E6 Network Connectivity: Phillips-Thorpe Theory of Constraints

Topic

The average coordination number of this glass—defined as the mean number of neighboring atoms or tetrahedra to which each atom in the network is connected—determines, according to Phillips-Thorpe constraint theory, the average number of mechanical constraints acting on each atom within that network; each bond and each angle between neighboring bonds imposes a geometric constraint on the relative positions of the atoms they connect. A higher average coordination number results, under this same theory, in a higher number of mechanical constraints per atom.

This number of mechanical constraints per atom, in turn, classifies the glass network into one of three possible categories: a flexible network, where the number of mechanical constraints per atom is less than the available degrees of freedom per atom, leaving the network with enough remaining degrees of freedom to deform without stretching bonds or opening bond angles; an over-constrained network, where the number of mechanical constraints per atom exceeds the available degrees of freedom, meaning the network cannot deform without straining its own bonds or angles; and, between these extremes, an isostatically rigid network, where the number of mechanical constraints per atom exactly equals the available degrees of freedom. Depending on which of these three cases applies regarding the number of mechanical constraints per atom, the entire network is classified as flexible, isostatically rigid, or over-constrained.

This classification of the network into one of the three categories ultimately predicts the glass-forming ability of the composition: an isostatically rigid network yields the best glass-forming ability because it possesses both sufficient rigidity to maintain a stable structure and just enough flexibility to rearrange itself without accumulating excessive internal stress during cooling. If the network deviates from this isostatically rigid state—shifting toward either the flexible or the over-constrained regime—that optimal glass-forming ability deteriorates.

Thus, this model—based on the average coordination number and the Phillips-Thorpe constraint theory—describes a network connectivity that is conceptually distinct from that of the crystalline ceramic described by another model in the same set: whereas in that ceramic the coordination number arises from the periodic packing of ions within a crystalline unit cell, here the average coordination number is derived by counting, atom by atom, the number of bridging and non-bridging oxygens connected to each tetrahedron within this amorphous network—consistent with the model in this set that describes the glass's amorphous structure. The fact that this network connectivity—quantified here in terms of constraints per atom—is identical to the connectivity determined by the modifier-to-former ratio in the model describing the glass's chemical composition means that both models describe the same network from different perspectives; furthermore, the absence of long-range order in this connectivity is precisely what is experimentally verified by yet another model in the set—the one describing the glass's spectroscopy and diffraction—through the amorphous halo observed in its diffraction pattern.

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