Equation of State Based Slip Spring Model for Entangled Polymer Dynamics

Learning Force Field Parameters from Differentiable Particle-Field Molecular Dynamics

We develop ∂-HylleraasMD (∂-HyMD), a fully end-to-end differentiable molecular dynamics software based on the Hamiltonian hybrid particle-field formalism, and use it to establish a protocol for automated optimization of force field parameters. ∂-HyMD is templated on the recently released HylleraaasMD software, while using the JAX autodiff framework as the main engine for the differentiable dynamics. ∂-HyMD exploits an embarrassingly parallel optimization algorithm by spawning independent simulations, whose trajectories are simultaneously processed by reverse mode automatic differentiation to calculate the gradient of the loss function, which is in turn used for iterative optimization of the force-field parameters. We show that parallel organization facilitates the convergence of the minimization procedure, avoiding the known memory and numerical stability issues of differentiable molecular dynamics approaches. We showcase the effectiveness of our implementation by producing a library of force field parameters for standard phospholipids, with either zwitterionic or anionic heads and with saturated or unsaturated tails. Compared to the all-atom reference, the force field obtained by ∂-HyMD yields better density profiles than the parameters derived from previously utilized gradient-free optimization procedures. Moreover, ∂-HyMD models can predict with good accuracy properties not included in the learning objective, such as lateral pressure profiles, and are transferable to other systems, including triglycerides.

Development of a Meshless Kernel-Based Scheme for Particle-Field Brownian Dynamics Simulations

We develop a meshless discretization scheme for particle-field Brownian dynamics simulations. The density is assigned on the particle level using a weighting kernel with finite support. The system's free energy density is derived from an equation of state (EoS) and includes a square gradient term. The numerical stability of the scheme is evaluated in terms of reproducing the thermodynamics (equilibrium density and compressibility) and dynamics (diffusion coefficient) of homogeneous samples. Using a reduced description to simplify our analysis, we find that numerical stability depends strictly on reduced reference compressibility, kernel range, time step in relation to the friction factor, and reduced external pressure, the latter being relevant under isobaric conditions. Appropriate parametrization yields precise thermodynamics, further improved through a simple renormalization protocol. The dynamics can be restored exactly through a trivial manipulation of the time step and friction coefficient. A semiempirical formula for the upper bound on the time step is derived, which takes into account variations in compressibility, friction factor, and kernel range. We test the scheme on realistic mesoscopic models of fluids, involving both simple (Helfand) and more sophisticated (Sanchez-Lacombe) equations of state.

Stimuli-responsive viscosity modifiers

Stimuli responsive viscosity modifiers entail an important class of materials which allow for smart material formation utilizing various stimuli for switching such as pH, temperature, light and salinity. They have seen applications in the biomedical space including tissue engineering and drug delivery, wherein stimuli responsive hydrogels and polymeric vessels have been extensively applied. Applications have also been seen in other domains like the energy sector and automobile industry, in technologies such as enhanced oil recovery. The chemistry and microstructural arrangements of the aqueous morphologies of dissolved materials are usually sensitive to the aforementioned stimuli which subsequently results in rheological sensitivity as well. Herein, we overview different structures capable of viscosity modification as well as go over the rheological theory associated with classical systems studied in literature. A detailed analysis allows us to explore correlations between commonly discussed models such as molecular packing parameter, tube reptation and stress relaxation with structural and rheological changes. We then present five primary mechanisms corresponding to stimuli responsive viscosity modification: (i) packing parameter modification via functional group conditioning and (ii) via dynamic bond formation, (iii) mesh formation by interlinking of network nodes, (iv) viscosity modification by chain conformation changes and (v) viscosity modification by particle jamming. We also overview several recent examples from literature that employ the concepts discussed to create novel classes of intriguing stimuli responsive structures and their corresponding rheological properties. Furthermore, we also explore systems that are responsive to multiple stimuli which can provide enhanced functionality and versatility by providing multi-level and precise actuation. Such systems have been used for programmed site-specific drug delivery.

