mol; R is the gas constant, 8.314 J/(molK); T is the absolute temperature, K. The temperature compensation effect is considered, For multicomponent system , the value of parameter k is assumed to be the linear addition of that of the binary systems M x O y -SiO 2 with the renormalized mole fractions of oxides M x O y as the weighting factors, The activation energy of viscosity in equation (2) is expressed as follows, (4) where n is the mole number and α describes the deforming ability of bond around the corresponding oxygen ion. The first, second, third, fourth, fifth, sixth terms in the denominator are the contributions of bridging oxygen , bonded with Si 4+ ion; oxygen , bonded with Al 3+ ion not charge compensated; bridging oxygen , bonded with Al 3+ ion charge compensated by cation i; non-bridging oxygen, bonded with Si 4+ ion and metal cation i; non-bridging oxygen , bonded with metal cation j and Al 3+ ion charge compensated by cation i; free oxygen , bonded with metal cation i, respectively.

Method of calculating the number of oxygen ion
Before using equations (1)～(4) to calculate viscosity, the numbers of different types of oxygen ions must be known. In our previous paper [21], five assumptions are suggested to calculate the numbers for CaO-MgO-Al 2 O 3 -SiO 2 melt.
Assumption I: The equilibrium constant for the reaction of free oxygen from basic oxide with bridging oxygen to generate non-bridging oxygen is infinite Assumption II: For system containing several basic oxides , the numbers of different types of oxygen ions can be calculated by the random mixing rule: firstly, calculate following Assumption (I) considering the system to be a pseudo-binary system; then multiply each term by the re-normalized mole fractions of basic oxides. Assumption IV: In the case of , when the extra M x O breaks the bridging oxygen, it is assumed that the bridging oxygen in and tetrahedron are equivalent, and the numbers of formed non-bridging oxygen bonded to Al 3+ ions and Si 4+ ions are proportional to the numbers of bridging oxygen in and tetrahedrons. Assumption V: Ca 2+ cation has higher priority of charge-compensating Al 3+ ion than Mg 2+ ion. Only after all the Ca 2+ ions have been exhausted, Mg 2+ ions will participate into the charge compensation.
Among them, Assumption V endows Ca 2+ ion an absolute priority over Mg 2+ ion when charge compensating Al 3+ ion, with which the viscosity variation of CaO-MgO-Al 2 O 3 -SiO 2 system can be well represented. However, the previous work can only deal with the case of aluminosilicate system contain two basic oxides: CaO and MgO, whereas, when the basic oxides are not CaO and MgO or when there are three or more basic oxides exist in more complex aluminosilicate system, how to establish the different priorities of different metal cations. This is one of the main tasks of the present study. Assumptions V will be revised into a more general form as follows.
Assumption V: Different metal cations have different abilities of charge-compensating Al 3+ ions. When there are several basic oxides in aluminosilicate melts, a strict priority order exists for different metal cations. In other words, only after all the cations with the higher priority have been exhausted to compensate Al 3+ ions, cations with the lower priority can participate the charge compensation. The order for different cations can be determined as follows.
Theoretically, the cation M z+ that has strong bond strength with O 2ion should possess weaker ability of compensating Al 3+ ion, for the reason that itself can form complex anion with oxygen ion, and strong repulsive force exists between cation M z+ and Al 3+ ion. So, in this study, parameter I expressed by the coulombic force between the cation and oxygen anion will be introduced to characterize the abilities of different cations, (5) where Q is the valence of M z+ ion are the radiuses of M z+ and oxygen ions, respectively. The ion radiuses (taken from the data compiled by Shannon [22]) of Mg 2+ , Ca 2+ , Na + , K + , Fe 2+ , Mn 2+ and O 2are 0.66Å, 0.99 Å, 0.97 Å, 1.33 Å, 0.74 Å, 0.8 Å and 1.44 Å, respectively. So the order of I for different cations is: K + <Na + <Ca 2+ <Mn 2+ <Fe 2+ <Mg 2+ . Therefore, the priority order of charge compensating Al 3+ ion is: K + >Na + >Ca 2+ >Mn 2+ >Fe 2+ >Mg 2+ . The order of K + >Na + >Ca 2+ >Mg 2+ has been proved by the evidences from thermochemical [23] and spectroscopic data [24,25]. However, it was pointed out that [17] for chemical bond with a high percentage of covalent bond, e.g. Fe-O (52.3%), Mn-O (41.0%) (close to the value of Si-O bond which is 55.3%), the bond strength can not be characterized simply by I. So the position of Fe 2+ and Mn 2+ in the order is only an approximation. The same approximation in the order was also suggested by Mysen [19].
Based on Assumption I ～ Assumption IV and Assumption V, the numbers of different types of oxygen ions can be calculated easily. The detailed calculating formulae for and M x O-Al 2 O 3 -SiO 2 systems are given before, and the calculating formulae for (M x O) i -(M x O) j -Al 2 O 3 -SiO 2 system can be obtained following the formulae of CaO-MgO-Al 2 O 3 -SiO 2 system [21]. In the following sections, model will be applied to aluminosilicate system containing FeO, MnO, MgO, CaO, Na 2 O and K 2 O. The literature data source utilized for the model parameters optimization and comparisons are shown in Table 1. The regressed parameters are shown in Tables 2 and 3, in which the parameters optimized in previous papers [17,20,21] are also included. Parameters for the present model are obtained as follows: parameters , are optimized according to the data of M x O-SiO 2 bianry system (have been accomplished in our previous work [17,20]); parameters and are optimized according to the data of M x O-Al 2 O 3 -SiO 2 system; while for the system (M two basic oxides, one more parameter is needed. Theoretically, no more parameter is needed for even higher order system.

