Nonlinear Elastic Potential Research Articles

Molecular dynamics simulations of the monomer density profiles of knotted ring polymer chains confined in a slit of two parallel walls with one attractive and another repulsive surface.

We have used Molecular Dynamics simulations to obtain the monomer density profiles for real linear and ring polymer chains of 360 monomers length with different topological structures such as simple knots: 31, 61, 91, 10124, complex knots 313151 and twisted knots with n = 10 and n = 20 in a slit geometry of two parallel walls with one attractive and another repulsive surface. We have used Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software to perform simulations with the Verlet integration algorithm. The interactions between monomers were simulated as Lennard-Jones 12-6 potential, for bonds we have used Finitely Extensible Nonlinear Elastic (FENE) potential and the interaction with the walls was taken into account via Lennard-Jones 9-3 potential. We observed that topologically complex polymers have lower monomer density profiles near the attractive wall, but at some distance in the direction to the repulsive wall this tendency changes to the opposite. We showed that most complex twisted knots have two maxima in narrow slits. In the wide slits we do not observe such relation for twisted knots at higher temperatures. These results are important for better understanding the nature of the depletion forces which arise in a slit geometry of two parallel walls with one attractive and one repulsive wall.

Controlling the Chain Folding for the Synthesis of Single-Chain Polymer Nanoparticles Using Thermoresponsive Polymers

Synthetic polymer single-chain nanoparticles (SCNPs) are an emerging new class of nanomaterials that possess similar folded structures as natural proteins. However, most SCNPs reported so far are p...

The incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids

We explore the behaviour of global-in-time weak solutions to a class of bead-spring chain models, with finitely extensible nonlinear elastic (FENE) spring potentials, for dilute polymeric fluids. In the models under consideration the solvent is assumed to be a compressible, isentropic, viscous, isothermal Newtonian fluid, confined to a bounded open domain in R3, and the velocity field is assumed to satisfy a complete slip boundary condition. We show that as the Mach number tends to zero the system is driven to its incompressible counterpart.

A Thermodynamic Constitutive Model for Saturated Sand.

This paper establishes a non-equilibrium thermodynamic constitutive model that can predict the undrained shear behavior of saturated sand. Starting from the basic laws of thermodynamics, the model does not require the classical concepts in elasto-plastic models, such as the yield function, the flow rule, and the hardening rule. It is also different from the existing thermodynamic constitutive models in soil mechanics literatures. The model does not use a complex nonlinear elastic potential as usually and introduces a coupling energy dissipative mechanism between the viscosity and elasticity relaxation, which is essential in granular materials. Then this model was used to simulate the undrained shear test of Toyoura sand. The model can predict the critical state, dilatancy-contraction and hardening-softening characteristics of sand during undrained triaxial shearing.

On convergent schemes for two-phase flow of dilute polymeric solutions

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a recent micro-macro bead-spring model for two-phase flow of dilute polymeric solutions derived by methods from nonequilibrium thermodynamics ([Grün, Metzger, M3AS 26 (2016) 823–866]). The model consists of Cahn-Hilliard type equations describing the evolution of the fluids and the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three spatial dimensions for the velocity and the pressure of the fluids with an elastic extra-stress tensor on the right-hand side in the momentum equation which originates from the presence of dissolved polymer chains. The polymers are modeled by dumbbells subjected to a finitely extensible, nonlinear elastic (FENE) spring-force potential. Their density and orientation are described by a Fokker-Planck type parabolic equation with a center-of-mass diffusion term. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of these finite element approximations converges towards a weak solution of the coupled Cahn-Hilliard-Navier-Stokes-Fokker-Planck system. To underline the practicality of the presented scheme, we provide simulations of oscillating dilute polymeric droplets and compare their oscillatory behaviour to the one of Newtonian droplets.

