Glass Liquidus Temperature Calculation
Modeling of glass liquidus temperatures using disconnected peak functions 
Abstract: Glass liquidus temperatures do not form a smooth continuous surface in the composition space, but sharp minima occur at eutectic points. Therefore, liquidus temperatures cannot be described using one low-order polynomial function in a wide composition space, which is possible for many other glass properties. In this work, a method is presented where several disconnected peak functions are fitted to a broad compositional liquidus surface. The fitting method allows free movement of the peak functions, thereby detecting continuous regions that coincide with one or several similar primary crystalline phase fields. Only one dominant peak is considered within a continuous region, allowing sharp minima at intersection points of two or several major primary phase fields. Linear regression is possible within one continuous region, including all advantages of linear modeling as compared to a non-linear approach, such as the neural networks method by Dreyfus et al. . The presented method is successfully applied to 1000 data from the SciGlass database in the six-component system SiO2-Na2O-CaO-Al2O3-MgO-K2O. Linear regression over the entire composition space is impossible, but a coefficient of determination, R², of 0.96 is obtained by using disconnected peak functions.
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Hrma et al. compared several liquidus temperature models . They came to the conclusion that models not considering primary crystalline phase fields  are often not reliable. Primary phase fields are taken into account by Dreyfus et al. , Rao et al. , Hanni et al. , and Fluegel .
Primary phase fields: Primary crystalline phase fields for glasses indicate the composition region (field) in which one specific phase will crystallize first (primary phase) upon cooling. The diagram below shows an example from the binary glass system SiO2-Li2O. Primary phase fields are marked by (1), (2), and (3), with (1) = SiO2 primary phase, (2) = Li2SiO3 primary phase, and (3) = Li4SiO4 primary phase . The 91 published data in the diagram are from SciGlass. The model fit was obtained using disconnected peak functions :
 A. Fluegel: "Modeling of glass liquidus temperatures using disconnected peak functions"; Presentation at ACerS 2007 Glass and Optical Materials Division Meeting, Rochester, NY, USA.
 C. Dreyfus, G. Dreyfus: "A machine learning approach to the estimation of the liquidus temperature of glass-forming oxide blends"; J. Non-Cryst. Solids, vol. 318, 2003, p 63-78.
Dreyfus et al. developed a large liquidus temperature model based on 1741 liquidus temperature data in the system SiO2-Na2O-CaO-Al2O3-MgO from SciGlass. The model is based on neural networks regression with 4-8 hidden neurons. Although the modeling procedure is described in detail, the model itself is not published, i.e., the weights in the neural network are not mentioned.
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 Q. Rao, G. F. Piepel, P. Hrma, J. V. Crum: "Liquidus temperatures of HLW glasses with zirconium-containing primary crystalline phases"; J. Non-Cryst. Solids, vol. 220, 1997, p 17-29.
 J. B. Hanni, E. Pressly, J. V. Crum, K. B. C. Minister, D. Tran, P. Hrma, J. D. Vienna: "Liquidus temperature measurements for modeling oxide glass systems relevant to nuclear waste vitrification"; J. Mater. Res., vol. 20, Dec 2005, no. 12, p 3346-3357.
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