Onset of Entanglements Revisited. Dynamical Analysis

Simulational Tests of the Rouse Model

An extensive review of literature simulations of quiescent polymer melts is given, considering results that test aspects of the Rouse model in the melt. We focus on Rouse model predictions for the mean-square amplitudes ⟨(Xp(0))2⟩ and time correlation functions ⟨Xp(0)Xp(t)⟩ of the Rouse mode Xp(t). The simulations conclusively demonstrate that the Rouse model is invalid in polymer melts. In particular, and contrary to the Rouse model, (i) mean-square Rouse mode amplitudes ⟨(Xp(0))2⟩ do not scale as sin-2(pπ/2N), N being the number of beads in the polymer. For small p (say, p≤3) ⟨(Xp(0))2⟩ scales with p as p-2; for larger p, it scales as p-3. (ii) Rouse mode time correlation functions ⟨Xp(t)Xp(0)⟩ do not decay with time as exponentials; they instead decay as stretched exponentials exp(-αtβ). β depends on p, typically with a minimum near N/2 or N/4. (iii) Polymer bead displacements are not described by independent Gaussian random processes. (iv) For p≠q, ⟨Xp(t)Xq(0)⟩ is sometimes non-zero. (v) The response of a polymer coil to a shear flow is a rotation, not the affine deformation predicted by Rouse. We also briefly consider the Kirkwood-Riseman polymer model.

Investigating anionic surfactant phase diagrams using dissipative particle dynamics: development of a transferable model

Dissipative particle dynamics (DPD) provides a powerful coarse-grained simulation technique for the study of a wide range of soft matter systems. Here, we investigate the transferability of DPD models to the prediction of anionic surfactant phase diagrams, taking advantage of fast parameter sweeps to optimise the choice of DPD parameters for these systems. Parameters are developed which provide a good representation of the phase diagrams of SDS (sodium dodecyl sulfate) and three different isomeric forms of LAS (linear alkylbenzene sulfonates) across an extensive concentration range. A high degree of transferability is seen, with parameters readily transferable to other systems, such as AES (alkyl ether sulfates). Excellent agreement is obtained with experimentally measured quantities, such as the lamellar layer spacing. Isosurfaces are produced from the surfactant head group, from which the second moment M of the isosurface normal distribution is calculated for different phase structures. Lyotropic liquid crystalline phases are characterised by a combination of the eigenvalues of M, radial distribution functions, and visual inspections.

Changes in Ion Concentrations upon the Binding of Short Polyelectrolytes on Phospholipid Bilayers: Computer Study Addressing Interesting Physiological Consequences

This computer study was inspired by the experimental observation of Y. Qian et al. published in ACS Applied Materials and Interfaces, 2018 that the short positively charged β-peptide chains and their oligomeric analogues efficiently suppress severe medical problems caused by antimicrobial drug-resistant bacteria despite them not penetrating the bacterial membrane. Our coarse-grained molecular dynamics (dissipative particle dynamics) simulations confirm the tentative explanation of the authors of the experimental study that the potent antimicrobial activity is a result of the entropically driven release of divalent ions (mainly magnesium ions essential for the proper biological function of bacteria) into bulk solution upon the electrostatic binding of β-peptides to the bacterial membrane. The study shows that in solutions containing cations Na⁺, Ca²⁺ and Mg²⁺, and anions Cl⁻, the divalent cations preferentially concentrate close to the membrane and neutralize the negative charge. Upon the addition of positively charged oligomer chains (models of β-peptides and their analogues), the oligomers electrostatically bind to the membrane replacing divalent ions, which are released into bulk solvent. Our simulations indicate that the entropy of small ions (which controls the behavior of synthetic polyelectrolyte solutions) plays an important role in this and also in other similar biologically important systems.

