Viscoelastic Properties of Polymer Melts from Equilibrium Molecular Dynamics Simulations

Versatilely Manipulating the Mechanical Properties of Polymer Nanocomposites by Incorporating Porous Fillers: A Molecular Dynamics Simulation Study

Polymer nanocomposites (PNCs) have been attracting myriad scientific and technological attention due to their promising mechanical and functional properties. However, there remains a need for an efficient method that can further strengthen the mechanical performance of PNCs. Here, we propose a strategy to design and fabricate novel PNCs by incorporating porous fillers (PFs) such as metal-organic frameworks with ultrahigh specific surface areas and tunable nanospaces to polymer matrices via coarse-grained molecular dynamics simulations. Three important parameters─the polymer chain stiffness (k), the interaction strength between the PF center and the end functional groups of polymer chains (ε_center end), and the PF weight fraction (w)─are systematically examined. First, attributed to the penetration of polymer chains into PFs at a strong ε_center end, the dimension of polymer chains such as the radius of gyration and the end-to-end distance increases greatly as a function of k compared to the case of the neat polymer system. The penetration of polymer chains is validated by characterizing the radial distribution function between end functional groups and filler centers, as well as the visualization of the snapshots. Also, the dispersion state of PFs tends to be good because of the chain penetration. Then, the glass transition temperature ratio of PNCs to that of the neat systems exhibits a maximum in the case of k = 5ε, indicating that the strongest interlocking between polymer chains and PFs occurs at intermediate chain stiffness. The polymer chain dynamics of PNCs decreases to a plateau at k = 5ε and then becomes stable, and the relative mobility to that of the neat system as well presents the same variation trend. Furthermore, the mechanical property under uniaxial deformation is thoroughly studied, and intermediates k, ε_center end, and w can bring about the best mechanical property. This is because of the robust penetration and interaction, which is confirmed by calculating the stress of every component of PNCs with and without end functional groups and PF centers as well as the nonbonded interaction energy change between different components. Finally, the optimal condition (k = 5.36ε, ε_center end = 5.29ε, and w = 6.54%) to design the PNC with superior mechanical behavior is predicted by Gaussian process regression, an active machine learning (ML) method. Overall, incorporating PFs greatly enhances the entanglements and interactions between polymer chains and nanofillers and brings effective mechanical reinforcements with lower filler weight fractions. We anticipate that this will provide new routes to the design of mechanically reinforced PNCs.

Molecular Dynamics Simulations of Polymer Nanocomposites Welding: Interfacial Structure, Dynamics and Strength

Polymer welding has received numerous scientific attention, however, the welding of polymer nanocomposites (PNCs) has not been studied yet. In this work, via coarse-grained molecular dynamics simulation, we focus our attention on investigating the welding interfacial structure, dynamics and strength by constructing the upper and lower layers of PNCs, by varying the polymer-nanoparticle (NP) interaction strength ε_NP-p . Remarkably, at low ε_NP-p , the NPs gradually migrate into the top and bottom surface layer perpendicular to the z direction during the adhesion process, while they are distributed in the middle region at high ε_NP-p . Meanwhile, the dimension of polymer chains is found to exhibit a remarkable anisotropy evidenced by the root-mean-square radius of gyration in the xy- (R_g,xy ) and z- (R_g,z ) component. The welding interdiffusion depth increases the fastest at low ε_NP-p, attributed to the high mobility of polymer chains and NPs. Lastly, although the mechanical properties of PNCs at high ε_NP-p is the strongest because of the presence of the NPs in the bulk region, the welding efficiency is the greatest at low ε_NP-p . Generally, our work could provide a fundamental understanding of the interfacial welding of PNCs, in hopes of guiding to design and fabricate excellent self-healable PNCs. This article is protected by copyright. All rights reserved.

Solute Diffusivity and Local Free Volume in Cross-Linked Polymer Network: Implication of Optimizing the Conductivity of Polymer Electrolyte

The diffusion of small molecules or ions within polymeric materials is critical for their applications, such as polymer electrolytes. Cross-linking has been one of the common strategies to modulate solute diffusivity and a polymer’s mechanical properties. However, various studies have shown different effects of cross-linking on altering the solute transports. Here, we utilized coarse-grained molecular dynamics simulation to systematically analyze the effects of cross-linking and polymer rigidity of solute diffusive behaviors. Above the glass transition temperature Tg, the solute diffusion followed the Vogel–Tammann–Fulcher (VTF) equation, D = D0 e−Ea/R(T−T0). Other than the conventional compensation relation between the activation energy Ea and the pre-exponential factor D0, we also identified a correlation between Ea and Vogel temperature T0. We further characterized an empirical relation between T0 and cross-linking density. Integrating the newly identified correlations among the VTF parameters, we formulated a relation between solute diffusion and the cross-linking density. The combined results proposed the criteria for the optimal solute diffusivity in cross-linked polymers, providing generic guidance for novel polymer electrolyte design.

