Calculation of glass properties

[
refractive index. [ [http://glassproperties.com/refractive_index/ Calculation of the Refractive Index of Glasses] ] ]

The calculation of glass properties (glass modeling) is used to predict glass properties of interest or glass behavior under certain conditions (e.g., during production) without experimental investigation, based on past data and experience, with the intention to save time, material, financial, and environmental resources, or to gain scientific insight. The combination of several glass models together with other relevant functions can be used for optimization and six sigma procedures. In the form of statistical analysis glass modeling can aid with accreditation of new data, experimental procedures, and measurement institutions (glass laboratories).

History

Historically, the calculation of glass properties is directly related to the founding of glass science. At the end of the 19th century the physicist Ernst Abbe developed equations that allow calculating the design of optimized optical microscopes in Jena, Germany, stimulated by co-operation with the optical workshop of Carl Zeiss. Before Ernst Abbe's time the building of microscopes was mainly a work of art and experienced craftsmanship, resulting in very expensive optical microscopes with variable quality. Now Ernst Abbe knew exactly how to construct an excellent microscope, but unfortunately, the required lenses and prisms with specific ratios of refractive index and dispersion did not exist. Ernst Abbe was not able to find answers to his needs from glass artists and engineers; glass making was not based on science at this time.Werner Vogel: "Glass Chemistry"; Springer-Verlag Berlin and Heidelberg GmbH & Co. K; 2nd revised edition (November 1994), ISBN 3540575723]

In 1879 the young glass engineer Otto Schott sent Abbe glass samples with a special composition (lithium silicate glass) that he had prepared himself and that he hoped to show special optical properties. Following measurements by Ernst Abbe, Schott's glass samples did not have the desired properties, and they were also not as homogeneous as desired. Nevertheless, Ernst Abbe invited Otto Schott to work on the problem further and to evaluate all possible glass components systematically. Finally, Schott succeeded in producing homogeneous glass samples, and he invented borosilicate glass with the optical properties Abbe needed. These inventions gave rise to the well-known companies Zeiss and Schott Glass (see also Timeline of microscope technology). Systematic glass research was born. In 1908, Eugene Sullivan founded glass research also in the United States (Corning, New York). [ [http://www.corning.com/inside_corning/our_heritage.aspx Eugene Sullivan and Corning Glass Works] ]

At the beginning of glass research it was most important to know the relation between the glass composition and its properties. For this purpose Otto Schott introduced the additivity principle in several publications for calculation of glass properties. [A. Winkelmann, O. Schott: "Über die Elastizität und über die Druckfestigkeit verschiedener neuer Gläser in ihrer Abhängigkeit von der chemischen Zusammensetzung"; Ann. Physik Chemie, vol. 51, 1894, p 697.] [A. Winkelmann, O. Schott: "Über thermische Widerstandscoefficienten verschiedener Gläser in ihrer Abhängigkeit von der chemischen Zusammensetzung", Ann. Physik Chemie, vol. 51, 1894, p 730] [A. Winkelmann, O. Schott: "Über die specifischen Wärmen verschieden zusammengesetzter Gläser", Ann. Physik Chemie, vol. 49, 1893, p 401.] This principle implies that the relation between the glass composition and a specific property is linear to all glass component concentrations, assuming an ideal mixture, with "C_i" and "b_i" representing specific glass component concentrations and related coefficients respectively in the equation below. The additivity principle is a simplification and only valid within narrow composition ranges as seen in the displayed diagrams for the refractive index and the viscosity. Nevertheless, the application of the additivity principle lead the way to many of Schott’s inventions, including optical glasses, glasses with low thermal expansion for cooking and laboratory ware (Duran), and glasses with reduced freezing point depression for mercury thermometers. Subsequently, English [S. English: "The effect of composition on the viscosity of glass"; J. Soc. Glass Technol., 1924, no. 8, p 205-48; 1925, no. 9, p 83-98; 1926, no. 10, p 52–66.] and Gehlhoff et al. [G. Gehlhoff, M. Thomas; Z. techn. Physik 6 (1925), p 544; Z. techn. Physik 7 (1926), p 105 and p 260; "Lehrbuch der technischen Physik", J. A. Barth-Verlag, Leipzig, 1924, p 376.] published similar additive glass property calculation models. Schott’s additivity principle is still widely in use today in glass research and technology. [T. Lakatos, L.-G. Johansson and B. Simmingsköld, "Viscosity temperature relations in the glass system SiO2-Al2O3-Na2O-K2O-CaO-MgO in the composition range of technical glasses"; Glass Technology, vol. 13, no. 3, June 1972, p 88-95.] ["High temperature glass melt property database for process modeling"; Eds.: Thomas P. Seward III and Terese Vascott; The American Ceramic Society, Westerville, Ohio, 2005, ISBN 1-57498-225-7]

