Accelerating the Design of Functional Glasses through Modeling

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Accelerating the Design of Functional Glasses through Modeling John C. Mauro, Adama Tandia, K. Deenamma Vargheese, Yihong Z. Mauro, and Morten M. Smedskjaer Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.6b01054 • Publication Date (Web): 03 Jun 2016 Downloaded from http://pubs.acs.org on June 7, 2016

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Chemistry of Materials

Accelerating the Design of Functional Glasses through Modeling John C. Mauro1, Adama Tandia1, K. Deenamma Vargheese1, Yihong Z. Mauro1, and Morten M. Smedskjaer2,* 1 2

Science and Technology Division, Corning Incorporated, Corning, USA

Department of Chemistry and Bioscience, Aalborg University, Aalborg, Denmark *

Corresponding author. [email protected] (M.M.S.)

ABSTRACT Functional glasses play a critical role in current and developing technologies. These materials have traditionally been designed empirically through trial-and-error experimentation. However, here we report recent advancements in the design of new glass compositions starting at the atomic level, which have become possible through an unprecedented level of understanding of glass physics and chemistry. For example, new damage-resistant glasses have been developed using models that predict both manufacturingrelated attributes (e.g., viscosity, liquidus temperature, and refractory compatibility), as well as the relevant end-use properties of the glass (e.g., elastic moduli, compressive stress, and damage resistance). We demonstrate how this approach can be used to accelerate the design of new industrial glasses for use in various applications. Through a combination of models at different scales, from atomistic through empirical modeling, it is now possible to decode the “glassy genome” and efficiently design optimized glass compositions for production at an industrial scale.

Keywords: glasses, computational design, modeling, damage resistance, glassy genome

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INTRODUCTION Functional glasses play a critical role for innovation in a range of industries, including energy, information technology and communications, defense, consumer electronics, transportation, and medicine. Research in functional glasses is at the interface of materials chemistry and physics, engineering, and industrial manufacturing. Advances in glass science and technology have been key enablers for human civilization throughout history.1 However, knowledge of the structure of glass and how it affects macroscopic properties remains challenging. This is because the structure of glass evolves across short- and intermediate-range orders on different length scales, which in turn have different influences on macroscopic properties. The local bonding and packing environments of cations and anions in glasses are composition dependent and typically differ from those in the compositionally analogous crystals. Moreover, nearly all elements of the Periodic Table are available for glass formation, creating infinite possibilities for glass structures.2 These complexities can obfuscate the understanding of composition-structure-property relationships, which are critically needed to design new functional glass compositions with tailored properties. The industrial design of glass compositions requires careful balancing of many parameters simultaneously, since changing the chemical composition affects all properties of the both the final glass and the glass-forming liquid. This includes product attributes such as density, thermal expansion, elastic moduli, chemical durability, relaxation behavior, and hardness, and also manufacturing-related attributes such as batch cost, melting temperatures, refractory compatibility, melt resistivity, and liquidus temperature. This is further complicated by the fact that the properties of glasses, as non-equilibrium systems, depend not only on the standard thermodynamic state variables (P, T, composition), but also on the complete thermal and pressure histories.3 Traditionally, new glass compositions have been developed through time-consuming trial-and-error experiments. For example, this has been done using the simple ideas that higher degree of connectivity and higher cation-oxygen bond strength results in an increase in physical properties such as glass transition temperature and hardness, but these rules are not strictly obeyed even by simple binary oxide glasses.4 Glass design through quantitatively predictive modeling is an alternative approach for accelerating the


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development of new optimized glass compositions. To do so, it is necessary to have an extensive set of modeling techniques available. Examples of such models are illustrated in Figure 1, which exemplifies the range of models from purely empirical to those incorporating detailed fundamental physics. The appropriate choice of modeling technique depends on the nature of the property under study, the availability of highquality data, and the level of physical understanding governing the relevant structure-property relationships in the material. Often a combination of multiple modeling approaches at different levels can give a more comprehensive picture of a given property. Of course, all models need to be validated by experiments, and models are often most effective when developed in close collaboration with experimentalists. Models incorporating a greater degree of physical understanding are often preferable for accurately predicting properties outside of the compositional ranges used for fitting and validation. The design of new damage-resistant glass compositions is especially challenging. Oxide glasses are brittle since they lack a stable shearing mechanism, but theoretically, glasses are among the strongest manmade materials. The practical strength of glass is much lower due to the presence of surface flaws and defects.5 Such flaws reduce the strength by concentrating the applied stress at the flaw tip, normally above 50-100 times the applied stress, and fracture occurs when the stress reaches the level to break atomic bonds. It is therefore of the outmost importance to improve and optimize the mechanical properties of glasses, such as hardness, resistance to crack initiation and propagation, and fracture energy.6 In addition to having obvious safety benefits for the public, improving the mechanical reliability of glass may also lead to new applications, e.g., thinner and stronger glass is essential to enable more fuel-efficient automobiles with reduced energy consumption and noxious emissions. Thinner and more damage-resistant glass materials can also improve the efficiency of energy conversion in photovoltaic solar panels and help enable the next generation of strong, lightweight, and energy efficient architectural and display glass.7 Ion exchange is a post-forming method used to chemically strengthen alkali-containing oxide glasses. The exchange of small monovalent cations (such as Na+) in a glass with larger monovalent cations (such as K+) from an external salt bath creates a compressive stress in the surface layer of the glass.8,9 The compressive stress enhances the mechanical performance of the glass, and ion-exchanged glasses have thus found a variety of applications, including use as scratch-resistant glass covers for personal electronic



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devices.6 The ion exchange properties of the glass are characterized by the magnitude of the compressive stress (CS) and the depth of the compressive layer (DOL) for a given salt bath treatment; these properties strongly depend on glass composition and thermal and pressure history.8-10 In addition to being optimized for high CS and DOL, glass compositions for ion exchange applications should ideally also possess favorable intrinsic mechanical properties. Glass compositions should also be suitable for production on an industrial scale, e.g., they need to have sufficiently high viscosity at the liquidus temperature, sufficiently low melting temperature, and suitable refractory compatibility. In this paper, we provide a concise overview of how ion-exchanged, damage-resistant oxide glasses can be efficiently designed using a range of modeling approaches. We first consider the modeling of manufacturing-related attributes (viscosity, liquidus temperature, and refractory compatibility), and then the customer-facing properties (DOL, CS, and resulting mechanical properties). Finally, we will conclude with an outlook on applications and future research topics.

