Atom Delocalization and Fluctuation Hole Formation in Amorphous Organic Polymers and Inorganic Glasses

Abstract

A viewpoint is developed that formation of a hole in amorphous substances corresponds to the atom delocalization process, i.e., its fluctuation limiting shift from the temporary equilibrium state. The molecular-kinetic processes proceeding in liquids and amorphous rigid solids do not depend on the classical van der Waals free volume, “void statistical space between atoms,” but mainly depend on the fluctuation dynamic free volume, which coincides with the fluctuation free volume caused by the atom delocalization. The atom delocalization process (local configuration structural change) plays the key role in the viscous flow of glass-forming liquids. The softening of glasses (at T_g) and their frozen reversible deformation (at 20°C) are characterized by the same molecular mechanism, namely, the atom delocalization. From the same positions as noted above, the effect of plasticity of inorganic glasses and amorphous organic polymers was considered.

Derivation of Formula (3)

A necessary condition for the active (excited) atom delocalization, i.e., its ultimate displacement from the equilibrium position, is the formation of the elementary fluctuation volume $\Delta {{v}_{{\text{e}}}}$ near it. The kinetic units of amorphous substances differ from each other mainly in the value of the fluctuation volume $\tilde {v}$, which is formed in their neighborhood. The number of particles dn with a fluctuation volume from $\tilde {v}$ to $\tilde {v} + d\tilde {v}$ is described by the function [16]

$$dn = A\exp \left( { - \frac{H}{{kT}}} \right)d\tilde {v} = A\exp \left[ { - \frac{{\left( {{{p}_{{\text{i}}}} + p} \right)\tilde {v}}}{{kT}}} \right]d\tilde {v},$$

where $H = ({{p}_{{\text{i}}}} + p)\tilde {v}$ is the enthalpy of the volume formation $\tilde {v}$. The sum of internal p_i and external p pressures (p_i + p) acts as a constant independent of $\tilde {v}$. After normalization

$$\int\limits_0^N {dn} = N,$$

making it possible to obtain the corresponding equation (N is the number of atoms)

$$A = \frac{{\left( {{{p}_{{\text{i}}}} + p} \right)N}}{{kT}},$$

the number of delocalized atoms N_e with $\tilde {v} \geqslant \Delta {{v}_{{\text{e}}}}$ is found

$$\begin{gathered} {{N}_{{\text{e}}}} = \int\limits_{\Delta {{v}_{{\text{e}}}}}^\infty {\frac{{\left( {{{p}_{{\text{i}}}} + p} \right)N}}{{kT}}\exp \left[ { - \frac{{\left( {{{p}_{{\text{i}}}} + p} \right)\tilde {v}}}{{kT}}} \right]} d\tilde {v} \\ = \;N\exp \left[ { - \frac{{\left( {{{p}_{{\text{i}}}} + p} \right)\Delta {{v}_{{\text{e}}}}}}{{kT}}} \right], \\ \end{gathered} $$

((A1))

from which formula (3) follows

$$\frac{{{{N}_{{\text{e}}}}}}{N} = \exp \left( { - \frac{{\Delta {{\varepsilon }_{{\text{e}}}} + p\Delta {{v}_{{\text{e}}}}}}{{kT}}} \right),$$

((A2))

where $\Delta {{\varepsilon }_{{\text{e}}}} = {{p}_{{\text{i}}}}\Delta {{v}_{{\text{e}}}}$ is the energy of the delocalized atoms.

At the usual atmospheric external pressure p ≈ 1 atm, in formula (A1), the internal pressure is much greater than the external one p_i ≫ p as solids and liquids have p_i ≈ (10⁴–10⁵) atm [3]. In this regard, equality (A2) may be approximated in this case as

$$\frac{{{{N}_{{\text{e}}}}}}{N} \cong \exp \left( { - \frac{{\Delta {{\varepsilon }_{{\text{e}}}}}}{{kT}}} \right).$$

((A3))