Chain contraction and nonlinear stress damping in primitive chain network simulations

Doi and Edwards (DE) proposed that the relaxation of entangled linear polymers under large deformation occurs in two steps: the fast chain contraction (via the longitudinal Rouse mode of the chain backbone) and the slow orientational relaxation (due to reptation). The DE model assumes these relaxation processes to be independent and decoupled. However, this decoupling is invalid for a generalized convective constraint release (CCR) mechanism that releases the entanglement on every occasion of the contraction of surrounding chains. Indeed, the decoupling does not occur in the sliplink models where the entanglement is represented by the binary interaction (hooking) of chains. Thus, we conducted primitive chain network simulations based on a multichain sliplink model to investigate the chain contraction under step shear. The simulation quantitatively reproduced experimental features of the nonlinear relaxation modulus G(t,γ). Namely, G(t,γ) was cast in the time-strain separable form, G(t,γ)=h(γ)G(t) with h(γ)=damping function and G(t)=linear modulus, but this rigorous separability was valid only at times t comparable to the terminal relaxation time, although a deviation from this form was rather small (within ±10%) at t>τ(R) (longest Rouse relaxation time). A molecular origin of this delicate failure of time-strain separability at t～τ(R) was examined for the chain contour length, subchain length, and subchain stretch. These quantities were found to relax in three steps, the fast, intermediate, and terminal steps, governed by the local force balance between the subchains, the longitudinal Rouse relaxation, and the reptation, respectively. The contributions of the terminal reptative mode to the chain length relaxation as well as the subchain length/stretch relaxation, not considered in the original DE model, emerged because the sliplinks (entanglement) were removed via the generalized CCR mechanism explained above and the reformation of the sliplinks was slow at around the chain center compared to the more rapidly fluctuating chain end. The number of monomers in the subchain were kept larger at the chain center than at the chain end because of the slow entanglement reformation at the center, thereby reducing the tension of the stretched subchain at the chain center compared to the DE prediction. This reduction of the tension at the chain center prevented completion of the length equilibration of subchains at t～τ(R) (which contradicts to the DE prediction), and it forces the equilibration to complete through the reptative mode at t?τ(R). The delicate failure of time-strain separability seen for G(t,γ) at t～τ(R) reflects this retarded length equilibration.     

On microphase segregation of interpenetrating polymer networks

The topological entanglements between subchains of two interpenetrating polymer networks are described in the simplest approximation supposing that the primitive path of each subchain is influenced due to the shift of one network relatively to the other. The entanglement contribution to the free energy of the networks is shown to behave as 1/q² for the state with deviation from uniform densities with the wave vector of order q. This contribution is shown to cause the microphase type of segregation.     

Simulation of the chemical potential of polymers

We propose a method for simulating the chemical potential of polymers based on Widom's formulation, which is an hybrid of known techniques. Thus, the test chain is divided into several subchains which are appended to each other during the simulation. The chemical potential of each subchain is calculated by growing a sample of subchains with the scanning method, which enables searching f steps ahead as compared to f = 1 used in the Rosenbluth procedure. The advantage of this method is that the number of subchains and f can be tuned to provide the most efficient simulation for given density, chain length, and other system conditions. The method is applied to a system of chains with excluded volume on the square lattice.     

Mesoscopic simulations of accelerated polymer drift in microfluidic capillaries

We use a mesoscopic simulation technique to study the transport of polymers in dilute solution flowing through a cylindrical tube. The simulations use an explicit solvent model to include all the relevant hydrodynamic couplings and a coarse grained ideal chain model for the polymers (appropriate for systems near the theta temperature). For the interactions between the solvent and the tube wall we use a novel method that ensures continuity of the stress at the interface. We show that the results for the polymer drift velocity are independent of the degree of coarse graining. Further, for the case where the size of the chains is small but not negligible compared to the tube radius, our results are in excellent agreement with experiment. However, they also show that in this regime, the "accelerated" drift, relative to the average solvent flow velocity, is described by the steric effect of the tube wall excluding the polymer center of mass from sampling the full cross section of the tube. Hydrodynamic interactions have a negligible influence in this regime. Consequently, the agreement between experiment and theories that approximates the former but includes the latter is fortunate. When the undisturbed polymer radius approaches or exceeds the tube radius, the hydrodynamic interactions do have a significant effect. They reduce the drift velocity, in qualitative agreement with theoretical predictions. The accelerated drift still approaches the maximum value, one would expect based on a Poiseuille flow but more slowly than if one neglects hydrodynamics. Finally, we propose an empirical fit that accurately describes data in the intermediate regime.     