M x O-Al 2 O 3 -SiO 2 ternary system
The viscosity data of FeO-Al 2 O 3 -SiO 2 and MnO-  Figure  1 and Figure 2, with the mean deviations 29.8% and 28.8%, respectively. The large deviations for these two systems may result from the reasons that both Fe and Mn are polyvalent metallic elements, and possible exsitences of Fe 3+ or Mn 3+ ions lead to the viscosity behavior departing from melts with pure FeO or MnO.   The data of Na 2 O-Al 2 O 3 -SiO 2 system are taken from Kou et al. [28] and Toplis et al. [29,30], with viscosity values varying from 10 to 4×10 5 poise. Parameters and are optimized in this system. The comparisons of calculated values with the measured values are shown in Figure 3, with the mean deviation 27.9%. For K 2 O-Al 2 O 3 -SiO 2 system, Urbain et al. [27] only measured one composition point with high content of SiO 2 and high viscosity; Mizoguchi et al. [31] measured composition points with high content of K 2 O and low viscosity. With the optimized parameters of and , viscosity of this system can be calculated. The comparisons between the estimated values and those by experiments are shown in Figure 4, with the mean deviation 31.8%. The large deviations for these two systems may result from large measure errors at high viscosity and evaporations of Na 2 O and K 2 O at high temperature. Viscosity data of CaO-FeO-Al 2 O 3 -SiO 2 are from Higgins et al. [32] and Kolesov et al. [33], while these for CaO-Na 2 O-Al 2 O 3 -SiO 2 and CaO-K 2 O-Al 2 O 3 -SiO 2 systems are from Sukenaga et al. [12]. The comparisons of calculated values with those by experiments are shown in Figures 5-7, with the mean deviations 28.1%, 24.7% and 23.0%, respectively.

CaO-FeO-Al 2 O 3 ternary system
The viscosity of CaO-FeO-Al 2 O 3 system are from Vidacak et al. [34]. When calculating viscosity of this system, no parameter needs to be optimized.     Figure 8, with the mean deviation 24.4%. In the following section, it can be seen that the existing viscosity models can not well represent the viscosity variation behavior of this system which is absence of SiO 2 .