RESULTS AND DISCUSSION Any glass melting and forming process requires knowledge and careful control of melt viscosity (which controls flow behavior and deformation), liquidus temperature (which defines lowest temperature before crystallization), electrical resistivity (to ensure safe and proper functionality of melting tanks), and refractory compatibility (to ensure stability of melting tanks over time and quality of glass product). Here we focus on the fusion draw process, also known as the overflow or downdraw fusion process, which enables industrial scale manufacturing of homogeneous, thin glass sheets with excellent flatness and surface quality.11 The process involves the flow of glass-forming liquid over a forming structure known as the “isopipe,” which is constructed using a refractory ceramic material such as zircon (ZrSiO4). The liquid overflows the top of the isopipe from both sides, fusing at the bottom of the isopipe to form a single sheet where only the interior of the final sheet has made direct contact with the isopipe, ensuring that both surfaces of the glass are of pristine quality.


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Viscosity. Accurate knowledge of the temperature (T) and composition (x) dependence of shear viscosity, η(T,x), is of critical importance for every step of industrial glass production, i.e., melting, fining, forming, and annealing. For example, for glass melts to be formable on the fusion draw process, the liquidus viscosity (viscosity at the liquidus temperature) must be sufficiently high to avoid crystallization. From the melting at high temperature to the final forming, viscosity varies by over 12 orders of magnitude. Viscosity is also sensitive to even minor changes in chemical composition, including oxide melts where small levels of impurities can significantly influence the flow behavior. However, measurements of η(T,x) for high temperature melts (i.e., in the low viscosity range) are challenging, whereas low temperature measurements are time consuming. Therefore, it is crucial to develop an accurate model of η(T,x). Considering first the temperature dependence of viscosity, the most popular viscosity model has historically been the Vogel-Fulcher-Tammann (VFT) equation, which was proposed in 1925.12 Although VFT has been successful in fitting temperature vs. viscosity data for a variety of oxide glass-forming liquids, it breaks down at low temperatures.12,13 To enable the efficient design of new glass compositions through modeling, it is important to have a model that covers the full range of temperatures with a limited number of variable parameters. This problem has recently been overcome through the introduction of the Mauro-YueEllison-Gupta-Allan (MYEGA) model,14

logη (T ) = logη ∞ + (12 − logη ∞ )

   Tg m − 1 − 1 . exp  T  T   12 − logη ∞

Tg

(1)

Here, Tg is the glass transition temperature, defined by

logη (Tg ) = 1012 Pa ⋅ s ,

(2)

m is the liquid fragility index, defined by15

m≡

d logη d (T / Tg )

,

(3)

T =Tg

and η∞ is the extrapolated infinite-temperature limit of liquid viscosity. The MYEGA equation is derived from the Adam-Gibbs relation16 and the Gupta-Mauro model of temperature-dependent constraints.17 It offers advantages over the VFT equation when fitting a variety of glass-forming liquids with the same



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number of fitting parameters, particularly regarding the low-temperature scaling of viscosity. This is illustrated in Figure 2, in which three viscosity models are fit to the high-temperature viscosity data only of 85 multicomponent silicate liquids. The models are then extrapolated to low temperatures to predict the 1011 Pa s isokom temperature. It is apparent that both the VFT and Avramov-Milchev (AM)18 expressions exhibit systematic errors, albeit in opposite directions. For the VFT model, this systematic error is a direct result of VFT’s spurious assumption of dynamic divergence at a finite temperature, which leads to an overly steep rise in viscosity at low temperatures. The current MYEGA viscosity model of Eq. (1) exhibits no such systematic error when performing low-temperature extrapolation. To predict the composition dependence of viscosity we again consider Eq. (1). Since η∞ has been shown to be a universal composition independent constant equal to 10-2.9 Pa s,19 we need to have accurate models for Tg(x) and m(x). While empirical models are useful for interpolating within the multicomponent composition space, they typically fail to extrapolate beyond the composition ranges used during fitting. Based on topological constraint theory, analytical models have previously been developed for Tg(x) and m(x) for various compositionally simple oxide compositions.20-25 Topological constraint theory of glass was originally developed by Phillips and Thorpe26,27 and assumes that each atom has three translational degrees of freedom that can be removed in the presence of rigid bond constraints (bond length or bond angle constraints). Depending on the relative difference between the network dimensionality (d = 3) and the average number of constraints per atom (n), the network is classified as floppy (n < d), isostatic (n = d), or stressed rigid (n > d). Gupta and Mauro generalized the approach by including an explicit temperature dependence of the constraints.17,20 At higher temperature, more thermal energy is available to overcome the activation barrier associated with a given constraint and the value of n decreases. By counting n as a function of composition and temperature, Tg(x) and m(x) can be calculated.17,20 Recently, new phenomenological models for Tg(x) and m(x) have been developed for multicomponent oxide glasses,28 for which analytical models of Tg(x) and m(x) do not exist. For Tg(x), we obtain28

Tg ( x) =

KR , d − ∑i xi ni / ∑ j x j N j

where


(4)

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 ∑ i x R ,i n i K R = Tg ( x R ) 3 −  ∑ j x R, j N j 

   

(5)

is a scaling constant calculated from the reference composition xR. xi is the mole fraction of oxide component i, ni is the number of rigid constraints contributed by i at the glass transition temperature, and Nj is the number of atoms in the oxide component. The specific values of ni are left as empirical fitting parameters. The composition dependence of liquid fragility can be predicted as28

∆C p , i   , m( x) = m0 1 + ∑ xi ∆S i  i 

(6)

where ∆Cp,i and ∆Si are the change in isobaric heat capacity at the glass transition and the entropy loss at the glass transition due to ergodic breakdown, respectively, for oxide component i. The value of the ratio ∆Cp,i/∆Si is left as an empirical fitting parameter for each viscosity-affecting component i. Figure 3 shows the validation of the model by comparison of predicted versus measured isokom temperatures for 7141 viscosity measurements on 760 different silicate liquids. The RMS error in isokom temperature is only 6.55 K. By using a combination of Eqs. (1), (4), and (6), we are thus able to extrapolate viscosity accurately in both temperature and composition. The success of this new model in explaining both the temperature and composition dependence of viscosity points to the underlying physical mechanism governing this property, viz., the connectivity of the glass-forming network. As the temperature of the glass-forming liquid is lowered during the glass formation process, more rigid bonds are formed. This reduces the number of degrees of freedom of the atoms in the system, thereby inhibiting flow. The same mechanism governs the composition dependence of viscosity: when compositional substitutions are made that increase the number of rigid bonds, the viscosity necessarily increases. The remarkable feature of the current model is that these effects can be accurately captured and quantitatively predicted using one unified approach.