Study on the molecular models commonly used in the simulation of dilute polymeric solutions in flow

Three molecular models, commonly used in the simulation of polymeric solutions and melts were employed to describe the rheological behaviour of dilute, elastic and constant viscosity solution formed by bead-and-bond chain molecules immersed into a soft-sphere solvent. The intermolecular interactions for the three models were calculated by the Lennard-Jones potential. The differences amongst the models proceeded from the intramolecular restrictions: the simplest one is a Freely-Joined-Chain (FJC) model with harmonic bond potentials, in the second model bonds are restricted by a finite extensible non-linear elastic (FENE) potential plus a repulsive WCA potential, and the third model is conformed by the United Atom (UA) approach which includes bond, flexion and torsion potentials. Both Couette and Poiseuille flows were simulated using Non-Equilibrium Molecular Dynamics. Deformation displayed by the three chain models; defined in terms of their radius of gyration was calculated and according to results it was found that for Couette flow, the three chains exhibit similar response to deformation. In Poiseulle flow, the FJC and FENE models behave similarly but the UA model presents a larger resistance to deformation. For both flow regimes, the forces involved to deform the chains were estimated in terms of the first normal stress differences. From these estimations it was found that the UA model depicted the stiffest chain, followed by the FENE model, and last the FJC model.

A molecular scale perspective: Monte Carlo simulation for rupturing of ultra thin polymer film melts

Monte Carlo simulation has been performed to study the rupturing process of thin polymer film under strong confinement. The change in mean square displacement; pair correlation function; density distribution; average bond length and microscopic viscosity are sampled by varying the molecular interaction parameters such as the strength and the equilibrium positions of the bonding, non-bonding potentials and the sizes of the beads. The variation in mean square angular displacement χθ=[<Δθ2>−<Δθ>2] fits very well to a function of type y(t)=A+B⁎e−tτ. This may help to study the viscous properties of the films and its dependence on different parameters. The ultra thin film annealed at high temperature gets ruptured and holes are created in the film mimicking spinodal dewetting. The pair correlation function and density profile reveal rich information about the equilibrium structure of the film. The strength and equilibrium bond length of finite extensible non-linear elastic potential (FENE) and non-bonding Morse potential have clear impact on microscopic rupturing of the film. The beads show Rouse or repetition motion forming rim like structures near the holes created inside the film. The higher order interaction as dipole–quadrupole may get prominence under strong confinement. The enhanced excluded volume interaction under strong confinement may overlap with the molecular dispersion forces. It can work to reorganize the molecules at the bottom of the scale and can imprint its signature in complex patterns evolved.

The impact of bonded interactions on the ground-state geometries of a small flexible polymer

Bonded interactions in coarse-grained models of elastic polymers are commonly represented by the finitely extensible nonlinear elastic (FENE) potential. In this study, we perform parallel multicanonical Monte Carlo simulations to examine the impact of an additional Lennard-Jones term in the bonded potential on the geometry of ground-state structures of a short polymer. Employing microcanonical inflection point analysis and conformational analysis, we construct a hyper-phase diagram and identify ground-state structures with two distinct geometries.

On micro–macro-models for two-phase flow with dilute polymeric solutions — modeling and analysis

By methods from nonequilibrium thermodynamics, we derive a diffuse-interface model for two-phase flow of incompressible fluids with dissolved noninteracting polymers. The polymers are modeled by dumbbells subjected to general elastic spring-force potentials including in particular Hookean and finitely extensible, nonlinear elastic (FENE) potentials. Their density and orientation are described by a Fokker–Planck-type equation which is coupled to a Cahn–Hilliard and a momentum equation for phase-field and gross velocity/pressure. Henry-type energy functionals are used to describe different solubility properties of the polymers in the different phases or at the liquid–liquid interface. Taking advantage of the underlying energetic/entropic structure of the system, we prove existence of a weak solution globally in time for the case of FENE-potentials. As a by-product in the case of Hookean spring potentials, we derive a macroscopic diffuse-interface model for two-phase flow of Oldroyd-B-type liquids.