Complex systems CaO-MgO-Na 2 O-Al 2 O 3 -SiO 2 system
The parameter is optimized in CaO-MgO-Na 2 O-Al 2 O 3 -SiO 2 system for the lack of viscosity data in MgO-Na 2 O-Al 2 O 3 -SiO 2 system. In this system, there are three types of cations: Na + , Ca 2+ and Mg 2+ ions, could compensate Al 3+ ion, and the priority order is Na + >Ca 2+ >Mg 2+ , thereby, oxygen ions exist. Considering the coressponding parameters have been optimized in CaO-Na 2 O-Al 2 O 3 -SiO 2 and CaO-MgO-Al 2 O 3 -SiO 2 systems, only parameters is optimized in this sytem.
Parts of the composition points of Sykes et al. [35] fulfill the condition: and, in which case all the Al 3+ ions are compensated Na + ions; the remaining Na + , Ca 2+ and Mg 2+ ions participate in forming non-bridging oxygen, but are not enough to break all the bridging oxygen. According to the five assumptions, numbers of different types of oxygen ions can be calculated as follows: Non-bridging oxygen bonded to Al 3+ ion: Non-bridging oxygen bonded to Si 4+ ion: Bridging oxygen: The other parts of composition points measured by Sykes et al. [35] and those measured by Kim et al. [36] fulfill the conditions: , and . In this case, all the Na + ions compensate Al 3+ ion; while Al 3+ ions not compensated by Na + ions are completed by Ca 2+ ions; the remaining Ca 2+ ions and Mg 2+ ions participate in forming the non-bridging oxygen, but are not enough to break all the bridging oxygen. The numbers of different types of oxygen ions can be calculated as follows: Non-bridging oxygen bonded to Ca 2+ ion: Non-bridging oxygen bonded to Mg 2+ ion: Bridging oxygen: According to the above equations, parameter can be optimized which are shown in Table 3. The comparisons of calculated viscosities with those measured by experiments are shown in Figure 9, with the mean deviation 25.7%.

CaO-MgO-FeO-Al 2 O 3 -SiO 2 system
The viscosity data of this system are taken from the work of Higgins et al. [32] and Kim et al. [37]. All the composition points fulfill the condition: and . In this case, all the Al 3+ ions are compensated by Ca 2+ ions; the remaining Ca 2+ , Mg 2+ and Fe 2+ ions participate in forming non-bridging oxygen, but can not break all the bridging oxygen. The numbers of different types of oxygen ions can be calculated following equations (6)(7)(8)(9)(10)(11)(12)(13). Calculate viscosity of this system with the optimized parameter in other systems, comparisons between the estimated values and those by experiments are shown in Figure 10, with the mean deviation 13.5%. Thereby, to some extent, the present model has strong extrapolation ability.