Single chain distribution analysis near a substrate using a combined method of three-dimensional imaging and SCF simulation

The effect of preferential wetting of one of the constituent block chains and corresponding block copolymer morphologies to a carbon substrate is studied from a molecular level. The single chain distribution of the block copolymer was estimated as a function of the distance from the substrate by a combined method of transmission electron microtomography (TEMT) and self-consistent field (SCF) simulation. The former provides three-dimensional (3D) morphological information of cylindrical microdomains near the surface, while the latter utilizes the 3D morphology to quantitatively determine the interaction between the block chains and substrate, which is further used to estimate the single chain distribution of one of the block chains, i.e., the subchain, of the matrix. It was found that the subchains in the vicinity of the wetting layer are substantially compressed, while the radius of gyration of the subchain at a distance L (L is the interlayer distance of the cylindrical microdomains from the substrate) has already reached the same value as that in the bulk, indicating that the propagation of the surface interaction is limited to one layer. The methodology developed in this study can be used not only to estimate the surface effect on polymer chains for a variety of different surfaces, but also to provide a means to understand complicated block copolymer morphologies from a molecular level.     

Energy-driven asymmetric partitioning of a semiflexible polymer between interconnected cavities

The distribution of a semiflexible chain in the volume of two interconnected spherical cavities of equal size has been investigated by using Monte Carlo simulations. The chain possessed an extension exceeding that of the cavity, leading to large probabilities of translocated states despite the entropic penalty of passing the narrow passage. Furthermore, an asymmetric state with unequal subchain lengths in the two cavities was more favorable than the symmetric state. The preference for the asymmetric state is driven by the bending energy. Basically, in the symmetric state both subchains are forced to be bent, whereas in the asymmetric case only one of the subchains must bend, leading to an overall smaller bending penalty and overall smaller free energy of the asymmetric state. These results are in contrast to the entropy-controlled partitioning of polymers into confinement and the symmetric translocation state appearing for flexible polymers.     

Theory of relaxation properties of two‐dimensional polymer networks, 2. Local dynamic characteristics

Using normal mode transformation obtained in Part 1 of this series^[1], the exact analytical expressions for the mean‐square displacements of junctions and non‐junction beads, the autocorrelation functions of the end‐to‐end chain vectors between neighboring junctions, and those of subchain vectors of a two‐dimensional regular network consisting of "bead and spring" Rouse chains are obtained. Contributions of intra‐ and interchain relaxation processes to the local dynamic characteristics considered are compared. The time behavior of dynamic quantities obtained is estimated for different scales of motions. The possibility of describing long‐time relaxation of a two‐dimensional network by a simplified coarse‐grained network model is demonstrated. It is shown that the local relaxation properties of a two‐dimensional polymer network (as well as a three‐dimensional network) on scales smaller than the average distance between cross‐links are very close to those of a single Rouse chain. The large‐scale collective relaxation of the polymer networks having a two‐dimensional connectivity differs considerably from that of the three‐dimensional networks.     

Simulation of polymer--polymer interdiffusion using the dynamic lattice liquid model

In this paper, we present computer simulation results concerning interdiffusion of fully compatible components in symmetric binary (AB) polymer mixtures in solutions. The simulation is performed in two dimensions using the algorithm based on the dynamic lattice liquid model. The solvent molecules are taken into account explicitly. The evolution of the concentration profiles in time at an interface is studied for chain lengths N=2,4,8,16 for three polymer concentrations phi=0.1,0.5,0.9. The tracer diffusion coefficients for polymer chains and for the solvent are obtained by monitoring the mean square displacements of their center of mass. The relationships between coefficients of interdiffusion and self-diffusion are tested.     

Relaxation phenomena in dense,glassy membranes prepared from a polymer solution,monitored by changes in dielectric properties

The drying of a solvent-cast polymer film is monitored in a non-invasive way, by measuring the changes in time of dielectric properties, using interdigitated or comb electrodes. Experimentally, the vitrification of the polymer solution is observed at a distinct time, followed by the slow evaporation of solvent from the glassy state. As the solvent diffusion coefficient is a strong function of the concentration in the polymer film, removal of residual solvent proceeds at a self-decreasing rate. With a simple model, solvent mass transfer coefficients in the glassy state are determined from the experiments. It is shown that volume relaxation may be slow compared to the drying process, when preparing thin solvent-cast membranes and using solvents that diffuse relatively fast through the glassy polymer film.     

