Ordering and crystallization

Ordering and crystallization > How do polymer crystals nucleate?

Atomistic MD simulation snapshot of a rotator phase in n-alkanes.

The mechanical and transport properties of semicrystalline polymers (e.g., polyethylene, polypropylene, ...) are greatly influenced by the morphology of lamellar crystallites in the material, which are themselves influenced by the nucleation rate. Which begs the question, how to polymer crystals nucleate?

Mounting experimental evidence suggests that in PE, and perhaps other polymers, nucleation occurs via an intermediate metastable phase, rather than the final crystal phase. In PE, this metastable phase is believed to be a "rotator" phase — roughly, the chains are aligned as in the crystal, but in some way free to rotate about their axes individually, thus disrupting the herringbone arrangement of the crystal packing.

The interface between a lamellar polymer crystal and its melt is a "carpet of loops".

The goal of this work is to ask, why would nucleation via rotator phase be observed? To answer this, we refer to nucleation theory, which says basically that the phase with the lowest nucleation barrier will nucleate first. The nucleation barrier is set by a compromise between large free energy difference between the melt and nucleating phase, and low interfacial tension between the melt and the nucleating phase.

Thus to see why a rotator phase would be preferred, we need to describe both the bulk structure of such phases, to see why they are closely competitive with the crystal phase, as well as the structure of the interface between the ordered (whether crystal or rotator) and melt phases, to see how the interfacial tension arises.

We have attacked these two problems with a combination of techniques. For the bulk free energy differences between crystal and rotator phases, we have relied heavily on thermodynamic data of Sirota et al. on long n-alkanes. For the interfacial tensions, we have constructed a lattice model of the "carpet of loops", using techniques first developed for grafted polymer brushes, to describe the energetic and entropic cost of the "interpolation" between aligned and isotropic chains that occurs in the interfacial region.

Ultimately, we are able to determine all model parameters from experiment, and predict the nucleation barrier and properties of the critical nucleus assuming either a) the nucleus is crystalline; or b) the nucleus is a metastable rotator phase. We find that the barrier for a rotator phase nucleus is slightly lower than that for nucleation via crystal, for all temperatures where homogeneous nucleation is ever observed.


Master plot of average segment lifetime τ(c) versus "centrality" c (centrality = size of smaller subtree) for animals with number of segments N=50 (blue circles), 100 (red squares), 200 (green stars), compared to analytical theory (black curve).

Our result depends critically on the near-stability of the rotator phase in PE, and relies crucially on the careful experiments on n-alkanes (= monodisperse, oligomeric PE). But "circumstantial" experimental evidence suggests that other polymers than PE nucleate via a mesophase as well. In collaboration with Mike Chung and Ralph Colby in the Materials Science and Engineering department, and Eric Sirota at ExxonMobil, we are looking for evidence of and data on mesophases in the next-most-common polymer, polypropylene.

Ordering and crystallization

Ordering and crystallization > Why does flow speed up polymer crystallization?

An interval of strong flow speeds up crystallization (measured by loss of transmitted light as the sample becomes opaque). From Kumaraswamy et al., Macromolecules 32, 7537 (1999).

Crystal nucleation in supercooled molten polymers proceeds hundreds of times faster if the melt is exposed to a brief period of fast shear flow. This is a commercially important phenomenon, which makes injection molding practical: the very act of injecting the molten polymer into the die, somehow "predisposes" it to rapid nucleation.

As a result, the "grain size" of the final material (set by the density of nucleation events) is much smaller than without the effect of the fast flow, leading ultimately to a stronger object.

However, flow effects on crystallization can also lead to the formation of an oriented "skin" on the surface of the sample, which can be detrimental to the mechanical properties of the finished object. There is then practical interest in the mechanism for and means of controlling flow-induced crystallization (FIC).

Ordering and crystallization

Ordering and crystallization > How do molecules rotate in a rotator phase?