Refractory compatibility. Reactions between the glass-forming liquid and the ceramic isopipe can result in defects during the glass manufacturing process. For example, reactions between glass and zircon refractory



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can result in the introduction of zirconia (ZrO2) defects in the resulting glass sheets through the following zircon dissociation reaction: ZrSiO4 ↔ ZrO2 + SiO2.29 The reaction between the glass-forming liquid and the zircon-containing forming structure is a strong function of temperature. We can define a zircon breakdown temperature (Tbrkdwn), above which the reaction of zircon with the glass-forming liquid becomes thermodynamically favorable toward dissociation into zirconia and silica. Silica is typically the primary glass forming oxide in industrial glasses and forms the backbone of the glassy network. From the perspective of zircon breakdown behavior, pure SiO2 is non-reactive with zirconcontaining forming structures. This is because the presence of SiO2 is favorable to drive the above reaction in the leftward direction, i.e., promoting the formation of ZrSiO4. However, the presence of most other oxides compromises the zircon breakdown performance and may eventually promote the dissociation of ZrSiO4 into ZrO2 and SiO2 and lead to formation of crystalline ZrO2 defects in the glass. In other words, Tbrkdwn is a strong function of the glass composition. Glass compositions having a high reactivity with ZrSiO4 will exhibit a relatively low value of Tbrkdwn, while more inert glass compositions exhibit higher values of Tbrkdwn. The zircon breakdown temperature for any particular glass composition will have a corresponding viscosity at that temperature, which is referred to as the zircon breakdown viscosity (ηbrkdwn) Both Tbrkdwn and the corresponding ηbrkdwn can be experimentally measured, but when designing glass compositions significant cost and time savings can be achieved by tuning the glass composition through modeling. The zircon breakdown temperature can be modeled using a simple empirical form,30 Tbrkdwn = Tb 0 +

∑x Z i

i

,

(7)

i

where xi is the mole fraction of oxide component i except SiO2. The parameters TbO (intercept) and Zi are fitting parameters chosen to optimize agreement with measured values of Tbrkdwn over a range of experimental data. An illustration of the good agreement among measured data and model predictions is shown in Figure 4. ηbrkdwn(x) can be predicted using a combination of Eq. (7) with the viscosity models introduced above.

In addition to ηbrkdwn, another important viscosity that needs to be considered in connection with reducing zirconia defects is the viscosity at which glass-forming liquid needs to be delivered to the forming structure (ηdelivery). As a representative value for a fusion process using an isopipe, ηdelivery is typically on the


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order of 3000-4000 Pa s, and thus has a corresponding delivery temperature (Tdelivery). Tdelivery can also be predicted using the model of Eq. (7). To minimize the level of zirconia defects in the final glass product, Tbrkdwn should be above Tdelivery, i.e., the glass-forming liquid should be delivered to the zircon-containing

forming structure at a temperature below that at which the reaction from ZrSiO4 to ZrO2 + SiO2 becomes favorable. Equivalently, the breakdown viscosity should be less than the delivery viscosity. This problem can be addressed through the modeling-aided design of glass compositions that fulfill the requirements with respect to ηbrkdwn. The zircon compatibility model captures the essential thermodynamics of the zircon breakdown temperature, which, as shown in Fig. 4, exhibits an essentially linear dependence on glass compositions. This indicates that each individual oxide component can be treated independently when calculating the zircon breakdown temperature. However, this result must be combined with the viscosity model of the previous section, since glass forming is performed at a constant delivery viscosity rather than at a constant temperature. Since the viscosity model incorporates a nonlinear dependence on both composition and temperature, the ultimate zircon breakdown viscosity is a strong nonlinear function of composition.

Liquidus temperature. Precise knowledge of liquidus temperature is critical for optimal glass formation on the fusion draw process. The liquidus temperature is the highest temperature at which crystals can exist in thermodynamic equilibrium for a given composition. It is critical to know the liquidus temperature of a composition in order to guarantee high viscosity at the liquidus to avoid crystallization. Since physics-based models of liquidus temperature for multicomponent oxide systems are not readily available, one can rely instead on using a database of high quality measurements to build semi-empirical models with artificial neural networks (ANN)31 or the genetic algorithm (GA). An ANN is an information processing paradigm inspired from the ways biological systems, such as the brain, process information through learning. Generally speaking, ANN modeling has features that allow for enhanced predictive capability, particularly for highly nonlinear properties like liquidus temperature. One key difference between classical regression and ANN is that the latter is a nonlinear function of both the variables and the fitting parameters. This, among other features, will provide additional degrees of freedom



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to the neural network method for achieving a robust fit. A single neuron (Fig. 5a) generates an output equal to an activated form of the linear combination of its weighted inputs. The number of neurons, their respective activation function, and the weighting parameters are part of the model development. The choice of numbers of hidden layers, the respective number of neurons on each layer, and the activation function for each neuron is problem dependent. A general observation is that increasing number of hidden layers can help improve fitting for highly nonlinear responses. We here seek to develop an ANN model through many iterative fittings to calibrate the parameters (w) of a function g ( x k , w) , nonlinear with respect to the parameters and the variables, that is designed to capture

( )

complex relationship between the variables x k and the property y kp that we are modeling. Because of the high nonlinearity of the function with respect to the parameters, there are in general multiple local minima of the cost function, J( w ), defined as M