DPD Modelling of the Self- and Co-Assembly of Polymers and Polyelectrolytes in Aqueous Media: Impact on Polymer Science

This review article is addressed to a broad community of polymer scientists. We outline and analyse the fundamentals of the dissipative particle dynamics (DPD) simulation method from the point of view of polymer physics and review the articles on polymer systems published in approximately the last two decades, focusing on their impact on macromolecular science. Special attention is devoted to polymer and polyelectrolyte self- and co-assembly and self-organisation and to the problems connected with the implementation of explicit electrostatics in DPD numerical machinery. Critical analysis of the results of a number of successful DPD studies of complex polymer systems published recently documents the importance and suitability of this coarse-grained method for studying polymer systems.

Comparison of equilibrium techniques for the viscosity calculation from DPD simulations

Dissipative Particle Dynamics (DPD) is a powerful mesoscopic modelling technique that is routinely used to predict complex fluid morphology and structural properties. While its ability to quickly scan the conformational space is well known, it is unclear if DPD can correctly calculate the viscosity of complex fluids. In this work, we estimate the viscosity of several unentangled polymer solutions using both the Einstein and Green-Kubo formulas. For this purpose, an Einstein relation is derived analogous to the revised Green-Kubo formula suggested by Jung and Schmid, J. Chem. Phys., 2016, 144, 204104. We show that the DPD simulations reproduce the dynamical behaviour predicted by the theory irrespectively of the values of the conservative and friction parameters used and estimate a Schmidt number compatible to that of a fluid system. Moreover, we observe that the Einstein method requires shorter trajectories to achieve the same statistical accuracy as the Green-Kubo formula. This work shows that DPD can confidently be used to calculate the viscosity of complex fluids and that the statistical accuracy of short trajectories can be improved by using our revised Einstein formula.

Multi-scale modeling of the polymer-filler interaction

We report mesoscopic simulations of the interaction between a silica nanoparticle and cis-1,4-polybutadiene chains with realistic coarse-(CG) grained models. The CG models are obtained with a bottom-up Bayesian method based on trajectory matching of atomistic configurations of the system. We then investigate the structural properties of the interfacial region as a function of the grafting density and polymer chain length. We take advantage of the realistic CG models to explore the dynamics of the nanoparticle over a period of 10 microseconds. We show that the dynamics of the nanoparticle is affected by the grafting density and the polymer chain length of the grafted chains.

Modeling of Entangled Polymer Diffusion in Melts and Nanocomposites: A Review

This review concerns modeling studies of the fundamental problem of entangled (reptational) homopolymer diffusion in melts and nanocomposite materials in comparison to experiments. In polymer melts, the developed united atom and multibead spring models predict an exponent of the molecular weight dependence to the polymer diffusion very similar to experiments and the tube reptation model. There are rather unexplored parameters that can influence polymer diffusion such as polymer semiflexibility or polydispersity, leading to a different exponent. Models with soft potentials or slip-springs can estimate accurately the tube model predictions in polymer melts enabling us to reach larger length scales and simulate well entangled polymers. However, in polymer nanocomposites, reptational polymer diffusion is more complicated due to nanoparticle fillers size, loading, geometry and polymer-nanoparticle interactions.

Multipoint segmental repulsive potential for entangled polymer simulations with dissipative particle dynamics

A model is developed for simulating entangled polymers by dissipative particle dynamics (DPD) using the segmental repulsive potential (SRP). In contrast to previous SRP models that define a single-point interaction on each bond, the proposed SRP model applies a dynamically adjustable multipoint on the bond. Previous SRP models could not reproduce the equilibrium properties of Groot and Warren's original DPD model [R. D. Groot and P. B. Warren, J. Chem. Phys. 107, 4423 (1997)] because the introduction of a single SRP induces a large excluded volume, whereas, the proposed multipoint SRP (MP-SRP) introduces a cylindrical effective excluded bond volume. We demonstrate that our MP-SRP model exhibits equilibrium properties similar to those of the original DPD polymers. The MP-SRP model parameters are determined by monitoring the number of topology violations, thermodynamic properties, and the polymer internal structure. We examine two typical DPD polymers with different bond-length distributions; one of them was used in the modified SRP model by Sirk et al. [J. Chem. Phys. 136, 134903 (2012)], whereas the other was used in the original DPD model. We demonstrate that for both polymers, the proposed MP-SRP model captures the entangled behaviors of a polymer melt naturally, by calculating the slowest relaxation time of a chain in the melt and the shear relaxation modulus. The results indicate that the proposed MP-SRP model can be applied to a variety of DPD polymers.