Entanglements via Slip Springs with Soft, Coarse-Grained Models for Systems Having Explicit Liquid-Vapor Interfaces

Recent advances in nano-rheology require that new techniques and models be developed to precisely describe the equilibrium and non-equilibrium characteristics of entangled polymeric materials and their interfaces at a molecular level. In this study, a slip-spring (SLSP) model is proposed to capture the dynamics of entangled polymers at interfaces, including those between liquids, liquids and vapors, and liquids and solids. The SLSP model employs a highly coarse-grained approach, which allows for comprehensive simulations of entire nano-rheological characterization systems using a particle-level description. The model relies on many-body dissipative particle dynamics (MDPD) non-bonded interactions, which permit explicit description of liquid-vapor interfaces; a compensating potential is introduced to ensure an unbiased representation of the shape of the liquid-vapor interface within the SLSP model. The usefulness of the proposed MDPD + SLSP approach is illustrated by simulating a capillary breakup rheometer (CaBR) experiment, in which a liquid droplet splits into two segments under the influence of capillary forces. We find that the predictions of the MDPD + SLSP model are consistent with experimental measurements and theoretical predictions. The proposed model is also verified by comparison to the results of explicit molecular dynamics simulations of an entangled polymer melt using a Kremer-Grest chain representation, both at equilibrium and far from equilibrium. Taken together, the model and methods presented in this study provide a reliable framework for molecular-level interpretation of high-polymer dynamics in the presence of interfaces.

Interaction of Macromolecular Chain with Phospholipid Membranes in Solutions: A Dissipative Particle Dynamics Simulation Study

The interaction between macromolecular chains and phospholipid membranes in aqueous solution was investigated using dissipative particle dynamics simulations. Two cases were considered, one in which the macromolecular chains were pulled along parallel to the membrane surfaces and another in which they were pulled vertical to the membrane surfaces. Several parameters, including the radius of gyration, shape factor, particle number, and order parameter, were used to investigate the interaction mechanisms during the dynamics processes by adjusting the pulling force strength of the chains. In both cases, the results showed that the macromolecular chains undergo conformational transitions from a coiled to a rod-like structure. Furthermore, the simulations revealed that the membranes can be damaged and repaired during the dynamic processes. The role of the pulling forces and the adsorption interactions between the chains and membranes differed in the parallel and perpendicular pulling cases. These findings contribute to our understanding of the interaction mechanisms between macromolecules and membranes, and they may have potential applications in biology and medicine.

On the equivalence of the hybrid particle-field and Gaussian core models

Hybrid particle-field molecular dynamics is a molecular simulation strategy, wherein particles couple to a density field instead of through ordinary pair potentials. Traditionally considered a mean-field theory, a momentum and energy-conserving hybrid particle-field formalism has recently been introduced, which was demonstrated to approach the Gaussian Core model potential in the grid-converged limit. Here, we expand on and generalize the correspondence between the Hamiltonian hybrid particle-field method and particle-particle pair potentials. Using the spectral procedure suggested by Bore and Cascella, we establish compatibility to any local soft pair potential in the limit of infinitesimal grid spacing. Furthermore, we document how the mean-field regime often observed in hybrid particle-field simulations is due to the systems under consideration, and not an inherent property of the model. Considering the Gaussian filter form, in particular, we demonstrate the ability of the Hamiltonian hybrid particle-field model to recover all structural and dynamical properties of the Gaussian Core model, including solid phases, a first-order phase transition, and anomalous transport properties. We quantify the impact of the grid spacing on the correspondence, as well as the effect of the particle-field filtering length scale on the emergent particle-particle correlations.

HylleraasMD: A Domain Decomposition-Based Hybrid Particle-Field Software for Multiscale Simulations of Soft Matter

We present HylleraasMD (HyMD), a comprehensive implementation of the recently proposed Hamiltonian formulation of hybrid particle-field molecular dynamics. The methodology is based on a tunable, grid-independent length-scale of coarse graining, obtained by filtering particle densities in reciprocal space. This enables systematic convergence of energies and forces by grid refinement, also eliminating nonphysical force aliasing. Separating the time integration of fast modes associated with internal molecular motion from slow modes associated with their density fields, we enable the first time-reversible, energy-conserving hybrid particle-field simulations. HyMD comprises the optional use of explicit electrostatics, which, in this formalism, corresponds to the long-range potential in particle-mesh Ewald. We demonstrate the ability of HyMD to perform simulations in the microcanonical and canonical ensembles with a series of test cases, comprising lipid bilayers and vesicles, surfactant micelles, and polypeptide chains, comparing our results to established literature. An on-the-fly increase of the characteristic coarse-grain length significantly speeds up dynamics, accelerating self-diffusion and leading to expedited aggregation. Exploiting this acceleration, we find that the time scales involved in the self-assembly of polymeric structures can lie in the tens to hundreds of picoseconds instead of the multimicrosecond regime observed with comparable coarse-grained models.