Molecular Dynamics Simulation of Lubricant Transfer at the Head-Disk Interface

Molecular dynamics simulation is used to investigate lubricant transfer at the head-disk interface as a function of local pressure change, disk velocity, air bearing design, lubricant bonding ratio, head-disk spacing, and finite width of air bearing. A coarse-grained bead-spring model is used in conjunction with the Lennard-Jones potential, a short-range attractive polar potential, and a finitely extensible nonlinear elastic potential. The results show that lubricant transfer increases as a function of local pressure change, disk velocity, and air bearing width, but decreases with an increase in air bearing convergence angle, lubricant bonding ratio, and slider-disk spacing.

Free energy evaluation in polymer translocation via Jarzynski equality

We perform, with the help of cloud computing resources, extensive Langevin simulations which provide free energy estimates for unbiased three dimensional polymer translocation. We employ the Jarzynski equality in its rigorous setting, to compute the variation of the free energy in single monomer translocation events. In our three-dimensional Langevin simulations, the excluded-volume and van der Waals interactions between beads (monomers and membrane atoms) are modeled through a repulsive Lennard-Jones (LJ) potential and consecutive monomers are subject to the Finite-Extension Nonlinear Elastic (FENE) potential. Analysing data for polymers with different lengths, the free energy profile is noted to have interesting finite size scaling properties.

Mesoscale modeling of shear-thinning polymer solutions

We simulate the linear and nonlinear rheology of two different viscoelastic polymer solutions, a polyisobutylene solution in pristane and an aqueous solution of hydroxypropylcellulose, using a highly coarse-grained approach known as Responsive Particle Dynamics (RaPiD) model. In RaPiD, each polymer has originally been depicted as a spherical particle with the effects of the eliminated degrees of freedom accounted for by an appropriate free energy and transient pairwise forces. Motivated by the inability of this spherical particle representation to entirely capture the nonlinear rheology of both fluids, we extended the RaPiD model by introducing a deformable particle capable of elongation. A Finite-Extensible Non-Linear Elastic potential provides a free energy penalty for particle elongation. Upon disentangling, this deformability allows more time for particles to re-entangle with neighbouring particles. We show this process to be integral towards recovering the experimental nonlinear rheology, obtaining excellent agreement. We show that the nonlinear rheology is crucially dependent upon the maximum elongation and less so on the elasticity of the particles. In addition, the description of the linear rheology has been retained in the process.

Study of the Thermal Decomposition of PFPEs Lubricants on a Thin DLC Film Using Finitely Extensible Nonlinear Elastic Potential Based Molecular Dynamics Simulation

Perfluoropolyethers (PFPEs) are widely used as hard disk lubricants for protecting carbon overcoat reducing friction between the hard disk interface and the head during the movement of head during reading and writing data in the hard disk. Due to temperature rise of PFPE Zdol lubricant molecules on a DLC surface, how polar end groups are detached from lubricant molecules during coating is described considering the effect of temperatures on the bond/break density of PFPE Zdol using the coarse-grained bead spring model based on finitely extensible nonlinear elastic potential. As PFPE Z contains no polar end groups, effects of temperature on the bond/break density (number of broken bonds/total number of bonds) are not so significant like PFPE Zdol. Effects of temperature on the bond/break density of PFPE Z on DLC surface are also discussed with the help of graphical results. How bond/break phenomenonaffects the end bead density of PFPE Z and PFPE Zdol on DLC surface is discussed elaborately. How the overall bond length of PFPE Zdol increases with the increase of temperature which is responsible for its decomposition is discussed with the help of graphical results. At HAMR condition, as PFPE Z and PFPE Zdol are not suitable lubricant on a hard disk surface, it needs more investigations to obtain suitable lubricant. We study the effect of breaking of bonds of nonfunctional lubricant PFPE Z, functional lubricants such as PFPE Zdol and PFPE Ztetrao, and multidented functional lubricants such as ARJ-DS, ARJ-DD, and OHJ-DS on a DLC substrate with the increase of temperature when heating of all of the lubricants on a DLC substrate is carried out isothermally using the coarse-grained bead spring model by molecular dynamics simulations and suitable lubricant is selected which is suitable on a DLC substrate at high temperature.