Simulations of Interfacial Tension of Liquid-Liquid Ternary Mixtures Using Optimized Parametrization for Coarse-Grained Models

In this work, liquid-liquid systems are studied by means of coarse-grained Monte Carlo simulations (CG-MC) and Dissipative Particle Dynamics (DPD). A methodology is proposed to reproduce liquid-liquid equilibrium (LLE) and to provide variation of interfacial tension (IFT), as a function of the solute concentration. A key step is the parametrization method based on the use of the Flory-Huggins parameter between DPD beads to calculate solute/solvent interactions. Parameters are determined using a set of experimental compositional data of LLE, following four different approaches. These approaches are evaluated, and the results obtained are compared to analyze advantages/disadvantages of each one. These methodologies have been compared through their application on six systems: water/benzene/1,4-dioxane,water/chloroform/acetone, water/benzene/acetic acid, water/benzene/2-propanol, water/hexane/acetone, and water/hexane/2-propanol. CG-MC simulations in the Gibbs (NVT) ensemble have been used to check the validity of parametrization approaches for LLE reproduction. Then, CG-MC simulations in the osmotic (μ_soluteN_solventP _zzT) ensemble were carried out considering the two liquid phases with an explicit interface. This step allows one to work at the same bulk concentrations as the experimental data by imposing the precise bulk phase compositions and predicting the interface composition. Finally, DPD simulations were used to predict IFT values for different solute concentrations. Our results on variation of IFT with solute concentration in bulk phases are in good agreement with experimental data, but some deviations can be observed for systems containing hexane molecules.

Comparison among multi-chain models for entangled polymer dynamics

Although lots of coarse-grained models have been proposed to trace the long-term behaviors of entangled polymers, compatibility among the different models has not been frequently discussed. In this study, some dynamical and static quantities, such as diffusion, relaxation modulus, chain dimension, and entanglement density, were examined for the multi-chain slip-link model (primitive chain network model) and the multi-chain slip-spring model, and the results were compared with those reported for the standard bead-spring model. For the diffusion, three models are compatible with scale-conversion parameters for units of length, time and bead (segment) number (or the molecular weight). The relaxation modulus is also compatible given that the model dependence can be accommodated by the entanglement density and the additional scale-conversion for the unit of modulus. The chain dimension is reasonably coincident with small deviations due to the weak non-Gaussianity of the models. Apart from these plausible compatibilities, significant discrepancies have been found for the inter-chain cross-correlations in the relaxation modulus.

Molecular dynamics simulation on the mechanical properties of natural-rubber-graft-rigid-polymer/rigid-polymer systems

Explanation of the experimental phenomenon of modified natural-rubber using a MD method.

Coarse-Graining the Liquid-Liquid Interfaces with the MARTINI Force Field: How Is the Interfacial Tension Reproduced?

We report two-phase coarse-grained (CG) simulations of organic-water liquid-liquid interfaces with the MARTINI force field. We discuss the ability of the CG force field to predict quantitatively the interfacial tension of alkanes-water, benzene-water, chloroform-water, and alcohol-water systems. The performance of the prediction of the interfacial tension is evaluated through its dependence on temperature and alkane length. This study contributes to the challenging discussion about the robustness and the transferability of the MARTINI force field to interfacial properties. We have also used the distributions of the molecules along the direction normal to the interface to investigate the composition of the interfacial region and to compare the simulated densities of the coexisting phases with experiments.