Soft Matter under Pressure: Pushing Particle–Field Molecular Dynamics to the Isobaric Ensemble

Hamiltonian hybrid particle-field molecular dynamics is a computationally efficient method to study large soft matter systems. In this work, we extend this approach to constant-pressure (NPT) simulations. We reformulate the calculation of internal pressure from the density field by taking into account the intrinsic spread of the particles in space, which naturally leads to a direct anisotropy in the pressure tensor. The anisotropic contribution is crucial for reliably describing the physics of systems under pressure, as demonstrated by a series of tests on analytical and monatomic model systems as well as realistic water/lipid biphasic systems. Using Bayesian optimization, we parametrize the field interactions of phospholipids to reproduce the structural properties of their lamellar phases, including area per lipid, and local density profiles. The resulting model excels in providing pressure profiles in qualitative agreement with all-atom modeling, and surface tension and area compressibility in quantitative agreement with experimental values, indicating the correct description of long-wavelength undulations in large membranes. Finally, we demonstrate that the model is capable of reproducing the formation of lipid droplets inside a lipid bilayer.

Response of Elementary Structural Transitions in Glassy Atactic Polystyrene to Temperature and Deformation

The effects of temperature, pressure, and imposed strain on the structural transition pathways of glassy atactic polystyrene (aPS) are studied for a wide range of conditions. By employing an atomistic description of the system, we systematically explore its free energy landscape, emphasizing connections between local free energy minima. A triplet of two minima connected to each other via a first-order saddle point provides the full description of each elementary structural relaxation event. The basis of the analysis is the potential energy landscape (PEL), where efficient methods for finding saddle points and exploring transition pathways have been developed. We then translate the stationary points of the PEL to stationary points of the proper free energy landscape that obeys the macroscopically imposed constraints (either stress- or strain-controlled). By changing the temperature under isobaric conditions (i.e., Gibbs energy landscape), we probe the temperature dependence of the transition rates of the subglass relaxations of aPS, thus obtaining their activation energies by fitting to the Arrhenius equation. The imposition of different strain levels under isothermic conditions allows us to estimate the apparent activation volume of every elementary transition. Our findings are in good agreement with experimental observations for the same system, indicating that both length- and time-scales of the structural transitions of glassy aPS can be obtained by proper free energy minimization of atomistically detailed configurations.

Effect of Slip-Spring Parameters on the Dynamics and Rheology of Soft, Coarse-Grained Polymer Models

Highly coarse-grained (hCG) linear polymer models allow for accessing long time and length scales by dissipative particle dynamics (DPD). This top-down strategy exploits the universal equilibrium behavior of long, flexible macromolecules by accounting only for the relevant interactions, such as molecular connectivity, and by parametrizing their strength via coarse-grained invariants, such as the mean-squared end-to-end distance. The description of the dynamics of long, entangled polymers, however, poses a challenge because (i) the noncrossability of the molecular backbones is not enforced by the soft interactions of an hCG model and (ii) the rheology involves multiple time and length scales, such as the Rouse-like dynamics on short scales and the reptation dynamics on long scales. One popular technique to effectively mimic the effect of entanglements in linear polymer melts via hCG models is slip-springs, and quantitative agreement with simulations that explicitly account for the noncrossability of molecular contours, experiments, and theoretical predictions has been achieved by identifying the time, length, and energy scales of the hCG model and adjusting the number of slip-springs per macromolecule. In the present work, we study how the spatial extent and the mobility of slip-springs affect the dynamics and discuss their implications in the choice of the degree of coarse-graining in computationally efficient hCG models.