Visualizing conformations of subchains by creating optical wavelength-sized periodically ordered structure in hydrogel

Thick single-crystalline fcc colloidal crystals exhibiting structural color are obtained by a solvent evaporation method from silica colloidal particle suspensions. A periodically ordered interconnecting porous structure can be imprinted in thermosensitive N-isopropylacrylamide (NIPA) gels by using the colloidal crystals as templates. The porous structure endows a structural color to the NIPA gels. We find that the peak position of the reflection spectra from the porous gels (lamdamax') is expressed as a function of the swelling degree and is synchronized with the change in the swelling degree. The color can be precisely tuned by simply changing the amount of the monomer and the cross-linker in the pre-gel solutions. We can estimate the linear expansion factor (> or =1) of the subchains by comparing the peak position at a given situation (lamdamax') and the reference state (lamdamax,0'), in which the subchains behave as Gaussian coils. Creating the periodically ordered structure, which is similar in size to the wavelength of optical light, in the gels allows us to determine the behavior of polymer chains by observing the structural color.     

Computer Simulation of Polymer Diffusion Under External Electric Field

Molecular dynamics simulations were carried out to investigate the diffusion of polymer chains in solvent under an external electric field. Polyethylene‐like polymer and methyl chloride molecule were chosen as solute and solvent. The external DC electric field strength was varied from 10 to 50 (4.3×10⁸ V/m). Both polar and non‐polar polymer chains in polar and non‐polar solvents were investigated. The simulation shows that the center of mass diffusion coefficient of polymer is sensitive to the polarities of polymer and solvent, field strength, polymer concentration and the density of the system. Various factors that affect the diffusion constant of the polymer are discussed. The present simulation is consistent with the results from light scattering experiments.     

Spherically Symmetric Solvent is Sufficient to Explain the LCST Mechanism in Polymer Solutions

The mechanism of the lower critical solution temperature (LCST) in thermoresponsive polymer solutions has been studied by means of a coarse‐grained single polymer chain simulation and a theoretical approach. The simulation model includes solvent explicitly and thus accounts for solvent interactions and entropy directly. The theoretical model consists of a single chain polymer in an implicit solvent where the effect of solvent is included through the intrapolymer solvophobic potential proposed by Kolomeisky and Widom. The results of this study indicate that the LCST behavior is determined by the competition between the mean energy difference between the bulk and bound solvent, and the entropy loss due to the bound solvent. At low temperatures, solvent molecules are bound to the polymer and the solvophobicity of the polymer is screened, resulting in a coiled state. At high temperatures the entropy loss due to bound solvent offsets the energy gain due to binding which causes the solvent molecules to unbind, leading to the collapse of the polymer chain to a globular state. Furthermore, the coarse‐grained nature of these models indicates that mean interaction energies are sufficient to explain LCST in comparison to specific solvent structural arrangements.

     

Molecular thermodynamics for swelling of a mesoscopic ionomer gel in 1 : 1 salt solutions

For a microphase-separated diblock copolymer ionic gel swollen in salt solution, a molecular-thermodynamic model is based on the self-consistent field theory in the limit of strongly segregated copolymer subchains. The geometry of microdomains is described using the Milner generic wedge construction neglecting the packing frustration. A geometry-dependent generalized analytical solution for the linearized Poisson-Boltzmann equation is obtained. This generalized solution not only reduces to those known previously for planar, cylindrical and spherical geometries, but is also applicable to saddle-like structures. Thermodynamic functions are expressed analytically for gels of lamellar, bicontinuous, cylindrical and spherical morphologies. Molecules are characterized by chain composition, length, rigidity, degree of ionization, and by effective polymer-polymer and polymer-solvent interaction parameters. The model predicts equilibrium solvent uptakes and the equilibrium microdomain spacing for gels swollen in salt solutions. Results are given for details of the gel structure: distribution of mobile ions and polymer segments, and the electric potential across microdomains. Apart from effects obtained by coupling the classical Flory-Rehner theory with Donnan equilibria, viz. increased swelling with polyelectrolyte charge and shrinking of gel upon addition of salt, the model predicts the effects of microphase morphology on swelling.     

Statics and dynamics of dense polymer systems studied by monte carlo simulation

Monte Carlo simulations of coarse–grained models of macromolecules offer a unique tool to study the interplay between coil conformations, thermodynamic properties, and chain configurational relaxation and diffusion. Two examples are discussed where the chain conformation strongly differs from a gaussian coil: (i) collapsed chains in a bad solvent, where anomalous diffusion occurs in the Rouse limit and the relaxation time increases at least with the third power of chain length. (ii) Expulsion of a chain from a semidilute polymer brush. The initially stretched chain contracts to a gaussian coil and the center of mass moves outward with constant velocity until it reaches the region of the “last blob” where crossover to diffusive behavior occurs.     