:Additivity Principle: $mbox\{Glass\; Property\}\; =\; b\_0\; +\; sum\_\{i=1\}^n\; b\_iC\_i$

Global models

[
alkali oxide, some properties show non-additive behavior. The image shows, that the viscosity of a glass is significantly decreased. [ [http://glassproperties.com/viscosity/mixed-alkali-effect-viscosity/ The Mixed-Alkali Effect for the Viscosity of Glasses] ] ]

[
accuracy of modern glass literature data for the density at 20°C in the binary system SiO2-Na2O. [http://glassproperties.com/errors/ Overview, Measurement Errors of Glass Properties] ] ]

Schott and many scientists and engineers afterwards applied the additivity principle to experimental data measured in their own laboratory within sufficiently narrow composition ranges (local glass models). This is most convenient because disagreements between laboratories and non-linear glass component interactions do not need to be considered. In the course of several decades of systematic glass research thousands of glass compositions were studied, resulting in millions of published glass properties, collected in glass databases. This huge pool of experimental data was not investigated as a whole, until Bottinga, [Y. Bottinga, D. F. Weill: "The viscosity of magmatic silicate liquids: a model for calculation", Am. J. Sci., vol. 272, May 1972, p 438-475.] , Kucuk [A. Kucuk, A. G. Clare, L. Jones: "An estimation of the surface tension of silicate glass melts at 1400°C using statistical analysis"; Glass Technol., vol. 40, Oct 1999, no. 5, p 149-153.] , Priven [A. I. Priven: "General Method for Calculating the Properties of Oxide Glasses and Glass-Forming Melts from their Composition and Temperature"; Glass Technology, vol. 45, Dec 2004, no. 6, p 244-254. ( [http://www.sciglass.info/Publications/Priven.pdf Full text article] )] , Choudhary [M. K. Choudhary, R. M. Potter: "Heat Transfer in Glass-Forming Melts"; Chapter 9 in: "Properties of Glass-Formation Melts" ed. by D. L. Pye, A. Montenaro, I. Joseph; CRC Press, Boca Raton, Florida, May 2005, ISBN 1-57444-662-2] , Mazurin [O. V. Mazurin, O. A. Prokhorenko: "Electrical conductivity of glass melts"; Chapter 10 in: "Properties of Glass-Forming Melts" ed. by D. L. Pye, I. Joseph, A. Montenaro; CRC Press, Boca Raton, Florida, 2005, ISBN 1-57444-662-2.] , and Fluegel [A. Fluegel: "Glass Viscosity Calculation Based on a Global Statistical Modeling Approach"; Glass Technol.: Europ. J. Glass Sci. Technol. A, vol. 48, 2007, no. 1, p 13-30. ( [http://glassproperties.com/viscosity/Viscosity_2006_AFluegel.pdf Full text article] )] [A. Fluegel: "Global Model for Calculating Room-Temperature Glass Density from the Composition"; J. Am. Ceram. Soc., vol. 90, no. 8, August 2008, p 2622-2625. ( [http://dx.doi.org/10.1111/j.1551-2916.2007.01751.x Full text article] )] published their global glass models, using various approaches. In contrast to the models by Schott the global models consider many independent data sources, making the model estimates more reliable. In addition, global models can reveal and quantify "non-additive" influences of certain glass component combinations on the properties, such as the "mixed-alkali effect" as seen in the diagram on the right, or the "boron anomaly". Global models also reflect interesting developments of glass property measurement accuracy, e.g., a decreasing accuracy of experimental data in modern scientific literature for some glass properties, shown in the diagram. They can be used for accreditation of new data, experimental procedures, and measurement institutions (glass laboratories). In the following sections (except melting enthalpy) "empirical" modeling techniques are presented, which seem to be a successful way for handling huge amounts of experimental data. The resulting models are applied in contemporary engineering and research for the calculation of glass properties.

Non-empirical ("deductive") glass models exist. [Milos B. Volf: "Mathematical Approach to Glass"; Glass Science and Technology, vol. 9, Elsevier, 1988, ISBN 0-444-98951-X] They are often not created to obtain reliable glass property predictions in the first place (except melting enthalpy), but to establish relations among several properties (e.g. atomic radius, atomic mass, chemical bond strength and angles, chemical valency, heat capacity) to gain scientific insight. In future, the investigation of property relations in deductive models may ultimately lead to reliable predictions for all desired properties, provided the property relations are well understood and all required experimental data are available.