Rotator phases in normal alkanes and polyethylene (PE) are ordered phases in which the molecules adopt parallel all-trans conformations (like pencils in a box), but (unlike the low-temperature crystal phase) the molecules are disordered by random rotations about their long axes.

View down molecular axis of idealized alkane phases. a) crystal; b) rotator R1 (random 90° rotations); c) rotator R2 (random herringbone directions).

These rotations do not happen by the molecule rotating all at once; rather, the molecule begins to twist at one end, which leads to a localized "twist soliton" that propagates from one end of the molecule to the other. These twist solitons occur spontaneously in rotator phases, but also play an important role in the plastic deformation of crystalline PE in response to external stress.

We are interested in characterizing twist solitons in crystal and rotator phases of PE, by means of both atomistic MD simulations and analytical theory. In simulations, we can "trap" a twist soliton on a chain bonded to itself across periodic boundary conditions in the z direction, allowing us to observe its average shape, and diffusive mobility along the chain.

We are also constructing analytical descriptions of the soliton shape, as the solution of an optimization problem: how best to rotate a chain from one favorable orientation to an equivalent orientation, when there are energy costs both for twisting or stretching a chain, and for being misaligned with neighboring chains?

For equilibrium properties, we want to know the soliton free energy as a function of temperature. These pointlike defects will in principle be present in some concentration at any temperature, stabilized by translational entropy; however, their concentration will be very low if their formation energy is many times kT.

Energy landscape in twist angle ψ and vertical offset z (in carbons), for displacement of one chain relative to crystalline neighbors. Black path is a twist soliton, connecting two energy minima separated by a π twist.

We expect in the rotator phases of alkanes and PE that twist solitons will have considerably lower free energies than in the better ordered, more tightly packed crystal phase. In the rotator phases of long alkanes, twist solitons may be prevalent enough that they contribute to the entropic stabilization of these partially ordered phases.

Ordering and crystallization

Ordering and crystallization > How does flow affect structure in complex fluids?

Complex fluids — fluids containing "stuff", which may be polymers, colloids, rodlike or platelike nanoparticles, aggregates or gels, viruses or cells — are exquisitely sensitive to flow, because the constituents are large and have long relaxation times relative to small molecules.

Flow is therefore a powerful and necessary tool for controlling structure of complex fluids, because the target structure (with desirable properties) may either be difficult to reach by quiescent annealing, or indeed only stabilized by flow itself. For example, macroscopically aligned surface layers of block copolymers may be useful for nanolithography, but quiescent annealing typically produces a useless polycrystalline jumble.

FlowSolve handles nonuniform flows and polymeric stresses, using Lagrangian finite elements. (Right, FlowSolve; left, data.)

What is a boon to the experimenter, is a challenge for the theorist, as the powerful tools of thermodynamic equilibrium are no longer valid for systems in flow. Because we seek to describe the interplay between flow and thermal fluctuations in creating order, we need models that describe the fluctuating dynamics in flow of complex fluid order parameters. Some progress has been made with analytical methods applied to such models. But severe limitations in those methods point to mesoscale numerical simulation as the best approach to these problems.

Well-chosen variables on a coarse grid (dots)
can represent a disordered lamellar phase.

Previous work of this kind suffers from three shortcomings:

Flow treated as 'imposed', with no chance for inhomogeneous order parameter to 'act back' to modify flow;

Noise either neglected, or treated very inefficiently, by straightforward explicit numerical integration;

Order parameters not chosen parsimoniously, so that the number of variables is inconveniently large.

Our current work in this area attacks all three of these challenges. Briefly, the approaches we are pursuing are:

Build on success of Lagrangian finite element method pioneered in FlowSolve code, to treat order parameter stresses;

Use state-of-the-art operator splitting techniques in combination with spectral methods, to evolve noisy equations with larger timesteps;

Employ mesoscale order parameters with 'built-in local order' to cut down on degrees of freedom.

With this combination of approaches, we hope to treat problems of flow effects on demixing polymer solutions, ordering and alignment of block copolymer mesophases, and related problems.

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