The Glass Transition
There is movement.
Thus: The world is not full.
There is empty space.
The nothing, the void, does exist.
Thus: The world consists of the existing,
the hard and full, and the void:
Of ‘atoms and the void’
-Democritus ca. 400 BC (trans. Karl Popper)
At the level of everyday experience, a glass appears different “in kind” from a liquid. Window glass is solid and brittle and does not flow (N.B., cathedral windows do NOT flow ). This implies that there is a “transition” of some sort occurring at an ideal glass transition temperature, T0, such that for temperatures below T0, the material is a glass while able T0, it is a liquid. The principle evidence for this view point is that the viscosity is reasonably well fit by the function Bexp(-A/(T-T0)) where both A and B are constants. Consequently, the viscosity is infinite for all T’s below T0, and since solids have an infinite viscosity, there must be a liquid to solid transition at T0.
There are several difficulties with the “different in kind” perspective. First, in order to conduct equilibrium experiments, the time scale of the experiment must be longer than the “relaxation time” of the glass, and since the relaxation time is proportional to the viscosity, the time needed to perform an experiment diverges to infinity as T0 is approached. Consequently, the lowest temperature where experiments can be conveniently conducted is the laboratory glass transition temperature, or simply the glass transition temperature, Tg, where T0 is often about 50°C below Tg. This means that T0 cannot be measured directly, but is the result of an extrapolation from T’s above Tg. A second difficulty is that there is no clear structural change at Tg (or T0). In contrast with the freezing transition where the nucleus and growth formation of crystals clearly indicates the transition, the liquid-glass transition shows no such signature. The Xray scattering function of a glass looks like that of the liquid. A third difficulty is that the language used to describe most other transitions, thermodynamics, is not directly applicable to the glass transition.
The alternate perspective, that the glass is nothing but a thick liquid, and is “different in degree”, as a number of advantages. It explains why the structure of a glass is that same as that of the liquid and what thermodynamics does not predict a transition. On the other hand, it cannot explain the existence of a non-zero T0.
We have analyzed computer simulation results for properties related to the viscosity by developing “scalar metrics” that collapse dynamical data over many different pressures and temperatures onto a single curve. An example of one such scalar metric is the packing fraction, [eta], which is the volume occupied by the molecules divided by the total volume. As implied by the above Democritus quote, one expects that at sufficiently high packing fraction, space will be full and nothing will move (i.e., the viscosity will be infinite). The following example, Fig. 1, presents the results  of Molecular Dynamics simulations of freely-jointed, 10-site chains. The dynamical variable of interest is the diffusion constant which approaches zero as the viscosity approaches infinity. This implies that the system dynamics can be “explained” by a functional dependence on the distance to the glass transition. We find that the powerlaw form works well and that lnD*~([eta]0-[eta])^2.0 for this molecular type.
Figure 1. Dimensionless diffusion coefficients, D, are plotted in a number of typical manners for the chain-center-of-mass of freely-jointed chains. The squares are for repulsive sites and the inverted triangles are for attractive sites. In figures A), B), and C), the diffusion coefficient is reduced by the site mass, m; Lennard-Jones well depth, [epsilon]; and Lennard-Jones site diameter, σ. In D), the diffusion coefficient is reduced by m; the chain length, N; the kinetic energy, kT; and the effective hard site diameter, d; where k is the Boltzmann constant and T is the temperature. In A) the reduced diffusion coefficient is plotted as a function of inverse temperature; in B), as a function of the volume per site (where V is the inverse density); in C), as a function of the packing fraction ([eta]=π[rho]d3/6); and, in D), as a function of reduced packing fraction where [eta]0 is the location of the ideal glass transition. Error-bars are smaller than symbol size.
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