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Statistics of modifier distributions in mixed network glasses
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10.1063/1.4773356
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1 Science and Technology Division, Corning Incorporated, Corning, New York 14831, USA
J. Chem. Phys. 138, 12A522 (2013)
/content/aip/journal/jcp/138/12/10.1063/1.4773356
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## Figures

FIG. 1.

Calculated fraction of modifiers associated with network former A as a function of ΔH/kT f , where ΔH = 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 ⟩.

FIG. 2.

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 ΔH/kT f . The mixed network former effect is clearly visible for ΔH/kT f > 0, where there is a nonlinear scaling of p A with [A]/([A] + [B]).

FIG. 3.

Calculated fraction of modifiers associated with network former A as a function of ΔH/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 ΔH/kT f = 0 or [M] = [A] + [B]. All other cases will lead to a mixed network former effect.

FIG. 4.

Calculated fraction of modifiers associated with network former A as a function of ΔH/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. , the modifiers are distributed proportionally between the two network formers in the limits of ΔH/kT f = 0 and [M] = [A] + [B].

FIG. 5.

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.

FIG. 6.

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

FIG. 7.

Relaxation of network speciation from a high fictive temperature (ΔH/kT f = 1) to a low fictive temperature (ΔH/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.

/content/aip/journal/jcp/138/12/10.1063/1.4773356
2013-01-07
2014-04-17

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