Heat- and shear-reversible networks in food: A review

While nature behaves like an irreversible network with respect to entropy and time, certain systems in nature exist that are, to some extent, reversible. The property of reversibility imparts unique benefits to systems that possess them, making them suitable for designing self-healing, stimuli-responsive, and smart materials that can be used in widely divergent fields. Reversible networks are currently being exploited for applications in tissue engineering, drug delivery, and soft robotics. They are also being utilized as low-calorie fat mimetics with melt-in-your-mouth textures, as well as being explored as potential scaffolds for three-dimensional (3D) printable food, among other applications. This review aims to gather representative examples of heat- and shear-reversible networks in the food science literature from the last 30 or so years, in other words, reversible food gels made either from linear biopolymers or from colloidal, particulate dispersions, including those that have been modified specifically to induce reversibility. An overview of the network mechanisms involved that impart reversibility, including a discussion of the strength and range of forces involved, will be highlighted. A model that explains why certain networks are thermoreversible while others are shear-reversible, and why others are both, will also be proposed. A fundamental understanding of these mechanisms will prove invaluable when designing reversible networks in the future, making possible the precise control of their properties, thus fostering innovative applications within the food industry and beyond.

Slip-Spring Hybrid Particle-Field Molecular Dynamics for Coarse-Graining Branched Polymer Melts: Polystyrene Melts as an Example

The topology of chains significantly modifies the dynamical properties of polymer melts. Here, we extend a recently developed efficient simulation method, namely the slip-spring hybrid particle-field (SS-hPF) model, to study the structural and dynamical properties of branched polymer melts over large spatial-temporal scales. In the coarse-grained SS-hPF simulation of polymers, the bonded potentials are derived by iterative Boltzmann inversion from the underlying fine-grained model. The nonbonded potentials are computed from a density functional field instead of pairwise interactions used in standard molecular dynamics simulations, which increases the computational efficiency by a factor of 10-20. The entangled dynamics is lost due to the soft-core nature of density functional field interactions. It is recovered by a multichain slip-spring model that is rigorously parametrized from existing experimental or simulation data. To quantitatively predict the relaxation and diffusion of branched polymers, which are dominated by arm retraction rather than chain reptation, the slip-spring algorithm is augmented to improve the polymer dynamics near the branch point. Multiple dynamical observables, e.g., diffusion coefficients, arm relaxations, and tube survival probabilities, are characterized in an example coarse-grained model of symmetric and asymmetric star-shaped polystyrene melts. Consistent dynamical behaviors are identified and compared with theoretical predictions. With a single rescaling factor, the prediction of diffusion coefficients agrees well with the available experimental measurements. In this work, an efficient approach is provided to build chemistry-specific coarse-grained models for predicting the dynamics of branched polymers.

Mobility of Polymer Melts in a Regular Array of Carbon Nanotubes

In polymer nanocomposites, mechanical properties essentially depend on the alignment of nanoparticles and polymers. In this work, we investigate an entangled polymer melt in a confinement computationally, in order to get an insight into the mobility behavior of the polymer chains. The confinement consists of nanotubes, arranged in a hexagonal array. We use dissipative particle dynamics, a fast, soft-core simulation method, and reintroduce entanglement dynamics via slip-springs. We observe a distinct influence of the confinement as diffusion is increased in the direction parallel to the nanotubes. Furthermore, we observe that an orientation of the polymers parallel to the nanotubes and chains are compressed in the direction orthogonal to their primitive path. The diffusion parallel to the nanotubes increases further as we increase the nanotube volume fraction in our systems. Moreover, we investigate the slip-spring distribution in the proximity of the nanotube surfaces of our fast and simple slip-spring model, which we find to coincide with results reported for more sophisticated and expensive methods. Our DPD model shows potential applicability to a wide range of polymer nanocomposites while preserving reptation behavior, which is typically lost due to the use of soft-core models.

Effects of Slip-Spring Parameters and Rouse Bead Density on Polymer Dynamics in Multichain Slip-Spring Simulations

