Discrete Chain Research Articles

Effect of flexibility of arms on the nematic ordering of V-shaped molecules

The nematic ordering of a melt of V-shaped molecules composed of two semiflexible arms connected by one end at an external angle α is inspected within the Landau theory of phase transitions. The conformational statistics of arms is described by a discrete worm-like chain model. It is found that for a sufficiently high stiffness of the arms the phase diagram contains the regions of stability of isotropic (I), prolate (NU+) and oblate (NU−) uniaxial, and biaxial (NB) nematic phases. The isotropic–uniaxial nematic and uniaxial nematic – biaxial nematic phase transitions are of the first and second order, respectively. At lowering the stiffness of arms the stability area of NB phase decreases; the re-entrant NB phase becomes possible in the certain range of angles α and the sequence of phase transitions I−NU−−NB−NU+−NB occurs. At a sufficiently low stiffness, the stability region of the nematic biaxial phase disappears completely, and phase diagrams display only the first order transitions between I−NU+, I−NU−, and NU+−NU− phases. A further decrease in the stiffness of arms leaves only the first order I−NU+ transition in the phase diagrams. The tricritical points appear at the curves of NU−−NB transition in the phase diagrams in certain intervals of values of the stiffness parameter.

Mathematical Analysis of the Effectiveness of Screening for Intracranial Aneurysms in First-Degree Relatives of Persons with Subarachnoid Hemorrhage

Population screening for aneurysms in patients with risk factors and preventive surgical treatment are beneficial according to numerous studies. One of the most significant risk factors is heredity, namely, the presence of first-degree relatives (FDR) with aneurysmal subarachnoid hemorrhage (aSAH). Nevertheless, there are still no generally accepted approaches or evidence bases regarding the benefits of the aneurysm screening strategy. Mathematical modeling of the dynamics of aneurysm development in the population was carried out using an algorithm implementing a discrete Markov's chain. To implement the model, all probabilities of events and distributions are taken from available literature sources. Three-dimensional time of flight noncontrast magnetic resonance angiography was chosen as a screening method. Patients underwent preventive surgical treatment if an aneurysm was detected. Screening and preventive treatment in the general population reduces the prevalence of aneurysms by 1.74% (3.44% in the FDR group) and the prevalence of aSAH by 14.36% (37.48% in the FDR group). Mortality due to aSAH was reduced by 14.44%. The number of disabilities also decreases. The occurrence of deep disability was reduced by 20.2% in the FDR group. Economic analysis of the part of the population consisting of FDRs showed annual savings of ies also decr CONCLUSIONS: The mathematical model demonstrated that screening and preventive treatment of cerebral aneurysms can reduce aSAH-associated morbidity and mortality. In the FDR group, there was decrease in the prevalence of aSAH and decrease in associated mortality. Screening for cerebral aneurysms is cost-effective.

High-Gain Traveling-Wave Parametric Amplifier Based on Three-Wave Mixing

We extend the theory for a Josephson-junction traveling-wave parametric amplifier (TWPA) operating in the three-wave-mixing regime and we propose a scheme for achieving high gain. The continuous three-mode model [P.K. Tien, J. Appl. Phys. 29, 1347 (1958)] is, on one hand, extended to describe a discrete chain of Josephson junctions at high frequencies close to the spectral cutoff where there is no up-conversion. On the other hand, we also develop a continuous multimode theory for the small frequency limit where the frequency dispersion is close to linear. We find that in both cases the gain is significantly reduced compared to the prediction by the continuous three-mode model as the result of increasingly strong dispersion at the high frequencies and generation of up-converted modes at the small frequencies. The developed theory is in quantitative agreement with simulations of the full solution of the dynamical equations. To recover the high gain, we propose to engineer a chain with dispersive features to form a two-band frequency spectrum and to place the pump frequency within the upper band close to the spectral cutoff. We prove that there exists a sweet spot, where the signal and the pump are phase matched, while the up-conversion is inhibited. This results in a high gain, which grows exponentially with the length of the TWPA.

Harmonic representatives in homology over arbitrary fields

We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW decompositions of orientable surfaces and examples of spaces arising from sampled data sets.

Elastic traits of the extensible discrete wormlike chain model.

Polymer models play the special role of elucidating the elementary features describing the physics of long molecules and become essential to interpret the measurements of their magnitudes. In this work the end-to-end distance of an extensible discrete wormlike chain polymer as a function of the applied force has been calculated both numerically and analytically, the latter as an effective approximation. The numerical evaluation uses the transfer matrix formalism to obtain an exact calculation of the partition function, while the analytic derivations generalize the simple phenomenological formulas largely used up to now. The obtained formulas are simple enough to be implemented in the fit analysis of experimental data of semiflexible extensible polymers, with the result that the elastic parameters obtained are compatible with previous measurements, and more, their accuracy strongly improves in a large range of chain extensibility.