Influence of molecular architecture on the entanglement network: topological analysis of linear, long- and short-chain branched polyethylene melts via Monte Carlo simulations

We present detailed results on the effect of chain branching on the topological properties of entangled polymer melts via an advanced connectivity-altering Monte Carlo (MC) algorithm. Eleven representative model linear, short-chain branched (SCB), and long-chain branched (LCB) polyethylene (PE) melts were employed, based on the total chain length and/or the longest linear chain dimension. Directly analyzing the entanglement [or the primitive path (PP)] network of the system via the Z-code, we quantified several important topological measures: (a) the PP contour length Lpp, (b) the number of entanglements Zes per chain, (c) the end-to-end length of an entanglement strand des, (d) the number of carbon atoms per entanglement strand Nes, and (e) the probability distribution for each of these quantities. The results show that the SCB polymer melts have significantly more compact overall chain conformations compared to the linear polymers, exhibiting, relative to the corresponding linear analogues, (a) ∼20% smaller values of 〈Lpp〉 (the statistical average of Lpp), (b) ∼30% smaller values of 〈Zes〉, (c) ∼20% larger values of 〈des〉, and (d) ∼50% larger values of 〈Nes〉. In contrast, despite the intrinsically smaller overall chain dimensions than those of the linear analogues, the LCB (H-shaped and A3AA3 multiarm) PE melts exhibit relatively (a) 7-8% larger values of 〈Lpp〉, (b) 6-11% larger values of 〈Zes〉 for the H-shaped melt and ∼2% smaller values of 〈Zes〉 for the A3AA3 multiarm, (c) 2-5% smaller values of 〈des〉, and (d) 7-11% smaller values of 〈Nes〉. Several interesting features were also found in the results of the probability distribution functions P for each topological measure.

Entanglements in polymer nanocomposites containing spherical nanoparticles

We investigate the polymer packing around nanoparticles and polymer/nanoparticle topological constraints (entanglements) in nanocomposites containing spherical nanoparticles in comparison to pure polymer melts using molecular dynamics (MD) simulations. The polymer-nanoparticle attraction leads to good dispersion of nanoparticles. We observe an increase in the number of topological constraints (decrease of total entanglement length Ne with nanoparticle loading in the polymer matrix) in nanocomposites due to nanoparticles, as evidenced by larger contour lengths of the primitive paths. An increase of the nanoparticle radius reduces the polymer-particle entanglements. These studies demonstrate that the interaction between polymers and nanoparticles does not affect the total entanglement length because in nanocomposites with small nanoparticles, the polymer-nanoparticles topological constraints dominate.

Prediction of structural and thermomechanical properties of polymers from multiscale simulations

We report mesoscale simulations of polymer melts and crosslinked polymer networks by using realistic coarse-grained (CG) models that are developed from atomistic simulations of polymer melts.

Pulling-force-induced elongation and alignment effects on entanglement and knotting characteristics of linear polymers in a melt

We employ a primitive path (PP) algorithm and the Gauss linking integral to study the degree of entanglement and knotting characteristics of linear polymer model chains in a melt under the action of a constant pulling force applied to selected chain ends. Our results for the amount of entanglement, the linking number, the average crossing number, the writhe of the chains and their PPs and the writhe of the entanglement strands all suggest a different response at the length scale of entanglement strands than that of the chains themselves and of the corresponding PPs. Our findings indicate that the chains first stretch at the level of entanglement strands and next the PP (tube) gets oriented with the "flow." These two phases of the extension and alignment of the chains coincide with two phases related to the disentanglement of the chains. Soon after the onset of external force the PPs attain a more entangled conformation, and the number of nontrivially linked end-to-end closed chains increases. Next, the chains disentangle continuously to attain an almost unentangled conformation. Using the linking matrix of the chains in the melt, we furthermore show that these phases are accompanied by a different scaling of the homogeneity of the global entanglement in the system. The homogeneity of the end-to-end closed chains first increases to a maximum and then decreases slowly to a value characterizing a completely unlinked system.

Simulating the Flow of Entangled Polymers

To optimize automation for polymer processing, attempts have been made to simulate the flow of entangled polymers. In industry, fluid dynamics simulations with phenomenological constitutive equations have been practically established. However, to account for molecular characteristics, a method to obtain the constitutive relationship from the molecular structure is required. Molecular dynamics simulations with atomic description are not practical for this purpose; accordingly, coarse-grained models with reduced degrees of freedom have been developed. Although the modeling of entanglement is still a challenge, mesoscopic models with a priori settings to reproduce entangled polymer dynamics, such as tube models, have achieved remarkable success. To use the mesoscopic models as staging posts between atomistic and fluid dynamics simulations, studies have been undertaken to establish links from the coarse-grained model to the atomistic and macroscopic simulations. Consequently, integrated simulations from materials chemistry to predict the macroscopic flow in polymer processing are forthcoming.

