Energetics of phase transformations in polyethylene

The molecular mechanisms involved in the orthorhombic-to-monoclinic phase transformation in polyethylene were investigated by the computer simulation of a structure–energy map based on empirically justified intermolecular potential functions. Stable packing structures for the orthorhombic and monoclinic form were isolated as relative minima, cohesive energies were determined from the energy minima, specific chain motions involved in the transformation were identified by the minimum energy path connecting the packing minima, and the activation energy for the transformation was determined from the energy barrier along the minimum energy transformation path. The packing structure parameters predicted from the energy map were in excellent agreement with unit cell dimensions observed near 0°K. The activation energy predicted for the transformation is relatively low (～0.5 kcal/mole of ethylene at 0°K and 0.25 kcal/mole of ethylene near the melting point, 411°K). Monoclinic packing was predicted to be slightly more stable than orthorhombic. Since this result is inconsistent with a large body of observations, we propose that the intramolecular energy of chain folds plays a dominant role in establishing chain-packing geometry. The inclusion of fold-transition energetics could give rise to transformation mechanisms which differ in details from those proposed in this work.     

Roots of Dehn twists

Margalit and Schleimer found examples of roots of the Dehn twist t _C about a nonseparating curve C in a closed orientable surface, that is, homeomorphisms h such that h ⁿ   =  t _C in the mapping class group. Our main theorem gives elementary number-theoretic conditions that describe the n for which an n ^th root of t _C exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, n ^th roots exist for all genera greater than (n − 2)(n − 1)/2. We also describe all n ^th roots having n greater than or equal to the genus.     

Evidence for α-helices in the gas phase: a case study using Melittin from honey bee venom

Gas phase methodologies are increasingly used to study the structure of proteins and peptides. A challenge to the mass spectrometrist is to preserve the structure of the system of interest intact and unaltered from solution into the gas phase. Small peptides are very flexible and can present a number of conformations in solution. In this work we examine Melittin a 26 amino acid peptide that forms the active component of honey bee venom. Melittin is haemolytic and has been shown to form an α-helical tetrameric structure by X-ray crystallography [M. Gribskov et al., The RCSB Protein Data Bank, 1990] and to be helical in high concentrations of methanol. Here we use ion mobility mass spectrometry, molecular dynamics and gas-phase HDX to probe its structure in the gas phase and specifically interrogate whether the helical form can be preserved. All low energy calculated structures possess some helicity. In our experiments we examine the peptide following nano-ESI from solutions with varying methanol content. Ion mobility gives collision cross sections (CCS) that compare well with values found from molecular modelling and from other reported structures, but with inconclusive results regarding the effect of solvent. There is only a slight increase in CCS with charge, showing minimal coloumbically driven unfolding. HDX supports preservation of some helical content into the gas phase and again shows little difference in the exchange rates of species sprayed from different solvents. The [M + 3H](3+) species has two exchanging populations both of which exhibit faster exchange rates than observed for the [M + 2H](2+) species. One interpretation for these results is that the time spent being analysed is sufficient for this peptide to form a helix in the 'ultimate' hydrophobic environment of a vacuum.     

Structures of ordered tungsten- or molybdenum-containing quaternary perovskite oxides

The room temperature crystal structures of six A₂MMoO₆ and A₂MWO₆ ordered double perovskites were determined from X-ray and neutron powder diffraction data. Ba₂MgWO₆ and Ba₂CaMoO₆ both adopt cubic symmetry (space group Fm3?m, tilt system a⁰a⁰a⁰). Ba₂CaWO₆ has nearly the same tolerance factor (t=0.972) as Ba₂CaMoO₆ (t=0.974), yet it surprisingly crystallizes with I4/m symmetry indicative of out-of-phase rotations of the MO₆ octahedra about the c-axis (a⁰a⁰c^?). Sr₂ZnMoO₆ (t=0.979) also adopts I4/m symmetry; whereas, Sr₂ZnWO₆ (t=0.976) crystallizes with monoclinic symmetry (P2₁/n) with out-of-phase octahedral tilting distortions about the a- and b-axes, and in-phase tilting about the c-axis (a^?a^?c⁺). Ca₂CaWO₆ (t=0.867) also has P2₁/n symmetry with large tilting distortions about all three crystallographic axes and distorted CaO₆ octahedra. Analysis of 93 double perovskites and their crystal structures showed that while the type and magnitude of the octahedral tilting distortions are controlled primarily by the tolerance factor, the identity of the A-cation acts as the secondary structure directing factor. When A=Ba²⁺ the boundary between cubic and tetragonal symmetries falls near t=0.97, whereas when A=Sr²⁺ this boundary falls somewhere between t=1.018 and t=0.992.     

Writing Elements of PSL(2, q) as Commutators

We prove that for q ≥ 13, an element A of SL(2, q) is the commutator of a generating pair if and only if A ≠ ?I and the trace of A is not 2. Consequently, when q is odd and q ≥ 13, every nontrivial element of PSL(2, q) is the commutator of a generating pair, and when q is even, an element of PSL(2, q) is the commutator of a generating pair if and only if its trace is not 0. The proof of these results also leads to an improved lower bound on the number of T-systems of generating pairs of PSL(2, q).     

Sequentially equidistant points in metric spaces

A sequence (z ₀,z ₁,z ₂,, ...,z _n, z _n+1) of points fromp=z ₀ toq=z _n+1 in a metric spaceX is said to besequentially equidistant ifd(z _i−1,z _i)=d(z _i,z _i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR ^m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results. Supported in part by NSF Grant DMS-8701666.     

Deformation mechanisms of constrained polymer chains. I. Geometrically induced correlations in the deformation response mechanisms of tie molecules

The geometrical constraints acting on sections of tie molecules in noncrystalline regions severely limit the number and type of available polymer chain conformations. It is shown that these constraints induce explicit correlations in the rotations about the backbone bonds. These correlated rotations, in turn, specify distinct structural conversion paths which define the molecular mechanisms underlying the deformation response of tie molecules. Application of these constraining relationships to highly oriented polyethylene shows that the kink and jog structures of tie molecules can be decomposed into combinations of three primary conformational building blocks. Each of the basic conformational subunits follow an explicit set of dihedral angle correlations and, consequently, imparts specific characteristics to the composite structure of tie molecules. It is proposed that the composite response characteristics of tie molecules can be described as linear combinations of the response characteristics of these three primary conformational subunits.