^{1}

### Abstract

The constituents of any network glass can be broadly classified as either network formers or network modifiers. Network formers, such as SiO_{2}, Al_{2}O_{3}, B_{2}O_{3}, P_{2}O_{5}, etc., provide the backbone of the glass network and are the primary source of its rigid constraints. Network modifiers play a supporting role, such as charge stabilization of the network formers or alteration of the network topology through rupture of bridging bonds and introduction of floppy modes. The specific role of the modifiers depends on which network formers are present in the glass and the relative free energies of modifier interactions with each type of network former site. This variation of free energy with modifier speciation is responsible for the so-called mixed network former effect, i.e., the nonlinear scaling of property values in glasses having fixed modifier concentration but a varying ratio of network formers. In this paper, a general theoretical framework is presented describing the statistical mechanics of modifier speciation in mixed network glasses. The model provides a natural explanation for the mixed network former effect and also accounts for the impact of thermal history and relaxation on glass network topology.

I would like to acknowledge many valuable conversations with Yihong Mauro.

I. INTRODUCTION

II. STATISTICAL THEORY

A. Multivariate hypergeometric distribution

B. Noncentral hypergeometric distribution

C. Incorporation of Boltzmann weighting factors

III. RESULTS

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Glasses
- 74.0
- Enthalpy
- 14.0
- Statistical mechanics models
- 9.0
- Borosilicate glasses
- 8.0
- Ozone
- 7.0

## Figures

Calculated fraction of modifiers associated with network former *A* as a function of Δ*H*/*kT* _{ f }, where Δ*H* = *H* _{ B } − *H* _{ A }, *k* is Boltzmann's constant, and *T* _{ f } is the fictive temperature of the glass. Here systems are considered where the modifier concentration equals half the total concentration of network formers, i.e., [*M*]/([*A*] + [*B*]) = 0.5. The concentrations of network formers *A* and *B* are adjusted to four different proportions, keeping the total concentration of network formers constant. All of the modifiers not associated with network former *A* are associated with network former *B*, such that ⟨*p* _{ A }⟩ + ⟨*p* _{ B }⟩ = 1, where ⟨*p* _{ A }⟩ is given by the curves in the figure and ⟨*p* _{ B }⟩ is obtained by 1 − ⟨*p* _{ A }⟩.

Calculated fraction of modifiers associated with network former *A* as a function of Δ*H*/*kT* _{ f }, where Δ*H* = *H* _{ B } − *H* _{ A }, *k* is Boltzmann's constant, and *T* _{ f } is the fictive temperature of the glass. Here systems are considered where the modifier concentration equals half the total concentration of network formers, i.e., [*M*]/([*A*] + [*B*]) = 0.5. The concentrations of network formers *A* and *B* are adjusted to four different proportions, keeping the total concentration of network formers constant. All of the modifiers not associated with network former *A* are associated with network former *B*, such that ⟨*p* _{ A }⟩ + ⟨*p* _{ B }⟩ = 1, where ⟨*p* _{ A }⟩ is given by the curves in the figure and ⟨*p* _{ B }⟩ is obtained by 1 − ⟨*p* _{ A }⟩.

Expectation value of the fraction of modifiers (⟨*p* _{ A }⟩) associated with network former *A* as a function of its molar fraction, [*A*]/([*A*] + [*B*]). Here it is assumed that the modifier concentration is maintained at half the total concentration of network formers, i.e., [*M*]/([*A*] + [*B*]) = 0.5. Calculations are performed for varying levels of Δ*H*/*kT* _{ f }. The mixed network former effect is clearly visible for Δ*H*/*kT* _{ f } > 0, where there is a nonlinear scaling of *p* _{ A } with [*A*]/([*A*] + [*B*]).

Expectation value of the fraction of modifiers (⟨*p* _{ A }⟩) associated with network former *A* as a function of its molar fraction, [*A*]/([*A*] + [*B*]). Here it is assumed that the modifier concentration is maintained at half the total concentration of network formers, i.e., [*M*]/([*A*] + [*B*]) = 0.5. Calculations are performed for varying levels of Δ*H*/*kT* _{ f }. The mixed network former effect is clearly visible for Δ*H*/*kT* _{ f } > 0, where there is a nonlinear scaling of *p* _{ A } with [*A*]/([*A*] + [*B*]).

