Dumbbell Model Research Articles

Impact of Solvent Quality on the Hysteresis in the Coil–Stretch Transition of Flexible Polymers in Good Solvents

Direct observations of long flexible polymers in elongational flow show a coil to stretch transition (CST) and hysteresis in this transition. In this article, solvent effects on the hysteresis of CST were examined using Brownian dynamics simulations of a dumbbell model with conformation dependent drag. A spring force relation that is able to capture the effects of solvent quality was used. Two methods of increasing the solvent quality were analyzed: (1) by increasing the number of Kuhn steps (NK) and keeping the excluded volume parameter (v/l3) constant, and (2) by increasing v/l3 and keeping NK constant. In the first case, the width of the hysteresis (ratio of upturn to downturn strain rates) increased with increase in NK. In the second case, the width of hysteresis decreased with increase in v/l3. This implies that molecules such as ss-DNA, which have v/l3 ≈ 1 in good solvents, will have less hysteresis than a molecule with the same NK in a Θ solvent.

NVU dynamics. III. Simulating molecules at constant potential energy

This is the final paper in a series that introduces geodesic molecular dynamics at constant potential energy. This dynamics is entitled NVU dynamics in analogy to standard energy-conserving Newtonian NVE dynamics. In the first two papers [T. S. Ingebrigtsen, S. Toxvaerd, O. J. Heilmann, T. B. Schrøder, and J. C. Dyre, J. Chem. Phys. 135, 104101 (2011)10.1063/1.3623585; T. S. Ingebrigtsen, S. Toxvaerd, T. B. Schrøder, and J. C. Dyre, J. Chem. Phys. 135, 104102 (2011)10.1063/1.3623586], a numerical algorithm for simulating geodesic motion of atomic systems was developed and tested against standard algorithms. The conclusion was that the NVU algorithm has the same desirable properties as the Verlet algorithm for Newtonian NVE dynamics, i.e., it is time-reversible and symplectic. Additionally, it was concluded that NVU dynamics becomes equivalent to NVE dynamics in the thermodynamic limit. In this paper, the NVU algorithm for atomic systems is extended to be able to simulate the geodesic motion of molecules at constant potential energy. We derive an algorithm for simulating rigid bonds and test this algorithm on three different systems: an asymmetric dumbbell model, Lewis-Wahnström o-terphenyl (OTP) and rigid SPC/E water. The rigid bonds introduce additional constraints beyond that of constant potential energy for atomic systems. The rigid-bond NVU algorithm conserves potential energy, bond lengths, and step length for indefinitely long runs. The quantities probed in simulations give results identical to those of Nosé-Hoover NVT dynamics. Since Nosé-Hoover NVT dynamics is known to give results equivalent to those of NVE dynamics, the latter results show that NVU dynamics becomes equivalent to NVE dynamics in the thermodynamic limit also for molecular systems.

A mesoscopic thermodynamical description of rheological models

We propose a simple and systematic description of the most used viscoelastic models by using a non-equilibrium thermodynamic approach, generally referred to as the internal variables theory. The main idea is to elevate the conformation tensor \(\textsf{C}\) describing the geometrical configuration of the macromolecules to the status of independent variable. It is shown that the sole knowledge of the Helmholtz free energy— whose explicit expression is provided by the kinetic theory—allows to derive the expressions of the polymeric stress tensor and the time evolution equation of the internal variable \(\textsf{C}\), which represents the main ingredients of the rheological models. As illustrations, the Hookean, FENE-P Giesekus, and Bird’s encapsulated dumbbell models are discussed. For completeness, the results are compared with these provided by extended irreversible thermodynamics.

Multi-scale molding and numerical simulation of the flow-induced crystallization of polymer

A numerical simulation for the isothermal flow-induced crystallization (FIC) of polymer melt in a simple shear flow is presented. The model is based on a two-phase suspension model established for crystallization and has the advantage of taking into account the polymer melt rheological behavior through the first normal stress difference calculation. A FENE dumbbell model and a rigid dumbbell model are used to describe the amorphous phase and the semi-crystalline phase, respectively. Crystallization is described as a spherulitical nucleation and growth process. The results show that the short-term shear has a large effect on the crystallization dynamics of polymer.

Optimized Cross-Slot Flow Geometry for Microfluidic Extensional Rheometry

A precision-machined cross-slot flow geometry with a shape that has been optimized by numerical simulation of the fluid kinematics is fabricated and used to measure the extensional viscosity of a dilute polymer solution. Full-field birefringence microscopy is used to monitor the evolution and growth of macromolecular anisotropy along the stagnation point streamline, and we observe the formation of a strong and uniform birefringent strand when the dimensionless flow strength exceeds a critical Weissenberg number Wi(crit) ≈ 0.5. Birefringence and bulk pressure drop measurements provide self-consistent estimates of the planar extensional viscosity of the fluid over a wide range of deformation rates (26 s(-1) ≤ ε ≤ 435 s(-1)) and are also in close agreement with numerical simulations performed by using a finitely extensible nonlinear elastic dumbbell model.

