Bead-spring Model Research Articles

Effect of simple shear on knotted polymer coils and globules.

We explore the effect of Couette flow on knotted linear polymer chains with extensive molecular dynamics simulations. Hydrodynamic interactions are accounted for using multi-particle collision dynamics. The polymer chain, originally containing a simple trefoil knot at rest, is described by a coarse-grained bead-spring model in a coil or globular state. We demonstrate that under shear existing loosely localized knots in polymer coils typically tighten to several segments beyond a certain shear rate threshold. At large shear rates, the polymer undergoes a tumbling-like motion during which knot sizes can fluctuate. In contrast, sheared knotted globules unwind into a convoluted pearl-necklace structure of sub-globules that folds back onto itself and in which knot types change over time.

Understanding the role of chain stiffness in the mechanical response of cross-linked polymer: Flexible vs. semi-flexible chains

Cross-linked polymers are widely used in structural, engineering, and biomedical applications due to their lightweight and superior properties. Although chain bending stiffness has been recognized to play an essential role in their thermodynamical and mechanical properties, how it influences these properties of cross-linked polymers with flexible or semi-flexible chains remains under debate. Here, we systematically explore its influences utilizing coarse-grained (CG) molecular dynamics (MD) simulations based on a bead-spring CG model. It is found that with chain bending stiffness increasing, both density and elastic moduli (i.e., shear modulus and tensile modulus) of cross-linked polymers first decrease slightly and then decrease significantly followed by a gradual increase, along with the polymer transition from a dense cross-linked thermoset to a highly porous fibrous network. The moduli of cross-linked polymers with flexible and semi-flexible chains exhibit distinct scaling laws with the density. For cross-linked polymers with flexible chains, their moduli increase significantly with increasing strain rate, which correlates to the change in potential energy of interchain interaction during deformation. However, the moduli display slight dependence on strain rate for porous cross-linked polymers with sufficiently stiff chains, where the intrachain interactions (i.e., bond stretching and angle bending energies) become dominant and independent of strain rate. Moreover, the elastic moduli exhibit scaling laws with Debye-Waller factor for both dense cross-linked thermosets with flexible chains and highly porous networks with stiff backbones. Our work facilitates a better understanding for mechanical properties and deformation mechanism of cross-linked polymers with variable chain bending stiffness at molecular level, shedding light on tailoring mechanical properties of cross-linked polymers via chain engineering.

Ultralow thermal conductivity of 2D Dion-Jacobson (PDA)(FA)n - 1PbnI3n + 1 perovskite films.

Two-dimensional (2D) perovskites exhibit enhanced thermal stability compared to three-dimensional perovskites, especially the emerging 2D Dion-Jacobson (DJ) phase perovskite. However, the heat transfer mechanisms in DJ phase perovskites are rarely reported. Herein, we determine thermal conductivities of (PDA)(FA)n - 1PbnI3n + 1 films with n = 1-6 by time-domain thermoreflectance. The measured results indicate that the thermal conductivities of these films are extremely low, showing a trend from decline to rise with increasing n values, and reaching to the lowest when n = 2. We measure the propagation of acoustic phonons in films with n = 1-3 by time-domain Brillouin scattering and find phonon velocity plays a key role in the thermal conductivity, which can be explained by the mismatch of spring constants between the inorganic layer and the organic layer using the bead-spring model. The gradually increasing thermal conductivity for larger n values is attributed to the gradual transformation of the grain orientation from horizontal to vertical, which is demonstrated by the grazing-incidence wide-angle x ray scattering (GIWAXS) results. Our work deepens the understanding of the thermal transport process in 2D DJ phase perovskite films and provides insights into thermal management solutions for their devices.

Impact of the unimodal molar mass distribution on the mechanical behavior of polymer nanocomposites below the glass transition temperature: A generic, coarse-grained molecular dynamics study

Polymer nanocomposites (PNCs) have shown great potential to meet the ever-growing requirements of modern engineering applications. Nowadays, molecular dynamics (MD) simulations are increasingly employed to complement experimental work and thereby gain a deeper understanding of the complex structure–property relations of PNCs. However, with respect to the thermoplastic’s mechanical behavior, the role of its average molar mass M¯n is rarely addressed, and many MD studies only consider uniform (monodispersed) polymers. Therefore, this contribution investigates the impact that M¯n and the dispersity Đ have on the stiffness and strength of PNCs through coarse-grained MD. To this end, we employed a Kremer–Grest bead–spring model and observed the expected increase in the mechanical performance of the neat polymer for larger M¯n. Our results indicated that the unimodal molar mass distribution does not impact the mechanical behavior in the investigated dispersity range 1.0≤Đ≤1.09. For the PNC, we obtained the same M¯n-dependence and Đ-independence of the mechanical properties over a wide range of filler sizes and contents. This contribution proves that even simple MD models can reproduce the experimentally well researched effect of the molar mass. Hence, this work is an important step in understanding the complex structure–property relations of PNCs, which is essential to unlock their full potential.