The multichain slip-spring (MCSS) model is one of the coarse-grained models of polymers developed in the niche between bead-spring models and tube type descriptions. In this model, polymers are represented by Rouse chains connected by virtual springs that temporally connect the chains, hop along the chain, and are constructed and annihilated at the chain ends. Earlier studies have shown that MCSS simulations can nicely reproduce entangled and unentangled polymer dynamics. However, the model parameters have been chosen arbitrarily, and their effects have not been reported. In this study, for the first time, we systematically investigated the effects of model parameters: fugacity of virtual springs, its intensity, and the Rouse bead density. We validated the employed simulation code by confirming that the statistics of the system follow the theoretical setup. Namely, the virtual spring density is correctly controlled, and polymer chains exhibit ideal chain statistics irrespective of the chosen parameter values. For diffusion and linear viscoelasticity, simulation results obtained for different parameters can be superposed with each other by conversion factors for the bead number per chain and units of length, time, and modulus. These conversion factors follow scaling laws concerning the number of Rouse segments between two consecutive anchoring points of virtual springs along the polymer chain. Besides, diffusion and viscoelasticity excellently agree with literature data for the standard bead-spring simulation. These results imply that the coarse-graining level for the MCSS model can be arbitrarily chosen and controlled by model parameters.

A slip-spring framework to study relaxation dynamics of entangled wormlike micelles with kinetic Monte Carlo algorithm

HYPOTHESIS

Wormlike micelles (WLMs) formed due to the self-assembly of amphiphiles in aqueous solution have similar viscoelastic properties as polymers. Owing to this similarity, in this work, it is postulated that kinetic Monte Carlo (kMC) sampling of slip-springs dynamics, which is able to model the rheology of polymers, can also be extended to capture the relaxation dynamics of WLMs.

THEORY

The proposed modeling framework considers the following relaxation mechanisms: reptation, union-scission, and constraint release. Specifically, each of these relaxation mechanisms is simulated as separate kMC events that capture the relaxation dynamics while considering the living nature of WLMs within the slip-spring framework. As a case study, the model is implemented to a system of sodium oleate and sodium chloride to predict the linear rheology and the characteristic relaxation times associated with the individual relaxation mechanisms at different pH and salt concentrations.

FINDINGS

Linear rheology predictions were found to be in good agreement with experimental data. Furthermore, the calculated relaxation times highlighted that reptation contributed to a continuous increase in viscosity while union-scission contributed to the decrease in viscosity of WLM solutions at a higher salinity and pH. This manifests the proposed model's capability to provide insights into the key processes governing WLM's rheology.

Knotting behaviour of polymer chains in the melt state for soft-core models with and without slip-springs

We analyse the knotting behaviour of linear polymer melts in two types of soft-core models, namely dissipative-particle dynamics and hybrid-particle-field models, as well as their variants with slip-springs which are added to recover entangled polymer dynamics. The probability to form knots is found drastically higher in the hybrid-particle-field model compared to its parent hard-core molecular dynamics model. By comparing the knottedness in dissipative-particle dynamics and hybrid-particle-field models with and without slip-springs, we find the impact of slip-springs on the knotting properties to be negligible. As a dynamic property, we measure the characteristic time of knot formation and destruction, and find it to be (i) of the same order as single-monomer motion and (ii) independent of the chain length in all soft-core models. Knots are therefore formed and destroyed predominantly by the unphysical chain crossing. This work demonstrates that the addition of slip-springs does not alter the knotting behaviour, and it provides a general understanding of knotted structures in these two soft-core models of polymer melts.

Combination of Hybrid Particle-Field Molecular Dynamics and Slip-Springs for the Efficient Simulation of Coarse-Grained Polymer Models: Static and Dynamic Properties of Polystyrene Melts

A quantitative prediction of polymer-entangled dynamics based on molecular simulation is a grand challenge in contemporary computational material science. The drastic increase of relaxation time and viscosity in high-molecular-weight polymeric fluids essentially limits the usage of classic molecular dynamics simulation. Here, we demonstrate a systematic coarse-graining approach for modeling entangled polymers under the slip-spring particle-field scheme. Specifically, a frequency-controlled slip-spring model, a hybrid particle-field model, and a coarse-grained model of polystyrene melts are combined into a hybrid simulation technique. Via a rigorous parameterization strategy to determine the parameters in slip-springs from existing experimental or simulation data, we show that the reptation behavior is clearly observed in multiple characteristics of polymer dynamics, mean-square displacements, diffusion coefficients, reorientational relaxation, and Rouse mode analysis, consistent with the predictions of the tube theory. All dynamical properties of the slip-spring particle-field models are in good agreement with classic molecular dynamics models. Our work provides an efficient and practical approach to establish chemical-specific coarse-grained models for predicting polymer-entangled dynamics.