Coarse-grained polyethylene: 1. The simplest model for the orthorhombic crystal

The coarse-grained model of polyethylene and alkanes (the united-atom model, in which each CH₂ group is represented by a single bead) was proposed several decades ago. It is widely applied in molecular dynamics simulations. For different tasks, the models with different geometrical and force parameters are used. Until now, it was thought that the coarse-grained model of polyethylene cannot reproduce the orthorhombic crystalline phase, which is typical of this polymer. In the present study, we analyze the simplest coarse-grained model of polyethylene. In this model, the Lennard-Jones potential (6–12) is adopted for van der Waals interactions between the beads of different chains. Of the bonded interactions, only the “valence” bonds between beads and the “bond” and “torsion” angles are taken into account, whereas the cross terms between them are disregarded. We consider the model variation in which the bead (the force center with the mass of a CH₂ group) is displaced from the center of the carbon atom and all the interactions, both bonded and nonbonded, are defined by the positions of these beads. For this model, we find the area of geometrical parameters (the displacement value and the van der Waals radius of the bead) in which all the three known crystalline phases of polyethylene are at equilibrium at low temperatures. We choose the force field constants for the model so that its oscillation spectrum reproduces the low-frequency part of the inelastic neutron scattering spectrum of the orthorhombic polyethylene. It proved to be that this choice can be made unambiguously. We compare the dispersion curves in the terahertz range with experimental data on the Raman scattering and infrared spectroscopy, and discuss the advantages and disadvantages of the analyzed simplest coarse model.     

Incompatibility in polymer solutions: Light scattering of two polymers in a single solvent

The fluctuation theory of light scattering for moderately concentrated polymer solutions is extended in a phenomenological fashion to include angle-dependent terms up to sin² (θ/2). The treatment is based on an uncoupling of the center of mass and segment contributions to the free enthalpy. In particular, equations are derived for three-component mixtures containing two polymer and one solvent component. These are applied near the region of incompatibility. The results are compared with Debye's theory of critical opalescence. It is found that the Debye l parameter should be of the order of magnitude of the radii of gyration of the polymer molecules. This is confirmed by some experiments.     

A macromolecule in a solvent: adaptive resolution molecular dynamics simulation

The authors report adaptive resolution molecular dynamics simulations of a flexible linear polymer in solution. The solvent, i.e., a liquid of tetrahedral molecules, is represented within a certain radius from the polymer's center of mass with a high level of detail, while a lower coarse-grained resolution is used for the more distant solvent. The high resolution sphere moves with the polymer and freely exchanges molecules with the low resolution region through a transition regime. The solvent molecules change their resolution and number of degrees of freedom on the fly. The authors show that their approach correctly reproduces the static and dynamic properties of the polymer chain and surrounding solvent.     

Entangled polymers in condensed phases

We present Monte Carlo results on a model of polymers in a condensed phase, over a range of monomer densities. We imagine cutting a cube out of the system. This cube will typically have several polymer molecules running through its interior, and starting and ending on the boundary. These subchains will be mutually entangled and we present a way to assess the extent of entanglement complexity as a function of the monomer density and the number of subchains in the cube. The model is a set of k self-avoiding and mutually avoiding walks, properly embedded in the cube.     

Coarse‐grained A‐graft‐B model of poly(lactic acid) for molecular dynamics simulations

A mesoscopic model of poly(lactic acid) is developed where the polymer is represented as an A‐graft‐B chain with monomer units consisting of two covalently connected beads. A coarse‐graining algorithm is proposed to convert an atomistic model of PLA into a coarse‐grained one. The developed model is based on atomistic simulations of oligolactides to take into account terminal groups correctly. It was used for coarse‐grained simulations of polylactide. Gyration radii and end to end distances of polymer chains as well as the density of the polymer melt are shown to be in a good agreement with those obtained from atomistic simulations. The thermal expansion coefficients of the OLA melts calculated using the coarse‐grained model are in reasonable agreement with those obtained from all‐atom molecular dynamics. The model provides a 17‐fold speedup compared with atomistic calculations. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 604–612     

Polarization and orientational order in polymer ferroelectric films based on vinylidene fluoride

Phase transitions and the development of orientational order are studied for three-dimensional polymer systems with the anisotropy of local intra- and interchain orientational-deformation interactions of chains with the dipole-type potential. In the proposed model of chains composed of elastically deformed segments with a fixed mean-square length (in the modified model of Gaussian subchains), there is a certain critical temperature at which the second-order phase transition from the isotropic state to the orientationally ordered state occurs. The temperature dependences of the parameter of the dipole order for thick films are calculated, and these dependences are compared with the corresponding dependences within the mean-field approximation according to the Ising model for ferromagnetics and within the Langevin continuum model for ferroelectric materials as well as with the experimental data on the thermal depolarization in the films based on the vinylidene fluoride-trifluoroethylene copolymer. The order parameter is calculated as a function of the film thickness (the length of chains) under certain boundary conditions imposed on film ends, and the calculated values are compared with the values predicted by the phenomenological theory and with the experimental data on the polarization distribution in the ferroelectric films based on vinylidene fluoride.