Anion Influence on Supramolecular Interactions in Exo-Coordinated Silver(I) Complexes with N2O2 Schiff Base Macrocycle

Silver(I) complexes with aza-oxa macrocyclic Schiff bases L (L = 1,5-diaza-2,4:7,8:16,17-tribenzo-9,15-dioxa-cyclooctadeca-1,5-dien) were prepared by the reaction of the corresponding macrocycle with four different silver salts (AgX; X = ClO4, PF6, SbF6 and BF4). In all four compounds, silver ions are exo-coordinated by two neighboring ligand molecules in linear and T-shaped geometries. Such a coordination mode results in the formation of infinite 1D polymeric chains. Compounds AgLClO4 and AgLBF4 are isostructural, and polymeric chains display 1D zigzag topology. In AgLPF6 there are three symmetrically unique Ag ions in the asymmetric unit of the compound. Two silver ions are linearly coordinated with two neighboring ligand molecules and are part of a discrete polymer chain. The third silver ion is coordinated with two ligand molecules and a methanol molecule in a T-shaped geometry. Such coordination geometry results in the formation of two discrete infinite polymer chains in the crystal structure. In the AgLSbF6 compound, the chain topology is a linear zigzag chain, but in this compound, there is a difference in the orientation of the Ag-N bond. The Ag-N-Ag bonds are in the trans position relative to the plane calculated through the ligand molecule, while the Ag-N bonds are in the cis position in all other compounds. Due to the presence of a bulky SbF6 anion, the ligand molecule is planar compared to other compounds. Considering intermolecular interactions, there is a huge variety of different interactions, mostly depending on the type of anion. A general supramolecular motif in all compounds is best described as 2D sheets of ligand–metal polymers with anions and solvent molecules sandwiched between them. In addition, the obtained compounds were characterized by IR spectroscopy and thermal analysis. The TG analysis indicates a rather surprising and considerable thermal stability of the prepared compounds, with some compounds thermally stable over 300 °C.

From the origins to the stream of consciousness and its neural correlates.

There are now dozens of very different theories of consciousness, each somehow contributing to our understanding of its nature. The science of consciousness needs therefore not new theories but a general framework integrating insights from those, yet not making it a still-born "Frankenstein" theory. First, the framework must operate explicitly on the stream of consciousness, not on its static description. Second, this dynamical account must also be put on the evolutionary timeline to explain the origins of consciousness. The Cognitive Evolution Theory (CET), outlined here, proposes such a framework. This starts with the assumption that brains have primarily evolved as volitional subsystems of organisms, inherited from primitive (fast and random) reflexes of simplest neural networks, only then resembling error-minimizing prediction machines. CET adopts the tools of critical dynamics to account for metastability, scale-free avalanches, and self-organization which are all intrinsic to brain dynamics. This formalizes the stream of consciousness as a discrete (transitive, irreflexive) chain of momentary states derived from critical brain dynamics at points of phase transitions and mapped then onto a state space as neural correlates of a particular conscious state. The continuous/discrete dichotomy appears naturally between the brain dynamics at the causal level and conscious states at the phenomenal level, each volitionally triggered from arousal centers of the brainstem and cognitively modulated by thalamocortical systems. Their objective observables can be entropy-based complexity measures, reflecting the transient level or quantity of consciousness at that moment.

Detailed dynamics of discrete Gaussian semiflexible chains with arbitrary stiffness along the contour.

We revisit a model of semiflexible Gaussian chains proposed by Winkler et al., solve the dynamics of the discrete description of the model, and derive exact algebraic expressions for some of the most relevant dynamical observables, such as the mean-square displacement of individual monomers, the dynamic structure factor, the end-to-end vector relaxation, and the shear stress relaxation modulus. The mathematical expressions for the dynamic structure factor are verified by comparing them with results from Brownian dynamics simulations, reporting an excellent agreement. Then, we generalize the model to linear polymer chains with arbitrary stiffness. In particular, we focus on the case of a linear polymer with stiffness that changes linearly from one end of the chain to the other, and we study the same dynamical functions previously presented. We discuss different approaches to check whether a polymer has constant or heterogeneous stiffness along its contour. Finally, we provide expressions for the Lagrangian multipliers for Gaussian chains with variable stiffness and bond length, as well as for chains with torsion-like interactions. Overall, this work presents a new insight into a well-known model for semiflexible chains and provides tools that can be exploited to explore a much broader class of polymers or compare the predictions of the model with simulations of coarse-grained semiflexible polymers.