(

(

J ( w) = ∑ y Pk − g x k , w

))

2

,

(8)

k =1

where y Pk is the measured property for the vector glass composition of rank k, while g ( x k , w) is the predicted property for the same vector at rank k, when using the vector parameter w at the same given state of the iteration. The training is done numerous times to sample different regions of the parameter configuration space. Each training run starts with initial random parameters, w, which are updated iteratively through small perturbations of their initial values for the minimization of the cost function J (w) . A robust least-square-error minimization algorithm, such as Levenberg-Marquardt,32,33 is required to achieve a reliable fit. For highly nonlinear responses like liquidus temperature, it is typical to train the best architecture thousand times and select the model with the lowest root mean square error (RMSE) on the validation set (σv). For the model development, we have used a two-step approach: design of the best architecture, with respect to the number of neurons (or model parameters), and choice of activation function, followed by training of the best architecture loaded with the optimal number of neurons and the best activation function. The choice of the best architecture was made while varying the number of hidden layers and the number of


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neurons on each layer and their respective activation function. All generated model architectures are trained a dozen times. We rank order them based on their complexity and the RMSE on the validation set, as illustrated in Figure 5b. The use of a validation set during the model development is critical since it allows the choice of a robust and parsimonious model that will prevent over-fitting and avoid the modeling of the data noise inside the model. We choose the final architecture to be the one with the lowest standard deviation of the validation set, which we now will subject to an extensive training. There are indeed many more parameters to be looked at and optimized during the training, such as the learning rate, which we have used but not described here. Most of them are outside the scope of this paper. The best architecture we found for the liquidus temperature for a set of 851 silicate oxide glasses with 8 oxides is based on three hidden layers, with (8,5,4) neurons with hyperbolic tangent as activation function. A potential danger with neural networks is over-fitting, which occurs when the model has so many parameters that it has memorized the data. Such model will have a very high R2 value in the data set used for training, but will behave poorly when predicting the response for unseen data. To avoid over-fitting, the use of a validation set during model development is required. Typically during the model development, we use 2/3 of the data for training and the remaining 1/3 for validation. A simple random split is not always satisfactory. One has to make sure that each set is a viable representative of the whole in terms of both the variables (inputs) and in terms of the response. To that effect, one could use the Kullback-Liebler34 divergence method between two probability distributions φ1 and φ2:

∆=

1 [Γ(φ1, φ2 ) + Γ(φ2 , φ1 )] , 2

(9)

+∞  φ ( x)  Γ(φ1 , φ2 ) = ∫ φ1 ( x ) ln 1 dx . −∞  φ2 ( x ) 

(10)

For data sets that are large enough we will also reserve a small subset, a fraction of the validation set, to be used for best model testing at the end of the model development. When developing predictive models, the objective should be to achieve the lowest errors on the training and validation sets simultaneously. Seeking to lower only the error on the training set without any consideration of the validation part, or failing to use a validation set in the process, will lead to models that



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are very accurate in repeating the training data, but that fail to make good predictions on unknown data. Once the best model architecture is found, the training procedure can be further refined by other validation methods, such as conventional cross-validation or the virtual leave-one-out technique as described in Ref. [35]. We here considered a set of 851 collected liquidus temperatures for silicate compositions with up to 8 different components. We used 568 data points for training and 283 data points for validation. The liquidus temperature measurements are from crucible melts with 72 h equilibration time. The R2 of the validation set (Fig. 6b) is similar to that of the training set (Fig. 6a), which is a good sign of a robust training procedure. Nevertheless, one has to be very careful in using such liquidus models too far from the data sets used for parameterization. Accurate modeling of liquidus temperature is known as one of the grand challenges in glass science,36 owing to its complicated nonlinear variation with composition. This nonlinear variation necessarily includes discontinuities in slope as the composition crosses phase boundaries. While no physically derived model can currently predict liquidus temperature for the complicated multi-component systems used for industrial glasses, the neural network approach has proved to be highly valuable in giving an accurate empirical description of this property. Further work is still required to connect this approach to the underlying thermodynamics of the system, viz., the variation in free energy of the liquid and crystalline phases with composition.

Diffusivity. Glass compositions for chemical strengthening should be designed with as fast K+-for-Na+ interdiffusion as possible. Although large values of DOL can also be achieved by increasing the ion exchange temperature, this simultaneously lowers the CS at the surface due to increased rate of stress relaxation at elevated temperature.8,9 A large value of DOL is required to embed surface flaws and defects in the compressive stress layer. For glasses designed with a fixed value of DOL (e.g., 50 µm), significant cost savings can be achieved by increasing the diffusion rate and thus shortening the ion exchange treatment time. To enable the composition dependent prediction of DOL, it is first important to understand the temperature dependence of DOL. The activation barrier for sodium-potassium interdiffusivity (∆HD), can be computed by assuming an Arrhenius temperature dependence,


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log D (T ) = log D(∞) −

∆H D , kT ln 10

(11)

where D is the mutual diffusivity and k is the Boltzmann’s constant. D can be calculated based on the measured K+ diffusion profile.37,38 Here we consider a set of 28 silicate glass compositions, which do not exhibit any clear composition dependence of ∆HD. Indeed, we find that this activation barrier for diffusive hopping is unaffected by the bulk viscous flow behavior over this range of compositions (Figure 7a). This is in agreement with literature findings that ionic diffusion and viscous flow are decoupled at low temperatures,4 i.e., the breakdown of the Stokes-Einstein relation. To predict the composition dependence of DOL, we consider a simple linear regression model. To do so, we calculate D by combining Eq. (11) with the following equation for relaxation of mutual diffusivity,39  t log D (t ) = log D (∞) + [log D (0) − log D (∞ )] exp −  τD

  , 

(12)

where t is the ion exchange duration and τD is the diffusivity relaxation time. We find that τD is typically around 8 h and thus independent of composition for this composition space, i.e., we assume τD = 8 h in our model calculations. The difference between log D(0) and log D(∞) in Eq. (12) has also been found to be relatively composition independent and equal to 0.1 (with D in units of cm2/s). Moreover, we set ∆HD equal to 1.12 eV, since this was the average value found in Figure 7b. Then, we calculate D(∞) using a simple linear regression model, D (∞ ) predicted = [ Al 2 O3 ]D(∞) Al2O3 + [ Na 2 O ]D (∞ ) Na2O + ... ,