Local existence of smooth solutions to the FENE dumbbell model

The author proves the local existence of smooth solutions to the finite extensible nonlinear elasticity (FENE) dumbbell model of polymeric flows in some weighted spaces if the non-dimensional parameter b > 2.

Librational characteristics of a dumbbell modeled tethered satellite under small, continuous, constant thrust

Tether drag deorbit technology with chemical propulsion is an effective method of active removal of space debris, which is completely different from the traditional applications of tethered satellites. During flight in a non-Keplerian orbit, the tethered satellite is impacted by a compelling active thrust and presents complicated librational characteristics. In this article, based on a dumbbell model of tethered satellite, analytical first-order solutions of the librational angles of a tethered satellite are derived under a small, continuous, constant thrust in the transversal, tangential, radial and binormal directions. The librational characteristics of a tethered satellite are revealed for two conditions: (1) in-plane orbit transfer of an abandoned satellite and (2) adjustment of the orbital inclination of a satellite. Furthermore, the ranges of designed system parameters, such as initial geocentric distance, mass of the system, mass ratio between base satellite and objective satellite, tether length, and thrust acceleration, are analyzed for an orbit transfer. Afterwards, linear quadratic regulator based tension controllers are designed to damp out the librational angles and to achieve attitude stability of a tethered satellite. Numerical simulation results show that complicated librational characteristics of a tethered satellite under small, continuous, constant thrust are strongly influenced by the designed system parameters. In addition, the librational angles are controllable by a designed inhibitive controller for a set of given system parameters, which is useful for flight safety in maneuvering.

Global existence of weak solutions to the FENE dumbbell model of polymeric flows

Systems coupling fluids and polymers are of great interest in many branches of sciences. One of the most classical models to describe them is the FENE (Finite Extensible Nonlinear Elastic) dumbbell model. We prove global existence of weak solutions to the FENE dumbbell model of polymeric flows. The main difficulty is the passage to the limit in a nonlinear term that has no obvious compactness properties. The proof uses many weak convergence techniques. In particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure. In addition, this quantity controls a rescaled defect measure of the gradient of the velocity.

선형 역덤벨 모델을 이용한 이족 보행 로봇의 기준 ZMP 궤적 생성 및 보행 구현

This paper presents reference ZMP trajectory generation and implementation for a biped robot via linear inverted dumbbell model (LIDM), which can consider the effect of external momentum on the center of mass (COM) of robot. Based on a reference ZMP trajectory derived by using LIDM, a base trajectory is proposed not only to make the locomotion of robot similar to that of human but also to facilitate its implementation and tuning. In order to realize a dynamic walking using the proposed trajectory, compliance, impedance and ZMP tracking controllers are adopted together. Extensive experiments show that the proposed locomotion of a biped robot is stable and also, similar to that of human. Further researches on balance recovery of a biped robot will be carried out to guarantee its robust locomotion in combination with the proposed trajectory.

Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd , d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. We perform a rigorous passage to the limit as first the spatial discretization parameter, and then the temporal discretization parameter tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data: a square-integrable and divergence-free initial velocity datum for the Navier–Stokes equation and a nonnegative initial probability density function ψ 0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M .

Isomorphs in Model Molecular Liquids

Isomorphs are curves in the phase diagram along which a number of static and dynamic quantities are invariant in reduced units (Gnan, N.; et al. J. Chem. Phys.2009, 131, 234504). A liquid has good isomorphs if and only if it is strongly correlating, i.e., if the equilibrium virial/potential energy fluctuations are more than 90% correlated in the NVT ensemble. Isomorphs were previously discussed with a focus on atomic systems. This paper generalizes isomorphs to liquids composed of rigid molecules and study the isomorphs of systems of small rigid molecules: the asymmetric dumbbell model, a symmetric inverse power-law dumbbell, and the Lewis–Wahnström o-terphenyl (OTP) model. For all model systems, the following quantities are found to a good approximation to be invariant along an isomorph: the isochoric heat capacity, the excess entropy, the reduced molecular center-of-mass self-part of the intermediate scattering function, and the reduced molecular center-of-mass radial distribution function. In agreement with theory, we also find that an instantaneous change of temperature and density from an equilibrated state point to an isomorphic state point leads to no relaxation. The isomorphs of the Lewis–Wahnström OTP model were found to be more approximative than those of the asymmetric dumbbell model; this is consistent with the OTP model being less strongly correlating. The asymmetric dumbbell and Lewis–Wahnström OTP models each have a “master isomorph”; i.e., the isomorphs have identical shape in the virial/potential energy phase diagram.