Polymers in turbulence: any better than dumbbells?

Polymer chains in turbulent flows are generally modelled as dumbbells, i.e. two beads joined by a nonlinear spring. The dumbbell only maps a single spatial configuration, described by the polymer end-to-end vector, thus a multi-bead FENE (finitely extensible nonlinear elastic) chain seems a natural improvement for a more accurate characterisation of the polymer spatial conformation. At a large Weissenberg number, a comparison with the more accurate Kuhn chain reveals that the multi-bead FENE chain drastically overestimates the probability of folded configurations. Surprisingly, the dumbbell turns out to be the only meaningful bead-spring model to coarse-grain a polymer macromolecule in turbulent pipe flows.

Collapse and expansion kinetics of a single polyelectrolyte chain with hydrodynamic interactions.

We investigate the collapse and expansion dynamics of a linear polyelectrolyte (PE) with hydrodynamic interactions. Using dissipative particle dynamics with a bead-spring PE model, long-range electrostatics, and explicit ions, we examine how the timescales of collapse tcol and expansion texp depend on the chain length N and obtain scaling relationships tcol ∼ Nα and texp ∼ Nβ. For neutral polymers, we derive values of α = 0.94 ± 0.01 and β = 1.97 ± 0.10. Interestingly, the introduction of electrostatic interaction markedly shifts α to α ≈ 1.4 ± 0.1 for salt concentrations within c = 10-4 to 10-2M. A reduction in the ion-to-monomer size ratio noticeably reduces α. On the other hand, the expansion scaling remains approximately constant, β ≈ 2, regardless of the salt concentration or ion size considered. We find β > α for all conditions considered, implying that expansion is always slower than collapse in the limit of long polymers. This asymmetry is explained by distinct kinetic pathways of collapse and expansion processes.

Effects of chain resolution on the configurational and rheological predictions of dilute polymer solutions in flow fields with hydrodynamic interactions

In a recent study, the resolution of a polymer chain model was shown to significantly affect rheological predictions from Brownian dynamics (BD) simulations [Kumar and Dalal, “Effects of chain resolution on the configurational and rheological predictions from Brownian dynamics simulations of an isolated polymer chain in flow,” J. Non-Newtonian Fluid Mech. 315, 105017 (2023)], even in the absence of hydrodynamic interactions (HI) and excluded volume. In this study, we investigate the effects of chain resolution in the presence of HI. Toward this, we perform BD simulations of a long polymer chain, with the discretization level varying from a single Kuhn step (bead–rod model) to several tens of Kuhn-steps (bead–spring model). The chain models were subjected to flow fields of uniaxial extension (purely stretching) and steady shear (equal rates of stretching and rotation). Broadly, our results indicate an amplification of the differences observed between the differently resolved bead–rod and bead–spring models, in the presence of HI. Interestingly, all rheological predictions qualitatively fall in two groups for extensional flow, with the predictions from the bead–spring model with HI being close to those of the bead–rod model without HI. This indicates significantly reduced sensitivity of coarser bead–spring models to HI, relative to the one resolved to a single Kuhn step. However, in shear flow, the bead–spring rheological predictions fall between those of the bead–rod model with and without HI, forming a third group. This is linked to the presence of stretched and coiled states in the ensemble for shear flow. HI effects are large for the coiled states and weak for the stretched states, thereby yielding predictions that are intermediate between those for no HI and dominant HI. Thus, quite surprisingly, the quality of predictions of the bead–spring models is strongly affected by the physics of the flow field, irrespective of the parameterization.

