Boundary layer description of directional polymer crystallisation.

Nearly fifty years ago Lovinger and Gryte suggested that the directional crystallization of a polymer was analogous to the quiescent isothermal crystallization experiment but at a supercooling where the crystal growth velocity was equal to the velocity of the moving front. Our experiments showed that this equivalence holds in a detailed manner at low directional velocities. To understand the underlying physics of these situations, we modeled the motion of a crystallization front in a liquid where the left side boundary is suddenly lowered below the melting point (Stefan's problem) but with the modification that the crystallization kinetics follow a version of the Avrami model. Our numerical results surprisingly showed that the results of the polymer analog track with the Stefan results which were derived for a simple liquid that crystallizes completely at its melting point; in particular, the position of the crystal growth-front evolved with time exactly as in the Stefan problem. The numerical solution also showed that the temperature in the immediate vicinity of the growth-front decreased with increasing front velocity, which is in line with Lovinger and Gryte's ansatz. To provide a clear theoretical understanding of these numerical results we derive a boundary layer solution to the governing coupled differential equations of the polymer problem. The analytical results are in agreement with our observations from experiments and numerical computations but show that this equivalence between the small molecule and polymer analog only holds in the limit where the crystallization enthalpy is much larger than the rate at which heat is conducted away in the polymer. In particular, in the context of the temperature profile, the enthalpy generated by the crystallisation process which is spread out over a narrow spatial region can be approximated as a point source whose location and temperature correspond to the Lovinger-Gryte ansatz.

Quantifying Nanoparticle Ordering Induced by Polymer Crystallization.

It has recently been established that polymer crystallization can preferentially place nanoparticles (NPs) into the amorphous domains of a lamellar semicrystalline morphology. The phenomenology of this process is clear: when the time for NP diffusion is shorter than the crystal growth time, then the NPs are rejected by the growing crystals and placed in the amorphous domains. However, since there is no quantitative characterization of this ordered NP state, we develop a correlation function analysis for small-angle X-ray scattering data, inspired by classical methods used for enunciating the local morphology of lamellar semicrystalline polymers. We show that when the spherulitic growth rate is slower than NP diffusion, then all the NPs are expelled from the crystals. As we increase the crystallization temperature, Tc, the long period characterizing the periodically repeating crystal-amorphous polymer structure, rcc, increases. This results in a smaller number of amorphous domains per unit volume-the number of NPs per amorphous domain thus increases. While the scattering contrast between the pure silica and the polymer is constant, these arguments predict that the apparent contrast between the NP-rich and the polymer-rich domains scale linearly with rcc, as we confirm from our experiments. These facts allow us to posit that the NPs become more efficiently packed in the interlamellar zone with increasing Tc until they form a fully filled monolayer. Above this temperature, NP multilayers form within each of the NP-rich domains. Our analysis approach, therefore, describes NP ordering that is achieved when driven by polymer crystallization.

Modeling polymer crystallisation induced by a moving heat sink.

Recent experimental work has shown that polymer crystallisation can be used to "move" and organize nanoparticles (NP). As a first effort at modeling this situation, we consider the classical Stefan problem but with the modification that polymer crystallisation does not occur at a single temperature. Rather, the rate of crystallisation is proportional to its subcooling, and here we employ a form inspired by the classical Avrami model to describe this functional form. Our results for the movement of the polymer crystallisation front, as defined as the point where the crystallinity is 50%, closely track the results of the classical Stefan problem. Thus, at this level of approximation, the crystallisation kinetics of the polymer do not cause qualitative changes to the physics of this situation. Inspired by this fact we study the more interesting situation where the directional recrystallisation of a polymer melt is considered, e.g., through the application of a moving heat sink over an initially molten polymer, reminiscent of a processing technique termed zone annealing. The polymer crystallisation shows that a steady state exists for a range of sink velocities. The solid-melt interface moves slightly ahead of the sink but at the same velocity. The steady-state distance between the sink and the interface decreases with increasing sink velocity - this is a consequence of the excess cooling provided by the sink over what is required to crystallise the melt. The most interesting new result is that the temperature of the crystal-melt interface decreases with increasing sink velocity. This is in line with the ansatz of Lovinger and Gryte who suggested that larger zone annealing velocities correspond to progressively larger effective undercoolings at which polymer crystallisation occurs.

Mechanisms of Directional Polymer Crystallization.

Zone annealing, a directional crystallization technique originally used for the purification of semiconductors, is applied here to crystalline polymers. Tight control over the final lamellar orientation and thickness of semicrystalline polymers can be obtained by directionally solidifying the material under optimal conditions. It has previously been postulated by Lovinger and Gryte that, at steady state, the crystal growth rate of a polymer undergoing zone annealing is equal to the velocity at which the sample is drawn through the temperature gradient. These researchers further implied that directional crystallization only occurs below a critical velocity, when crystal growth rate dominates over nucleation. Here, we perform an analysis of small-angle X-ray scattering, differential scanning calorimetry, and cross-polarized optical microscopy of zone-annealed poly(ethylene oxide) to examine these conjectures. Our long period data validate the steady-state ansatz, while an analysis of Herman's orientation function confirms the existence of a transitional region around a critical velocity, v crit, where there is a coexistence of oriented and isotropic domains. Below v crit, directional crystallization is achieved, while above v crit, the mechanism more closely resembles that of conventional isotropic isothermal crystallization.

Effects of Hairy Nanoparticles on Polymer Crystallization Kinetics.

We previously showed that nanoparticles (NPs) could be ordered into structures by using the growth rate of polymer crystals as the control variable. In particular, for slow enough spherulitic growth fronts, the NPs grafted with amorphous polymer chains are selectively moved into the interlamellar, interfibrillar, and interspherulitic zones of a lamellar morphology, specifically going from interlamellar to interspherulitic with progressively decreasing crystal growth rates. Here, we examine the effect of NP polymer grafting density on crystallization kinetics. We find that while crystal nucleation is practically unaffected by the presence of the NPs, spherulitic growth, final crystallinity, and melting point values decrease uniformly as the volume fraction of the crystallizable polymer, poly(ethylene oxide) or PEO, ϕPEO, decreases. A surprising aspect here is that these results are apparently unaffected by variations in the relative amounts of the amorphous polymer graft and silica NPs at constant ϕ, implying that chemical details of the amorphous defect apparently only play a secondary role. We therefore propose that the grafted NPs in this size range only provide geometrical confinement effects which serve to set the crystal growth rates and melting point depressions without causing any changes to crystallization mechanisms.