425 乗り心地解析を可能とする新しい人体モデルの開発

This paper presents theoretical investigation concerning modeling of a human body, seat, steering wheel and pedals system. We have developed a new model of human body that the evaluation of ride comfort is able to estimate. The model of human body consists of six parts: the head, body, thigh, shin, upper arm and fore arm. Each part of the human body is composed with the same fundamental element which is called as a dumbbell model. The dumbbell model is the one that two mass is connected with a flexible bar that has the coefficient of spring, dumper, and plasticity. The movement of the dumbbell mode is considered in the vertical and horizontal direction and rotational movement at the point of center. Each part of the dumbbell model has different parameters and is connected to compose human body. Connected point of each part is considered the friction moment that depends on displacement, relative angle, and relative angular velocity. The functional model is applied in modeling. The analytical result is compared with the measurement result and it agreed extremely well. Some results are presented in the form of parametric graphs. It is shown that the model of human body that has newly developed is able to apply to evaluate the ride comfort.

The Cauchy--Dirichlet Problem for the Finitely Extensible Nonlinear Elastic Dumbbell Model of Polymeric Fluids

The finitely extensible nonlinear elastic (FENE) dumbbell model consists of the incompressible Navier--Stokes equation for the solvent and the Fokker--Planck equation for the polymer distribution. In such a model, the polymer elongation cannot exceed a limit $m_0$ which yields all interesting features of solutions near this limit. This work is concerned with the sharpness of boundary conditions in terms of the dimensionless parameter $b=\frac{Hm_0^2}{k_BT}$. Through a careful analysis of the Fokker--Planck operator coupled with the Navier--Stokes equation, we establish a local well-posedness for the full coupled FENE dumbbell model under a class of Dirichlet-type boundary conditions dictated by the parameter $b$. For each $b>0$ we identify a sharp boundary requirement for the underlying density distribution, while the sharpness follows from the existence result for each specification of the boundary behavior. It is shown that the probability density governed by the Fokker--Planck equation approaches zero near the boundary, necessarily faster than the distance function $d$ for $b>2$, faster than $d|ln d|$ for $b=2$, and as fast as $d^{b/2}$ for $0<b<2$. Moreover, the sharp boundary requirement for $b\geq 2$ is also sufficient for the distribution to be a probability density.

On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

We propose a new weighted weak formulation for the Fokker–Planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well-posedness in weighted Sobolev spaces. We also propose simple and efficient semi-implicit time-discretization schemes which are unconditionally stable, i.e., the step size of time marching does not depend on the number of the bases used in configurational space. We then restrict ourselves to the two-dimensional case and construct two Fourier–Jacobi spectral-Galerkin algorithms which enjoy the following properties: (i) they are unconditionally stable, spectrally accurate, and of optimal computational complexity; (ii) they conserve the volume and provide accurate approximation to higher-order moments of the distribution function; and (iii) they can be easily extended to coupled nonhomogeneous systems. Numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress.

An Entropy Satisfying Conservative Method for the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model

In this paper, we propose an entropy satisfying conservative method to solve the Fokker–Planck equation of the finitely extensible nonlinear elastic dumbbell model for polymers, subject to homogeneous fluids. Both semidiscrete and fully discrete schemes satisfy all three desired properties—(i) mass conservation, (ii) positivity preserving, and (iii) entropy satisfying—in the sense that these schemes satisfy discrete entropy inequalities for both the physical entropy and the quadratic entropy. These ensure that the computed solution is a probability density and the schemes are entropy stable and preserve the equilibrium solutions. We also prove convergence of the numerical solution to the equilibrium solution as time becomes large. Zero flux at boundary is naturally incorporated, and boundary behavior is resolved sharply. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme and the effects of some canonical homogeneous flows.