Writhe and mutual entanglement combine to give the entanglement length

We propose a method to estimate N(e), the entanglement length, that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions. This estimate uses the writhe of the chains, the writhe of the primitive paths, and the number of kinks in the chains in a melt. An advantage of this method is that it works for both linear and ring chains, works under all periodic boundary conditions, does not require knowing the contour length of the primitive paths, and does not rely on a smooth set of data. We apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies.

Entangled triblock copolymer gel: morphological and mechanical properties

The morphological and mechanical properties of entangled ABA triblock copolymer gels, where solvent were selective to the midblock, were studied as a function of polymer concentration using a novel dissipative particle dynamics model which includes a modified segmental repulsive potential that restricts chain crossing. Morphological properties, such as micelle size, distance between micelles, and the bridge fraction, were calculated as a function of concentration. Although the micelle size was shown to have a strong dependence on concentration, the bridge fraction and distance between micelles were shown to plateau at moderate concentrations. Deformation under uni-axial tension was also performed to extract the cross-link and entanglement contribution to the elastic modulus. Scaling results qualitatively agree with other theoretical predications.

An enhanced entangled polymer model for dissipative particle dynamics

We develop an alternative polymer model to capture entanglements within the dissipative particle dynamics (DPD) framework by using simplified bond-bond repulsive interactions to prevent bond crossings. We show that structural and thermodynamic properties can be improved by applying a segmental repulsive potential (SRP) that is a function of the distance between the midpoints of the segments, rather than the minimum distance between segments. The alternative approach, termed the modified segmental repulsive potential (mSRP), is shown to produce chain structures and thermodynamic properties that are similar to the softly repulsive, flexible chains of standard DPD. Parameters for the mSRP are determined from topological, structural, and thermodynamic considerations. The effectiveness of the mSRP in capturing entanglements is demonstrated by calculating the diffusion and mechanical properties of an entangled polymer melt.

Systematic coarse-graining of the dynamics of entangled polymer melts: the road from chemistry to rheology

For optimal processing and design of entangled polymeric materials it is important to establish a rigorous link between the detailed molecular composition of the polymer and the viscoelastic properties of the macroscopic melt. We review current and past computer simulation techniques and critically assess their ability to provide such a link between chemistry and rheology. We distinguish between two classes of coarse-graining levels, which we term coarse-grained molecular dynamics (CGMD) and coarse-grained stochastic dynamics (CGSD). In CGMD the coarse-grained beads are still relatively hard, thus automatically preventing bond crossing. This also implies an upper limit on the number of atoms that can be lumped together (up to five backbone carbon atoms) and therefore on the longest chain lengths that can be studied. To reach a higher degree of coarse-graining, in CGSD many more atoms are lumped together (more than ten backbone carbon atoms), leading to relatively soft beads. In that case friction and stochastic forces dominate the interactions, and action must be undertaken to prevent bond crossing. We also review alternative methods that make use of the tube model of polymer dynamics, by obtaining the entanglement characteristics through a primitive path analysis and by simulation of a primitive chain network. We finally review super-coarse-grained methods in which an entire polymer is represented by a single particle, and comment on ways to include memory effects and transient forces.

NMR observation of entangled polymer dynamics: tube model predictions and constraint release

We use a benchtop NMR technique to monitor entangled polymer melt dynamics over 5 decades in time covering regimes II-IV of the tube model. We confirm the familiar molar mass scaling exponents of the regime transitions, but we observe a mass-dependent time scaling exponent describing segmental fluctuations in the constrained-Rouse regime II up to high molecular weights. Local chain motions are thus governed by modes that are much less restricted than predicted by the tube model. Diluting protonated chains in deuterated invisible matrix chains, we prove that the exponent is determined by the matrix molecular weight, suggesting constraint-release effects as the origin.