Polymeric surfactants at liquid–liquid interfaces: Dependence of structural and thermodynamic properties on copolymer architecture

Polymeric surfactants are amphiphilic molecules with two or more different types of monomers. If one type of monomer interacts favorably with a liquid, and another type of monomer interacts favorably with another, immiscible liquid, then polymeric surfactants adsorb at the interface between the two liquids and reduce the interfacial tension. The effects of polymer architecture on the structural and thermodynamic properties of the liquid–liquid interface are studied using molecular simulations. The interface is modeled with a non-additive binary Lennard-Jones fluid in the two-phase region of the phase diagram. Block and gradient copolymer surfactants are represented with coarse-grained, bead-spring models, where each component of the polymer favors one or the other liquid. Gradient copolymers have a greater concentration at the interface than do block copolymers because the gradient copolymers adopt conformations partially aligned with the interface. The interfacial tension is determined as a function of the surface excess of polymeric surfactant. Gradient copolymers are more potent surfactants than block copolymers because the gradient copolymers cross the dividing surface multiple times, effectively acting as multiple individual surfactants. For a given surface excess, the interfacial tension decreases monotonically when changing from a block to a gradient architecture. The coarse-grained simulations are complemented by all-atom simulations of acrylic-acid/styrene copolymers at the chloroform-water interface, which have been studied in experiments. The agreement between the simulations (both coarse-grained and atomistic) and experiments is shown to be excellent, and the molecular-scale structures identified in the simulations help explain the variation of surfactancy with copolymer architecture.

Existence of large-data global weak solutions to kinetic models of nonhomogeneous dilute polymeric fluids

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $ \mathbb{R}^d $, $ d = 2 $ or $ 3 $. The class of models under consideration involves the Navier–Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker–Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.

Polymer Modeling Reveals Interplay between Physical Properties of Chromosomal DNA and the Size and Distribution of Condensin-Based Chromatin Loops.

Transient DNA loops occur throughout the genome due to thermal fluctuations of DNA and the function of SMC complex proteins such as condensin and cohesin. Transient crosslinking within and between chromosomes and loop extrusion by SMCs have profound effects on high-order chromatin organization and exhibit specificity in cell type, cell cycle stage, and cellular environment. SMC complexes anchor one end to DNA with the other extending some distance and retracting to form a loop. How cells regulate loop sizes and how loops distribute along chromatin are emerging questions. To understand loop size regulation, we employed bead-spring polymer chain models of chromatin and the activity of an SMC complex on chromatin. Our study shows that (1) the stiffness of the chromatin polymer chain, (2) the tensile stiffness of chromatin crosslinking complexes such as condensin, and (3) the strength of the internal or external tethering of chromatin chains cooperatively dictate the loop size distribution and compaction volume of induced chromatin domains. When strong DNA tethers are invoked, loop size distributions are tuned by condensin stiffness. When DNA tethers are released, loop size distributions are tuned by chromatin stiffness. In this three-way interaction, the presence and strength of tethering unexpectedly dictates chromatin conformation within a topological domain.

Bead-Spring Simulation of Ionomer Melts-Studying the Effects of Chain-Length and Associating Group Fraction on Equilibrium Structure and Extensional Flow Behavior.

Ionomers are associative polymers with diverse applications ranging from selective membranes and high-performance adhesives to abrasion- and chemical-resistant coatings, insulation layers, vacuum packaging, and foamed sheets. Within equilibrium melt, the ionic or associating groups are known to form thermally reversible, associative clusters whose presence can significantly affect the system's mechanical, viscoelastic, and transport properties. It is, thus, of great interest to understand how to control such clusters' size distribution, shape, and stability through the designed choice of polymer architecture and the ionic groups' fraction, arrangement, and interaction strength. In this work, we represent linear associating polymers using a Kremer-Grest type bead-spring model and perform large-scale MD simulations to explore the effect of polymer chain-length (l) and fraction (fs) of randomly placed associating groups on the size distribution and stability of formed clusters. We consider different chain-lengths (below and above entanglement), varying fractions of associating groups (represented by 'sticky' beads) between 5 and 20%, and a fixed sticky-sticky nonbond interaction strength of four times that between regular non-associating beads. For all melts containing associating groups the equilibrium structure factor S(q) displays a signature ionomer peak at low wave vector q whose intensity increases with increasing fs and l. The average cluster size Nc increases with fs. However, the effect of chain-length on Nc appears to be pronounced only at higher values of fs. Under extensional flows, the computed stress (and viscosity) is higher at higher fs and l regardless of strain rate. Beyond a critical strain rate, we observe fragmentation of the associative clusters, which has interesting effects on the stress/viscous response.

