Bond constraint theory applied to complex phosphate glasses
While the mystery of the glassy state and its fundamental relation to the glass transition temperature (Tg) is often touted as the main driving force behind research on how it depends on the glass’ chemical composition, this also elicits a great deal of interest from the glass industry since the Tg is a very important parameter in virtually every modern manufacturing process. This is the background the brought about the Temperature Dependent Bond Constraint Theory (TBCT) by Gupta and Mauro in 2009. The TDBCT is based on the Bond Constraint Theory originally developed by Phillips and Thorpe to help elucidate the composition dependency of the glass forming ability of chalcogenide glasses. By abstracting the glass network as a static mechanical scaffold, they found that the glass compositions with greater glass forming ability generally are “isostatic”, where the network has no excess of dangling bonds (or floppy modes – a “floppy network”) and no redundant bonds (a “stressed-rigid network”), corresponding with an average coordination number of 2.4. Gupta and Mauro extended the theory by introducing the concept of temperature dependency to the constraints, which are organized in hierarchical order and become broken at certain temperatures. This allows the theory to treat the problem of the compositional dependency of the glass transition by linking the appearance of the floppy modes (or broken constraints) to the system’s configurational entropy. The greatest appeal of the TDBCT is its simplicity: with just the knowledge of how the glass structure evolves with changing chemical composition one could easily model the glass transition temperature. But it also depends on several assumptions in order to be applied, some of which are stronger than others. In order to evaluate how the TDBCT holds against closer scrutiny we based our analysis on phosphate glasses, which not only have very precise and easy to calculate evolution of the phosphate network with increasing modifier concentration, but also a plethora of reliable experimental data available in literature. This allows us to subtract the influence of the glass network from the experimental number of constraints and focus on the effect of other variables. We find that for binary phosphate glasses up until the metaphosphate composition the influence of the constraints added by the modifiers are of paramount importance to the overall behaviour of the glass. These constraints are not tied to the coordination number of the first coordination shell around the modifier, but are instead determined by the strength of the electrostatic interactions between the modifier and the surrounding non-bridging oxygens. Coupled with that, we also found that the modifier contribution depends on whether it is located in an “isolated” site (meaning that the majority of the surrounding oxygens are double-bonded to the phosphorus) or a “crosslinking” site (where the majority of the oxygens are non-bridging), and, in the case of mixed alkali ultra- and metaphosphates, whether or not one can find different modifiers in the immediate vicinity. In addition to that, experimental measurements of the glass transition temperature of silver metaphosphate – silver halide glasses are much higher than expected from theoretical estimations; this effect is attributed to the conformation change the phosphate network goes through, transitioning from primarily chains to a mixture of chains and rings. When analyzing the viscosity and glass transition temperature of binary alkali borate glasses there are some inconsistencies that can be attributed to the glass system not complying to one of the base assumptions of the TDBCT: the average energy barrier associated with cooperative motion, represented by B(x) in the Adam-Gibbs viscosity equation, is not held constant throughout the whole compositional range. The same behaviour could also be discerned in binary alkali silicate glasses, accounting for the observed severe drop on the glass transition temperature with the addition of relative small amounts of modifiers. Finally, the outcome of the current development of the Temperature Dependent Bond Constraint Theory emphasizes its ambivalent character. On one hand, the TBCT has been shown to be a powerful model that is easy to apply and to expand, allowing it to model more complex glass compositions; on the other hand, reasonable results are only guaranteed through judiciously selecting a glass system that complies with the underlying theoretical assumptions, and the expansions to the theory highlight its empirical nature, since the additional parameters can’t be calculated from first principles nor have any clear physical meaning.
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