Dynamics of glassy systems > Why is Tg in polymer thin films different from its bulk value?

A variety of experimental findings seem to agree that the glass transition temperature T_{g} in a thin polymer film is typically lower near a free surface,
and often higher near a strongly attractive substrate, than the bulk T_{g} value. This finding is of practical importance in for example photoresists for microelectronics,
when very fine-scale patterns are rendered in thin polymer films that need to be robustly rigid in later processing steps.

The effects of the free surface or substrate on the glass transition appear to extend many nanometers away into the adjacent material. Glass transitions are not well understood, but are hypothesized to result from dynamical arrest of dense liquids, resulting from increasingly hindered local motion, in some cooperative fashion. Why then should the effects of free surfaces extend many nanometers away? Is this a polymeric effect, or not? Does the range depend on polymer molecular weight? Experiments are not clear on this key point.

Two theoretical ideas have been suggested to account for these free surface and substrate effects on glass transitions. One, proposed by Long and Lequeux, makes no explicit reference to polymer connectivity; one, proposed by de Gennes, is explicitly polymeric. In collaboration with Prof. Jane Lipson (Department of Chemistry, Dartmouth College), we have constructed quantitative theories for local glass transition temperatures, which can be compared to experiment. Both approaches at present have shortcomings - assuming the published experimental results themselves have no artifacts.

The first approach, suggested originally by Long and Lequeux, is to assert that a sample is glassy when slowly relaxing regions percolate across the system. Free surfaces suppress percolation of glassy regions, because some paths that would in an infinite sample have led to connection across the system (red cluster, top figure) are interrupted by the missing half-space. Strongly attractive substrates enhance percolation of glassy regions by provide a "short circuit" path to slowly relaxing regions immediately adjacent to the substrate (green cluster, bottom figure).

We carried out averages over a randomly occupied lattice, with occupation probability representing the increasing prevalence of slowly relaxing regions as the temperature decreased,
in order to make quantitative calculations of the range of effects on T_{g} at free surfaces and substrates. With a reasonable (nm-sized) definition of the "local region" (lattice site),
we find that although the percolation model can reasonably account for the range of substrate effects, the effect of a free surface is too short-ranged to account for experiments.

The second approach, suggested by de Gennes, addresses the effect of a free surface, by hypothesizing that it serves as a source of "free volume" or "mobility" that can travel,
carried by "kinks", along the backbone of polymer chains, at temperatures well below the bulk T_{g}, for a considerable distance along the chain.
We have developed a quantitative theory for both freestanding polymer films and the upper regions of thick supported films, based on this physical idea.

The explicitly polymeric mechanism has important qualitative consequences, independent of specific implementation.
First, the length along the chain that kinks can travel decreases as temperature is decreased, which leads both to a maximum range of effect (set by a parameter in the model),
and to greater depression of T_{g} nearer to a free surface. Second, the mechanism leads to molecular weight dependence of the free surface effect.
Kink transmission is interrupted by the end of a chain, so free surface effects cannot persist further than of order R_{g} (radius of gyration).

Finally, in films thin compared to R_{g}, molecular weight dependence would be expected to be absent. This is certainly a robust feature of our calculations,
but not of ellipsometric measurements of T_{g} for thin freestanding films. Resolving this important discrepancy between experiment and the only extant
microscopic theoretical model that accounts for molecular weight dependence, is a subject of continuing work.

- J.E.G. Lipson and S.T. Milner, "Percolation model of interfacial eﬀects in polymeric glasses", Eur. Phys. J. B 72, 133–137 (2009);
- S. T. Milner and J. E. G. Lipson, "Delayed glassiﬁcation model for free-surface suppression of Tg in polymer glasses", to be published;
- J. E. G. Lipson and S. T. Milner, "Local and average glass transitions in polymer thin ﬁlms", to be published.)

Dynamics of glassy systems > 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 very simplest example of a liquid that can become glassy is a theorist's playground, the "hard sphere" liquid, in which the molecules are replaced by impenetrable spherical particles. Only the volume fraction of particles matters to describe what state we are in. Here we can ask, how does the mobility of the particles diminish as the volume fraction increases?