Solving the constitutive equation of dilute polymeric flows: A general Fokker–Planck approach for linear elastic dumbbell models

We present a mesoscopic numerical solver for the constitutive equation of dilute polymer solutions, as described by the Fokker–Planck equation of bead-spring dumbbell models with linear spring forces, such as Oldroyd-B, FENE-P (finitely extensible nonlinear elastic with Peterlin closure), and C-FENE-P (charged FENE-P). The configuration density function of the Fokker–Planck equation is expanded into a series of Hermite polynomials, and we show that only a second order in the expansion is needed to describe the evolution of the polymer-induced stresses. The polymer-induced stress tensor is given exactly through a discrete representation of the configuration distribution function, which is obtained from a finite set of Gauss–Hermite quadrature points. In addition, we derive the conformation tensor equation, which serves as a mathematical quality check for the method. We solve the time evolution of the extra stress tensor for homogeneous flows of dilute polyelectrolyte solutions, conceptualized by the C-FENE-P model, which is a generalization of the FENE-P dumbbell model. The results are shown to be in excellent agreement with analytical and semi-analytical reference results for simple shear and extensional flows. While in this paper the focus is on linear connector forces, we believe that the principles of the derivation are extendable to other force laws.

Predicting scale-dependent chromatin polymer properties from systematic coarse-graining

Simulating chromatin is crucial for predicting genome organization and dynamics. Although coarse-grained bead-spring polymer models are commonly used to describe chromatin, the relevant bead dimensions, elastic properties, and the nature of inter-bead potentials are unknown. Using nucleosome-resolution contact probability (Micro-C) data, we systematically coarse-grain chromatin and predict quantities essential for polymer representation of chromatin. We compute size distributions of chromatin beads for different coarse-graining scales, quantify fluctuations and distributions of bond lengths between neighboring regions, and derive effective spring constant values. Unlike the prevalent notion, our findings argue that coarse-grained chromatin beads must be considered as soft particles that can overlap, and we derive an effective inter-bead soft potential and quantify an overlap parameter. We also compute angle distributions giving insights into intrinsic folding and local bendability of chromatin. While the nucleosome-linker DNA bond angle naturally emerges from our work, we show two populations of local structural states. The bead sizes, bond lengths, and bond angles show different mean behavior at Topologically Associating Domain (TAD) boundaries and TAD interiors. We integrate our findings into a coarse-grained polymer model and provide quantitative estimates of all model parameters, which can serve as a foundational basis for all future coarse-grained chromatin simulations.

Mode selectivity of dynamically induced conformation in many-body chainlike bead-spring models.

We consider conformation of a chain consisting of beads connected by stiff springs, where the conformation is determined by the bending angles between the consecutive two springs. Stability of a conformation is determined intrinsically by a potential energy function depending on the bending angles. However, effective forces induced by excited springs can change the stability, and a conformation can stay around a local maximum or a saddle of the bending potential. A stabilized conformation was named the dynamically induced conformation in a previous work on a three-body system [Y. Y. Yamaguchi etal., Phys. Rev. E 105, 064201 (2022)2470-004510.1103/PhysRevE.105.064201]. A remarkable fact is that the stabilization by the spring motion depends on the excited normal modes, which depend on a conformation. We extend analyses of the dynamically induced conformation in many-body chainlike bead-spring systems. Simple rules are that the lowest-eigenfrequency mode contributes to the stabilization and that the higher the eigenfrequency is, the more the destabilization emerges. We verify theoretical predictions by performing numerical simulations.

Hybrid cellular Potts and bead-spring modeling of cells in fibrous extracellular matrix

The mechanical interaction between cells and the extracellular matrix (ECM) is fundamental to coordinate collective cell behavior in tissues. Relating individual cell-level mechanics to tissue-scale collective behavior is a challenge that cell-based models such as the cellular Potts model (CPM) are well-positioned to address. These models generally represent the ECM with mean-field approaches, which assume substrate homogeneity. This assumption breaks down with fibrous ECM, which has nontrivial structure and mechanics. Here, we extend the CPM with a bead-spring model of ECM fiber networks modeled using molecular dynamics. We model a contractile cell pulling with discrete focal adhesion-like sites on the fiber network and demonstrate agreement with experimental spatiotemporal fiber densification and displacement. We show that at high network cross-linking, contractile cell forces propagate over at least eight cell diameters, decaying with distance with power law exponent n= 0.35 – 0.65 typical of viscoelastic ECMs. Further, we use in silico atomic force microscopy to measure local cell-induced network stiffening consistent with experiments. Our model lays the foundation for investigating how local and long-ranged cell-ECM mechanobiology contributes to multicellular morphogenesis.