(13)

which includes the weighted contributions from the different oxides to D(∞). We minimize the sum of squared errors by changing these values and obtain a very good prediction of the measured DOL values (Figure 8). The most important physical insight from this model is the breakdown of the Stokes-Einstein relation between viscosity and diffusivity. While these properties are intimately connected in the high temperature liquid state, this connection breaks down at low temperatures. The reason for this breakdown is that the viscosity of the network is governed by the topology of the entire glass network, whereas the diffusivity of a given species is governed by the local topology around that species. In a glass designed for optimal chemical



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strengthening, the alkali ions have a low connectivity to the rest of the network, thereby enabling high mobility of the alkali species within a rigid glass network.

Compressive stress. The compressive stress (CS) is given by, CS =

BE (C surf − C bulk ) 1 −ν

=

BE∆C , 1 −ν

(14)

where B is the network dilation coefficient, E is Young’s modulus, υ is Poisson’s ratio, and Csurf and Cbulk are the potassium concentration at surface and bulk, respectively. CS is created at the surface as a result of ion exchange depends on both ion exchange temperature and time, but unlike DOL, which increases with both temperature and time, CS decreases with both temperature and time as a result of stress relaxation. A simple exponential function can be accurately fitted to the time evolution of compressive stress. Extrapolating the compressive stress back to t = 0 gives the compressive stress as a result of purely elastic strain with no stress relaxation. As stress relaxes, elastic strain is converted to plastic strain. The strain ε can be then calculated as the product of CS and (1-υ)/E. Using CS(0), we obtain the purely elastic strain, which is plotted in Figure 9(a) as a function of CS(0). As expected, the linear elastic strain increases approximately linearly with the initial compressive stress. By assuming that 90% of available sodium ions are exchanged at the surface, we can also calculate the lattice dilation coefficient, since ε(0) is equal to the product of B and ∆C(0). A plot of the lattice dilation coefficient as a function of the linear elastic strain is shown in Figure 9(b). Three sets of glasses are shown in the figure, which have different Na2O concentrations in the base glasses. For each set of glasses, B increases linearly with the elastic strain. At higher Na2O concentration, we find that B is smaller for a given elastic strain, i.e., the created compressive stress at a higher elastic strain is smaller. In other words, stress relaxation is facilitated with increasing alkali concentration in the glass. It is generally believed that a high strain point or glass transition temperature is required for obtaining a high compressive stress due to ion exchange. However, when considering the compressive stress extrapolated to zero time, there is no influence of stress relaxation and we thus find a relatively weak correlation between the initial compressive stress and the glass transition temperature (Figure 10(a)). Interestingly, if we transform this plot into a network dilation description, we find a linear dependence of


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network dilation coefficient on glass transition temperature (Figure 10(b)). This shows that network dilation is controlled by the bulk glass network topology, since the glass transition temperature is quantitatively linked with the topological constraints of the glass network.17 Therefore, a prediction of the composition dependence of the network dilation coefficient from constraint theory is possible using the viscosity approach above. Ultimately, this leads to a prediction of the composition dependence of the compressive stress created by ion exchange. The importance of this insight is that the compressive stress and depth of layer of a chemically strengthened glass are controlled by two different aspects of the underlying glass structure. Depth of layer is governed by alkali inter-diffusion, which depends on the local topology of the glass network around the alkali ions. On the other hand, compressive stress is governed by the overall topology of the glass network, having a similar physical origin as viscosity. The downside of this correlation is that the glasses with high compressive stress tend toward higher viscosities, which require higher temperatures for melting.

Elastic moduli. Young’s modulus is one of the key properties that needs to be tailored for designing damage-resistant glasses. This requires an accurate predictive model for Young’s modulus over a wide composition range. One approach is to build predictive empirical models for Young’s modulus from existing data. However, a challenge in building accurate empirical models is the availability of quality data over the interested composition range. Quite often the available data is clustered around a certain composition space, and the model fails outside of this region. This situation can be overcome if existing physics-based models are available for computing the properties of interest. The elastic moduli of the glassy systems can be computed using molecular dynamics (MD) simulations. Such calculation involves generating the glass structure followed by a stress (σ) vs. strain (ε) analysis from which curve the Young’s modulus can be extracted as the slope of σ(ε) at the origin. Figure 11 shows an example of such curve for an aluminosilicate glass composition. Depending on the system size considered in the study such calculation can easily take couple of hours to days. In this study, we have coupled the results from MD with empirical models to develop a robust model for elastic moduli. To that end, 250 glass



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compositions are selected from the composition space we are interested in using a Design of Experiments based approach. Glass structures at 300 K are generated using the procedure described elsewhere.40-42 The elastic modulus is calculated as the slope of the σ(ε) curve in the linear elastic regime (up to a strain of 0.03). This procedure was repeated for all the 250 glasses to create a database for elastic moduli. A neural network model was developed based on this database using a similar procedure as described for liquidus temperature. For validation purposes, we selected a subset of the Young’s modulus data computed by MD to which we added a set of data from measurements. Figure 12 shows the model prediction for this validation set. The predicted elastic moduli from MD and neural network model are found to agree well with measured values. This modeling approach shows that a combination of a physics-based model (molecular dynamics) with an empirical model (neural networks) can offer accurate predictions of elastic moduli even in compositions regimes that have not yet been explored experimentally. While empirical models alone have limited ability to extrapolate to new composition spaces, when combined with specially targeted molecular dynamics simulations, new data can be generated to expand the applicability of the empirical model. This can be accomplished with a high degree of accuracy without the need to melt every glass in the lab.