Electrophoretic stretching of tethered polymer chains by travelling-wave electric fields: tunable stretching, expedited coil–stretch transition, and a new paradigm of dynamic molecular probing

The use of low-frequency travelling wave fields in electrophoretic stretching of charged polymer chains is investigated theoretically. We find, using a simple elastic dumbbell model, that the stretching behavior with and without fluctuations can show a number of distinctive features that cannot be seen in usual steady or alternating current electric fields. In the deterministic study without fluctuations, we show that an end-tethered charged polymer chain can be pulled by asymmetric strokes generated by a travelling wave field and kept extended along the wave. It is found that while the averaged chain extension can be increased by raising the field strength, it can be decreased by increasing the wave speed. This is a new example that stretching a polymer chain can be realized in a time periodic field with zero mean. As the free energy landscape here acts like a vibrating harmonic oscillator having double or multiple wells, a stretched chain trapped in an energy groove created by a travelling wave field can hop back to the lower energy coil state due to fluctuations. In this stochastic study, we develop a theory and carry out Brownian dynamics simulations to show that as long as the wave speed does not exceed the damping threshold and fluctuations do not prevail to diminish stretching, increasing the wave speed can help the chain maintain its stretch by preventing it from hopping to the coil state. In addition, two distinct hopping kinetics, Arrhenius and Kramers modes, can exist to govern the respective coil–stretch transitions in the double-well and multiple-well scenarios, depending on the extent of stretching. These features are the results of cooperative effects of travelling wave fields and fluctuations, and further manifested by tongue-like coil–stretch phase diagrams. Applications of the present stretching to dynamic molecular probing are also illustrated by monitoring and regulating the molecular transport over a cyclically stretching polymer chain at the nanoscale or single-molecule level.

The Conformation of the Poly(ethylene glycol) Chain in Mono-PEGylated Lysozyme and Mono-PEGylated Human Growth Hormone

Covalent conjugation of poly(ethylene glycol) or "PEGylation" has proven an effective strategy to improve pharmaceutical protein efficacy by hindering recognition by proteases, inhibitors, and antibodies and by retarding renal clearance. Because it determines the strength and range of intermolecular steric forces and the hydrodynamic properties of the conjugates, the configuration of protein-conjugated PEG chains is the key factor determining how PEGylation alters protein in vivo circulation time. Mono-PEGylated proteins are typically described as having a protective PEG shroud wrapped around the protein, but recent dynamic light scattering studies suggested that conjugates adopt a dumbbell configuration, with a relatively unperturbed PEG random coil adjacent to the globular protein. We used small-angle neutron scattering (SANS) to distinguish between the dumbbell model and the shroud model for chicken-egg lysozyme and human growth hormone covalently conjugated to a single 20 kDa PEG chain. The SANS contrast variation technique was used to isolate the PEG portion of the conjugate. Scattering intensity profiles were well described by the dumbbell model and inconsistent with the shroud model.

Thermodynamics of water treeing

The paper examines the models proposed by Zeller for the development of water trees within the polyethylene (PE) insulation of underground distribution cables. Those models are of two different geometries idealizing a water tree: a prolate spheroid and ageometry where two spheres are linked by a narrow conducting channel. In these models a dissolved species, within a solution permeating the water tree, enhances the conductivity of the solution, distorting the electric field and increasing the Gibbs' free energy. Consequently, the solute species have a chemical potential that is greater than that in the absence of an electric field and this potential is suggested as leading to reaction of the solute with PE. Zeller used sodium chloride as the solute and did not take into account (nor suggest) any reaction product in solution. Consequently, the chemical potential difference between reactants and products, the true driving force for reaction, was not obtainable from this previous work. The present paper suggests alternative species (ferric and ferrous iron) that can serve as reactant and product in a plausible reaction with PE. Calculation of chemical potential differences, based on both aspheroidal model and a uniform property dumbbell model, suggests that these species might indeed be involved in PE degradation but only at low solute concentrations. The paper concludes by hypothesizing how such low concentrations may occasionally occur.

Work fluctuations in an elastic dumbbell model of polymers in planar elongational flow

We use a path-integral approach to calculate the distribution P(w,t) of the fluctuations in the work w at time t of a polymer molecule (modeled as an elastic dumbbell in a viscous solvent) that is acted on by an elongational flow field having a flow rate ̇γ. We find that P(w,t) is non-Gaussian and that, at long times, the ratio P(w,t)/P(-w,t) is equal to exp[w/(k(B)T)], independent of ̇γ. On the basis of this finding, we suggest that polymers in elongational flows satisfy a fluctuation theorem.

Deformation of a flexible polymer in a random flow with long correlation time

The effects induced by long temporal correlations of the velocity gradients on the dynamics of a flexible polymer are investigated by means of theoretical and numerical analysis of the Hookean and finitely extensible nonlinear elastic (FENE) dumbbell models in a random renewing flow. For Hookean dumbbells, we show that long temporal correlations strongly suppress the Weissenberg-number dependence of the power-law tail characterising the probability density function (PDF) of the elongation. For the FENE model, the PDF becomes bimodal, and the coil–stretch transition occurs through the simultaneous drop and rise of the two peaks associated with the coiled and stretched configurations, respectively.