Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 297

Implicit in the original PCT and BCT theories was the notion that constraints are fixed for good – either intact (= 1) or broken (= 0) – and that they do not vary with temperature (T). Thermal energy was implicitly neglected in the original theories which were thus valid only at T = 0 K. To remedy this problem, Gupta [5] introduced the concept of a T‐dependent bond constraint. He argued that, if E_i is the energy of a certain class of bonds, then the value of the corresponding constraint h_i(T) should be expressed by a Boltzmann expression:

(12)

where k_B is the Boltzmann constant. Note that the value of h_i always lies in the interval [0,1], being zero in the high‐temperature limit, equal to 1 at sufficiently low temperatures, and decreasing monotonically with increasing T. Physically, a fractional value of a bond constraint means that only a fraction of ith type of bonds are intact at a given instant. One may associate a characteristic temperature T_i for the ith type of constraint as follows:

(13)

so that this constraint can be considered effectively as broken (= 0) for T > T_i and intact (=1) for T < T_i. For both stretching and bending constraints, this formalism has been validated by comparisons of the standard deviations of the partial distributions calculated for glassy and crystalline alkali disilicates as a function of temperature in MD simulations [9].

The average degree of freedom per vertex, f(T), in the network thus becomes T‐dependent and (for d = 3) is given by

(14)

Since h_i is a decreasing function of T, f(T) always increases with temperature.