We would like to be able to distinguish configurations in which spheres can "move about", from configurations in which although particles may rattle about in the "cage" formed by their neighbors, nothing can move. We have found it useful to define a mobile particle as one which, with all others frozen, can "hop" — can move in such a way as to change at least one "Voronoi neighbor" (a mathematician’s definition of 'nearest neighbor'). If no particles can change neighbors, clearly nothing can really change — we are in a glass.

It turns out to be possible to geometrically construct which particles in a configuration can hop, based on a recent construction for the "free volume" of a particle — with all other particles frozen in place, the volume over which the particle’s center can translate without overlapping another particle. If a particle’s free volume is "hemmed in" entirely by its (Voronoi) neighbors, it cannot hop; contrariwise, if it can "escape its cage" while exploring its free volume, it can change neighbors, and so hop.

With computer simulations, we have determined the "hopping fraction" h(Φ) of particles able to hop, as a function of system volume fraction Φ. Remarkably, h(Φ) is well described by a Vogel-Fulcher dependence on volume fraction, vanishing at Φ = 0.654 ± 0.004 (very near random close packing). This suggests our purely geometric definition of mobile particles is dynamically relevant.

The potentially hopping particles have a probability distribution P(vf) for their free volume vf that behaves differently from that of all particles in the system. Both distributions are quite broad for a given Φ, and both move towards smaller vf values as the system becomes more dense. The hopping particles, naturally, tend to be more prevalent among particles with higher vf.

It is interesting to consider the fraction H(vf) of particles with a given free volume that able to hop. This is an s-shaped function, zero for small vf and unity for large vf. It can be interpreted as the integral of the distribution P(vb) of "hopping barriers" vb; the probability that a particle with vf can hop, equals the probability that its hopping barrier vb is smaller than its free volume vf. The shape of the barrier distribution appears to be constant as the system becomes more dense.

Dynamics of glassy systems > How do forces propagate in jammed granular solids?

Granular materials - sandpiles, silos full of grain, dense colloidal suspensions - can behave as solids, bearing forces across the sample, as long as sufficient load is applied to "jam" the particles together. As the load is progressively decreased, fewer particles are pressed into load-bearing contact. At a special point, called the isostatic point (or "point J", for jamming), exactly enough contacts between particles are formed, such that on average every particle has 2d contacts in d dimensions. At the isostatic point, there are just enough interparticle forces present for it to be possible to satisfy the requirement for static equilibrium, that zero net force acts on each particle.

As the isostatic point is approached, the forces acting between particles become correlated in ways peculiar to granular materials. The network of forces between particles becomes more and more constrained; to change the force carried by a single contact, an increasing number of nearby forces must also be changed. It is likely that the range ξ over which changes in forces would be correlated, diverges somehow as point J is reached.

Force chains in systems slightly above jamming have been studied experimentally and in simulations. One beautiful set of experiments (from Prof. Behringer at Duke University) views a pack of plastic beads under load, illuminated with polarized green light and viewed through a crossed polarizer. Particles bearing stress become birefringent and therefore visible through the crossed polarizers.

Simulations have been carried out in which particles are explicitly modeled as slightly deformable spheres. But when we deal with the geometric complexities of particles or molecules acting cooperatively, it often proves useful to develop a simpler "discrete" or "lattice" model of the phenomenon, that captures essential features while simplifying the geometry.

Here, we adapt and extend earlier work by Blighe et al., who introduced a lattice model of force chains. Blighe proposed a 2d hexagonal lattice of particles, each able to communicate a central force with integer-valued strength to neighboring particles. Different force arrangements that satisfy static equilibrium (forces balance on all particles) can be explored with "wheel moves", in which forces on the six radial bonds emanating from a particle are changed by Δ, while at the same time the six bonds connecting its neighbors are changed by -Δ. Wheel moves are not allowed to produce negative forces (the forces between particles can only push, not pull).

We use these lattice simulations to explore configurations near point J, by biasing a Monte Carlo simulation to favor bonds bearing zero force, thus "starving" the system of particle contacts. With a means to generate such configurations in hand, we can ask questions about the range of correlations among force changes that neither make nor break existing particle contacts (i.e., neither creates nor destroys zero bonds). In this way, we can explore with our lattice model the nature of the singularities near the isostatic point.