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.

Mechanical and Self-Healing Behavior of Matrix-Free Polymer Nanocomposites Constructed via Grafted Graphene Nanosheets

Through molecular dynamics (MD) simulation, the structure and mechanical properties of matrix-free polymer nanocomposites (PNCs) constructed via polymer-grafted graphene nanosheets are studied. The dispersion of graphene sheets is characterized by the radial distribution function (RDF) between graphene sheets. We observe that a longer polymer chain length L_g leads to a relatively better dispersion state attributed to the formation of a better brick-mud structure, effectively screening the van der Waals interactions between sheets. By tuning the interaction strength ε_end-end between end functional groups of grafted chains, we construct physical networks with various robustness characterized by the formation of the fractal clusters at high ε_end-end values. The effects of ε_end-end and L_g on the mechanical properties are examined, and the enhancement of the stress-strain behavior is observed with the increase of ε_end-end and L_g. Structural evolution during deformation is quantified by calculating the orientation of the graphene sheets and their distribution, the stress decomposition, and the size of the clusters formed between end groups and their distribution. Then, we briefly study the effects of time and temperature on the self-healing behavior of these unique PNCs in the rubbery state. Lastly, the self-healing kinetics is quantitatively analyzed. In general, this work can provide some rational guidelines to design and fabricate matrix-free PNCs with both excellent mechanical and self-healing properties.

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.

Correlation between morphology and anisotropic transport properties of diblock copolymers melts

Molecular simulations of coarse-grained diblock copolymers (DBP) were conducted to study the effect of segregation strength and morphology on transport properties. It was found that in the strong segregation limit (i.e., high χN, where χ is the Flory-Huggins parameter and N is the degree of polymerization), the presence of the DBP interfaces imposes topological constraints similar to those of entanglements as manifested in the rheological signature of the polymer (i.e., a plateau modulus). Furthermore, compared to the behavior of isotropic melts, the crossover from Rouse to reptation scaling of the self-diffusion coefficient (D) parallel to the DBP interface takes place at a smaller N, an effect that depends on temperature and is more pronounced in the Lamellae morphology than in the hexagonal cylinder morphology. Additionally, it is shown that for an entangled melt (i.e., N ≫ Ne where Ne is the entanglement length) block retraction is instrumental for chains to diffuse parallel to the interface of lamellar layers. Lastly, it is found that the anisotropic viscosity of different morphologies is mostly affected by the orientation of the chains relative to the shear flow direction, exhibiting reduced values when chains align in the neutral or flow directions.

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 mechanism of viscoelastic polymer enhanced oil recovery in nanopores

Polymer flooding is a promising chemical enhanced oil recovery (EOR) method, which realizes more efficient extraction in porous formations characterized with nanoscale porosity and complicated interfaces. Understanding the molecular mechanism of viscoelastic polymer EOR in nanopores is of great significance for the advancement of oil exploitation. Using molecular dynamics simulations, we investigated the detailed process of a viscoelastic polymer displacing oil at the atomic scale. We found that the interactions between polymer chains and oil provide an additional pulling effect on extracting the residual oil trapped in dead-end nanopores, which plays a key role in increasing the oil displacement efficiency. Our results also demonstrate that the oil displacement ability of polymer can be reinforced with the increasing chain length and viscoelasticity. In particular, a polymer with longer chain length exhibits stronger elastic property, which enhances the foregoing pulling effect. These findings can help to enrich our understanding on the molecular mechanism of polymer enhanced oil recovery and provide guidance for oil extraction engineering.

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.

Static and dynamic properties of large polymer melts in equilibrium

We present a detailed study of the static and dynamic behaviors of long semiflexible polymer chains in a melt. Starting from previously obtained fully equilibrated high molecular weight polymer melts [G. Zhang et al., ACS Macro Lett. 3, 198 (2014)], we investigate their static and dynamic scaling behaviors as predicted by theory. We find that for semiflexible chains in a melt, results of the mean square internal distance, the probability distributions of the end-to-end distance, and the chain structure factor are well described by theoretical predictions for ideal chains. We examine the motion of monomers and chains by molecular dynamics simulations using the ESPResSo++ package. The scaling predictions of the mean squared displacement of inner monomers, center of mass, and relations between them based on the Rouse and the reptation theory are verified, and related characteristic relaxation times are determined. Finally, we give evidence that the entanglement length Ne,PPA as determined by a primitive path analysis (PPA) predicts a plateau modulus,GN (0)=45(ρkBT/Ne), consistent with stresses obtained from the Green-Kubo relation. These comprehensively characterized equilibrium structures, which offer a good compromise between flexibility, small Ne, computational efficiency, and small deviations from ideality, provide ideal starting states for future non-equilibrium studies.

