# Calculation of glass properties

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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. It was first practised at the end of the 19th century by A. Winkelmann and O. Schott. 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. [2]

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. [2] 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). [3]

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. [4] [5] [6] 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 Ci and bi 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 [7] and Gehlhoff et al. [8] published similar additive glass property calculation models. Schott’s additivity principle is still widely in use today in glass research and technology. [9] [10]

Additivity Principle:   ${\displaystyle {\mbox{Glass Property}}=b_{0}+\sum _{i=1}^{n}b_{i}C_{i}}$

## Global models

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, [13] Kucuk, [14] Priven, [15] Choudhary, [16] Mazurin, [17] and Fluegel [18] [19] 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 adjacent diagram, 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. [20] 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 GE-SYSTEM [21] SciGlass [22] and Interglad, [23] sometimes combined with the finite element method. For estimating the melting enthalpy thermodynamic databases are used.

### Linear regression

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. [24]

${\displaystyle {\mbox{Glass Property}}=b_{0}+\sum _{i=1}^{n}\left(b_{i}C_{i}+\sum _{k=i}^{n}b_{ik}C_{i}C_{k}\right)}$

### Non-linear regression

The liquidus temperature has been modeled by non-linear regression using neural networks [26] and disconnected peak functions. [25] The disconnected peak functions approach is based on the observation that within one primary crystalline phase field linear regression can be applied [27] 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. [28]

### Finite element method

For modeling of the glass flow in a glass melting furnace the finite element method is applied commercially, [29] [30] 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. [21] [31] This can be performed, e.g., by simplex search, or in a spreadsheet as follows:

1. Listing of the desired properties;
2. Entering of models for the reliable calculation of properties based on the glass composition, including a formula for estimating the production costs;
3. Calculation of the squares of the differences (errors) between desired and calculated properties;
4. Reduction of the sum of square errors using the Solver option [32] 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. [33] The combination of several glass models together with further relevant technological and financial functions can be used in six sigma optimization.

## Related Research Articles

Glass is a non-crystalline, often transparent amorphous solid, that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenching) of the molten form; some glasses such as volcanic glass are naturally occurring. The most familiar, and historically the oldest, types of manufactured glass are "silicate glasses" based on the chemical compound silica, the primary constituent of sand. Soda-lime glass, containing around 70% silica, accounts for around 90% of manufactured glass. The term glass, in popular usage, is often used to refer only to this type of material, although silica-free glasses often have desirable properties for applications in modern communications technology. Some objects, such as drinking glasses and eyeglasses, are so commonly made of silicate-based glass that they are simply called by the name of the material.

Melting, or fusion, is a physical process that results in the phase transition of a substance from a solid to a liquid. This occurs when the internal energy of the solid increases, typically by the application of heat or pressure, which increases the substance's temperature to the melting point. At the melting point, the ordering of ions or molecules in the solid breaks down to a less ordered state, and the solid "melts" to become a liquid.

Magma is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural satellites. Besides molten rock, magma may also contain suspended crystals and gas bubbles.

The melting point of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends on pressure and is usually specified at a standard pressure such as 1 atmosphere or 100 kPa.

Carl Zeiss was a German scientific instrument maker, optician and businessman who founded the workshop of Carl Zeiss in 1846, which is still in business today as Carl Zeiss AG. Zeiss gathered a group of gifted practical and theoretical opticians and glass makers to reshape most aspects of optical instrument production. His collaboration with Ernst Abbe revolutionized optical theory and practical design of microscopes. Their quest to extend these advances brought Otto Schott into the enterprises to revolutionize optical glass manufacture. The firm of Carl Zeiss grew to one of the largest and most respected optical firms in the world.

Borosilicate glass is a type of glass with silica and boron trioxide as the main glass-forming constituents. Borosilicate glasses are known for having very low coefficients of thermal expansion, making them more resistant to thermal shock than any other common glass. Such glass is subjected to less thermal stress and can withstand temperature differentials without fracturing of about 165 °C (297 °F). It is commonly used for the construction of reagent bottles and flasks as well as lighting, electronics and cookware.

ZBLAN is the most stable, and consequently the most used, fluoride glass, a subcategory of the heavy metal fluoride glass (HMFG) group. Typically its composition is 53% ZrF4, 20% BaF2, 4% LaF3, 3% AlF3 and 20% NaF. ZBLAN is not a single material but rather has a spectrum of compositions, many of which are still untried. The biggest library in the world of ZBLAN glass compositions is currently owned by Le Verre Fluore, the oldest company working on HMFG technology. Hafnium fluoride is chemically similar to zirconium fluoride, and is sometimes used in place of it.

Hot melt adhesive (HMA), also known as hot glue, is a form of thermoplastic adhesive that is commonly sold as solid cylindrical sticks of various diameters designed to be applied using a hot glue gun. The gun uses a continuous-duty heating element to melt the plastic glue, which the user pushes through the gun either with a mechanical trigger mechanism on the gun, or with direct finger pressure. The glue squeezed out of the heated nozzle is initially hot enough to burn and even blister skin. The glue is sticky when hot, and solidifies in a few seconds to one minute. Hot melt adhesives can also be applied by dipping or spraying, and are popular with hobbyists and crafters both for affixing and as an inexpensive alternative to resin casting.

