Glasses are both practically important materials and theoretically fascinating systems. They are fascinating because 1) they are rigid solids without being crystalline, and 2) they become solid not abruptly but gradually as the temperature is lowered, by becoming progressively more viscous until we can no longer say they are liquids. They are practically important, because window glass, optical fibers, amorphous silicon for electronics, and many polymer materials including polystyrene and “vinyl” polymers (PVC) are glasses.
We have performed simulations both of idealized “hard-sphere” models for fundamental studies of the glass transition, and atomistic simulations of real polymers to understand how free surfaces and plasticizers shift the glass transition in these materials.
Geometric criterion for Tg
“What makes dense, nearly-glassy liquids so sluggish?”
As liquids become dense or cold, one of two things eventually happen: either they crystallize, or they become glassy. The former is familiar — the molecules adopt a lattice of positions, and the system behaves as a solid. The latter is mysterious — the molecular positions still “look like a liquid”, but nothing much moves, at least on the timescale of our patience.
The simplest example of a liquid that can become glassy is a theorist’s playground, the “hard sphere” liquid, in which molecules are replaced by impenetrable spherical particles. Only the volume fraction of particles matters to describe the state we are in. Here we can ask, how does particle mobility diminish as volume fraction increases?
For a system to be liquid, particles must be able to somehow change their neighbors. Otherwise, can only rattle about in the “cage” formed by their immediate neighbors, and we have a glass.
Top: a particle can “hop” to change neighbors, if there is room enough next to the particle for a new particle to be inserted. Bottom: even if a particle cannot hop, it can change neighbors by moving in its “cage” to come close to a particle on the edge of its shell of neighbors.
To analyze which particle motions in what arrangements can change neighbors, it is helpful to consider moving only one particle at a time, with the others held fixed. If the system is not too dense, a particle can can sometimes “hop”, into an adjacent empty space large enough to hold a particle, thereby changing some neighbors (figure, top).
When the system is yet more dense, hopping events are very rare, because there just aren’t any empty spaces to hop into. Still there are ways to rearrange neighbors — as in a very crowded party, when people move closer to someone at the edge of their circle, and away from others behind them.
We can give a mathematical definition to this kind of motion, which we call a “T1 event”. And we have programmed a computer to count, in a given arrangement of particles, which particles can add or lose a neighbor, by the effect of a T1 event. Those that are so hemmed in they cannot change neighbors, we call “T1-inactive”.
As the system becomes more dense, the number of T1-inactive particles increase. Eventually they are common enough that T1-inactive particles are neighbors of other T1-inactive particles, such that a cluster of T1-inactive particles spans the system. When such a cluster spans the system, we say it “percolates”. The “percolation threshold” is when the particles are concentrated enough that a large system essentially always percolates.
As the particle concentration increases, the probability of particles to be T1-inactive increases, such that eventually clusters of T1-inactive particles percolate across the system. This happens at the same concentration at which particle diffusion appears to plummet to zero.
Remarkably, we find that the percolation threshold for T1-inactive particles, is practically coincident with the concentration at which the particles no longer diffusion in our computer simulations. That is, percolation of T1-inactive particles corresponds to the onset of the glass.
Zhou, Y. and Milner, S. T. “T1 Process and Dynamics in Glass-Forming Hard-Sphere Liquids” Soft Matter 11, no. 13 (2015): 2700–2705. doi:10.1039/C4SM02459A
Plasticizers and local Tg shifts
“How do plasticizers soften glassy polymers?”
Commercial glassy polymers like polyvinylchloride (PVC) come in an amazing variety of forms— from sturdy white pipes for plumbing, to pliable vinyl for upholstery and children’s toys. This variety of material properties depends on “plasticizers” — small-molecule additives that shift the glass transition of the polymer, to make it softer around room temperature than it is as a pure material.
The plasticizer molecules, which for PVC are most commonly alkyl phthalates, somehow add “mobility” to the polymer. Although these additives have been used commercially for many decades, microscopic understanding of how they work is lacking. This is an important question, because phthalates have been implicated as endocrine disruptors and are being phased out of many products, so designing replacement molecules is of pressing interest.
Snapshot of atomistic simulation of PVC plasticized by DEHP (diethylhexylphthalate).
To explore how plasticizers work microscopically, we simulated a melt of PVC with various levels of added phthalates, and observed how readily the polymers diffused over very short distances (Angstroms) and timescales (nanoseconds), as we cooled the system towards the glass transition. That turns out to be enough to assess the effectiveness and learn about mechanism of the plasticizers.
We found that the shift in the mobility of a given PVC segment was determined by how many plasticizer molecules were nearby, within a few Angstroms away. In fact, the mobility shift of each segment was a sum of the shifts caused by each plasticizer molecule, with the size of the effect diminishing with distance over a few nanometers.
Left: local Tg shift versus distance to nearest plasticizer, for a family of different plasticizer concentrations. Black dots are predictions, based on sum of contributions from each nearby plasticizer molecule (cartoon at right).
This suggests a physical picture in which each plasticizer molecule “loosens up” the polymer in its immediate vicinity, which transitively “loosens” up the polymer immediately beyond, but with diminishing effect. It will be interesting to see whether other plasticizer molecules behave similarly in simulations, and whether they have the same range of effect but different magnitudes depending on how effective they are.
Zhou, Y. and Milner, S. T. “Average and Local T-G Shifts of Plasticized PVC From Simulations” Macromolecules 51, no. 10 (2018): 3865–3873. doi:10.1021/acs.macromol.8b00271