Tailor-designed boron speciation. Boric oxide (B2O3) is widely used as a network forming constituent in many damage-resistant glasses due to its contribution to high glass forming ability and low melting temperature, and for its favorable impact on thermal, mechanical, and optical properties. B2O3 consists of corner-sharing BO3/2 triangles (BIII), a large fraction of which combine to form three-membered boroxol ring units.43-45 Upon addition of modifier oxide to B2O3, there are two possibilities: (a) creation of a non-bridging oxygen (NBO), rupturing the linkage between two trigonally coordinated BIII groups, or (b) conversion of boron from three-fold coordinated (trigonal boron, BIII) to a four-fold coordinated (tetrahedral boron, BIV) state without the creation of a NBO.4 Due to their differences in molar volume and rigidity, BIII and BIV have significantly different influences on the glass structure and properties. The K+-for-Na+ interdiffusivity decreases significantly upon addition of B2O3 to a silicate glass, but this is primarily due to the presence of BIV units.46,47 The generated compressive stress also decreases


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monotonically with increasing concentration of tetrahedral boron in sodium boroaluminosilicate glasses.47 As such, BIII units are more favorable than BIV for promoting fast and effective ion exchange, i.e., tailoring the boron speciation thus becomes important. The reason for adding B2O3 to damage-resistant glasses despite its negative impact on diffusivity and compressive stress lies in its effect on the mechanical properties. For example, this can be quantified through the indentation (or crack initiation) threshold value. This is the Vickers indentation load at which median or radial cracks extend from the corners of the indent impressions at the glass surface in 50% of indents. Indentation threshold increases with the addition of trigonally boron increases. However, the incorporation of tetrahedrally coordinated boron can have a negative impact on indentation threshold. Trigonal boron creates a more open glass structure that is able to densify upon mechanical compression, whereas tetrahedral boron creates a more rigid network that is less able to densify. The importance of glass densification on the damage-resistance is illustrated in Figure 13 for two glasses exhibiting lateral cracking (Figure 13(a)) and densification (Figure 13(b)) upon deformation and scratching. It is thus clear that damage-resistant glasses should be designed in a way to maintain boron in the threefold coordinated state in order to improve diffusivity and indentation threshold. To do so, both the effects of chemical composition and thermal history on boron speciation should be accounted for. We have recently developed such a predictive model based on statistical mechanics.48 The model accounts for the competition between BIII-to-BIV conversion and formation of NBOs on the SiO4 tetrahedra, with a preference for the former, upon modifier addition in borosilicate glasses. NBOs can form on the BIII groups, but most of the NBOs are associated with SiO4 tetrahedra. In the high-B2O3 glasses, the BIII-to-BIV conversion is much more favorable compared to NBO-on-SiO4 formation. The free energy associated with NBO formation on SiO4 takes an intermediate value compared to the BIII-to-BIV conversion and NBO-on-BIII formation. This value appears close to the energy associated with BIII-to-BIV conversion, i.e., there are competing enthalpic and entropic effects at play, in which the NBO formation on SiO4 becomes entropically favored when more SiO4 units are available. According to this two-state model, the modifiers can be used for either BIII-to-BIV conversion or NBO-on-SiO4 formation when there is limited modifier concentration. The role of the modifiers is determined by the enthalpy difference between the two states (the enthalpic effect governed by ∆H, a fitting parameter) and the number of available boron vs. silicon sites (the entropic effect governed by



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fictive temperature and [SiO2] vs. [B2O3]). The model has also been validated for various sodium boroaluminosilicate glasses.49,50 This model points to the importance of both the initial topology of the glass network and the ability of the glass structure to transform into different configurations. In the case of borosilicate glasses designed with a large initial fraction of threefold-coordinated boron, application of pressure can induce the boron to convert to a higher density tetrahedral coordination. The ability to undergo this topological change effectively creates a new plastic mode of deformation that can increase the damage resistance of a glass to help avoid brittle fracture.

CONCLUSIONS We have shown how a variety of models, ranging from empirical ones to those grounded in fundamental physics, can be used to design new functional glass compositions producible on an industrial scale. Such model-driven development of industrial glass compositions has already shown successful. For example, Corning® Gorilla® Glass 3, an ion-exchanged aluminosilicate glass, was designed to have maximum compressive stress by adjusting the glass composition under the subjection of certain constraints (e.g., liquidus viscosity, zircon breakdown viscosity, ion exchange time, damage resistance etc.). Using the approach presented herein, the modeled composition was indeed confirmed to be the optimal one out of more than 50 compositions subsequently melted.6 These and other prominent advances in understanding the dependence of composition and processing on glass manufacturability, mechanics, and other properties have provided a solid foundation for designing new compositions. This is line with the Materials Genome Initiative, which aims to accelerate the development of new advanced materials through modeling. As such, our “glass genome” approach presented herein can be used to accelerate the design of new functional glasses for use in various applications. Nonetheless, the challenge of improving the mechanical properties of industrial glasses remains and some of the more pressing open questions are: i) How can we control flaw chemistry and stress fields? ii) Can we induce brittle-to-ductile transition in oxide glasses through variation of Poisson’s ratio? iii) What is


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the relation between crack propagation, fracture toughness, scratching behavior, and frangibility? iv) What is the fundamental mechanics controlling crack tip formation and propagation at the nanoscale? v) What is the ultimate achievable strength and toughness that can be attained by using microstructured glasses, glassceramics, and composites? A fundamental understanding of these issues will be essential in the design and development of mechanically reliable oxide glasses capable of withstanding harsh chemo-mechanical environments.

AUTHOR INFORMATION Corresponding author. *E-mail: [email protected] Notes. The authors declare no competing financial interest.

ACKNOWLEDGEMENTS We are grateful for valuable discussions with D.C. Allan, M.J. Dejneka, A.J. Ellison, L.L. Hepburn, T.J. Kiczenski, A.L. Rovelstad, and R.E. Youngman of Corning Incorporated.