Determination of linear viscoelastic properties of an entangled polymer melt by probe rheology simulations

Particle rheology is used to extract the linear viscoelastic properties of an entangled polymer melt from molecular dynamics simulations. The motion of a stiff, approximately spherical particle is tracked in both passive and active modes. We demonstrate that the dynamic modulus of the melt can be extracted under certain limitations using this technique. As shown before for unentangled chains [Karim et al., Phys. Rev. E 86, 051501 (2012)PLEEE81539-375510.1103/PhysRevE.86.051501], the frequency range of applicability is substantially expanded when both particle and medium inertia are properly accounted for by using our inertial version of the generalized Stokes-Einstein relation (IGSER). The system used here introduces an entanglement length d_{T}, in addition to those length scales already relevant: monomer bead size d, probe size R, polymer radius of gyration R_{g}, simulation box size L, shear wave penetration length Δ, and wave period Λ. Previously, we demonstrated a number of restrictions necessary to obtain the relevant fluid properties: continuum approximation breaks down when d≳Λ; medium inertia is important and IGSER is required when R≳Λ; and the probe should not experience hydrodynamic interaction with its periodic images, L≳Δ. These restrictions are also observed here. A simple scaling argument for entangled polymers shows that the simulation box size must scale with polymer molecular weight as M_{w}^{3}. Continuum analysis requires the existence of an added mass to the probe particle from the entrained medium but was not observed in the earlier work for unentangled chains. We confirm here that this added mass is necessary only when the thickness L_{S} of the shell around the particle that contains the added mass, L_{S}>d. We also demonstrate that the IGSER can be used to predict particle displacement over a given timescale from knowledge of medium viscoelasticity; such ability will be of interest for designing nanoparticle-based drug delivery.

Trajectories of probe spheres in generalized linear viscoelastic complex fluids

We have developed a fast simulation that generates a random walk of an isolated probe sphere in a generalized linear viscoelastic complex fluid over a highly extended dynamic range. We introduce a coupled harmonically bound Brownian particle (c-HBBP) model, in which the relaxation modes of the viscoelastic medium are treated as harmonic wells. These wells are coupled to the probe sphere and perform Brownian motion in bound harmonic potentials corresponding to the next-longer relaxation mode, according to the relaxation spectrum of the viscoelastic material. We implement this c-HBBP model by generating variable temporal step sizes that have a uniform distribution in logarithmic time. We create and analyze trajectories for several different viscoelastic complex fluids: a polymer system at its gel point, a dense emulsion system, a blend of two monodisperse polystyrene polymers for which the relaxation spectrum has been measured, and a model anisotropic soft system that shows dense emulsion-like and gel-point behaviors along two orthogonal directions. Except for unusual viscoelastic materials, such as the polymer system at its gel point, the generated trajectories are neither self-similar nor self-affine. The resulting mean square displacements predicted by the c-HBBP model are consistent with the single-particle generalized Stokes-Einstein relation of linear passive microrheology.

Distinct mechanical properties of nanoparticle-tethering polymers

Nanoparticle-tethering polymers exhibit enhanced mechanical properties relative to neat polymers and nanoparticle/polymer blends.

Computational study of the propagation of the longitudinal velocity in a polymer melt contained within a cylinder using a scale-bridging method

The "constitutive equation"-free scale-bridging method connecting nonequilibrium molecular dynamics and continuum fluid mechanics, that had hitherto been applied only to a parallel-plates geometry, is extended to study the flow of a polymer melt in a cylindrical pipe subject to a velocity in the direction parallel to the cylinder's axis. The system, initially at rest, is given a velocity at the cylinder's surface, and the evolution of the velocity profile within the fluid is studied, along with the time taken for the velocity to propagate toward the cylinder's axis. The said time of propagation is found to increase with the boundary velocity-a fact in contrast with the case of a Newtonian fluid for which the time of propagation is expected to be independent of the boundary velocity. For a fixed value of the boundary velocity, the propagation time is found to increase with the cylinder radius according to a power law with an exponent that is smaller than the corresponding exponent for a Newtonian fluid. For the lower values of the boundary velocity and the lower values of the radius studied, a velocity overshoot is observed at the cylinder's axis-a manifestation of elastic behavior of the fluid.