Calculated fraction of modifiers associated with network former *A* as a function of Δ*H*/*kT* _{ f } for systems with different modifier concentrations and a fixed ratio of network formers, [*A*]/([*A*] + [*B*]) = 0.25. The modifiers are distributed proportionally between network formers *A* and *B* in the limits of Δ*H*/*kT* _{ f } = 0 or [*M*] = [*A*] + [*B*]. All other cases will lead to a mixed network former effect.

Calculated fraction of modifiers associated with network former *A* as a function of Δ*H*/*kT* _{ f } for systems with different modifier concentrations and a fixed ratio of network formers, [*A*]/([*A*] + [*B*]) = 0.25. The modifiers are distributed proportionally between network formers *A* and *B* in the limits of Δ*H*/*kT* _{ f } = 0 or [*M*] = [*A*] + [*B*]. All other cases will lead to a mixed network former effect.

Calculated fraction of modifiers associated with network former *A* as a function of Δ*H*/*kT* _{ f } for systems with different modifier concentrations and an equal fraction of network formers *A* and *B*, i.e., [*A*]/([*A*] + [*B*]) = 0.5. As in Fig. 3 , the modifiers are distributed proportionally between the two network formers in the limits of Δ*H*/*kT* _{ f } = 0 and [*M*] = [*A*] + [*B*].

Calculated fraction of modifiers associated with network former *A* as a function of Δ*H*/*kT* _{ f } for systems with different modifier concentrations and an equal fraction of network formers *A* and *B*, i.e., [*A*]/([*A*] + [*B*]) = 0.5. As in Fig. 3 , the modifiers are distributed proportionally between the two network formers in the limits of Δ*H*/*kT* _{ f } = 0 and [*M*] = [*A*] + [*B*].

Fraction of modifiers associated with each of three network formers (*A*, *B*, and *C*) as a function of (*H* _{ C } − *H* _{ A })/*kT* _{ f }. Here we have set (*H* _{ B } − *H* _{ A })/*kT* _{ f } = 1 and fixed the concentrations of network formers and modifiers as indicated in the figure. In this system having three network formers, a complicated mixed network former effect can be observed due to (i) the difference in enthalpies of formation associated with bonding between the network modifier and each of the network formers, and (ii) the fact that the modifier concentration is less than the total concentration of network formers.

Fraction of modifiers associated with each of three network formers (*A*, *B*, and *C*) as a function of (*H* _{ C } − *H* _{ A })/*kT* _{ f }. Here we have set (*H* _{ B } − *H* _{ A })/*kT* _{ f } = 1 and fixed the concentrations of network formers and modifiers as indicated in the figure. In this system having three network formers, a complicated mixed network former effect can be observed due to (i) the difference in enthalpies of formation associated with bonding between the network modifier and each of the network formers, and (ii) the fact that the modifier concentration is less than the total concentration of network formers.

Probability density of *p* _{ A }, the fraction of modifiers associated with network former *A*. Probability density functions are shown for four different levels of Δ*H*/*kT* _{ f } and fixed concentrations of network formers and modifiers as indicated in the figure. In the limit of Δ*H*/*kT* _{ f } → 0, the modifier statistics follow a hypergeometric distribution. In the limit of high Δ*H*/*kT* _{ f }, the distribution collapses to a Dirac delta function.

Probability density of *p* _{ A }, the fraction of modifiers associated with network former *A*. Probability density functions are shown for four different levels of Δ*H*/*kT* _{ f } and fixed concentrations of network formers and modifiers as indicated in the figure. In the limit of Δ*H*/*kT* _{ f } → 0, the modifier statistics follow a hypergeometric distribution. In the limit of high Δ*H*/*kT* _{ f }, the distribution collapses to a Dirac delta function.

Relaxation of network speciation from a high fictive temperature (Δ*H*/*kT* _{ f } = 1) to a low fictive temperature (Δ*H*/*kT* _{ f } = 4). Here it is assumed that structural relaxation follows a stretched exponential decay with a relaxation time τ and stretching exponent β = 3/7.

Relaxation of network speciation from a high fictive temperature (Δ*H*/*kT* _{ f } = 1) to a low fictive temperature (Δ*H*/*kT* _{ f } = 4). Here it is assumed that structural relaxation follows a stretched exponential decay with a relaxation time τ and stretching exponent β = 3/7.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content