Discussion
(1) According to the results of our previous paper [21], the model can be extrapolated to Al 2 O 3 , CaO-Al 2 O 3 systems without optimizing new parameters. In this work, the model can also well describe the composition and temperature dependences of viscosity for CaO-FeO-Al 2 O 3 system. Therefore, the present model can be applied to system without SiO 2 .
(2) From the values of parameters shown in Table  2, it can be seen that the following orders exist: , , . Therefore, the deforming ability of bonds around the non-bridging oxygen bonded with different cations increase following the order: Mg 2+ <Ca 2+ <Na + <K + . While, the order is opposed for the bridging oxygen bonded with Al 3+ ions compensated by different cations: Mg 2+ >Ca 2+ >Na + >K + . It is known that the basicity order for these four oxides is K 2 O>Na 2 O>CaO>MgO. Therefore, basic oxide with a higher basicity leads to the forming of a weaker non-bridging oxygen bond, which decreases viscosity; while forming a stronger bridging oxygen bond in a more stability tetrahedron structure by charge compensating Al 3+ ion, which increases viscosity. So basic oxide plays a paradox role in influencing viscosity when Al 2 O 3 exists.
(3) It is found by Sukenaga et al. [12] that the addition of K 2 O to CaO-Al 2 O 3 -SiO 2 slag increases viscosity. However, all the theoretical models can not interpret this abnormal phenomenon, and only give a decreasing tendency. Nakamoto et al. [4] claimed that their model was a progressive for this problem relative to other models. However, it can only decrease the decreasing tendency, but still can not give an increasing tendency. Figure 11 shows the variations of viscosity for CaO-K 2 O-Al 2 O 3 -SiO 2 quaternary system with content of K 2 O, in which the theoretical lines estimated by Nakamoto et al. model and the present model are also given. It can be seen that the present model can well describe this phenomenon.
According to the description of aluminosilicate melt structure in our model, the reason for the increase of viscosity may be that when adding K 2 O to CaO-Al 2 O 3 -SiO 2 system, K + ion will substitute the position of Ca 2+ ion to compensate Al 3+ ion for its higher priority relative to Ca 2+ ion. The displaced Ca 2+ ions will form more non-bridging oxygen ions, and this factor decreases viscosity. But as the contents of basic oxides ( ) are not very high, and there are still enough bridging oxygen ions (including those bonded with Al 3+ ion and Si 4+ ion). In the case that content of K 2 O is smaller than that of Al 2 O 3 (the case of Sukenaga et al. [12]), all the added K + ion will substitute the position of Ca 2+ to compensate Al 3+ ion, Therefore, parts of the bridging oxygen bonded with Al 3+ ion will transform from . According to parameters values shown in Table 2 , the bond around is stronger than that around , thus substitution of K + ion for Ca 2+ ion leads to the forming of more stability bridging oxygen bond, and this factor increases viscosity. When the increasing tendency (resulting from the change of bridging oxygen type) exceeds the decreasing tendency (resulting from the increase of the non-bridging oxygen number), viscosity increases, which may be the cause of experiment finding by Sukenaga et al. [12].
It is conceivable that when the content of Al 2 O 3 is very small relative to the content of K 2 O addition, the decreasing tendency of viscosity will be dominated, since in this case most of the added K + ions do not form bridging oxygen but non-bridging oxygen ( ), thereby, viscosity may decrease. In the limiting case of content of Al 2 O 3 equal to 0, the addition of K 2 O to CaO-SiO 2 melts will obviously decrease viscosity for the decrease of absolute content of SiO 2 and forming of more non-bridging oxygen. In another cases that the contents of basic oxides ( ) is very high to break all the bridging oxygen or only few bridging oxygen ions remaining, the viscosity may also decrease with the addition of K 2 O, because it is the forming of bridging oxygen increasing viscosity, while its number is very limited in this case.
(4) In order to see clearly the accuracy of the present model relative to other models, the mean deviations of our model and models of Riboud et al. [1], NPL [2], Nakamoto et al. [4], Ray et al. [5], Urbain [6], Kondratiev et al. [7] and Iida et al. [10] are given in Table 4, from which it can be seen that our model has strong ability for viscosity estimation, especially for complex aluminosilicate systems and systems absence of SiO 2 .

Conclusions
A structurally based viscosity model is proposed for aluminosilicate melt containing MgO, CaO, FeO, MnO, Na 2 O and K 2 O, in which a simple method of calculating the numbers of different types of oxygen ions is proposed, to describe influence of structure on viscosity. When several basic oxides exist, the priority order for different cations when compensating Al 3+ ions is suggested: K + >Na + >Ca 2+ >Mn 2+ >Fe 2+ >Mg 2+ , according to the coulombic force between the cation and oxygen anion. The values of the optimized parameters indicate that existence of Al 2 O 3 can lead to two inverse influences of basic oxides on viscosity. The basic oxide with a higher basicity can decrease viscosity by forming weaker non-bridging oxygen bond; while it can also increase viscosity by forming stronger bridging oxygen bond by charge compensating Al 3+ ion. The present viscosity model can extrapolate its application range to the system without SiO 2 , and give well results for complex aluminosilicate system. Furthermore, a satisfy interpretation can be given by the present model to the abnormal phenomenon of viscosity increasing when adding K 2 O to CaO-Al 2 O 3 -SiO 2 melt.