Crown glass is a type of optical glass used in lenses and other optical components. It has relatively low refractive index (≈1.52) and low dispersion. Crown glass is produced from alkali-lime silicates containing approximately 10% potassium oxide and is one of the earliest low dispersion glasses.

Friedrich Otto Schott was a German chemist, glass technologist, and the inventor of borosilicate glass. Schott systematically investigated the relationship between the chemical composition of the glass and its properties. In this way, he solved fundamental problems in glass properties, identifying compositions with optical properties that approach the theoretical limit. Schott's findings were a major advance in the optics for microscopy and optical astronomy. His work has been described as "a watershed in the history of glass composition".

Soda–lime glass, also called soda–lime–silica glass, is the most prevalent type of glass, used for windowpanes and glass containers for beverages, food, and some commodity items. Some glass bakeware is made of soda-lime glass, as opposed to the more common borosilicate glass. Soda–lime glass accounts for about 90% of manufactured glass.

Fractional crystallization, or crystal fractionation, is one of the most important geochemical and physical processes operating within crust and mantle of a rocky planetary body, such as the Earth. It is important in the formation of igneous rocks because it is one of the main processes of magmatic differentiation. Fractional crystallization is also important in the formation of sedimentary evaporite rocks.

The liquidus temperature, TL or Tliq, specifies the temperature above which a material is completely liquid, and the maximum temperature at which crystals can co-exist with the melt in thermodynamic equilibrium. It is mostly used for impure substances (mixtures) such as glasses, alloys and rocks.

Glass databases are a collection of glass compositions, glass properties, glass models, associated trademark names, patents etc. These data were collected from publications in scientific papers and patents, from personal communication with scientists and engineers, and other relevant sources.

Ceramic chemistry studies the relationship between the physical properties of fired ceramics and ceramic glazes and their chemistry. Although ceramic technicians have long understood many of these relationships, the advent of computer software to automate the conversion from batch to formula and analysis has brought this science within the reach of many more people. Physical properties of glazes in fired products are directly related to the chemistry. Properties of glass melts like viscosity and surface tension are also principally products of chemistry.

The glass–liquid transition, or glass transition, is the gradual and reversible transition in amorphous materials from a hard and relatively brittle "glassy" state into a viscous or rubbery state as the temperature is increased. An amorphous solid that exhibits a glass transition is called a glass. The reverse transition, achieved by supercooling a viscous liquid into the glass state, is called vitrification.

In condensed matter physics and physical chemistry, the terms viscous liquid, supercooled liquid, and glassforming liquid are often used interchangeably to designate liquids that are at the same time highly viscous, can be or are supercooled, and able to form a glass.

In glass physics, fragility characterizes how rapidly the dynamics of a material slow down as it is cooled toward the glass transition: materials with a higher fragility have a relatively narrow glass transition temperature range, while those with low fragility have a relatively broad glass transition temperature range. Physically, fragility may be related to the presence of dynamical heterogeneity in glasses, as well as to the breakdown of the usual Stokes–Einstein relationship between viscosity and diffusion.

Glass frit bonding, also referred to as glass soldering or seal glass bonding, describes a wafer bonding technique with an intermediate glass layer. It is a widely used encapsulation technology for surface micro-machined structures, e.g., accelerometers or gyroscopes. This technique utilizes low melting glass and therefore provides various advantages including that viscosity of glass decreases with an increase of temperature. The viscous flow of glass has effects to compensate and planarize surface irregularities, convenient for bonding wafers with a high roughness due to plasma etching or deposition. A low viscosity promotes hermetically sealed encapsulation of structures based on a better adaption of the structured shapes. Further, the coefficient of thermal expansion (CTE) of the glass material is adapted to silicon. This results in low stress in the bonded wafer pair. The glass has to flow and wet the soldered surfaces well below the temperature where deformation or degradation of either of the joined materials or nearby structures occurs. The usual temperature of achieving flowing and wetting is between 450 and 550 °C.

The glass forming ability of gallium(III) sulfide and lanthanum sulfide was discovered in 1976 by Loireau-Lozac’h, Guittard, and Flahut. This family of chalcogenide glasses, referred to as gallium lanthanum sulfide (Ga-La-S) glasses, have a wide region of glass formation centred about the 70Ga2S3:30La2S3 composition and can readily accept other modifiers into their structure. This means that Ga-La-S can be compositionally adjusted to give a wide variety of optical and physical properties. Optically, Ga-La-S has a high refractive index, a transmission window covering most of the visible wavelengths and extending to about 10 µm and a low maximum phonon energy, approx. 450 cm−1. Thermally, the refractive index of Ga-La-S glasses has a strong temperature dependence and low thermal conductivity, which results in strong thermal lensing. However, the high glass transition temperature of Ga-La-S makes it resistant to thermal damage, it has good chemical durability and unlike many chalcogenides which are based on arsenic, its glass components are non-toxic. A clear advantage over other chalcogenides is its high lanthanum content which allows excellent rare-earth solubility and dispersion of the ions in the glass matrix for active devices. Ga-La-S can exist in both glassy and crystalline phases, in a glassy phase, it is a semiconductor with a bandgap of 2.6 eV corresponding to a wavelength of 475 nm; consequently Ga-La-S glass takes a deep orange colour. As with all chalcogenides the phase of the bulk is determined by two key factors; the material composition and the rate at which the molten material is cooled. These variables can be controlled to manipulate the final phase of the material.

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