Variation of Spring Stiffness, Monomeric Friction, and Brownian Intensity in the Simulation System of Unentangled Melt under Steady Flow

Understanding the nonlinear dynamics of an unentangled polymer melt from the bead-spring chain model requires knowledge of the variation of spring stiffness, monomeric friction, and Brownian intensity. In this work, these nonequilibrium (NE) parameters are quantified in the model simulation systems of unentangled melt under steady-flow conditions. We focus on a particular version of the model with undefined NE parameters. Some expressions to relate the NE parameters with the rheological and structural properties are pointed out. The spring stiffening parameter rκ is proven to be easily accessible in the planar flows. The distribution of the stiffening effect along the polymer contour is identified. The alignment-induced reduction of monomeric friction is confirmed in the simulation systems, but the anisotropic feature of friction is also emphasized. The constraint on the cooperative variation of spring stiffness and friction is proven to be an important property of extensional flow. The variation of Brownian intensity is found to be decoupled with the variation of friction, which means the violation of the fluctuation–dissipation (FD) theorem. The NE parameters of the simulation systems are further compared with those from the experimental systems.

Dynamics of flexible fibers in confined shear flows at finite Reynolds numbers

We carry out a numerical study on the dynamics of a single non-Brownian flexible fiber in two-dimensional confined simple shear (Couette) flows at finite Reynolds numbers. We employ the bead-spring model of flexible fibers to extend the fluid particle dynamics (FPD) method that was originally developed for rigid particles in viscous fluids. We implement the extended FPD method using a multiple-relaxation-time scheme of the lattice Boltzmann method. The numerical scheme is validated first by a series of benchmark simulations that involve fluid–solid coupling. The method is then used to study the dynamics of flexible fibers in Couette flows. We only consider the highly symmetric cases where the fibers are placed on the symmetry center of Couette flows, and we focus on the effects of the fiber stiffness, the confinement strength, and the finite Reynolds number (from 1 to 10). A diagram of the fiber shape is obtained. For fibers under weak confinement and a small Reynolds number, three distinct tumbling orbits have been identified: (1) Jeffery orbits of rigid fibers—the fibers behave like rigid rods and tumble periodically without any visible deformation; (2) S-turn orbits of slightly flexible fibers—the fiber is bent to an S-shape and is straightened again when it orients to an angle of around 45° relative to the positive x-direction; and (3) S-coiled orbits of fairly flexible fibers—the fiber is folded to an S-shape and tumbles periodically and steadily without being straightened anymore during its rotation. Moreover, the fiber tumbling is found to be hindered by increasing either the Reynolds number or the confinement strength, or both.

Sliding Dynamics of Slide-Ring Polymers Based on the Bead-Spring Model

Sliding Dynamics of Slide-Ring Polymers Based on the Bead-Spring Model

Effect of surfactants on the elasticity of the liquid-liquid interface.

We investigated the effect of surfactants on an interface between two kinds of liquids by molecular dynamics (MD) simulation. We adopted the simple bead-spring model with two atoms as the surfactants. We controlled the interfacial tension of the surfactant adsorbed on the interface by changing the bond length. Although the interface's structure changed depending on the magnitude of the interfacial tension, the interface was stable even under conditions where the interfacial tension was virtually zero. The Fourier spectrum of the fluctuations of the surface structure showed a crossover from q2 to q4 when the interfacial tension was almost zero, where q is the wavenumber. This crossover means that the bending rigidity is dominant for the restoring force when the surfactant molecules are sufficiently absorbed on the interface and the interfacial tension is almost zero, whereas the interfacial tension is dominant when the interfacial tension is a finite value.

Multiscale SPH simulations of viscoelastic injection molding processes based on bead-spring chain model

In this paper we present an extension of a multiscale smoothed particle hydrodynamics (SPH) method to transient viscoelastic injection molding processes. Specifically, we use the method presented by Xu and Yu, J. Non-Newtonian Fluid Mech. 229 (2016) pp. 27–42, in which the bead-spring chain model is employed to describe the viscoelastic behavior of the fluid without resort to a closed-form constitutive equation. Moreover, to resolve the heavy computation and the time-consuming problem, an efficient parallel algorithm, including parallelizations of both SPH particles and Brownian configuration fields, is developed. To validate the algorithm, we first simulate the viscoelastic Couette flow based on Hookean dumbbell, FENE dumbbell, and FENE chain models. The SPH results are compared with the analytical solutions or those obtained by other methods. The parallel performances of the algorithm are analyzed. Then, we extend the method to the challenging injection molding problem. A number of numerical examples including the injection molding of an F-shaped cavity in two dimensions and a door handle in two and three dimensions are investigated. Some molecular information such as the molecular stretch, the orientation angle, and the mean configuration thickness are displayed. It is found that the injection molding processes could be successfully investigated by SPH from a micro perspective. The proposed parallel algorithm is essential for the efficient simulation of injection molding processes. The maximum speedup can reach more than 130 for the case of the door handle in three dimensions.