The effect of temperature jumps during polymer crystallization

Temperature changes during the growth of lamellar polymer crystals give rise to steps on the surface of the crystals. It has recently been suggested that these steps could provide important insights into the mechanism of polymer crystallization. In particular, a characterization of the profiles of these steps might reveal the fixed-point attractor that underlies a recently proposed crystallization mechanism. Here we examine this hypothesis by performing simulations of such temperature jumps using the Sadler-Gilmer model. We find that for this model the step profiles do reveal the fixed-point attractor. However, for temperature decreases they also reflect the rounding of the crystal edge that occurs in this model and for temperature increases they also reflect the fluctuations in the thickness present in the crystal. We discuss the implications of these results for the interpretation of experimental step profiles.Comment: 8 pages, 7 figures, revte

The effect of chain stiffness on the phase behaviour of isolated homopolymers

We have studied the thermodynamics of isolated homopolymer chains of varying stiffness using a lattice model. A complex phase behaviour is found; phases include chain-folded `crystalline' structures, the disordered globule and the coil. It is found, in agreement with recent theoretical calculations, that the temperature at which the solid-globule transition occurs increases with chain stiffness, whilst the $\theta$-point has only a weak dependence on stiffness. Therefore, for sufficiently stiff chains there is no globular phase and the polymer passes directly from the solid to the coil. This effect is analogous to the disappearance of the liquid phase observed for simple atomic systems as the range of the potential is decreased.Comment: 10 pages, 10 figures, revte

Canonical Basis and Macdonald Polynomials

In the basic representation of $U_q(\hat{sl}(2))$ realized via the algebra of symmetric functions we compare the canonical basis with the basis of Macdonald polynomials with $q=t^2$. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the transition matrix are necessarily bar invariant. We also discuss the positivity and integrality of these coefficients. For level $k$, we expect a similar relation between the canonical basis and Macdonald polynomials with $q^2=t^{k}.$Comment: 25 pages, Latex2e. Advances in Math, to appea

The mechanism of thickness selection in the Sadler-Gilmer model of polymer crystallization

Recent work on the mechanism of polymer crystallization has led to a proposal for the mechanism of thickness selection which differs from those proposed by the surface nucleation theory of Lauritzen and Hoffman and the entropic barrier model of Sadler and Gilmer. This has motivated us to reexamine the model used by Sadler and Gilmer. We again find a fixed-point attractor which describes the dynamical convergence of the crystal thickness to a value just larger than the minimum stable thickness, l_min. This convergence arises from the combined effect of two constraints on the length of stems in a layer: it is unfavourable for a stem to be shorter than l_min and for a stem to overhang the edge of the previous layer. The relationship between this new mechanism and the explanation given by Sadler and Gilmer in terms of an entropic barrier is discussed. We also examine the behaviour of the Sadler-Gilmer model when an energetic contribution from chain folds is included.Comment: 15 pages, 13 figures, revte

Kinetic Monte Carlo simulations of the growth of polymer crystals

Based upon kinetic Monte Carlo simulations of crystallization in a simple polymer model we present a new picture of the mechanism by which the thickness of lamellar polymer crystals is constrained to a value close to the minimum thermodynamically stable thickness, l_{min}. The free energetic costs of the polymer extending beyond the edges of the previous crystalline layer and of a stem being shorter than l_{min} provide upper and lower constraints on the length of stems in a new layer. Their combined effect is to cause the crystal thickness to converge dynamically to a value close to l_{min} where growth with constant thickness then occurs. This description contrasts with those given by the two dominant theoretical approaches. However, at small supercoolings the rounding of the crystal profile does inhibit growth as suggested in Sadler and Gilmer's entropic barrier model.Comment: 12 pages, 13 figures, revte

The Dirac Sea

We give an alternate definition of the free Dirac field featuring an explicit construction of the Dirac sea. The treatment employs a semi-infinite wedge product of Hilbert spaces. We also show that the construction is equivalent to the standard Fock space construction.Comment: 7 page

Pulse propagation in discrete systems of coupled excitable cells

Propagation of pulses in myelinated fibers may be described by appropriate solutions of spatially discrete FitzHugh-Nagumo systems. In these systems, propagation failure may occur if either the coupling between nodes is not strong enough or the recovery is too fast. We give an asymptotic construction of pulses for spatially discrete FitzHugh-Nagumo systems which agrees well with numerical simulations and discuss evolution of initial data into pulses and pulse generation at a boundary. Formulas for the speed and length of pulses are also obtained.Comment: 16 pages, 10 figures, to appear in SIAM J. Appl. Mat

The self-assembly of DNA Holliday junctions studied with a minimal model

In this paper, we explore the feasibility of using coarse-grained models to simulate the self-assembly of DNA nanostructures. We introduce a simple model of DNA where each nucleotide is represented by two interaction sites corresponding to the phosphate-sugar backbone and the base. Using this model, we are able to simulate the self-assembly of both DNA duplexes and Holliday junctions from single-stranded DNA. We find that assembly is most successful in the temperature window below the melting temperatures of the target structure and above the melting temperature of misbonded aggregates. Furthermore, in the case of the Holliday junction, we show how a hierarchical assembly mechanism reduces the possibility of becoming trapped in misbonded configurations. The model is also able to reproduce the relative melting temperatures of different structures accurately, and allows strand displacement to occur.Comment: 13 pages, 14 figure

Fermionization, Convergent Perturbation Theory, and Correlations in the Yang-Mills Quantum Field Theory in Four Dimensions

We show that the Yang-Mills quantum field theory with momentum and spacetime cutoffs in four Euclidean dimensions is equivalent, term by term in an appropriately resummed perturbation theory, to a Fermionic theory with nonlocal interaction terms. When a further momentum cutoff is imposed, this Fermionic theory has a convergent perturbation expansion. To zeroth order in this perturbation expansion, the correlation function $E(x,y)$ of generic components of pairs of connections is given by an explicit, finite-dimensional integral formula, which we conjecture will behave as $$E(x,y) \sim |x - y|^{-2 - 2 d_G},$$ \noindent for $|x-y|>>0,$ where $d_G$ is a positive integer depending on the gauge group $G.$ In the case where $G=SU(n),$ we conjecture that $$d_G = {\rm dim}SU(n) - {\rm dim}S(U(n-1) \times U(1)),$$ \noindent so that the rate of decay of correlations increases as $n \to \infty.$Comment: Minor corrections of notation, style and arithmetic errors; correction of minor gap in the proof of Proposition 1.4 (the statement of the Proposition was correct); further remark and references adde

Sequence-dependent thermodynamics of a coarse-grained DNA model

We introduce a sequence-dependent parametrization for a coarse-grained DNA model [T. E. Ouldridge, A. A. Louis, and J. P. K. Doye, J. Chem. Phys. 134, 085101 (2011)] originally designed to reproduce the properties of DNA molecules with average sequences. The new parametrization introduces sequence-dependent stacking and base-pairing interaction strengths chosen to reproduce the melting temperatures of short duplexes. By developing a histogram reweighting technique, we are able to fit our parameters to the melting temperatures of thousands of sequences. To demonstrate the flexibility of the model, we study the effects of sequence on: (a) the heterogeneous stacking transition of single strands, (b) the tendency of a duplex to fray at its melting point, (c) the effects of stacking strength in the loop on the melting temperature of hairpins, (d) the force-extension properties of single strands and (e) the structure of a kissing-loop complex. Where possible we compare our results with experimental data and find a good agreement. A simulation code called oxDNA, implementing our model, is available as free software.Comment: 15 page