Hybrid particle-field molecular dynamics under constant pressure

Hybrid particle-field methods are computationally efficient approaches for modeling soft matter systems. So far, applications of these methodologies have been limited to constant volume conditions. Here, we reformulate particle-field interactions to represent systems coupled to constant external pressure. First, we show that the commonly used particle-field energy functional can be modified to model and parameterize the isotropic contributions to the pressure tensor without interfering with the microscopic forces on the particles. Second, we employ a square gradient particle-field interaction term to model non-isotropic contributions to the pressure tensor, such as in surface tension phenomena. This formulation is implemented within the hybrid particle-field molecular dynamics approach and is tested on a series of model systems. Simulations of a homogeneous water box demonstrate that it is possible to parameterize the equation of state to reproduce any target density for a given external pressure. Moreover, the same parameterization is transferable to systems of similar coarse-grained mapping resolution. Finally, we evaluate the feasibility of the proposed approach on coarse-grained models of phospholipids, finding that the term between water and the lipid hydrocarbon tails is alone sufficient to reproduce the experimental area per lipid in constant-pressure simulations and to produce a qualitatively correct lateral pressure profile.

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.

Studying polymer solutions with particle-based models linked to classical density functionals: co-non-solvency

We demonstrate the potential of hybrid particle-based models, where interactions are introduced through functionals of local order parameters, in describing multicomponent polymer solutions. The link to a free-energy-like functional is advantageous for controlling the thermodynamics of the model. We focus on co-non-solvency - the collapse of polymer chains in dilute mixtures with two miscible good solvents, having different affinities towards the polymer. We employ a simple model where polymers and solvents are represented, respectively, by worm-like chains and single particles. Non-bonded interactions are captured by a polynomial which is third order in local densities and can, therefore, describe liquid-vapour coexistence. The parameterisation of the functional benefits from an elementary mean-field approximation to the statistical mechanics of the model. The model provides a framework for Monte Carlo simulations using a particle-to-mesh algorithm. Studies with conventional generic bead-spring and all-atom models have demonstrated that co-non-solvency is caused by preferential binding of the better solvent (termed cosolvent) with polymer. Hence, segmental loops bridged by cosolvent molecules are formed, initiating polymer collapse. The mesoscopic hybrid model differs conceptually from the conventional microscopic descriptions. Yet, it reproduces the same co-non-solvency mechanism supporting its universality. Films of adsorbed ternary solutions, showing co-non-solvency in the dilute regime, are considered at high concentrations. In this case, chains do not collapse. The properties of loops and tails of the adsorbed polymer agree with early theoretical predictions obtained for concentrated binary solutions.

Slip Spring-Based Mesoscopic Simulations of Polymer Networks: Methodology and the Corresponding Computational Code

In previous work by the authors, a new methodology was developed for Brownian dynamics/kinetic Monte Carlo (BD/kMC) simulations of polymer melts. In this study, this methodology is extended for dynamical simulations of crosslinked polymer networks in a coarse-grained representation, wherein chains are modeled as sequences of beads, each bead encompassing a few Kuhn segments. In addition, the C++ code embodying these simulations, entitled Engine for Mesoscopic Simulations for Polymer Networks (EMSIPON) is described in detail. A crosslinked network of cis-1,4-polyisoprene is chosen as a test system. From the thermodynamic point of view, the system is fully described by a Helmholtz energy consisting of three explicit contributions: entropic springs, slip springs and non-bonded interactions. Entanglements between subchains in the network are represented by slip springs. The ends of the slip springs undergo thermally activated hops between adjacent beads along the chain backbones, which are tracked by kinetic Monte Carlo simulation. In addition, creation/destruction processes are included for the slip springs at dangling subchain ends. The Helmholtz energy of non-bonded interactions is derived from the Sanchez⁻Lacombe equation of state. The isothermal compressibility of the polymer network is predicted from equilibrium density fluctuations in very good agreement with the underlying equation of state and with experiment. Moreover, the methodology and the corresponding C++ code are applied to simulate elongational deformations of polymer rubbers. The shear stress relaxation modulus is predicted from equilibrium simulations of several microseconds of physical time in the undeformed state, as well as from stress-strain curves of the crosslinked polymer networks under deformation.

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.