Dynamic fracture regimes forinitially prestressed elastic chains.

We study the propagation of a bridge crack in an anisotropic multi-scale system involving two discrete elastic chains that are interconnected by links and possess periodically distributed inertia. The bridge crack is represented by the destruction of every other link between the two elastic chains, and this occurs with a uniform speed. This process is assumed to be sustained by energy provided to the system through its initial configuration, corresponding to the alternating application of compression and tension to neighbouring links. The solution, based on the Wiener–Hopf technique and presented in Ayzenberg-Stepanenko et al. (Ayzenberg-Stepanenko et al. 2014 Proc. R. Soc. A 470, 20140121 (doi:10.1098/rspa.2014.0121)) is used to compute the profile of the medium undergoing failure. We investigate when this solution, representing the steady failure process, is physically acceptable. It is shown that the analytical solution is not always physically applicable and can be used to determine the onset of non-steady failure regimes. These arise from the presence of critical deformations in the wake of the crack front at the sites of the intact links. Additionally, we demonstrate that the structural integrity of the discrete elastic chains can significantly alter the range of speeds for which the bridge crack can propagate steadily.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

Almost Sharp Bounds on the Number of Discrete Chains in the Plane

The following generalisation of the Erdős Unit Distance problem was recently suggested by Palsson, Senger, and Sheffer. For a fixed sequence δ = (δ1, …, δk) of k distances, a (k + 1)-tuple (p1, …, pk+1) of distinct points in ℝd is called a k-chain if ∥pj − pj+1∥ = δj for every 1 ≤ j ≤ k. What is the maximum number C (n) of k-chains in a set of n points in ℝd? Improving the results of Palsson, Senger, and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3, and propose further generalisations.

Well-tempered metadynamics applied to field-theoretic simulations of diblock copolymer melts.

Well-tempered metadynamics (WTMD) is applied to field-theoretic simulations (FTS) to locate the order-disorder transition (ODT) in incompressible melts of diblock copolymer with an invariant polymerization index of N̄=104. The polymers are modeled as discrete Gaussian chains with N = 90 monomers, and the incompressibility is treated by a partial saddle-point approximation. Our implementation of WTMD proves effective at locating the ODT of the lamellar and cylindrical regions, but it has difficulty with that of the spherical and gyroid regions. In the latter two cases, our choice of order parameter cannot sufficiently distinguish the ordered and disordered states because of the similarity in microstructures. The gyroid phase has the added complication that it competes with a number of other morphologies, and thus, it might be beneficial to extend the WTMD to multiple order parameters. Nevertheless, when the method works, the ODT can be located with impressive accuracy (e.g., ΔχN ∼ 0.01).

Quench dynamics of noninteracting fermions with a delta impurity

We study the out-of-equilibrium dynamics of noninteracting fermions in one dimension and in continuum space, in the presence of a delta impurity potential at the origin whose strength g is varied at time t = 0. The system is prepared in its ground state with g = g 0 = +∞, with two different densities and Fermi wave-vectors k L and k R on the two half-spaces x > 0 and x < 0 respectively. It then evolves for t > 0 as an isolated system, with a finite impurity strength g. We compute exactly the time dependent density and current. For a fixed position x and in the large time limit t → ∞, the system reaches a non-equilibrium stationary state (NESS). We obtain analytically the correlation kernel, density, particle current, and energy current in the NESS, and characterize their relaxation, which is algebraic in time. In particular, in the NESS, we show that, away from the impurity, the particle density displays oscillations which are the non-equilibrium analog of the Friedel oscillations. In the regime of ‘rays’, x/t = ξ fixed with x, t → ∞, we compute the same quantities and observe the emergence of two light cones, associated to the Fermi velocities k L and k R in the initial state. Interestingly, we find non trivial quantum correlations between two opposite rays with velocities ξ and −ξ which we compute explicitly. We extend to a continuum setting and to a correlated initial state the analytical methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the context of a discrete fermionic chain with an impurity. We also generalize our results to an initial state at finite temperature, recovering, via explicit calculations, some predictions of conformal field theory in the low energy limit.