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REFERENCES 1. Vogt, T.; Shinbrot, T. Editorial: Overlooking Glass? Phys. Rev. Appl. 2015, 3, 050001. 2. Zanotto, E. D.; Coutinho, F. A. B. How Many Non-crystalline Solids Can Be Made From All the Elements of the Periodic Table? J. Non-Cryst. Solids 2004, 347, 285-288. 3. Ediger, M. D.; Harrowell, P. Perspective: Supercooled Liquids and Glasses. J. Chem. Phys. 2012, 137, 080901. 4. Varshneya, A. K. Fundamentals of Inorganic Glasses; Society of Glass Technology: Sheffield, 2006. 5. Wiederhorn, S. M.; Fett, T.; Guin, J. P.; Ciccotti, M. Griffith Cracks at the Nanoscale. Int. J. Appl. Glass Sci. 2013, 4, 76-86. 6. Wondraczek, L.; Mauro, J. C.; Eckert, J.; Kühn, U.; Horbach, J.; Deubener, J.; Rouxel, T. Towards Ultrastrong Glasses. Adv. Mater. 2011, 23, 4578-4586. 7. Mattos, L. Usable Glass Strength Coalition: Patience, Perseverance, and Progress. Am. Ceram. Soc. Bull.

2013, 91 [4], 22-29. 8. Varshneya, A. K. The Physics of Chemical Strengthening of Glass: Room for a New View. J. Non-Cryst. Solids 2010, 356, 2289-2294. 9. Varshneya, A. K. Chemical Strengthening of Glass: Lessons Learned and yet to be Learned. Int. J. Appl. Glass Sci. 2010, 1, 131-142. 10. Svenson, M. N.; Thirion, L. M.; Youngman, R. E.; Mauro, J. C.; Rzoska, S. J.; Bockowski, M.; Smedskjaer, M. M. Pressure-induced Changes in Interdiffusivity and Compressive Stress in Chemically Strengthened Glass. ACS Appl. Mater. Interfaces 2014, 6, 10436-10444. 11. Käfer, D.; He, M.; Li, J.; Pambianchi, M. S.; Feng, J.; Mauro, J. C.; Bao, Z. Ultra-Smooth and UltraStrong Ion-Exchanged Glass as Substrates for Organic Electronics. Adv. Funct. Mater. 2013, 23, 32333238. 12. Scherer, G. W. Editorial Comments on a Paper by Gordon S. Fulcher. J. Am. Ceram. Soc. 1992, 75, 1060-1062.


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1963, 11, 431-441. 34. Kullback, S.; Liebler, R. A. On Information and Sufficiency. Ann. Math. Stat. 1951, 22, 79-86. 35. Vapnik, V. N. The Nature of Statistical Learning Theory; Springer: Berlin, 1995. 36. Mauro, J. C. Grand Challenges in Glass Science. Front. Mater. 2014, 1, 20. 37. Varshneya, A.K.; Milberg, M. E. Ion Exchange in Sodium Borosilicate Glasses. J. Am. Ceram. Soc.

1974, 57, 165-169. 38. Smedskjaer, M. M.; Zheng, Q. J.; Mauro, J. C.; Potuzak, M.; Mørup, S.; Yue, Y. Z. Sodium Diffusion in Boroaluminosilicate Glasses. J. Non-Cryst. Solids 2011, 357, 3744-3750. 39. Allan, D. C.; Ellison, A. J.; Mauro, J. C. Methods for Producing Ion-exchangeable Glasses. US Patent 8,720,226 2014. 40. Vargheese, K. D.; Tandia, A.; Mauro, J. C. Molecular Dynamics Simulations of Ion-exchanged Glass. J. Non-Cryst. Solids 2014, 403, 107-112. 41. Tandia, A.; Vargheese, K. D.; Mauro, J. C. Elasticity of Ion Stuffing in Chemically Strengthened Glass. J. Non-Cryst. Solids 2012, 358, 1569-1574.


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42. Tandia, A.; Vargheese, K. D.; Mauro, J. C.; Varshneya, A. K. Atomistic Understanding of the Network Dilation Anomaly in Ion-exchanged Glass. J. Non-Cryst. Solids 2012, 358, 316-320. 43. Jellison, G. E.; Panek, L. W.; Bray, P. J.; Rouse, G. B. Determinations of Structure and Bonding in Vitreous B2O3 by means of B10, B11, and O17 NMR. J. Chem. Phys. 1977, 66, 802-812. 44. Micoulaut, M.; Kerner, R.; dos Santos-Loff, D. M. Statistical Modelling of Structural and Thermodynamical Properties of Vitreous B2O3. J. Phys.: Condens. Matter 1995, 7, 8035-8052. 45. Youngman, R. E.; Haubrich, S. T.; Zwanziger, J. W.; Janicke, M. T.; Chmelka, B. F. Short-and Intermediate-Range Structural Ordering in Glassy Boron Oxide. Science 1995, 269, 1416-1420. 46. Smedskjaer, M. M.; Mauro, J. C.; Yue, Y. Z. Cation Diffusivity and the Mixed Network Former Effect in Borosilicate Glasses. J. Phys. Chem. B 2015, 119, 7106-7115. 47. Dejneka, M. J.; Mauro, J. C.; Potuzak, M.; Smedskjaer, M. M.; Youngman, R. E. Chemicallystrengthened Borosilicate Glass Articles. US Patent 9,145,333 2015. 48. Smedskjaer, M. M.; Mauro, J. C.; Youngman, R. E.; Hogue, C. L.; Potuzak, M.; Yue, Y. Z. Topological Principles of Borosilicate Glass Chemistry. J. Phys. Chem. B 2011, 115, 12930-12946. 49. Zheng, Q. J.; Youngman, R. E.; Hogue, C. L.; Mauro, J. C.; Smedskjaer, M. M.; Yue, Y. Z. Structure of Boroaluminosilicate Glasses: Impact of [Al2O3]/[SiO2] Ratio on the Structural Role of Sodium. Phys. Rev. B 2012, 86, 054203. 50. Smedskjaer, M. M.; Youngman, R. E.; Mauro, J. C. Principles of Pyrex® Glass Chemistry: Structure– Property Relationships. Appl. Phys. A 2014, 116, 491-504.



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FIGURES Figure 1. Overview of modeling techniques, from purely empirical models to those firmly grounded in fundamental physics.


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Figure 2. Temperature dependence of liquid viscosity of 85 multicomponent silicate liquids. Predicted vs. measured isokom temperatures found by extrapolation of fitted high temperature data points to lower temperatures using three different models, as described in the text. Data are taken from Ref. [14].