Defects in nonlinear elastic crystals: differential geometry, finite kinematics, and second‐order analytical solutions

A differential geometric description of crystals with continuous distributions of lattice defects and undergoing potentially large deformations is presented. This description is specialized to describe discrete defects, i.e., singular defect distributions. Three isolated defects are considered in detail: the screw dislocation, the wedge disclination, and the point defect. New analytical solutions are obtained for elastic fields of these defects in isotropic solids of finite extent, whereby terms up to second order in strain, involving elastic constants up to third order, are retained in the stress components. The strain measure used in the nonlinear elastic potential – a symmetric function, expressed in material coordinates, of the inverse deformation gradient – differs from that used in previous solutions for crystal defects, and is thought to provide a more realistic depiction of mechanics of large deformation than previous theory involving third‐order Lagrangian elastic constants and the Green strain tensor. For the screw dislocation and wedge disclination, effects of core pressure and/or possible contraction along the defect line are considered, and radial displacement contributions arise that are absent in the linear elastic solution, affecting dilatation. Stress components are shown to differ from those of linear elastic solutions near defect cores. Volume change from point defects is strongly affected by elastic nonlinearity.

Equivalence of continuum and discrete analytic sensitivity methods for nonlinear differential equations

This paper presents results for the study of equivalence between the total form continuum sensitivity equation (CSE) method and the discrete analytic method of shape design sensitivity analysis. For the discrete analytic method, the sensitivity equations are obtained by taking analytic derivatives of the discretized equilibrium equations with respect to the shape design parameters. For the CSE method, the equilibrium equations are firstly differentiated to form a set of linear continuous sensitivity equations and then discretized for solving the shape sensitivities. The sensitivity equations can be derived by taking the material derivatives (total form) or the partial derivatives (local form) of the equilibrium equations. The total form CSE method is shown for the first time to be equivalent, after finite element discretization, to the discrete analytic method for nonlinear second-order differential equations of a particular form with design dependent loads when they use the same: (1) finite element discretization, (2) numerical integration of element matrices, (3) design velocity fields that are linear with respect to the design variable and (4) shape functions for domain transformation and for design velocity field calculations. The shape sensitivity equations for three-dimensional geometric nonlinear elastic structures and linear potential flow are derived by using both total form CSE and discrete analytic method to show the equivalence of the two methods for these specific examples. The accuracy of shape sensitivity analysis is verified by potential flow around an airfoil and a joined beam with an airfoil under gust load. The results show that analytic sensitivity results are consistent with the complex step results.

Study of the Spreading of Perfluoropolyether Lubricants on a Diamond-Like Carbon Film

Nonpolar perfluoropolyether (PFPE) Z and PFPE Zdol with polar end groups are widely used as hard disk lubricants for protecting carbon overcoats by reducing friction between the hard disk and head during movement of the head while reading and writing data on the hard disk. We investigate the spreading phenomenon of PFPE Z and PFPE Zdol on a thin diamond-like carbon (DLC) film adopting molecular dynamics (MD) simulations based on a coarse-grained bead-spring model to describe the thickness profiles and molecular movement, which evolve with time and temperature. In the present article, the hard disk surface was considered as a DLC and the position of its carbon atoms was obtained by heating and quenching the face-centered cubic (FCC) or body-centered cubic (BCC) diamond structures by MD simulation using the Tersoff potential. To simulate PFPE Z and PFPE Zdol on a thin DLC film using a coarse-grained bead-spring model based on finitely extensible nonlinear elastic potential and nonbonded potential, the original DLC thin film was compressed to half of its original configuration in all three spatial dimensions. How PFPE Z and PFPE Zdol on the DLC surface spread with time are briefly discussed. How the spreading profile of PFPE Z and PFPE Zdol on the DLC film spreads laterally and on the DLC film as a circular shape beyond its original position is also discussed. The effect of temperature on the film thickness and spreading area of PFPE Z and PFPE Zdol on the DLC film is also discussed. We show that the time dependence for the spreading of PFPE Z and PFPE Zdol droplets deviated from the expected proportionality to the square root of time in their spreading profiles with time. The model-calculated spreading rate of PFPE Z and PFPE Zdol on a thin DLC film increased inversely with absolute temperature as expected.