Discrete Giant Polymeric Chain with Precise Sequence and Regio-configuration: A Concise Multiblock Model System

The vastly expanding chemical and architectural parameter space of multiblock copolymers brings both opportunities and challenges for tailoring the structure and properties of a synthetic polymer. In this study, we reported a precise and concise multiblock model system to amplify the subtle structural variations and resolve the underlying phase principles. A series of giant polymeric chains based on polyhedral oligomeric silsesquioxane nanoparticles with an exact block number, regio-configuration, and sequence were designed and prepared. A close scrutiny of these sequence-/regio-isomers revealed that the interplay of geometric constraints, regio-configurations, and collective interactions dictates the self-assembled structures. Zigzag-packed lamellae with the layer normal perpendicular to the backbone were observed for the alternating chains, while a head-to-head arrangement with the lamellar normal along the chain direction was recognized in the case of block chains. Due to the geometric constraint, the alternating chains were exceedingly sensitive to the regio-regularity, leading to dramatically different phase stabilities. This work reveals the critical contribution of the block sequence and regio-configuration to phase behaviors, providing deeper insights toward rational structural engineering of multiblock copolymers.

Stationary scattering solution and logic operations in X-coupled tight-binding chains

The realization of logical operations on X-shaped structures consisting of two discrete input chains, which are linearly connected to two output chains through a central coupling region, is investigated. We provide the exact stationary scattering solution for the harmonic waves propagating in the proposed structure and report the full transmission spectrum. We show that a number of Boolean logic operations can be implemented using a digitization scheme of the output signal based on the amplitude modulation of the transmission coefficients. In particular, we show that the relative phase between input harmonics can be used as a control parameter to select the desired logic operation. A detailed study of the contrast ratio is used to identify optimal regions in the parameter space for the operation of each logical function.

Diagnosis of inhomogeneous one-dimensional discrete chains by photon scattering

Among the ways to reduce radiation doses (increasing the distance from the source to the person, reducing the time spent in radiation fields, reducing the radiation dose rate of the source) an important role is played by shielding the source of ionizing radiation.Currently, the use of screens is limited by the small selection of materials and the inconvenience of their use, which complicates their operation, installation and replacement, or disassembly.The propagation of a plane wave in a one-dimensional force chain with Hertzian contacts is analyzed in a linear approximation for cases with the presence of incorporated single and double impurities (defects). The algorithm of analytical solution of control equations and finding of reflection and transmission coefficients of incident radiation is determined. In terms of phase shifts, the frequency criterion of multi-modes of generation of transparent modes for inhomogeneous chains (with double impurities) similar to the Ramsauer-Townsend effect in the scattering of electrons by inert gas atoms is established. The obtained theoretical results are consistent with the data obtained from alternative sources. Possibilities of application of terahertz spectroscopy of radiation transmission for structural diagnostics and detection of in homogeneities, impurities and defects in discrete micro-mechanical (granular) systems are discussed. In particular, preliminary experimental data on scattering of terahertz radiation on samples of granular materials with different degrees of packing, from low to densely packed, which clearly indicate the dependence of the transmittance on the degree of packing and the formation of the corresponding band structure of spectra. The obtained results are also discussed in terms of applications to the creation of the element base of photonic circuitry in the sub millimeter range using decorated with impurities and structured low-dimensional discrete power circuits.

A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems

Abstract An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on the continuous level. Examples include the Maxwell–Stefan systems for gas mixtures, tumor-growth models and systems for the fabrication of thin-film solar cells. The proposed numerical scheme preserves the structure of the continuous equations, namely the entropy dissipation inequality as well as the non-negativity of the concentrations and the volume-filling constraints. The discrete entropy structure is a consequence of a new vector-valued discrete chain rule. The existence of discrete solutions, their positivity, and the convergence of the scheme is proved. The numerical scheme is implemented for a one-dimensional Maxwell–Stefan model and a two-dimensional thin-film solar cell system. It is illustrated that the convergence rate in space is of order two and the discrete relative entropy decays exponentially.

A coupled CFD-DEM numerical simulation of formation and evolution of sealing zones