Methods

Glass properties and glass behavior during production can be calculated through statistical analysis of glass databases such as SciGlass [ [http://www.sciglass.info/ SciGlass] ] and Interglad, [ [http://www.newglass.jp/interglad_6/gaiyo/info_e.html Interglad] ] sometimes combined with the finite element method. For estimating the melting enthalpy thermodynamic databases are used.

Linear regression

[
₂-Na₂O. Dummy variables can be used to quantify systematic differences of whole dataseries from one investigator.]

If the desired glass property is not related to crystallization (e.g., liquidus temperature) or phase separation, linear regression can be applied using common polynomial functions up to the third degree. Below is an example equation of the second degree. The "C"-values are the glass component concentrations like Na₂O or CaO in percent or other fractions, the "b"-values are coefficients, and "n" is the total number of glass components. The glass main component silica (SiO₂) is excluded in the equation below because of over-parametrization due to the constraint that all components sum up to 100%. Many terms in the equation below can be neglected based on correlation and significance analysis. Systematic errors such as seen in the picture are quantified by dummy variables. Further details and examples are available in an online tutorial by Fluegel. [A. Fluegel: [http://glassproperties.com/principle/Statistics_Glass_2007.pdf Statistical Regression Modeling of Glass Properties - A Tutorial] ]

:$mbox\{Glass\; Property\}\; =\; b\_0\; +\; sum\_\{i=1\}^n\; left(\; b\_iC\_i\; +\; sum\_\{k=i\}^n\; b\_\{ik\}C\_iC\_k\; ight)$

Non-linear regression

[
₂-Na₂O-CaO using disconnected peak functions based on 237 experimental data from 28 investigators. Error = 15°C. [http://glassproperties.com/liquidus/ Glass Liquidus Temperature Calculation using disconnected peak functions] ] ]

The liquidus temperature has been modeled by non-linear regression using neural networks [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.] and disconnected peak functions. The disconnected peak functions approach is based on the observation that within one primary crystalline phase field linear regression can be applied [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"; Journal of Materials Science, vol. 20, 2005, no. 12, p 3346-3357. ( [http://dx.doi.org/10.1557/JMR.2005.0424 Full text article] )] and at eutectic points sudden changes occur.

Glass melting enthalpy

The glass melting enthalpy reflects the amount of energy needed to convert the mix of raw materials (batch) to a melt glass. It depends on the batch and glass compositions, on the efficiency of the furnace and heat regeneration systems, the average residence time of the glass in the furnace, and many other factors. A pioneering article about the subject was written by Carl Kröger in 1953. [Carl Kröger: "Theoretischer Wärmebedarf der Glasschmelzprozesse (Theoretical heat demand of glass melting processes)"; Glastechnische Berichte, vol. 26, 1953, no. 7, p 202-214. (in German language)] More recently, R. Conradt at RWTH Aachen, Germany, is a leading expert in this field. [ [http://www.ghi.rwth-aachen.de/www/index.html Institut für Gesteinshüttenkunde] ]

Finite element method

For modeling of the glass flow in a glass melting furnace the finite element method is applied commercially, [ [http://www.gsl.cz/en/products/furnace-design.html Glass Service, Furnace Design] ] [Brochure: [http://www.fluent.com/solutions/brochures/glass.pdf Flow modeling software for the glass industry] , Fluent Inc.] based on data or models for viscosity, density, thermal conductivity, heat capacity, absorption spectra, and other relevant properties of the glass melt. The finite element method may also be applied to glass forming processes.

Optimization

It is often required to optimize several glass properties simultaneously, including production costs. [ [http://glassproperties.com/optimization/ Glass property optimization] ] This can be performed, e.g., by simplex search, or in a spreadsheet as follows:
# Listing of the desired properties;
# Entering of models for the reliable calculation of properties based on the glass composition, including a formula for estimating the production costs;
# Calculation of the squares of the differences (errors) between desired and calculated properties;
# Reduction of the sum of square errors using the Solver option [ [http://www.solver.com/suppstdsolver.htm Excel Solver] ] in Microsoft Excel with the glass components as variables. Other software (e.g. Microcal Origin) can also be used to perform these optimizations.It is possible to weight the desired properties differently. Basic information about the principle can be found in an article by Huff et al. [N. T. Huff, A. D. Call: "Computerized Prediction of Glass Compositions from Properties"; J. Am. Ceram. Soc., vol. 56, 1973, p 55–57.] The combination of several glass models together with further relevant technological and financial functions can be used in six sigma optimization.

ee also

*Glass batch calculation

References