Nonlinear dynamic vibration absorbers with a saturation

The behavior of a new type of nonlinear dynamic vibration absorber is studied. A distinctive characteristic of the proposed absorber is the impossibility to extend the system to infinity. The mathematical formulation is based on a finite extensibility nonlinear elastic potential to model the saturable nonlinearity. The absorber is attached to a single degree-of-freedom linear/nonlinear oscillator subjected to a periodic external excitation. In order to solve the equations of motion and to analyze the frequency-response curves, the method of averaging is used. The performance of the FENE absorber is evaluated considering a variation of the nonlinearity of the primary system, the damping and the linearized frequency of the absorber and the mass ratio. The numerical results show that the proposed absorber has a very good efficiency when the nonlinearity of the primary system increases. When compared with a cubic nonlinear absorber, for a large nonlinearity of the primary system, the FENE absorber shows a better effectiveness for the whole studied frequency range. A complete absence of quasi-periodic oscillations is also found for an appropriate selection of the parameters of the absorber. Finally, direct integrations of the equations of motion are performed to verify the accuracy of the proposed method.

Existence of global weak solutions to finitely extensible nonlinear bead–spring chain models for dilute polymers with variable density and viscosity

We prove the existence of global-in-time weak solutions to a general class of coupled bead–spring chain models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids with noninteracting polymer chains, with finitely extensible nonlinear elastic (FENE) spring potentials. The class of models under consideration involves the unsteady incompressible Navier–Stokes equations with variable density and density-dependent dynamic viscosity in a bounded domain in Rd, d=2 or 3, for the density, 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 by the Kramers expression 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 and a nonlinear density-dependent drag coefficient. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With initial density ρ0∈[ρmin,ρmax] for the continuity equation, where ρmin>0; a square-integrable and divergence-free initial velocity datum u˜0 for the Navier–Stokes equation; and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M associated with the spring potential in the model, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t↦(ρ(t),u˜(t),ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ρ(0),u˜(0),ψ(0))=(ρ0,u˜0,ψ0), such that t↦ρ(t)∈[ρmin,ρmax], t↦u˜(t) belongs to the classical Leray space and t↦ψ(t) has bounded relative entropy with respect to M and t↦ψ(t)/M has integrable Fisher information (w.r.t. the Gibbs measure dν:=M(q˜)dq˜dx˜) over any time interval [0,T], T>0.

Internucleosomal Interactions Mediated by Histone Tails Allow Distant Communication in Chromatin

Action across long distances on chromatin is a hallmark of eukaryotic transcriptional regulation. Although chromatin structure per se can support long-range interactions, the mechanisms of efficient communication between widely spaced DNA modules in chromatin remain a mystery. The molecular simulations described herein suggest that transient binary internucleosomal interactions can mediate distant communication in chromatin. Electrostatic interactions between the N-terminal tails of the core histones and DNA enhance the computed probability of juxtaposition of sites that lie far apart along the DNA sequence. Experimental analysis of the rates of communication in chromatin constructs confirms that long-distance communication occurs efficiently and independently of distance on tail-containing, but not on tailless, chromatin. Taken together, our data suggest that internucleosomal interactions involving the histone tails are essential for highly efficient, long-range communication between regulatory elements and their targets in eukaryotic genomes.

Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (J. Non-Newtonian Fluid Mech. 139:153-176, 2006) for the numerical solution of high-dimensional Fokker-Planck equations featuring in Navier-Stokes-Fokker-Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson's equation on a rectangular domain in R^2, subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Leli\`evre and Maday (Const. Approx. 30:621-651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173-187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein-Uhlenbeck operator of the kind that appears in Fokker-Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D = D_1 x ... x D_N contained in R^(N d), where each set D_i, i = 1, ..., N, is a bounded open ball in R^d, d = 2, 3.