Lost circulation can be one of the most troublesome situations during drilling, especially in naturally fractured formations. This situation may lead to serious economic losses because of the loss of expensive drilling fluid into the formation and nonproductive time spent on regaining drilling ROP. Granular lost circulation materials (LCMs) are the most commonly used to prevent and cure lost circulations. However, poor understanding of how granular LCM works at a micro-scale has limited their effectivity. In this paper, we developed a coupled CFD-DEM model by combining computational fluid dynamics with discrete element methods to simulate the sealing process of LCM in a wedge-shaped fracture by tracking the motion of each individual particle. Formation and evolution of both the sealing zone and the resulting force chain network were investigated by combining micro process visualization and analysis of flow characteristic curves of LCM suspension injection. The results show that injected LCM particles eventually result in three situations after transport in the fracture, depending on their size and concentration: 1) no bridging no sealing, 2) bridging without sealing, and 3) bridging with sealing. The fracture sealing zone is formed by either single-particle bridging or dual/multi-particles bridging of LCMs. A successful sealing comprises four stages: 1) LCM suspension uniform flow, 2) unstable bridging, 3) sealing zone formation and growth, and 4) fluid flow through the porous sealing zone. These situations and evolutionary stages are clearly reflected in the morphological changes of the flow curve. The formation and evolution of the force chain network also include four stages: 1) discrete force chain dynamic initiation, 2) small force chain network unstable formation, 3) discrete force chain network aggregation, 4) force chain network stable propagation. Strong force chains are in the front section and weak force chains are in the middle and end section of the sealing zone. The formation and collapse of force chains within bridging particles determines if a bridging structure and a sealing zone can be formed and stable. Dual-particles bridging formed sealing zone is weaker than single-particle bridging one because of the weak point in the strong force chain in the fracture width's direction. This research provides a better insight into the process of fracture sealing by granular LCMs, which contributes to improve efficiency of lost circulation control jobs.

Closed-form existence conditions for bandgap resonances in a finite periodic chain under general boundary conditions.

Bragg scattering in periodic media generates bandgaps, frequency bands where waves attenuate rather than propagate. Yet, a finite periodic structure may exhibit resonance frequencies within these bandgaps. This is caused by boundary effects introduced by the truncation of the nominal infinite medium. Previous studies of discrete systems determined existence conditions for bandgap resonances, although the focus has been limited to mainly periodic chains with free-free boundaries. In this paper, we present closed-form existence conditions for bandgap resonances in discrete diatomic chains with general boundary conditions (free-free, free-fixed, fixed-free, or fixed-fixed), odd or even chain parity (contrasting or identical masses at the ends), and the possibility of attaching a unique component (mass and/or spring) at one or both ends. The derived conditions are consistent with those theoretically presented or experimentally observed in prior studies of structures that can be modeled as linear discrete diatomic chains with free-free boundary conditions. An intriguing case is a free-free chain with even parity and an arbitrary additional mass at one end of the chain. Introducing such an arbitrary mass underscores a transition among a set of distinct existence conditions, depending on the type of chain boundaries and parity. The proposed analysis is applicable to linear periodic chains in the form of lumped-parameter models, examined across the frequency spectrum, as well as continuous granular media models, or similar configurations, examined in the low-frequency regime.

Propagation Characteristics of Flexural Wave in One-Dimensional Phononic Crystals Based on Lattice Dynamics Model

In this paper, we establish discrete flexural lattice chain models of Bragg and locally resonant phononic crystals by setting mass defect atoms and local resonant elements on the flexural lattice chain. The bandgap characteristics of flexural wave in phononic crystals are studied by establishing the governing equations of the model. The results from models show that with the change of the mass ratio of defective atoms to normal atoms, the bandgap of the flexural wave produced by Bragg scattering shows a certain rule. When the local resonant bandgap and Bragg scattering bandgap are close to each other, the two bandgaps will be coupled to form a wider flexural wave bandgap. The effect of axial strain on bending wave propagation is only the shift of bandgap position. The effect of material damping on the propagation of a bending wave is only energy dissipation at high frequency. In addition, we use finite element simulation to calculate the bandgap of flexural wave in phononic crystals with mass defects, and the results are consistent with lattice chain model. This shows that lattice chain model can effectively guide the bandgap design of phononic crystals. This comprehensive study may help to elucidate the rule of bandgap generation of flexural wave in one-dimensional phononic crystals.

On the dissipation at a shock wave in an elastic bar

This paper aims to relate the energy dissipated at a shock wave in a nonlinearly elastic bar to the energy in the oscillations in two related dissipationless, dispersive systems. Three, one-dimensional, dynamic impact problems are studied: Problem 1 concerns a nonlinearly elastic bar, Problem 2 a discrete chain of particles, and Problem 3 a continuum with a strain gradient term in the constitutive relation. In the impact problem considered, the free boundary of each initially quiescent body is subjected to a sudden velocity, that is then held constant for all subsequent time. There is energy dissipation at the shock in Problem 1, but Problems 2 and 3 are conservative. Problem 1 is solved analytically, Problem 2 numerically, and an approximate solution to Problem 3 is constructed analytically. The rate of increase of the oscillatory energy in Problems 2 and 3 are calculated and compared with the dissipation rate at the shock in Problem 1. The results indicate that the former is a good qualitative measure of the latter. The quantitative agreement is satisfactory at larger impact speeds but less so at smaller speeds, some possible reasons for which are discussed.