Determination of viscoelastic properties by analysis of probe-particle motion in molecular simulations

We present a technique for the determination of viscoelastic properties of a medium by tracking the motion of an embedded probe particle by using molecular dynamics simulations. The approach involves the analysis of the simulated particle motion by continuum theory; it is shown to work in both passive and active modes. We demonstrate that, for passive rheology, an analysis based on the generalized Stokes-Einstein relationship is not adequate to obtain the values of the viscoelastic moduli over the frequency range studied. For both passive and active modes, it is necessary to account for the medium and particle inertia when analyzing the particle motion. For a polymer melt system consisting of short chains, the values calculated from the proposed approach are in good quantitative agreement with previous literature results that were obtained using completely different simulation approaches. The proposed particle rheology simulation technique is general and could provide insight into the characterization of the mechanical properties in biological systems, such as cellular environments and polymeric systems, such as thin films and nanocomposites that exhibit spatial variation in properties over the nanoscale.

Mobility of Nonsticky Nanoparticles in Polymer Liquids

We use scaling theory to derive the time dependence of the mean-square displacement 〈Δr2〉 of a spherical probe particle of size d experiencing thermal motion in polymer solutions and melts. Particles with size smaller than solution correlation length ξ undergo ordinary diffusion (〈Δr2 (t)〉 ~ t) with diffusion coefficient similar to that in pure solvent. The motion of particles of intermediate size (ξ < d < a), where a is the tube diameter for entangled polymer liquids, is sub-diffusive (〈Δr2 (t)〉 ~ t1/2) at short time scales since their motion is affected by sub-sections of polymer chains. At long time scales the motion of these particles is diffusive and their diffusion coefficient is determined by the effective viscosity of a polymer liquid with chains of size comparable to the particle diameter d. The motion of particles larger than the tube diameter a at time scales shorter than the relaxation time τ e of an entanglement strand is similar to the motion of particles of intermediate size. At longer time scales (t > τ e ) large particles (d > a) are trapped by entanglement mesh and to move further they have to wait for the surrounding polymer chains to relax at the reptation time scale τrep. At longer times t > τrep, the motion of such large particles (d > a) is diffusive with diffusion coefficient determined by the bulk viscosity of the entangled polymer liquids. Our predictions are in agreement with the results of experiments and computer simulations.

Dynamic evolution in coarse-grained molecular dynamics simulations of polyethylene melts

We test a coarse-grained model assigned based on united atom simulations of C50 polyethylene to seven chain lengths ranging from C76 to C300. The prior model accurately reproduced static and dynamic properties. For the dynamics, the coarse-grained time evolution was scaled by a constant value [t=alphatCG] predictable based on the difference in intermolecular interactions. In this contribution, we show that both static and dynamic observables have continued accuracy when using the C50 coarse-grained force field for chains representing up to 300 united atoms. Pair distribution functions for the longer chain systems are unaltered, and the chain dimensions present the expected N0.5 scaling. To assess dynamic properties, we compare diffusion coefficients to experimental values and united atom simulations, assign the entanglement length using various methods, examine the applicability of the Rouse model as a function of N, and compare tube diameters extracted using a primitive path analysis to experimental values. These results show that the coarse-grained model accurately reproduces dynamic properties over a range of chain lengths, including systems that are entangled.

Multiscale modeling of polymer rheology

We propose a simulation method which can be used to readily parallelize simulations on systems with a large spatial extent. We simulate small parts of the system with independent molecular dynamics simulations, and only occasionally pass information between them through a constitutive model free continuum approach. We illustrate the power of this method in the case of a polymeric fluid undergoing rapid one-dimensional shear flow. Since we show that this flow problem cannot be modeled by using a steady-state constitutive model, this method offers the unique capability for accessing the nonlinear viscoelasticity of complex fluids.

Analysis of uncertainties in polymer viscoelastic properties obtained from equilibrium computer simulations

We critically evaluate the uncertainties in the stress autocorrelation function obtained from equilibrium molecular dynamics simulation of model polymer melts. This quantity is central to evaluating transport properties, e.g., the complex modulus and the viscosity. In contrast to the intuitive expectation that simulations have to be run five to six orders of magnitude longer than the chain relaxation time to reduce uncertainties to acceptable levels, our analysis shows that the majority of the uncertainty is associated with rapidly oscillating bonded interactions. These fluctuations occur on time scales which are approximately 10(4) times shorter than the relaxation time of a chain of length 80. Consequently, the effects of these oscillations on the stress autocorrelation function can be dramatically reduced by (i) conducting long simulations (typically 10(6) times longer than the bond relaxation times or only 10(2) chain relaxation times) and (ii) by performing running averages with time windows whose time scales are much longer than these oscillations. Conducting such long simulations also allows for the accurate determination of the melt viscosity and the low-frequency complex modulus, but performing running averages do not impact these quantities since they are time integrals of the stress autocorrelation function.