M. G. Brereton

THE STATISTICAL-MECHANICS OF A MELT OF POLYMER RINGS

Topological restrictions are introduced into a melt of flexible polymer rings by specifying the linking number between each pair of rings. Attention is focused on the configurational properties of a single ring in the melt, where the winding number between each pair of rings is zero. The configurations of all the other chains are averaged out in the partition sum. The effect of the topological constraint on the remaining chain is expressed as a product of two configurational weighting factors which involve unusual geometrical properties of the configuration. One is a phase factor depending on the torsion of the configuration and the other is a Boltzmann-like factor with the self-inductance of the configuration in the role of the interaction energy. The term based on the torsion has the remarkable property of transforming random walk-like configurations into those of stiff rods, while the inductance term promotes a transition to a completely collapsed state. Our preliminary results suggest that the actual configuration of the loop is a non-trivial balance between these opposing tendencies with the size R of the loop scaling with the length as R2 approximately L1-1(3 pi /) approximately L0.89.

Rouse and Reptation Dynamics of Linear Polybutadiene Chains Studied by 2H NMR Transverse Relaxation

Deuterium NMR has been used to investigate two different types of dynamics of linear polybutadiene chains in the melt. The transverse relaxations of short Rouse chains of molecular weight 640- 3000 were biexponential, which has been attributed to separate decays of the methylene and methine deuterons. Interpretation of the transverse relaxation rates using a model for Rouse dynamics, combined with molecular simulations, gave the shortest Rouse unit as approximately 4.4 monomers and the shortest Rouse time as 8.3 × 10 -7 s. The reptation dynamics of higher molecular weight entangled chains were investigated using ABA isotopic triblock copolymers, of total molecular weight 14000-135000, where A is protonated polybutadiene of molecular weight greater than the entanglement molecular weight and B is a deuterated block. These polymers were specifically synthesized so that the fast motion of the Rouselike chain ends should not complicate the signal. The fundamental parameters found for the Rouse chain were used in the reptation model, assuming fast dynamics, and gave an entanglement molecular weight, M e , of 5380 or approximately 21 Rouse units. This M e is more than twice the conventional value, obtained from rheology, and is more suggestive of the critical molecular weight M c , consistent with previous NMR work. The theoretical analysis used in this work is based on the assumption that the chain dynamics are fast on the time scale set by the NMR deuterium quadrupolar interaction. The highest molecular weight samples were found to not satisfy this criterion and indicate the molecular weight at which a new theoretical approach is needed.

A proton NMR investigation of crosslinks and entanglements in polyethylene networks and star polymers

Abstract This paper investigates the ability of proton transverse NMR relaxation to study the structure and dynamics of polyethylene melts. Commercial linear samples of similar M n and various M w have been examined, together with samples crosslinked by electron beam irradiation. Apart from the linear sample of lowest M w , all the free induction decays (FIDs) are very similar and this is attributed to the presence of free chain ends. The magnitude of this problem is illustrated by generating theoretical FIDs from networks of different crosslink density and reptating linear chains of different entanglement spacings, with and without free chain ends. A new result for the FID using the reptation spectrum is presented and used to fit data from the linear polyethylenes. This yields an “NMR average” molar mass, similar to the accepted value and provides information on a samples polydispersity. Three-arm polyethylene star polymers, of low polydispersity and various arm lengths, have also been studied. The large variation in the chain dynamics from the core to the free end of the star arm is not reflected in the FIDs, which can all be well-fitted by double exponentials. This highlights the importance of specific isotopic labelling in studying such systems.

The statistical mechanics of a concatenated polymer chain

The elastic and configurational properties of mesoscopic structures, which have been formed by linking a finite number of polymer loops to each other, are considered. For simplicity it is assumed that the linking numbers between pairs of loops provides a sufficient description of the topological structure. To conserve the topology the statistical mechanical calculation of the partition sum is restricted to the configuration space labelled by the specified linking numbers. An expansion of the partition sum is proposed where the linking numbers, corresponding to a particular structure, select a sub-set of terms from the expansion. The case of a linear concatenated chain of loops is studied and it is shown that the first non-zero contribution from this expansion is already able to describe the chain as a re-scaled Gaussian chain capable of supporting stress. This confirms that the approximation maintains the structural integrity of the chain. The effective step length and step length distribution function are also identified together with the elastic properties. The relationship of linking numbers to the Abelian Chern-Simons gauge field theory is used to illustrate essential parts of the calculation. The relevant term in the partition sum is represented in terms of Feynman diagrams describing the two-particle scattering amplitude for vector gauge bosons. It is shown that the expected re-scaled Gaussian chain result comes from the dominance in the s-channel of a massive scalar boson representing the structure of an individual loop in the chain.