Predicted Isokom Temperature (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1150 1125

VFT Avramov-Milchev MYEGA y=x

1100 1075 1050 1025 1000 1000

1025

1050

1075

1100

1125

Measured Isokom Temperature (K)


1150


Figure 3. Composition dependence of liquid viscosity. Predictions are made using a combination of Eqs. (4)(6). Experimental data points include 7141 viscosity measurements on 760 different silicate compositions. The RMS error in isokom temperature is 6.55 K. Data taken from Ref. [28].

Predicted Isokom Temperature (K)

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2000 RMS error = 6.55 K

1800 1600 1400 1200 1000 1000

1200

1400

1600

1800

Measured Isokom Temperature (K)


2000

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Figure 4. Composition dependence of zircon breakdown temperature (Tbrkdwn). Tbrkdwn has been modelled using Eq. (7). The RMS error in Tbrkdwn is 12.3 K.

1300 RMS error = 12.3 K

Predicted Tbrkdwn (oC)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1200

1100

1000

900 900

1000

1100

1200 o

Measured Tbrkdwn ( C)


1300


Figure 5. Modeling of liquidus temperature using artificial neural networks. (a) A neuron is a nonlinear bounded function g ( x, w) = Ω({x1 , x2 ,..., xn };{w1 , w2 ,..., wp }) , where {x1 , x2 ,..., xn } are the variables,

{w1 , w2 ,..., w p } are the parameters or the weights of the neuron, and Ω is its activation function. In this example, we choose Ω( x) = tanh(x) . (b) Training of generated model architectures. The best model should have the lowest complexity and the smallest root mean square error (RMSE) on the validation set.

(a)

x1 w1 n

x2 . . .

xn

∑x w i

w2 wn

k

i

i =1

 n  g ( x, w) ANN = βk tanh ∑ wi xi   i =1 

Neuron k: Activation Function: tanh (x)

(b)

Training Set Validation Set

RMSE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Model Complexity / Number of iterations


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Figure 6. Composition dependence of liquidus temperature for a set of 851 silicate compositions with up to 8 oxide components. We used 568 data points for training (a) and 283 data points for validation (b).

Predicted Liquidus Temperature (oC)

(a) 1350 1300

Training set R2 = 0.947 RMS error = 10.4 K

1250 1200 1150 1100 1050 1050

1100

1150

1200

1250

1300

1350

o

Measured Liquidus Temperature ( C) (b)

Predicted Liquidus Temperature (oC)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1350 1300

Validation set R2 = 0.947 RMS error = 10.5 K

1250 1200 1150 1100 1050 1050

1100

1150

1200

1250

1300

Measured Liquidus Temperature (oC)


1350


Figure 7. (a) Temperature dependence of the mutual diffusivity for three different glass compositions, showing constant slope and thus activation barrier for sodium-potassium interdiffusivity (∆HD). (b) ∆HD as a function of the activation barrier for viscous flow calculated from the measured viscosity data. (a)

log [Mutual Diffusivity (cm2/s)]

-9.3

Glass 1 Glass 2 Glass 3

-9.6 -9.9 -10.2 -10.5 -10.8 1.38

1.41

1.44

1.47

1.50

1.53

1.56

-1

1000/T [K ] (b)

Activation Barrier for Mutual Diffusivity (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.3

1.2 1.12 eV

1.1

1.0

0.9 5.3

5.4

5.5

5.6

5.7

5.8

5.9

Activation Barrier for Viscous Flow (eV)


6.0

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Figure 8. Comparison of measured and predicted values of DOL for 28 silicate glass compositions.

150

Model DOL (µm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


120 90 60 30 0 0

30

60

90

120

Measured DOL (µm)


150


Figure 9. (a) Linear elastic strain as a function of the initial compressive stress at 410 °C for 28 silicate compositions. (b) Network (lattice) dilation coefficient as a function of the linear elastic strain for three sets of glasses with different contents of Na2O in the base glasses. (a)

Linear Elastic Strain (%)

1.15 1.10 1.05 1.00 0.95 0.90 850

900

950

1000

1050

Initial Compressive Stress (MPa)

(b) Network Dilation Coefficient (ppm/mol%-K2O)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10000 Fixed [Na2O]

9500 9000 Na2O

8500 8000

Na2O

7500 7000 0.90

0.95

1.00

1.05

1.10

Linear Elastic Strain (%)


1.15

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Figure 10. (a) Initial compressive stress at 410 °C as a function of the glass transition temperature. (b) Network (lattice) dilation coefficient as a function of the glass transition temperature. (a)

Initial Compressive Stress (MPa)

1050

1000

950

900

850 580

600

620

640

660

680

o

Glass Transition Temperature ( C)

(b) Network Dilation Coefficient (ppm/mol%-K2O)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10000 9500 9000 8500 8000 7500 7000 580

600

620

640

660

Glass Transition Temperature (oC)


680


Figure 11. Example stress strain curve for aluminosilicate glass computed at 300 K. Inset: elastic modulus is calculated as the slope of this curve in the linear elastic regime (strain up to 0.03). 10

Stress (GPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3

8

2

6

1

4

0 0.00

R2 = 0.987

0.01

0.02

0.03

2 0 0.0

0.2

0.4

0.6

0.8

Axial strain (%)


1.0

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Figure 12. Composition dependence of Young’s modulus. The measured and predicted moduli are in good agreement for a range of different glass compositions.

Predicted Young's modulus (GPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


100

R2 = 0.991 RMS error = 0.64 GPa

80

60

40 40

60

80

100

Measured Young's modulus (GPa)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 13. Damage resistance of two different ion-exchanged glasses subjected to deformation and scratching with Vickers indenter. (a) Glass exhibiting lateral cracking, with many starter flaws in deformed region and high residual stress. (b) Glass exhibiting densification to absorb the force, resulting in no starter flaws in deformed region and lower residual stress. Scale bar = 500 µm. (a)

(b)


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ToC Image Predicted Young's modulus (GPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


R2 = 0.991

100 RMS error = 0.64 GPa

80

60

40 40

60

80

100

Measured Young's modulus (GPa)


Accelerating the Design of Functional Glasses through Modeling

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